Research ArticleProperties of Neutron Noise Induced by LocalizedPerturbations in an SFR

Hoai-Nam Tran

Institute of Research and Development Duy Tan University K725 Quang Trung Da Nang 550000 Vietnam

Correspondence should be addressed to Hoai-Nam Tran tranhoainam4dtueduvn

Received 31 March 2015 Revised 6 June 2015 Accepted 10 June 2015

Academic Editor Keith E Holbert

Copyright copy 2015 Hoai-Nam Tran This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Investigation of the properties of neutron noise induced by localized perturbations in a sodium-cooled fast reactor has beenperformed using a multigroup neutron noise simulatorThree representations of the noise source associated with the perturbationsof absorption fission and scattering cross sections respectively were assumed to be located at the first fuel ring around the centralassemblyThe energy- and space-dependent noise that is the amplitude and the phase was calculated in awide range of frequenciesfor example 01ndash100HzThe results show that in the important energy range (gt10 keV) where the noise amplitude is significant thephase is almost constant with energy at the calculated frequencies despite the source types At low frequencies the variation of thephase is negligibly small at a large distance from the source The perturbation in several fast groups has a significant contributionand dominates the amplitude and the phase of the induced noise

1 Introduction

Online diagnostics for monitoring the operating status oflight water reactors (LWRs) based on analyzing the detectorsignals of neutron noise were deployed widely in variouscountries [1ndash5] Measurement of the neutron noise in fastreactors one of the next generation nuclear systems anda test facility has also been conducted [3 6 7] Howeverthe knowledge and experience of the neutron noise in fastreactors in both measurement and simulation are very littlecompared to that of LWRs Most neutron detectors used inLWRs such as ionization chamber and fission chamber aresensitive to thermal neutron In many cases for simplicityone can investigate the neutron noise in the thermal groupto represent the behaviour of the detector noise Thus recentnumerical development for simulating the neutron noise inLWRs is based on two-group diffusion theory [8ndash11] Nev-ertheless numerical simulation still remains a challenge toreproduce and interpret themeasurement data for improvingcore surveillance It is a more difficult task for a fast reactorwith less knowledge and experience

Similar to static calculations in deterministic method theneutron noise in a fast reactor should be calculated based ona multigroup model In measurement the detector signal isa combination of the energy-dependent noise with the crosssection of the detector as a weighting function In a sodium-cooled fast reactor (SFR) a fission chamber consisting offissionable material for example 235U or 242Pu coated onthe inner wall of the chamber has a major potential for in-core fast neutron detection [12] Thus the energy-dependentcross section of the detector is complicated This is also oneof the difficulties for numerical simulation in predicting orinterpreting the measurement phenomena To simulate theneutron noise in fast reactors with hexagonal fuel assem-blies a neutron noise simulator was developed based onmultigroup diffusion theory [13 14] The tool consists of twomodules a static module for solving the eigenvalue problemof a static state and a noise module for solving the neutronnoise equation with a given source in a frequency domainAn application was performed for investigating the neutronnoise behaviour induced by periodic core deformation effectin a large SFR [15]

Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2015 Article ID 140979 14 pageshttpdxdoiorg1011552015140979

2 Science and Technology of Nuclear Installations

In a realistic fluctuation of a system the noise sourcecan be modelled via a linear combination of the fluctuationsof all cross section types in the first order approximationThe contribution of the perturbation of each cross sectiontype in the total source depends on a specific scenario Forinstance the vibration of a control rod in an LWR can bemodelled as the vibration of the absorption cross sectionwhile to simulate the noise induced by fuel vibration it isnecessary to consider all cross section types including thefission cross section The problem in an SFR is even morecomplicated since the fluctuation of cross sections is stronglyenergy-dependent and is considered in a multigroup theoryTherefore prior to assessing realistic scenarios of fluctuationsin an SFR it is worth investigating the properties of theneutron noise induced by the fluctuations of absorption scat-tering and fission cross sections separately This is becausethese fluctuations lead to different properties of the noisesources respectively and as a result different properties ofthe induced noise

The present paper aims at investigating the properties ofneutron noise induced by localized perturbations in a largeSFR core using the noise simulator Three representations ofthe noise sources associated with the localized perturbationsof absorption fission and scattering cross sections respec-tively were assumed to be located at the center of the coreThe space- and energy-dependent neutron noise has beencalculated in a wide range of frequencies for example 01ndash100Hz

The paper is organized as follows Section 2 presentsbriefly the principles of the neutron noise equation inmultigroup diffusion theory which was solved in a frequencydomain in the noise simulator Section 3 describes the coremodel of a large SFR and the assumption of local per-turbations as the noise source Results and discussion onthe properties of the energy- and space-dependent noise atthe calculated frequencies are also presented Finally someconcluding remarks are given in Section 4

2 Principles of the Neutron Noise Simulator

The basis of the neutron noise equation is the assumptionof small stationary fluctuations of the system that is theaveraged value of a time-dependent quantity over time isequal to the static value Assume that all time-dependentterms119883(r 119905) can be split into a stationary component1198830(r)which corresponds to the value at the steady state plus a smallfluctuation 120575119883(r 119905) as

119883 (r 119905) = 1198830 (r) + 120575119883 (r 119905) (1)

By assuming the small fluctuations the first order noiseis taken into account products of fluctuation terms canbe neglected from time-dependent diffusion equations andthe result is a linear equation for the fluctuation of theflux Subtracting the static equation and after performing aFourier transform of all time-dependent terms the first order

space- and frequency-dependent neutron noise equation inmultigroup diffusion theory is written as follows

minusnabla sdot119863119892nabla120575120601119892 (r 120596) + Σ119905119892 (120596) 120575120601119892 (r 120596)

=

1119896eff120594119892 (120596)sum

1198921015840

]Σ1198911198921015840 (120596) 1205751206011198921015840 (r 120596)

+ sum

1198921015840 =119892

Σ1199041198921015840rarr119892 (120596) 1205751206011198921015840 (r 120596) + 119878119892 (r 120596)

(2)

where 119892 denotes the energy group 120575120601119892 is the neutron noise ingroup 119892 119863119892 is the diffusion coefficient in group 119892 and ]Σ119891119892is the production cross section in group 119892 Σ119905119892(120596) is definedas

Σ119905119892 (120596) = Σ119886119892 (120596) + sum

1198921015840 =119892

Σ119904119892rarr1198921015840 (3)

with

Σ119886119892 (120596) = Σ119886119892 +

119894120596

120592119892

(4)

where Σ119886119892 is the absorption cross section in group 119892 Σ1199041198921015840rarr119892is the scattering cross section from group 1198921015840 to group 119892 120592119892 isthe velocity of neutron in group119892 and120594119892(120596) is the frequency-dependent fission energy spectrum which is obtained fromthe equation of delayed neutron as

120594119892 (120596) = 120594119892 minussum

119895

120594119889

119892119895

119894120596120573

120582119895 + 119894120596

(5)

The last term in (2) 119878119892(r 120596) denotes the noise sourcein group 119892 which is calculated via the fluctuations of themacroscopic cross sections and the static flux 120601119892 as follows

119878119892 (r 120596) = minus 120575Σ119886119892 (120596) 120601119892 (r) minus sum1198921015840 =119892

120575Σ119904119892rarr1198921015840120601119892 (r)

+ sum

1198921015840 =119892

120575Σ1199041198921015840rarr119892 (120596) 1206011198921015840 (r)

+

1119896eff120594119892 (120596)sum

1198921015840

120575]Σ1198911198921015840 (120596) 1206011198921015840 (r)

(6)

where 120575Σ120572(120596) with 120572 = 119886 119891 119904 represents the fluctuationof macroscopic cross sections In (6) the fluctuation of thediffusion coefficient is neglectedThe neutron noise equation(2) is an inhomogeneous equation with an external sourceof which all quantities are frequency-dependent that iscomplex quantities In order to solve the noise equation itis necessary to define a noise source Therefore one needs toknow the characteristics of the static state such as the 119896eff and

Science and Technology of Nuclear Installations 3

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Richard Sanchezi

j

u

Figure 1 Spatial discretization of a hexagonal system in a 60-degree domain

the static flux120601119892(r) to calculate the noise source according to(6) This means that the solution of the static equation is alsorequiredThe simulator was implemented with two modulesa staticmodule solving the static equation and a noisemodulesolving the noise equation Finite difference approach is usedfor the spatial discretization of the systemwith hexagonal fuelassemblies where a hexagonal assembly is radially dividedinto triangular right prisms Therefore in a 3D modeleach fundamental node has five interfaces including twoequilateral triangular bases and three rectangular sides Moredetailed description about the noise simulator can be seen in[14]

Figure 1 displays the spatial discretization of a hexagonalsystem in a 60-degree domain In this figure the (119894119895)coordinates are used to handle the triangular finemeshes andthe (119906V) coordinates are used to handle the hexagonal coarsemeshes A power iterative solution procedure is implementedfor solving the balance equations of both the static and thenoise equations A coarse mesh finite difference (CMFD)method is employed for accelerating the convergence ofboth the static and the noise solutions in which a coarsemesh is radially defined as a hexagonal assembly In previousworks benchmarking calculations for the static state of theESFR core were performed and had a good agreement withERANOS [14] Noise calculations in a two-group model hadalso a good agreement with analytical solutions [13] Theseresults give certain assurance for the noise calculations andfurther investigation of the noise behaviour in fast reactors

3 Calculations of the Neutron Noise in an SFR

31 Core Model In the present work numerical calculationshave been performed based on a 2D model of Europeansodium-cooled fast reactor (ESFR) [16] using the multigroupnoise simulator [14] The core is designed with an outputpower of 3600MWtThe active core consists of two radial fuelregions with a height of 10m and an equivalent diameter of47m The inner fuel region consists of 201 fuel assemblies ofmixed U-Pu oxide fuel with Pu enrichment of 14 wt Theouter fuel region consists of 252 fuel assemblies of mixed U-Pu oxide with Pu enrichment of 16 wt The radial reflectorregion consists of 234 steel assemblies The operation of thecore is controlled by 24 control rods and 9 safety rods Thecross section data and the kinetic parameters of a 3Dmodel atthe beginning of cycle were taken from previous works whichwere performed for verifying the tool and for simulatingperiodic core deformation of the ESFR core [14 15] The dataare then processed for a 2D model in this calculation for thepurpose of investigating the properties of the neutron noiseon an axial planeThe radial configuration of 16th of the coreconstructed by 128 assemblies is shown in Figure 2

32 Modelling of the Noise Source Numerical calculationshave been performed based on 33 energy groups and 8groups of delayed neutron precursors using the neutronnoise simulator to investigate the properties of the space-and frequency-dependent neutron noise In general realistic

4 Science and Technology of Nuclear Installations

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Radial reflector assemblyPower generation control rodInner fuel assembly

Outer fuel assembly

Emergency control rod

Figure 2 Radial configuration of 16th of the ESFR core

minus001

minus0008

minus0006

minus0004

minus0002

0

0002

0004

0006

0008

001

Neutron energy (eV)

Noi

se so

urce

s (au

)

10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

Figure 3 Three representations of the noise source associated with the perturbations of absorption fission and scattering cross sectionsdenoted as 119878abs 119878fis and 119878scat respectively are assumed to be located at the core center

Science and Technology of Nuclear Installations 5

0

1

2

3

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Neutron energy (eV)

002040608

1

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

10minus1

100

101

102

103

104

105

106

107

SabsSfis

|120575120601|

(au

)

|120575120601|

(au

)

Sscat40

(a)

minus15

minus1

minus05

0

05

1

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2

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3

35

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 4 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 10Hz

0

02

04

06

08

1

12

14

16

18

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)

SabsSfis

002040608

1

Neutron energy (eV)

10minus1

100

101

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|120575120601|

(au

)

Sscat40

(a)

minus4

minus3

minus2

minus1

0

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2

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

SabsSfisSscat

(b)

Figure 5 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 100Hz

fluctuations in a nuclear system can be modelled via thefluctuations of the macroscopic cross sections and thenthe total noise source which is a linear combination of allfluctuations is calculated according to (6) using the static fluxand the fluctuations of the cross sections Therefore priorto assessing realistic fluctuations it is worth investigating

the properties of the noise induced by the perturbation ofeach cross section type separately

Assume that the noise source is originated from thesmall fluctuation of sodium density in the first fuel ringaround the central assembly This leads to the perturbationsof the macroscopic cross sections such as the absorption

6 Science and Technology of Nuclear Installations

0

05

1

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2

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3

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Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

Position (88) f = 10Hz

(a)

minus05

0

05

1

15

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3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

Position (88) f = 100Hz

(b)

Figure 6 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 10Hz

Position (88) f = 100Hz

0

005

01

015

02

025

03

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05

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

(a)

Position (88) f = 100Hz

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 7 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 100Hz

fission and scattering cross sections taking into account thespectrum shift in this area These perturbations are thenrepresented by three types of the noise sources associatedwith the perturbations of absorption fission and scatteringcross sections denoted as 119878120572 120572 = abs fis scat respectivelyWith this assumption the noise sources are calculated using

(6) and displayed in Figure 3 Since the amplitude of the119878scat is greater than that of the 119878abs and 119878fis by a factor of40 it dominates the behaviour of the total induced neutronnoise Therefore to investigate the total noise it is sufficientto investigate the noise induced by 119878scat in this case Howevera general perturbation could be sufficiently represented by

Science and Technology of Nuclear Installations 7

0

01

02

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Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

|120575120601|

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f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(a)

minus3

minus2

minus1

0

1

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Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(b)

Figure 8 Amplitude (a) and phase (b) of the noise at the position (1414) in the reflector as a function of neutron energyThenoise is calculatedat several frequencies of 01 to 100Hz

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 6

0 50 100 150 200 250 300 3500

02

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1

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Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 6

minus1

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus05

0

Distance from core center (cm)

(b)

Figure 9 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878abs

a combination of them Hence investigation of the behaviourof the noise induced by the three noise sources separatelyis needed It can be seen in Figure 3 that the three sourceshave different properties as a function of neutron energyThe source associated with the perturbation of the fissioncross section 119878fis has an opposite phase to that associatedwith the perturbation of the absorption cross section 119878abswhereas the source associated with the perturbation of thescattering cross section 119878scat oscillates with energy groups Itis because a downscattering reaction in a certain group leadsto a negative source in this group but a positive source inlower energy groups

The amplitude of the noise sources at energy lt01 keV isnegligibly small compared to that in higher energy due to thefact that the static flux in this energy range is smaller by threeorders of magnitude or greater Since most of the neutronsgenerated by fission are fast neutrons the source 119878fis isnegligibly small up to 100 keVThe perturbations are assumedto be uniformly distributed in the first fuel ring around thecentral assembly Due to the symmetrical property the noisecalculations have been performed in a 60-degree domainthat is 16th of the core at several frequencies of 01 1010 and 100Hz to investigate the properties of the space-and frequency-dependent noise In the calculated domain as

8 Science and Technology of Nuclear Installations

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 20

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 20

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0

Distance from core center (cm)

(b)

Figure 10 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878abs

0 50 100 150 200 250 300 3500

05

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Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

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(a)

0 50 100 150 200 250 300 35005

1

15

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Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 6

(b)

Figure 11 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878fis

shown in Figure 2 the perturbation is located in assembly(22)

33 Neutron Noise Induced by Localized Perturbations

331 Energy-Dependent Noise Numerical calculations havebeen performed to investigate the noise behaviour in awide range of frequencies with the noise sources given inFigure 3 The neutron noise can be described as a sum of two

components known as a local and a global component Theformer is a particular solution of the inhomogeneous equa-tion (2) which dominates the locations close to the sourcewhereas the latter is related to the point-kinetic response ofthe reactor Since the location (33) is close to the sourcethe local component of the noise has more effect Figure 4illustrates the amplitude and the phase of the inducedneutronnoise as a function of energy at a frequency of 10Hz(low frequency) at the location of assembly (33) (close to

Science and Technology of Nuclear Installations 9

0 50 100 150 200 250 300 3500

05

1

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Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 20

(a)

0 50 100 150 200 250 300 35005

1

15

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Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 20

(b)

Figure 12 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878fis

0100

200300 0

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0

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|120575120601|

(au

)

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)

(a)

0100

200300 0

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minus02

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e of120575

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ad)

y(cm

)

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(b)

Figure 13 Amplitude (a) and phase (b) of the space-dependent noise in group 6 induced by 119878scat at 10Hz

the source) Figure 5 shows the same quantities as Figure 4but at a frequency of 100Hz (high frequency) Similar to theneutron spectrum the amplitude of the induced noise is onlysignificant in the energy range gt10 keV Comparing amongthe three cases at the energy lt100 keV the noise induced by119878abs and 119878fis is slightly thermalized compared to that inducedby 119878scat The out-of-phase behaviour of the 119878scat in successivegroups in this energy range leads to the cancellation inthe induced noise The normalized amplitude of the noiseinduced by 119878abs at energylt100 keV is slightly greater than thatinduced by 119878fis despite the significantly greater magnitude of

119878abs in this energy rangeThismeans that the source at energygt100 keV has more significant contribution to the inducednoise over the energy range

The phase of the noise is almost constant with energyat a frequency of 10Hz except some variation of the phasein the case of 119878scat in the thermal and epithermal energiesas shown in Figure 4 At a higher frequency (100Hz) thevariation of the phase is larger in the thermal and epithermalenergy ranges but in the important range (gt10 keV) thephase remains constant with energy The noise induced by119878abs and 119878fis is out-of-phase due to the out-of-phase behaviour

10 Science and Technology of Nuclear Installations

0100

200300 0

100

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|120575120601|

(au

)

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(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

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|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

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Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

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5

10

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25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

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FuelsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

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International Journal ofPhotoenergy

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Nuclear InstallationsScience and Technology of

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Solar EnergyJournal of

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Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

2 Science and Technology of Nuclear Installations

In a realistic fluctuation of a system the noise sourcecan be modelled via a linear combination of the fluctuationsof all cross section types in the first order approximationThe contribution of the perturbation of each cross sectiontype in the total source depends on a specific scenario Forinstance the vibration of a control rod in an LWR can bemodelled as the vibration of the absorption cross sectionwhile to simulate the noise induced by fuel vibration it isnecessary to consider all cross section types including thefission cross section The problem in an SFR is even morecomplicated since the fluctuation of cross sections is stronglyenergy-dependent and is considered in a multigroup theoryTherefore prior to assessing realistic scenarios of fluctuationsin an SFR it is worth investigating the properties of theneutron noise induced by the fluctuations of absorption scat-tering and fission cross sections separately This is becausethese fluctuations lead to different properties of the noisesources respectively and as a result different properties ofthe induced noise

The present paper aims at investigating the properties ofneutron noise induced by localized perturbations in a largeSFR core using the noise simulator Three representations ofthe noise sources associated with the localized perturbationsof absorption fission and scattering cross sections respec-tively were assumed to be located at the center of the coreThe space- and energy-dependent neutron noise has beencalculated in a wide range of frequencies for example 01ndash100Hz

The paper is organized as follows Section 2 presentsbriefly the principles of the neutron noise equation inmultigroup diffusion theory which was solved in a frequencydomain in the noise simulator Section 3 describes the coremodel of a large SFR and the assumption of local per-turbations as the noise source Results and discussion onthe properties of the energy- and space-dependent noise atthe calculated frequencies are also presented Finally someconcluding remarks are given in Section 4

2 Principles of the Neutron Noise Simulator

The basis of the neutron noise equation is the assumptionof small stationary fluctuations of the system that is theaveraged value of a time-dependent quantity over time isequal to the static value Assume that all time-dependentterms119883(r 119905) can be split into a stationary component1198830(r)which corresponds to the value at the steady state plus a smallfluctuation 120575119883(r 119905) as

119883 (r 119905) = 1198830 (r) + 120575119883 (r 119905) (1)

By assuming the small fluctuations the first order noiseis taken into account products of fluctuation terms canbe neglected from time-dependent diffusion equations andthe result is a linear equation for the fluctuation of theflux Subtracting the static equation and after performing aFourier transform of all time-dependent terms the first order

space- and frequency-dependent neutron noise equation inmultigroup diffusion theory is written as follows

minusnabla sdot119863119892nabla120575120601119892 (r 120596) + Σ119905119892 (120596) 120575120601119892 (r 120596)

=

1119896eff120594119892 (120596)sum

1198921015840

]Σ1198911198921015840 (120596) 1205751206011198921015840 (r 120596)

+ sum

1198921015840 =119892

Σ1199041198921015840rarr119892 (120596) 1205751206011198921015840 (r 120596) + 119878119892 (r 120596)

(2)

where 119892 denotes the energy group 120575120601119892 is the neutron noise ingroup 119892 119863119892 is the diffusion coefficient in group 119892 and ]Σ119891119892is the production cross section in group 119892 Σ119905119892(120596) is definedas

Σ119905119892 (120596) = Σ119886119892 (120596) + sum

1198921015840 =119892

Σ119904119892rarr1198921015840 (3)

with

Σ119886119892 (120596) = Σ119886119892 +

119894120596

120592119892

(4)

where Σ119886119892 is the absorption cross section in group 119892 Σ1199041198921015840rarr119892is the scattering cross section from group 1198921015840 to group 119892 120592119892 isthe velocity of neutron in group119892 and120594119892(120596) is the frequency-dependent fission energy spectrum which is obtained fromthe equation of delayed neutron as

120594119892 (120596) = 120594119892 minussum

119895

120594119889

119892119895

119894120596120573

120582119895 + 119894120596

(5)

The last term in (2) 119878119892(r 120596) denotes the noise sourcein group 119892 which is calculated via the fluctuations of themacroscopic cross sections and the static flux 120601119892 as follows

119878119892 (r 120596) = minus 120575Σ119886119892 (120596) 120601119892 (r) minus sum1198921015840 =119892

120575Σ119904119892rarr1198921015840120601119892 (r)

+ sum

1198921015840 =119892

120575Σ1199041198921015840rarr119892 (120596) 1206011198921015840 (r)

+

1119896eff120594119892 (120596)sum

1198921015840

120575]Σ1198911198921015840 (120596) 1206011198921015840 (r)

(6)

where 120575Σ120572(120596) with 120572 = 119886 119891 119904 represents the fluctuationof macroscopic cross sections In (6) the fluctuation of thediffusion coefficient is neglectedThe neutron noise equation(2) is an inhomogeneous equation with an external sourceof which all quantities are frequency-dependent that iscomplex quantities In order to solve the noise equation itis necessary to define a noise source Therefore one needs toknow the characteristics of the static state such as the 119896eff and

Science and Technology of Nuclear Installations 3

1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 9

12

34

56

78

910

111

213

14

171

8151

6

2

1

3

4

5

6

7

8

9

12

34

56 7

8 910

1112

1314

1516 17

18 1920

2122 23

2425

12

34

56 7

8 910

1112

1314

1516 17

18 1920

2122 23

241

23

45

6 78 9

1011

1213

1415

16 1718 19

2021

22 23

12

34

56 7

8 910

1112

1314

1516 17

18 1920

2122

12

34

56

78

910

1112

1314

1516 17

18 1920

21

12

34

56 7

8 910

1112

1314

1516

1718 19

201

23

45

6 78 9

1011

1213

1415

1617

18 19

12

34

56 7

89

10 1112

1314

1516 17

181

23

45

6 78

910 11

1213

141

23

45

6 78

910

12

34

56

12

34

5

1

78

9

1112

13

1516 17

Richard Sanchezi

j

u

Figure 1 Spatial discretization of a hexagonal system in a 60-degree domain

the static flux120601119892(r) to calculate the noise source according to(6) This means that the solution of the static equation is alsorequiredThe simulator was implemented with two modulesa staticmodule solving the static equation and a noisemodulesolving the noise equation Finite difference approach is usedfor the spatial discretization of the systemwith hexagonal fuelassemblies where a hexagonal assembly is radially dividedinto triangular right prisms Therefore in a 3D modeleach fundamental node has five interfaces including twoequilateral triangular bases and three rectangular sides Moredetailed description about the noise simulator can be seen in[14]

Figure 1 displays the spatial discretization of a hexagonalsystem in a 60-degree domain In this figure the (119894119895)coordinates are used to handle the triangular finemeshes andthe (119906V) coordinates are used to handle the hexagonal coarsemeshes A power iterative solution procedure is implementedfor solving the balance equations of both the static and thenoise equations A coarse mesh finite difference (CMFD)method is employed for accelerating the convergence ofboth the static and the noise solutions in which a coarsemesh is radially defined as a hexagonal assembly In previousworks benchmarking calculations for the static state of theESFR core were performed and had a good agreement withERANOS [14] Noise calculations in a two-group model hadalso a good agreement with analytical solutions [13] Theseresults give certain assurance for the noise calculations andfurther investigation of the noise behaviour in fast reactors

3 Calculations of the Neutron Noise in an SFR

31 Core Model In the present work numerical calculationshave been performed based on a 2D model of Europeansodium-cooled fast reactor (ESFR) [16] using the multigroupnoise simulator [14] The core is designed with an outputpower of 3600MWtThe active core consists of two radial fuelregions with a height of 10m and an equivalent diameter of47m The inner fuel region consists of 201 fuel assemblies ofmixed U-Pu oxide fuel with Pu enrichment of 14 wt Theouter fuel region consists of 252 fuel assemblies of mixed U-Pu oxide with Pu enrichment of 16 wt The radial reflectorregion consists of 234 steel assemblies The operation of thecore is controlled by 24 control rods and 9 safety rods Thecross section data and the kinetic parameters of a 3Dmodel atthe beginning of cycle were taken from previous works whichwere performed for verifying the tool and for simulatingperiodic core deformation of the ESFR core [14 15] The dataare then processed for a 2D model in this calculation for thepurpose of investigating the properties of the neutron noiseon an axial planeThe radial configuration of 16th of the coreconstructed by 128 assemblies is shown in Figure 2

32 Modelling of the Noise Source Numerical calculationshave been performed based on 33 energy groups and 8groups of delayed neutron precursors using the neutronnoise simulator to investigate the properties of the space-and frequency-dependent neutron noise In general realistic

4 Science and Technology of Nuclear Installations

1691610

16111612

16131614

16151616

1516

1416

1316

1216

1116

1016

916815

814

915914

1015

1115

1215

1315

15151514

15131512

15111510

159158

148149

14101411

14121413

14141415

1314

1214

1114

1014

137138

1391310

1311

13121313

1213

1113

1013

913

813

713

127

128129

12101211

1212

1112

1012

912

812

712

116117

118119

11101111

1011

611

711

811

911

106

107108

1091010

910

810

710

610

95

9697

9899

89

79

69

59

85

8687

88

78

68

58

7475

7677

67

57

47

6465

66

56

46

5354

55

45

35

4344

3433

23

32

2211

Radial reflector assemblyPower generation control rodInner fuel assembly

Outer fuel assembly

Emergency control rod

Figure 2 Radial configuration of 16th of the ESFR core

minus001

minus0008

minus0006

minus0004

minus0002

0

0002

0004

0006

0008

001

Neutron energy (eV)

Noi

se so

urce

s (au

)

10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

Figure 3 Three representations of the noise source associated with the perturbations of absorption fission and scattering cross sectionsdenoted as 119878abs 119878fis and 119878scat respectively are assumed to be located at the core center

Science and Technology of Nuclear Installations 5

0

1

2

3

4

5

6

Neutron energy (eV)

002040608

1

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

10minus1

100

101

102

103

104

105

106

107

SabsSfis

|120575120601|

(au

)

|120575120601|

(au

)

Sscat40

(a)

minus15

minus1

minus05

0

05

1

15

2

25

3

35

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 4 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 10Hz

0

02

04

06

08

1

12

14

16

18

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)

SabsSfis

002040608

1

Neutron energy (eV)

10minus1

100

101

102

103

104

105

106

107

|120575120601|

(au

)

Sscat40

(a)

minus4

minus3

minus2

minus1

0

1

2

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

SabsSfisSscat

(b)

Figure 5 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 100Hz

fluctuations in a nuclear system can be modelled via thefluctuations of the macroscopic cross sections and thenthe total noise source which is a linear combination of allfluctuations is calculated according to (6) using the static fluxand the fluctuations of the cross sections Therefore priorto assessing realistic fluctuations it is worth investigating

the properties of the noise induced by the perturbation ofeach cross section type separately

Assume that the noise source is originated from thesmall fluctuation of sodium density in the first fuel ringaround the central assembly This leads to the perturbationsof the macroscopic cross sections such as the absorption

6 Science and Technology of Nuclear Installations

0

05

1

15

2

25

3

35

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

Position (88) f = 10Hz

(a)

minus05

0

05

1

15

2

25

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

Position (88) f = 100Hz

(b)

Figure 6 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 10Hz

Position (88) f = 100Hz

0

005

01

015

02

025

03

035

04

045

05

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

(a)

Position (88) f = 100Hz

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 7 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 100Hz

fission and scattering cross sections taking into account thespectrum shift in this area These perturbations are thenrepresented by three types of the noise sources associatedwith the perturbations of absorption fission and scatteringcross sections denoted as 119878120572 120572 = abs fis scat respectivelyWith this assumption the noise sources are calculated using

(6) and displayed in Figure 3 Since the amplitude of the119878scat is greater than that of the 119878abs and 119878fis by a factor of40 it dominates the behaviour of the total induced neutronnoise Therefore to investigate the total noise it is sufficientto investigate the noise induced by 119878scat in this case Howevera general perturbation could be sufficiently represented by

Science and Technology of Nuclear Installations 7

0

01

02

03

04

05

06

07

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(a)

minus3

minus2

minus1

0

1

2

3

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(b)

Figure 8 Amplitude (a) and phase (b) of the noise at the position (1414) in the reflector as a function of neutron energyThenoise is calculatedat several frequencies of 01 to 100Hz

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 6

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

16

18

2

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 6

minus1

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus05

0

Distance from core center (cm)

(b)

Figure 9 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878abs

a combination of them Hence investigation of the behaviourof the noise induced by the three noise sources separatelyis needed It can be seen in Figure 3 that the three sourceshave different properties as a function of neutron energyThe source associated with the perturbation of the fissioncross section 119878fis has an opposite phase to that associatedwith the perturbation of the absorption cross section 119878abswhereas the source associated with the perturbation of thescattering cross section 119878scat oscillates with energy groups Itis because a downscattering reaction in a certain group leadsto a negative source in this group but a positive source inlower energy groups

The amplitude of the noise sources at energy lt01 keV isnegligibly small compared to that in higher energy due to thefact that the static flux in this energy range is smaller by threeorders of magnitude or greater Since most of the neutronsgenerated by fission are fast neutrons the source 119878fis isnegligibly small up to 100 keVThe perturbations are assumedto be uniformly distributed in the first fuel ring around thecentral assembly Due to the symmetrical property the noisecalculations have been performed in a 60-degree domainthat is 16th of the core at several frequencies of 01 1010 and 100Hz to investigate the properties of the space-and frequency-dependent noise In the calculated domain as

8 Science and Technology of Nuclear Installations

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 20

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 20

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0

Distance from core center (cm)

(b)

Figure 10 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878abs

0 50 100 150 200 250 300 3500

05

1

15

2

25

3

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 6

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 6

(b)

Figure 11 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878fis

shown in Figure 2 the perturbation is located in assembly(22)

33 Neutron Noise Induced by Localized Perturbations

331 Energy-Dependent Noise Numerical calculations havebeen performed to investigate the noise behaviour in awide range of frequencies with the noise sources given inFigure 3 The neutron noise can be described as a sum of two

components known as a local and a global component Theformer is a particular solution of the inhomogeneous equa-tion (2) which dominates the locations close to the sourcewhereas the latter is related to the point-kinetic response ofthe reactor Since the location (33) is close to the sourcethe local component of the noise has more effect Figure 4illustrates the amplitude and the phase of the inducedneutronnoise as a function of energy at a frequency of 10Hz(low frequency) at the location of assembly (33) (close to

Science and Technology of Nuclear Installations 9

0 50 100 150 200 250 300 3500

05

1

15

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 20

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 20

(b)

Figure 12 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878fis

0100

200300 0

100

200

300

0

10

20

30

|120575120601|

(au

)

x (cm)

y(cm

)

(a)

0100

200300 0

100

200

300

minus02

minus015

minus01

minus005

0

Phas

e of120575

120601(r

ad)

y(cm

)

x (cm)

(b)

Figure 13 Amplitude (a) and phase (b) of the space-dependent noise in group 6 induced by 119878scat at 10Hz

the source) Figure 5 shows the same quantities as Figure 4but at a frequency of 100Hz (high frequency) Similar to theneutron spectrum the amplitude of the induced noise is onlysignificant in the energy range gt10 keV Comparing amongthe three cases at the energy lt100 keV the noise induced by119878abs and 119878fis is slightly thermalized compared to that inducedby 119878scat The out-of-phase behaviour of the 119878scat in successivegroups in this energy range leads to the cancellation inthe induced noise The normalized amplitude of the noiseinduced by 119878abs at energylt100 keV is slightly greater than thatinduced by 119878fis despite the significantly greater magnitude of

119878abs in this energy rangeThismeans that the source at energygt100 keV has more significant contribution to the inducednoise over the energy range

The phase of the noise is almost constant with energyat a frequency of 10Hz except some variation of the phasein the case of 119878scat in the thermal and epithermal energiesas shown in Figure 4 At a higher frequency (100Hz) thevariation of the phase is larger in the thermal and epithermalenergy ranges but in the important range (gt10 keV) thephase remains constant with energy The noise induced by119878abs and 119878fis is out-of-phase due to the out-of-phase behaviour

10 Science and Technology of Nuclear Installations

0100

200300 0

100

200

3000

2

4

6

8

|120575120601|

(au

)

x (cm)y (cm)

(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

Distance from core center (cm)

Group 6

|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

15

2

25

3

35

Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

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International Journal of

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FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

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Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Science and Technology of Nuclear Installations 3

1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 9

12

34

56

78

910

111

213

14

171

8151

6

2

1

3

4

5

6

7

8

9

12

34

56 7

8 910

1112

1314

1516 17

18 1920

2122 23

2425

12

34

56 7

8 910

1112

1314

1516 17

18 1920

2122 23

241

23

45

6 78 9

1011

1213

1415

16 1718 19

2021

22 23

12

34

56 7

8 910

1112

1314

1516 17

18 1920

2122

12

34

56

78

910

1112

1314

1516 17

18 1920

21

12

34

56 7

8 910

1112

1314

1516

1718 19

201

23

45

6 78 9

1011

1213

1415

1617

18 19

12

34

56 7

89

10 1112

1314

1516 17

181

23

45

6 78

910 11

1213

141

23

45

6 78

910

12

34

56

12

34

5

1

78

9

1112

13

1516 17

Richard Sanchezi

j

u

Figure 1 Spatial discretization of a hexagonal system in a 60-degree domain

the static flux120601119892(r) to calculate the noise source according to(6) This means that the solution of the static equation is alsorequiredThe simulator was implemented with two modulesa staticmodule solving the static equation and a noisemodulesolving the noise equation Finite difference approach is usedfor the spatial discretization of the systemwith hexagonal fuelassemblies where a hexagonal assembly is radially dividedinto triangular right prisms Therefore in a 3D modeleach fundamental node has five interfaces including twoequilateral triangular bases and three rectangular sides Moredetailed description about the noise simulator can be seen in[14]

Figure 1 displays the spatial discretization of a hexagonalsystem in a 60-degree domain In this figure the (119894119895)coordinates are used to handle the triangular finemeshes andthe (119906V) coordinates are used to handle the hexagonal coarsemeshes A power iterative solution procedure is implementedfor solving the balance equations of both the static and thenoise equations A coarse mesh finite difference (CMFD)method is employed for accelerating the convergence ofboth the static and the noise solutions in which a coarsemesh is radially defined as a hexagonal assembly In previousworks benchmarking calculations for the static state of theESFR core were performed and had a good agreement withERANOS [14] Noise calculations in a two-group model hadalso a good agreement with analytical solutions [13] Theseresults give certain assurance for the noise calculations andfurther investigation of the noise behaviour in fast reactors

3 Calculations of the Neutron Noise in an SFR

31 Core Model In the present work numerical calculationshave been performed based on a 2D model of Europeansodium-cooled fast reactor (ESFR) [16] using the multigroupnoise simulator [14] The core is designed with an outputpower of 3600MWtThe active core consists of two radial fuelregions with a height of 10m and an equivalent diameter of47m The inner fuel region consists of 201 fuel assemblies ofmixed U-Pu oxide fuel with Pu enrichment of 14 wt Theouter fuel region consists of 252 fuel assemblies of mixed U-Pu oxide with Pu enrichment of 16 wt The radial reflectorregion consists of 234 steel assemblies The operation of thecore is controlled by 24 control rods and 9 safety rods Thecross section data and the kinetic parameters of a 3Dmodel atthe beginning of cycle were taken from previous works whichwere performed for verifying the tool and for simulatingperiodic core deformation of the ESFR core [14 15] The dataare then processed for a 2D model in this calculation for thepurpose of investigating the properties of the neutron noiseon an axial planeThe radial configuration of 16th of the coreconstructed by 128 assemblies is shown in Figure 2

32 Modelling of the Noise Source Numerical calculationshave been performed based on 33 energy groups and 8groups of delayed neutron precursors using the neutronnoise simulator to investigate the properties of the space-and frequency-dependent neutron noise In general realistic

4 Science and Technology of Nuclear Installations

1691610

16111612

16131614

16151616

1516

1416

1316

1216

1116

1016

916815

814

915914

1015

1115

1215

1315

15151514

15131512

15111510

159158

148149

14101411

14121413

14141415

1314

1214

1114

1014

137138

1391310

1311

13121313

1213

1113

1013

913

813

713

127

128129

12101211

1212

1112

1012

912

812

712

116117

118119

11101111

1011

611

711

811

911

106

107108

1091010

910

810

710

610

95

9697

9899

89

79

69

59

85

8687

88

78

68

58

7475

7677

67

57

47

6465

66

56

46

5354

55

45

35

4344

3433

23

32

2211

Radial reflector assemblyPower generation control rodInner fuel assembly

Outer fuel assembly

Emergency control rod

Figure 2 Radial configuration of 16th of the ESFR core

minus001

minus0008

minus0006

minus0004

minus0002

0

0002

0004

0006

0008

001

Neutron energy (eV)

Noi

se so

urce

s (au

)

10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

Figure 3 Three representations of the noise source associated with the perturbations of absorption fission and scattering cross sectionsdenoted as 119878abs 119878fis and 119878scat respectively are assumed to be located at the core center

Science and Technology of Nuclear Installations 5

0

1

2

3

4

5

6

Neutron energy (eV)

002040608

1

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

10minus1

100

101

102

103

104

105

106

107

SabsSfis

|120575120601|

(au

)

|120575120601|

(au

)

Sscat40

(a)

minus15

minus1

minus05

0

05

1

15

2

25

3

35

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 4 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 10Hz

0

02

04

06

08

1

12

14

16

18

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)

SabsSfis

002040608

1

Neutron energy (eV)

10minus1

100

101

102

103

104

105

106

107

|120575120601|

(au

)

Sscat40

(a)

minus4

minus3

minus2

minus1

0

1

2

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

SabsSfisSscat

(b)

Figure 5 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 100Hz

fluctuations in a nuclear system can be modelled via thefluctuations of the macroscopic cross sections and thenthe total noise source which is a linear combination of allfluctuations is calculated according to (6) using the static fluxand the fluctuations of the cross sections Therefore priorto assessing realistic fluctuations it is worth investigating

the properties of the noise induced by the perturbation ofeach cross section type separately

Assume that the noise source is originated from thesmall fluctuation of sodium density in the first fuel ringaround the central assembly This leads to the perturbationsof the macroscopic cross sections such as the absorption

6 Science and Technology of Nuclear Installations

0

05

1

15

2

25

3

35

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

Position (88) f = 10Hz

(a)

minus05

0

05

1

15

2

25

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

Position (88) f = 100Hz

(b)

Figure 6 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 10Hz

Position (88) f = 100Hz

0

005

01

015

02

025

03

035

04

045

05

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

(a)

Position (88) f = 100Hz

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 7 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 100Hz

fission and scattering cross sections taking into account thespectrum shift in this area These perturbations are thenrepresented by three types of the noise sources associatedwith the perturbations of absorption fission and scatteringcross sections denoted as 119878120572 120572 = abs fis scat respectivelyWith this assumption the noise sources are calculated using

(6) and displayed in Figure 3 Since the amplitude of the119878scat is greater than that of the 119878abs and 119878fis by a factor of40 it dominates the behaviour of the total induced neutronnoise Therefore to investigate the total noise it is sufficientto investigate the noise induced by 119878scat in this case Howevera general perturbation could be sufficiently represented by

Science and Technology of Nuclear Installations 7

0

01

02

03

04

05

06

07

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(a)

minus3

minus2

minus1

0

1

2

3

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(b)

Figure 8 Amplitude (a) and phase (b) of the noise at the position (1414) in the reflector as a function of neutron energyThenoise is calculatedat several frequencies of 01 to 100Hz

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 6

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

16

18

2

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 6

minus1

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus05

0

Distance from core center (cm)

(b)

Figure 9 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878abs

a combination of them Hence investigation of the behaviourof the noise induced by the three noise sources separatelyis needed It can be seen in Figure 3 that the three sourceshave different properties as a function of neutron energyThe source associated with the perturbation of the fissioncross section 119878fis has an opposite phase to that associatedwith the perturbation of the absorption cross section 119878abswhereas the source associated with the perturbation of thescattering cross section 119878scat oscillates with energy groups Itis because a downscattering reaction in a certain group leadsto a negative source in this group but a positive source inlower energy groups

The amplitude of the noise sources at energy lt01 keV isnegligibly small compared to that in higher energy due to thefact that the static flux in this energy range is smaller by threeorders of magnitude or greater Since most of the neutronsgenerated by fission are fast neutrons the source 119878fis isnegligibly small up to 100 keVThe perturbations are assumedto be uniformly distributed in the first fuel ring around thecentral assembly Due to the symmetrical property the noisecalculations have been performed in a 60-degree domainthat is 16th of the core at several frequencies of 01 1010 and 100Hz to investigate the properties of the space-and frequency-dependent noise In the calculated domain as

8 Science and Technology of Nuclear Installations

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 20

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 20

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0

Distance from core center (cm)

(b)

Figure 10 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878abs

0 50 100 150 200 250 300 3500

05

1

15

2

25

3

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 6

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 6

(b)

Figure 11 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878fis

shown in Figure 2 the perturbation is located in assembly(22)

33 Neutron Noise Induced by Localized Perturbations

331 Energy-Dependent Noise Numerical calculations havebeen performed to investigate the noise behaviour in awide range of frequencies with the noise sources given inFigure 3 The neutron noise can be described as a sum of two

components known as a local and a global component Theformer is a particular solution of the inhomogeneous equa-tion (2) which dominates the locations close to the sourcewhereas the latter is related to the point-kinetic response ofthe reactor Since the location (33) is close to the sourcethe local component of the noise has more effect Figure 4illustrates the amplitude and the phase of the inducedneutronnoise as a function of energy at a frequency of 10Hz(low frequency) at the location of assembly (33) (close to

Science and Technology of Nuclear Installations 9

0 50 100 150 200 250 300 3500

05

1

15

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 20

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 20

(b)

Figure 12 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878fis

0100

200300 0

100

200

300

0

10

20

30

|120575120601|

(au

)

x (cm)

y(cm

)

(a)

0100

200300 0

100

200

300

minus02

minus015

minus01

minus005

0

Phas

e of120575

120601(r

ad)

y(cm

)

x (cm)

(b)

Figure 13 Amplitude (a) and phase (b) of the space-dependent noise in group 6 induced by 119878scat at 10Hz

the source) Figure 5 shows the same quantities as Figure 4but at a frequency of 100Hz (high frequency) Similar to theneutron spectrum the amplitude of the induced noise is onlysignificant in the energy range gt10 keV Comparing amongthe three cases at the energy lt100 keV the noise induced by119878abs and 119878fis is slightly thermalized compared to that inducedby 119878scat The out-of-phase behaviour of the 119878scat in successivegroups in this energy range leads to the cancellation inthe induced noise The normalized amplitude of the noiseinduced by 119878abs at energylt100 keV is slightly greater than thatinduced by 119878fis despite the significantly greater magnitude of

119878abs in this energy rangeThismeans that the source at energygt100 keV has more significant contribution to the inducednoise over the energy range

The phase of the noise is almost constant with energyat a frequency of 10Hz except some variation of the phasein the case of 119878scat in the thermal and epithermal energiesas shown in Figure 4 At a higher frequency (100Hz) thevariation of the phase is larger in the thermal and epithermalenergy ranges but in the important range (gt10 keV) thephase remains constant with energy The noise induced by119878abs and 119878fis is out-of-phase due to the out-of-phase behaviour

10 Science and Technology of Nuclear Installations

0100

200300 0

100

200

3000

2

4

6

8

|120575120601|

(au

)

x (cm)y (cm)

(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

Distance from core center (cm)

Group 6

|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

15

2

25

3

35

Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

4 Science and Technology of Nuclear Installations

1691610

16111612

16131614

16151616

1516

1416

1316

1216

1116

1016

916815

814

915914

1015

1115

1215

1315

15151514

15131512

15111510

159158

148149

14101411

14121413

14141415

1314

1214

1114

1014

137138

1391310

1311

13121313

1213

1113

1013

913

813

713

127

128129

12101211

1212

1112

1012

912

812

712

116117

118119

11101111

1011

611

711

811

911

106

107108

1091010

910

810

710

610

95

9697

9899

89

79

69

59

85

8687

88

78

68

58

7475

7677

67

57

47

6465

66

56

46

5354

55

45

35

4344

3433

23

32

2211

Radial reflector assemblyPower generation control rodInner fuel assembly

Outer fuel assembly

Emergency control rod

Figure 2 Radial configuration of 16th of the ESFR core

minus001

minus0008

minus0006

minus0004

minus0002

0

0002

0004

0006

0008

001

Neutron energy (eV)

Noi

se so

urce

s (au

)

10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

Figure 3 Three representations of the noise source associated with the perturbations of absorption fission and scattering cross sectionsdenoted as 119878abs 119878fis and 119878scat respectively are assumed to be located at the core center

Science and Technology of Nuclear Installations 5

0

1

2

3

4

5

6

Neutron energy (eV)

002040608

1

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

10minus1

100

101

102

103

104

105

106

107

SabsSfis

|120575120601|

(au

)

|120575120601|

(au

)

Sscat40

(a)

minus15

minus1

minus05

0

05

1

15

2

25

3

35

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 4 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 10Hz

0

02

04

06

08

1

12

14

16

18

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)

SabsSfis

002040608

1

Neutron energy (eV)

10minus1

100

101

102

103

104

105

106

107

|120575120601|

(au

)

Sscat40

(a)

minus4

minus3

minus2

minus1

0

1

2

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

SabsSfisSscat

(b)

Figure 5 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 100Hz

fluctuations in a nuclear system can be modelled via thefluctuations of the macroscopic cross sections and thenthe total noise source which is a linear combination of allfluctuations is calculated according to (6) using the static fluxand the fluctuations of the cross sections Therefore priorto assessing realistic fluctuations it is worth investigating

the properties of the noise induced by the perturbation ofeach cross section type separately

Assume that the noise source is originated from thesmall fluctuation of sodium density in the first fuel ringaround the central assembly This leads to the perturbationsof the macroscopic cross sections such as the absorption

6 Science and Technology of Nuclear Installations

0

05

1

15

2

25

3

35

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

Position (88) f = 10Hz

(a)

minus05

0

05

1

15

2

25

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

Position (88) f = 100Hz

(b)

Figure 6 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 10Hz

Position (88) f = 100Hz

0

005

01

015

02

025

03

035

04

045

05

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

(a)

Position (88) f = 100Hz

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 7 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 100Hz

fission and scattering cross sections taking into account thespectrum shift in this area These perturbations are thenrepresented by three types of the noise sources associatedwith the perturbations of absorption fission and scatteringcross sections denoted as 119878120572 120572 = abs fis scat respectivelyWith this assumption the noise sources are calculated using

(6) and displayed in Figure 3 Since the amplitude of the119878scat is greater than that of the 119878abs and 119878fis by a factor of40 it dominates the behaviour of the total induced neutronnoise Therefore to investigate the total noise it is sufficientto investigate the noise induced by 119878scat in this case Howevera general perturbation could be sufficiently represented by

Science and Technology of Nuclear Installations 7

0

01

02

03

04

05

06

07

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(a)

minus3

minus2

minus1

0

1

2

3

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(b)

Figure 8 Amplitude (a) and phase (b) of the noise at the position (1414) in the reflector as a function of neutron energyThenoise is calculatedat several frequencies of 01 to 100Hz

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 6

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

16

18

2

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 6

minus1

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus05

0

Distance from core center (cm)

(b)

Figure 9 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878abs

a combination of them Hence investigation of the behaviourof the noise induced by the three noise sources separatelyis needed It can be seen in Figure 3 that the three sourceshave different properties as a function of neutron energyThe source associated with the perturbation of the fissioncross section 119878fis has an opposite phase to that associatedwith the perturbation of the absorption cross section 119878abswhereas the source associated with the perturbation of thescattering cross section 119878scat oscillates with energy groups Itis because a downscattering reaction in a certain group leadsto a negative source in this group but a positive source inlower energy groups

The amplitude of the noise sources at energy lt01 keV isnegligibly small compared to that in higher energy due to thefact that the static flux in this energy range is smaller by threeorders of magnitude or greater Since most of the neutronsgenerated by fission are fast neutrons the source 119878fis isnegligibly small up to 100 keVThe perturbations are assumedto be uniformly distributed in the first fuel ring around thecentral assembly Due to the symmetrical property the noisecalculations have been performed in a 60-degree domainthat is 16th of the core at several frequencies of 01 1010 and 100Hz to investigate the properties of the space-and frequency-dependent noise In the calculated domain as

8 Science and Technology of Nuclear Installations

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 20

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 20

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0

Distance from core center (cm)

(b)

Figure 10 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878abs

0 50 100 150 200 250 300 3500

05

1

15

2

25

3

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 6

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 6

(b)

Figure 11 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878fis

shown in Figure 2 the perturbation is located in assembly(22)

33 Neutron Noise Induced by Localized Perturbations

331 Energy-Dependent Noise Numerical calculations havebeen performed to investigate the noise behaviour in awide range of frequencies with the noise sources given inFigure 3 The neutron noise can be described as a sum of two

components known as a local and a global component Theformer is a particular solution of the inhomogeneous equa-tion (2) which dominates the locations close to the sourcewhereas the latter is related to the point-kinetic response ofthe reactor Since the location (33) is close to the sourcethe local component of the noise has more effect Figure 4illustrates the amplitude and the phase of the inducedneutronnoise as a function of energy at a frequency of 10Hz(low frequency) at the location of assembly (33) (close to

Science and Technology of Nuclear Installations 9

0 50 100 150 200 250 300 3500

05

1

15

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 20

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 20

(b)

Figure 12 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878fis

0100

200300 0

100

200

300

0

10

20

30

|120575120601|

(au

)

x (cm)

y(cm

)

(a)

0100

200300 0

100

200

300

minus02

minus015

minus01

minus005

0

Phas

e of120575

120601(r

ad)

y(cm

)

x (cm)

(b)

Figure 13 Amplitude (a) and phase (b) of the space-dependent noise in group 6 induced by 119878scat at 10Hz

the source) Figure 5 shows the same quantities as Figure 4but at a frequency of 100Hz (high frequency) Similar to theneutron spectrum the amplitude of the induced noise is onlysignificant in the energy range gt10 keV Comparing amongthe three cases at the energy lt100 keV the noise induced by119878abs and 119878fis is slightly thermalized compared to that inducedby 119878scat The out-of-phase behaviour of the 119878scat in successivegroups in this energy range leads to the cancellation inthe induced noise The normalized amplitude of the noiseinduced by 119878abs at energylt100 keV is slightly greater than thatinduced by 119878fis despite the significantly greater magnitude of

119878abs in this energy rangeThismeans that the source at energygt100 keV has more significant contribution to the inducednoise over the energy range

The phase of the noise is almost constant with energyat a frequency of 10Hz except some variation of the phasein the case of 119878scat in the thermal and epithermal energiesas shown in Figure 4 At a higher frequency (100Hz) thevariation of the phase is larger in the thermal and epithermalenergy ranges but in the important range (gt10 keV) thephase remains constant with energy The noise induced by119878abs and 119878fis is out-of-phase due to the out-of-phase behaviour

10 Science and Technology of Nuclear Installations

0100

200300 0

100

200

3000

2

4

6

8

|120575120601|

(au

)

x (cm)y (cm)

(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

Distance from core center (cm)

Group 6

|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

15

2

25

3

35

Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

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FuelsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

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International Journal ofPhotoenergy

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Nuclear InstallationsScience and Technology of

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Solar EnergyJournal of

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Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

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High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Science and Technology of Nuclear Installations 5

0

1

2

3

4

5

6

Neutron energy (eV)

002040608

1

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

10minus1

100

101

102

103

104

105

106

107

SabsSfis

|120575120601|

(au

)

|120575120601|

(au

)

Sscat40

(a)

minus15

minus1

minus05

0

05

1

15

2

25

3

35

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 4 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 10Hz

0

02

04

06

08

1

12

14

16

18

Neutron energy (eV)

Normalization

10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)

SabsSfis

002040608

1

Neutron energy (eV)

10minus1

100

101

102

103

104

105

106

107

|120575120601|

(au

)

Sscat40

(a)

minus4

minus3

minus2

minus1

0

1

2

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

SabsSfisSscat

(b)

Figure 5 Amplitude (a) and phase (b) of the noise at the location of assembly (33) (close to the source) as a function of neutron energyThenoise is calculated at a frequency of 100Hz

fluctuations in a nuclear system can be modelled via thefluctuations of the macroscopic cross sections and thenthe total noise source which is a linear combination of allfluctuations is calculated according to (6) using the static fluxand the fluctuations of the cross sections Therefore priorto assessing realistic fluctuations it is worth investigating

the properties of the noise induced by the perturbation ofeach cross section type separately

Assume that the noise source is originated from thesmall fluctuation of sodium density in the first fuel ringaround the central assembly This leads to the perturbationsof the macroscopic cross sections such as the absorption

6 Science and Technology of Nuclear Installations

0

05

1

15

2

25

3

35

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

Position (88) f = 10Hz

(a)

minus05

0

05

1

15

2

25

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

Position (88) f = 100Hz

(b)

Figure 6 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 10Hz

Position (88) f = 100Hz

0

005

01

015

02

025

03

035

04

045

05

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

(a)

Position (88) f = 100Hz

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 7 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 100Hz

fission and scattering cross sections taking into account thespectrum shift in this area These perturbations are thenrepresented by three types of the noise sources associatedwith the perturbations of absorption fission and scatteringcross sections denoted as 119878120572 120572 = abs fis scat respectivelyWith this assumption the noise sources are calculated using

(6) and displayed in Figure 3 Since the amplitude of the119878scat is greater than that of the 119878abs and 119878fis by a factor of40 it dominates the behaviour of the total induced neutronnoise Therefore to investigate the total noise it is sufficientto investigate the noise induced by 119878scat in this case Howevera general perturbation could be sufficiently represented by

Science and Technology of Nuclear Installations 7

0

01

02

03

04

05

06

07

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(a)

minus3

minus2

minus1

0

1

2

3

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(b)

Figure 8 Amplitude (a) and phase (b) of the noise at the position (1414) in the reflector as a function of neutron energyThenoise is calculatedat several frequencies of 01 to 100Hz

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 6

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

16

18

2

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 6

minus1

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus05

0

Distance from core center (cm)

(b)

Figure 9 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878abs

a combination of them Hence investigation of the behaviourof the noise induced by the three noise sources separatelyis needed It can be seen in Figure 3 that the three sourceshave different properties as a function of neutron energyThe source associated with the perturbation of the fissioncross section 119878fis has an opposite phase to that associatedwith the perturbation of the absorption cross section 119878abswhereas the source associated with the perturbation of thescattering cross section 119878scat oscillates with energy groups Itis because a downscattering reaction in a certain group leadsto a negative source in this group but a positive source inlower energy groups

The amplitude of the noise sources at energy lt01 keV isnegligibly small compared to that in higher energy due to thefact that the static flux in this energy range is smaller by threeorders of magnitude or greater Since most of the neutronsgenerated by fission are fast neutrons the source 119878fis isnegligibly small up to 100 keVThe perturbations are assumedto be uniformly distributed in the first fuel ring around thecentral assembly Due to the symmetrical property the noisecalculations have been performed in a 60-degree domainthat is 16th of the core at several frequencies of 01 1010 and 100Hz to investigate the properties of the space-and frequency-dependent noise In the calculated domain as

8 Science and Technology of Nuclear Installations

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 20

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 20

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0

Distance from core center (cm)

(b)

Figure 10 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878abs

0 50 100 150 200 250 300 3500

05

1

15

2

25

3

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 6

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 6

(b)

Figure 11 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878fis

shown in Figure 2 the perturbation is located in assembly(22)

33 Neutron Noise Induced by Localized Perturbations

331 Energy-Dependent Noise Numerical calculations havebeen performed to investigate the noise behaviour in awide range of frequencies with the noise sources given inFigure 3 The neutron noise can be described as a sum of two

components known as a local and a global component Theformer is a particular solution of the inhomogeneous equa-tion (2) which dominates the locations close to the sourcewhereas the latter is related to the point-kinetic response ofthe reactor Since the location (33) is close to the sourcethe local component of the noise has more effect Figure 4illustrates the amplitude and the phase of the inducedneutronnoise as a function of energy at a frequency of 10Hz(low frequency) at the location of assembly (33) (close to

Science and Technology of Nuclear Installations 9

0 50 100 150 200 250 300 3500

05

1

15

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 20

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 20

(b)

Figure 12 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878fis

0100

200300 0

100

200

300

0

10

20

30

|120575120601|

(au

)

x (cm)

y(cm

)

(a)

0100

200300 0

100

200

300

minus02

minus015

minus01

minus005

0

Phas

e of120575

120601(r

ad)

y(cm

)

x (cm)

(b)

Figure 13 Amplitude (a) and phase (b) of the space-dependent noise in group 6 induced by 119878scat at 10Hz

the source) Figure 5 shows the same quantities as Figure 4but at a frequency of 100Hz (high frequency) Similar to theneutron spectrum the amplitude of the induced noise is onlysignificant in the energy range gt10 keV Comparing amongthe three cases at the energy lt100 keV the noise induced by119878abs and 119878fis is slightly thermalized compared to that inducedby 119878scat The out-of-phase behaviour of the 119878scat in successivegroups in this energy range leads to the cancellation inthe induced noise The normalized amplitude of the noiseinduced by 119878abs at energylt100 keV is slightly greater than thatinduced by 119878fis despite the significantly greater magnitude of

119878abs in this energy rangeThismeans that the source at energygt100 keV has more significant contribution to the inducednoise over the energy range

The phase of the noise is almost constant with energyat a frequency of 10Hz except some variation of the phasein the case of 119878scat in the thermal and epithermal energiesas shown in Figure 4 At a higher frequency (100Hz) thevariation of the phase is larger in the thermal and epithermalenergy ranges but in the important range (gt10 keV) thephase remains constant with energy The noise induced by119878abs and 119878fis is out-of-phase due to the out-of-phase behaviour

10 Science and Technology of Nuclear Installations

0100

200300 0

100

200

3000

2

4

6

8

|120575120601|

(au

)

x (cm)y (cm)

(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

Distance from core center (cm)

Group 6

|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

15

2

25

3

35

Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

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FuelsJournal of

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Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombustionJournal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

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Nuclear InstallationsScience and Technology of

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Solar EnergyJournal of

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Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

6 Science and Technology of Nuclear Installations

0

05

1

15

2

25

3

35

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

Position (88) f = 10Hz

(a)

minus05

0

05

1

15

2

25

3

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

Position (88) f = 100Hz

(b)

Figure 6 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 10Hz

Position (88) f = 100Hz

0

005

01

015

02

025

03

035

04

045

05

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat40

|120575120601|

(au

)

(a)

Position (88) f = 100Hz

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

SabsSfisSscat

Phas

e of120575

120601(r

ad)

(b)

Figure 7 Amplitude (a) and phase (b) of the noise at the position of assembly (88) (far from the source) as a function of neutron energyThe noise is calculated at a frequency of 100Hz

fission and scattering cross sections taking into account thespectrum shift in this area These perturbations are thenrepresented by three types of the noise sources associatedwith the perturbations of absorption fission and scatteringcross sections denoted as 119878120572 120572 = abs fis scat respectivelyWith this assumption the noise sources are calculated using

(6) and displayed in Figure 3 Since the amplitude of the119878scat is greater than that of the 119878abs and 119878fis by a factor of40 it dominates the behaviour of the total induced neutronnoise Therefore to investigate the total noise it is sufficientto investigate the noise induced by 119878scat in this case Howevera general perturbation could be sufficiently represented by

Science and Technology of Nuclear Installations 7

0

01

02

03

04

05

06

07

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(a)

minus3

minus2

minus1

0

1

2

3

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(b)

Figure 8 Amplitude (a) and phase (b) of the noise at the position (1414) in the reflector as a function of neutron energyThenoise is calculatedat several frequencies of 01 to 100Hz

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 6

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

16

18

2

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 6

minus1

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus05

0

Distance from core center (cm)

(b)

Figure 9 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878abs

a combination of them Hence investigation of the behaviourof the noise induced by the three noise sources separatelyis needed It can be seen in Figure 3 that the three sourceshave different properties as a function of neutron energyThe source associated with the perturbation of the fissioncross section 119878fis has an opposite phase to that associatedwith the perturbation of the absorption cross section 119878abswhereas the source associated with the perturbation of thescattering cross section 119878scat oscillates with energy groups Itis because a downscattering reaction in a certain group leadsto a negative source in this group but a positive source inlower energy groups

The amplitude of the noise sources at energy lt01 keV isnegligibly small compared to that in higher energy due to thefact that the static flux in this energy range is smaller by threeorders of magnitude or greater Since most of the neutronsgenerated by fission are fast neutrons the source 119878fis isnegligibly small up to 100 keVThe perturbations are assumedto be uniformly distributed in the first fuel ring around thecentral assembly Due to the symmetrical property the noisecalculations have been performed in a 60-degree domainthat is 16th of the core at several frequencies of 01 1010 and 100Hz to investigate the properties of the space-and frequency-dependent noise In the calculated domain as

8 Science and Technology of Nuclear Installations

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 20

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 20

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0

Distance from core center (cm)

(b)

Figure 10 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878abs

0 50 100 150 200 250 300 3500

05

1

15

2

25

3

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 6

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 6

(b)

Figure 11 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878fis

shown in Figure 2 the perturbation is located in assembly(22)

33 Neutron Noise Induced by Localized Perturbations

331 Energy-Dependent Noise Numerical calculations havebeen performed to investigate the noise behaviour in awide range of frequencies with the noise sources given inFigure 3 The neutron noise can be described as a sum of two

components known as a local and a global component Theformer is a particular solution of the inhomogeneous equa-tion (2) which dominates the locations close to the sourcewhereas the latter is related to the point-kinetic response ofthe reactor Since the location (33) is close to the sourcethe local component of the noise has more effect Figure 4illustrates the amplitude and the phase of the inducedneutronnoise as a function of energy at a frequency of 10Hz(low frequency) at the location of assembly (33) (close to

Science and Technology of Nuclear Installations 9

0 50 100 150 200 250 300 3500

05

1

15

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 20

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 20

(b)

Figure 12 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878fis

0100

200300 0

100

200

300

0

10

20

30

|120575120601|

(au

)

x (cm)

y(cm

)

(a)

0100

200300 0

100

200

300

minus02

minus015

minus01

minus005

0

Phas

e of120575

120601(r

ad)

y(cm

)

x (cm)

(b)

Figure 13 Amplitude (a) and phase (b) of the space-dependent noise in group 6 induced by 119878scat at 10Hz

the source) Figure 5 shows the same quantities as Figure 4but at a frequency of 100Hz (high frequency) Similar to theneutron spectrum the amplitude of the induced noise is onlysignificant in the energy range gt10 keV Comparing amongthe three cases at the energy lt100 keV the noise induced by119878abs and 119878fis is slightly thermalized compared to that inducedby 119878scat The out-of-phase behaviour of the 119878scat in successivegroups in this energy range leads to the cancellation inthe induced noise The normalized amplitude of the noiseinduced by 119878abs at energylt100 keV is slightly greater than thatinduced by 119878fis despite the significantly greater magnitude of

119878abs in this energy rangeThismeans that the source at energygt100 keV has more significant contribution to the inducednoise over the energy range

The phase of the noise is almost constant with energyat a frequency of 10Hz except some variation of the phasein the case of 119878scat in the thermal and epithermal energiesas shown in Figure 4 At a higher frequency (100Hz) thevariation of the phase is larger in the thermal and epithermalenergy ranges but in the important range (gt10 keV) thephase remains constant with energy The noise induced by119878abs and 119878fis is out-of-phase due to the out-of-phase behaviour

10 Science and Technology of Nuclear Installations

0100

200300 0

100

200

3000

2

4

6

8

|120575120601|

(au

)

x (cm)y (cm)

(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

Distance from core center (cm)

Group 6

|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

15

2

25

3

35

Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Science and Technology of Nuclear Installations 7

0

01

02

03

04

05

06

07

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

|120575120601|

(au

)Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(a)

minus3

minus2

minus1

0

1

2

3

4

Neutron energy (eV)10minus1 100 101 102 103 104 105 106 107

Phas

e of120575

120601(r

ad)

Position (1414) Sabs

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

(b)

Figure 8 Amplitude (a) and phase (b) of the noise at the position (1414) in the reflector as a function of neutron energyThenoise is calculatedat several frequencies of 01 to 100Hz

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 6

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

16

18

2

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 6

minus1

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus05

0

Distance from core center (cm)

(b)

Figure 9 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878abs

a combination of them Hence investigation of the behaviourof the noise induced by the three noise sources separatelyis needed It can be seen in Figure 3 that the three sourceshave different properties as a function of neutron energyThe source associated with the perturbation of the fissioncross section 119878fis has an opposite phase to that associatedwith the perturbation of the absorption cross section 119878abswhereas the source associated with the perturbation of thescattering cross section 119878scat oscillates with energy groups Itis because a downscattering reaction in a certain group leadsto a negative source in this group but a positive source inlower energy groups

The amplitude of the noise sources at energy lt01 keV isnegligibly small compared to that in higher energy due to thefact that the static flux in this energy range is smaller by threeorders of magnitude or greater Since most of the neutronsgenerated by fission are fast neutrons the source 119878fis isnegligibly small up to 100 keVThe perturbations are assumedto be uniformly distributed in the first fuel ring around thecentral assembly Due to the symmetrical property the noisecalculations have been performed in a 60-degree domainthat is 16th of the core at several frequencies of 01 1010 and 100Hz to investigate the properties of the space-and frequency-dependent noise In the calculated domain as

8 Science and Technology of Nuclear Installations

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 20

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 20

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0

Distance from core center (cm)

(b)

Figure 10 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878abs

0 50 100 150 200 250 300 3500

05

1

15

2

25

3

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 6

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 6

(b)

Figure 11 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878fis

shown in Figure 2 the perturbation is located in assembly(22)

33 Neutron Noise Induced by Localized Perturbations

331 Energy-Dependent Noise Numerical calculations havebeen performed to investigate the noise behaviour in awide range of frequencies with the noise sources given inFigure 3 The neutron noise can be described as a sum of two

components known as a local and a global component Theformer is a particular solution of the inhomogeneous equa-tion (2) which dominates the locations close to the sourcewhereas the latter is related to the point-kinetic response ofthe reactor Since the location (33) is close to the sourcethe local component of the noise has more effect Figure 4illustrates the amplitude and the phase of the inducedneutronnoise as a function of energy at a frequency of 10Hz(low frequency) at the location of assembly (33) (close to

Science and Technology of Nuclear Installations 9

0 50 100 150 200 250 300 3500

05

1

15

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 20

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 20

(b)

Figure 12 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878fis

0100

200300 0

100

200

300

0

10

20

30

|120575120601|

(au

)

x (cm)

y(cm

)

(a)

0100

200300 0

100

200

300

minus02

minus015

minus01

minus005

0

Phas

e of120575

120601(r

ad)

y(cm

)

x (cm)

(b)

Figure 13 Amplitude (a) and phase (b) of the space-dependent noise in group 6 induced by 119878scat at 10Hz

the source) Figure 5 shows the same quantities as Figure 4but at a frequency of 100Hz (high frequency) Similar to theneutron spectrum the amplitude of the induced noise is onlysignificant in the energy range gt10 keV Comparing amongthe three cases at the energy lt100 keV the noise induced by119878abs and 119878fis is slightly thermalized compared to that inducedby 119878scat The out-of-phase behaviour of the 119878scat in successivegroups in this energy range leads to the cancellation inthe induced noise The normalized amplitude of the noiseinduced by 119878abs at energylt100 keV is slightly greater than thatinduced by 119878fis despite the significantly greater magnitude of

119878abs in this energy rangeThismeans that the source at energygt100 keV has more significant contribution to the inducednoise over the energy range

The phase of the noise is almost constant with energyat a frequency of 10Hz except some variation of the phasein the case of 119878scat in the thermal and epithermal energiesas shown in Figure 4 At a higher frequency (100Hz) thevariation of the phase is larger in the thermal and epithermalenergy ranges but in the important range (gt10 keV) thephase remains constant with energy The noise induced by119878abs and 119878fis is out-of-phase due to the out-of-phase behaviour

10 Science and Technology of Nuclear Installations

0100

200300 0

100

200

3000

2

4

6

8

|120575120601|

(au

)

x (cm)y (cm)

(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

Distance from core center (cm)

Group 6

|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

15

2

25

3

35

Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

8 Science and Technology of Nuclear Installations

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sabs group 20

0 50 100 150 200 250 300 3500

02

04

06

08

1

12

14

Distance from core center (cm)

(a)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sabs group 20

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0

Distance from core center (cm)

(b)

Figure 10 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878abs

0 50 100 150 200 250 300 3500

05

1

15

2

25

3

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 6

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 6

(b)

Figure 11 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878fis

shown in Figure 2 the perturbation is located in assembly(22)

33 Neutron Noise Induced by Localized Perturbations

331 Energy-Dependent Noise Numerical calculations havebeen performed to investigate the noise behaviour in awide range of frequencies with the noise sources given inFigure 3 The neutron noise can be described as a sum of two

components known as a local and a global component Theformer is a particular solution of the inhomogeneous equa-tion (2) which dominates the locations close to the sourcewhereas the latter is related to the point-kinetic response ofthe reactor Since the location (33) is close to the sourcethe local component of the noise has more effect Figure 4illustrates the amplitude and the phase of the inducedneutronnoise as a function of energy at a frequency of 10Hz(low frequency) at the location of assembly (33) (close to

Science and Technology of Nuclear Installations 9

0 50 100 150 200 250 300 3500

05

1

15

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 20

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 20

(b)

Figure 12 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878fis

0100

200300 0

100

200

300

0

10

20

30

|120575120601|

(au

)

x (cm)

y(cm

)

(a)

0100

200300 0

100

200

300

minus02

minus015

minus01

minus005

0

Phas

e of120575

120601(r

ad)

y(cm

)

x (cm)

(b)

Figure 13 Amplitude (a) and phase (b) of the space-dependent noise in group 6 induced by 119878scat at 10Hz

the source) Figure 5 shows the same quantities as Figure 4but at a frequency of 100Hz (high frequency) Similar to theneutron spectrum the amplitude of the induced noise is onlysignificant in the energy range gt10 keV Comparing amongthe three cases at the energy lt100 keV the noise induced by119878abs and 119878fis is slightly thermalized compared to that inducedby 119878scat The out-of-phase behaviour of the 119878scat in successivegroups in this energy range leads to the cancellation inthe induced noise The normalized amplitude of the noiseinduced by 119878abs at energylt100 keV is slightly greater than thatinduced by 119878fis despite the significantly greater magnitude of

119878abs in this energy rangeThismeans that the source at energygt100 keV has more significant contribution to the inducednoise over the energy range

The phase of the noise is almost constant with energyat a frequency of 10Hz except some variation of the phasein the case of 119878scat in the thermal and epithermal energiesas shown in Figure 4 At a higher frequency (100Hz) thevariation of the phase is larger in the thermal and epithermalenergy ranges but in the important range (gt10 keV) thephase remains constant with energy The noise induced by119878abs and 119878fis is out-of-phase due to the out-of-phase behaviour

10 Science and Technology of Nuclear Installations

0100

200300 0

100

200

3000

2

4

6

8

|120575120601|

(au

)

x (cm)y (cm)

(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

Distance from core center (cm)

Group 6

|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

15

2

25

3

35

Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Science and Technology of Nuclear Installations 9

0 50 100 150 200 250 300 3500

05

1

15

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

|120575120601|

(au

)

Sfis group 20

(a)

0 50 100 150 200 250 300 35005

1

15

2

25

3

35

Distance from core center (cm)

f = 01Hzf = 10Hz

f = 10Hzf = 100Hz

Phas

e of120575

120601(r

ad)

Sfis group 20

(b)

Figure 12 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878fis

0100

200300 0

100

200

300

0

10

20

30

|120575120601|

(au

)

x (cm)

y(cm

)

(a)

0100

200300 0

100

200

300

minus02

minus015

minus01

minus005

0

Phas

e of120575

120601(r

ad)

y(cm

)

x (cm)

(b)

Figure 13 Amplitude (a) and phase (b) of the space-dependent noise in group 6 induced by 119878scat at 10Hz

the source) Figure 5 shows the same quantities as Figure 4but at a frequency of 100Hz (high frequency) Similar to theneutron spectrum the amplitude of the induced noise is onlysignificant in the energy range gt10 keV Comparing amongthe three cases at the energy lt100 keV the noise induced by119878abs and 119878fis is slightly thermalized compared to that inducedby 119878scat The out-of-phase behaviour of the 119878scat in successivegroups in this energy range leads to the cancellation inthe induced noise The normalized amplitude of the noiseinduced by 119878abs at energylt100 keV is slightly greater than thatinduced by 119878fis despite the significantly greater magnitude of

119878abs in this energy rangeThismeans that the source at energygt100 keV has more significant contribution to the inducednoise over the energy range

The phase of the noise is almost constant with energyat a frequency of 10Hz except some variation of the phasein the case of 119878scat in the thermal and epithermal energiesas shown in Figure 4 At a higher frequency (100Hz) thevariation of the phase is larger in the thermal and epithermalenergy ranges but in the important range (gt10 keV) thephase remains constant with energy The noise induced by119878abs and 119878fis is out-of-phase due to the out-of-phase behaviour

10 Science and Technology of Nuclear Installations

0100

200300 0

100

200

3000

2

4

6

8

|120575120601|

(au

)

x (cm)y (cm)

(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

Distance from core center (cm)

Group 6

|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

15

2

25

3

35

Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

10 Science and Technology of Nuclear Installations

0100

200300 0

100

200

3000

2

4

6

8

|120575120601|

(au

)

x (cm)y (cm)

(a)

0100

200300 0

100200

300minus3

minus2

minus1

0

Phas

e of120575

120601(r

ad)

x (cm)y (cm)

(b)

Figure 14 Amplitude (a) and phase (b) of the space-dependent noise in group 20 induced by 119878scat at 10Hz

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

40

45

50

Distance from core center (cm)

Group 6

|120575120601|

(au

)

S1scatS2scatTotal

(a)

0 50 100 150 200 250 300 350minus05

0

05

1

15

2

25

3

35

Distance from core center (cm)

Group 6

Phas

e of120575

120601(r

ad)

S1scatS2scatTotal

(b)

Figure 15 Amplitude (a) and phase (b) of the noise in group 6 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

of the two sources whereas the phases of the noise induced by119878abs and 119878scat are approximately equal in the important energyrange (gt10 keV) despite the oscillation of 119878scat with energy

Figures 6 and 7 show the same quantities as Figures 4 and5 but at the location of assembly (88) in the midcore (farfrom the source) Figure 8 shows the amplitude and phase ofthe noise at position (1414) in the reflector as a function ofenergy at several frequencies At the positions far from thesource the local component has no effect and the normalizednoise amplitudes are identical despite the source types Asshown in Figures 6ndash8 the phase at the positions is almost

constant with energy at low frequenciesThis behaviour is thesame with the three sources

At a high frequency the variation of the phase in thethermal and epithermal energy ranges increases but inthe important energy range (gt10 keV) the phase remainsconstant (see Figures 7 and 8) Principally the frequencyaffects the phase of the noise via the ratio 120596120592 in (4) and theterm related to the delayed neutrons in (5) In the fast energythat is high velocity of neutron 120596120592 becomes small and hasless effect on the phase of the induced noise even at a highfrequencyThe phase of the noise induced by 119878fis is opposite to

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Science and Technology of Nuclear Installations 11

S1scatS2scatTotal

|120575120601|

(au

)

0 50 100 150 200 250 300 3500

5

10

15

20

25

Distance from core center (cm)

Group 20

(a)

S1scatS2scatTotal

Phas

e of120575

120601(r

ad)

0 50 100 150 200 250 300 350minus3

minus2

minus1

0

1

2

3

4

Distance from core center (cm)

Group 20

(b)

Figure 16 Amplitude (a) and phase (b) of the noise in group 20 along the core radius induced by 119878scat at 10Hz 119878scat is assumed to be dividedinto two terms 1198781scat and 119878

2scat representing the sources in the first six fast groups and in the rest of energy range respectively

that induced by 119878abs similar to that found near the source dueto the out-of-phase behaviour of the two sources The phaseinduced by 119878scat at a position far from the source is approxi-mately equal to that induced by 119878abs over the energy range

332 Space-Dependent Noise Figures 9 and 10 display theamplitude and the phase of the space-dependent noise alongthe core radius in group 6 (a fast group in the energy range821times105 to 135times106 eV) and group 20 (an epithermal groupin the energy range 749times102 to 123times103 eV) induced by 119878absrespectively Figures 11 and 12 show the same quantities butinduced by 119878fis One can see that the noise amplitude in thefast group decreases with the increase of the frequency Thelocal peak of the noise at the position of the source decreasesfast with the distance from the source This behaviour issimilar to that found in LWRs Variation of the phase alongthe core diameter at frequencies of 001ndash10Hz ismuch smallercompared to that in a frequency of 100Hz As illustrated inFigures 9 and 11 the noises induced by 119878abs and 119878fis have quitesimilar behaviour but with opposite phases This indicatesthat the source in the energy range lt100 keV has less effectthan that at higher energy

Figures 13 and 14 show the amplitude and phase of thespace-dependent noise in group 6 (fast group) and group 20(epithermal group) respectively induced by 119878scat at 10HzA local peak can be seen at the source position in the fastgroup but in the epithermal group it is a local depth Thisis because the out-of-phase behaviour of 119878scat in successiveenergy groups leads to the cancellation in the total noiseThebehaviour can be seen more clearly in Figures 15 and 16

In order to evaluate the contribution of the perturbationat different energy it is assumed that the source 119878scat is divided

into two components one corresponds to the perturbationin the six fast groups (highest energy) denoted as 1198781scat andanother corresponds to the source in the rest of the energyrange denoted as 1198782scat The former has positive value overenergy while the latter oscillates with neutron energy asdisplayed in Figure 3 Calculations were then performed forthe noise induced by the two source components separatelyand compared with the total noise Figures 15 and 16 displaythe comparison of the amplitude and the phase of the noise ingroups 6 and 20 respectively It can be seen that the two noisecomponents are out-of-phase Thus they cancel together inthe total noise Since the amplitude of the noise of 1198781scat isgreater than that of 1198782scat the total noise has the same phasewith that of 1198781scatThis indicates that the properties of the totalnoise are dominated by the perturbation in a few fast groups

Calculation of the detector noise has also been performedwith the use of two detector types (fission chambers) consist-ing of 235U and 242Pu fissionable materials respectively asused in [15] The responses of the two detectors denoted as120575120601det as functions of detector positions along the core radiuswere obtained Figure 17 shows the amplitude and phase ofthe 235U detector noise with the frequencies of 1Hz and100Hz Figure 18 shows the same quantity but with the 242Pudetector Again one can see that the phase of the detectornoise induced by 119878abs and 119878scat is approximate and opposite tothat induced by 119878fis Since the

235U detector is more sensitiveto thermal and epithermal neutrons it recognizes well thenoise in the reflector region The 242Pu detector highlightsthe noise in the fuel region and the local peak at the sourceposition

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

12 Science and Technology of Nuclear Installations

0 100 200 3000

200

400

600

800

1000

1200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 1Hz

(b)

0 100 200 3000

50

100

150

200

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

235U detector 100Hz

(c)

0 100 200 300minus4

minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

235U detector 100Hz

(d)

Figure 17 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 235U detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

4 Conclusions

The properties of the neutron noise induced by localizedperturbations in an SFR have been investigated Three rep-resentations of the noise sources were assumed to be locatedat the core center Numerical calculations were performedin a wide range of frequencies Similar to the static flux theinduced noise amplitude is significant at energy gt1 keV Inthis energy range the phase of the noise is almost constantwith energy despite the source types This can be explainedby the low effect of 120596120592 ratio in the high energy rangeVariation of the phase decreases with the increase of the

distance from the source and is negligibly small at lowfrequencies Calculation of the noise induced by the twosource components of the perturbation of the scattering crosssection shows that the perturbation in several fast groups hasa significant contribution and dominates the amplitude andthe phase of the induced noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Science and Technology of Nuclear Installations 13

0 100 200 3000

20

40

60

80

100

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 1Hz

(a)

0 100 200 300minus1

0

1

2

3

4

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 1Hz

(b)

0 100 200 3000

10

20

30

40

50

Distance from core center (cm)

|120575120601

det|

(au

)

SabsSfisSscat

242Pu detector 100Hz

(c)

0 100 200 300minus3

minus2

minus1

0

1

2

3

Distance from core center (cm)

Phas

e of120575

120601de

t(r

ad)

SabsSfisSscat

242Pu detector 100Hz

(d)

Figure 18 Amplitude (a and c) and phase (b and d) of the neutron noise calculated with the 242Pu detector at the frequencies of 1Hz (a andb) and 100Hz (c and d)

Acknowledgments

A part of calculations in this work was performed duringthe authorrsquos stay at Chalmers University of Technology Thisresearch is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10304-201479

References

[1] J A Thie ldquoCore motion monitoringrdquo Nuclear Technology vol45 pp 5ndash45 1979

[2] H M Hashemian ldquoOn-line monitoring applications in nuclearpower plantsrdquo Progress in Nuclear Energy vol 53 no 2 pp 167ndash181 2011

[3] J Gourdon and R Casejuane ldquoOff-line and on-line noiseanalysis for core surveillance in French LMFBR lsquoANABELrsquordquoProgress in Nuclear Energy vol 9 pp 365ndash373 1982

[4] T Umeda K Chiba S Ebata Y Ando and H SakamotoldquoExperience of on-line surveillance at onagawa-1 bwr plantrdquoProgress in Nuclear Energy vol 21 pp 35ndash41 1988

[5] D Wach ldquoVibration neutron noise and acoustic monitoring inGerman LWRsrdquoNuclear Engineering and Design vol 129 no 2pp 129ndash150 1991

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

14 Science and Technology of Nuclear Installations

[6] J A Thie ldquoFast flux test facility noise data managementrdquoProgress in Nuclear Energy vol 21 pp 173ndash180 1988

[7] G le Guillou R Berger and M Brunet ldquoBoiling detection infast reactors by noise analysis studies performed in FrancerdquoProgress in Nuclear Energy vol 1 no 2ndash4 pp 409ndash426 1977

[8] C Demaziere ldquoCORE SIM a multi-purpose neutronic tool forresearch and educationrdquo Annals of Nuclear Energy vol 38 no12 pp 2698ndash2718 2011

[9] V Larsson C Demaziere I Pazsit and H N Tran ldquoNeutronnoise calculations using the analytical nodal method and com-parisons with analytical solutionsrdquo Annals of Nuclear Energyvol 38 no 4 pp 808ndash816 2011

[10] V Larsson and C Demaziere ldquoA coupled neutronicsthermal-hydraulics tool for calculating fluctuations in PressurizedWaterReactorsrdquo Annals of Nuclear Energy vol 43 pp 68ndash76 2012

[11] S A Hosseini and N Vosoughi ldquoNeutron noise simulation byGFEM and unstructured triangle elementsrdquo Nuclear Engineer-ing and Design vol 253 pp 238ndash258 2012

[12] P Filliatre C Jammes B Geslot and L Buiron ldquoIn vesselneutron instrumentation for sodium-cooled fast reactors typelifetime and locationrdquo Annals of Nuclear Energy vol 37 no 11pp 1435ndash1442 2010

[13] H N Tran and C Demaziere ldquoNeutron noise calculations inhexagonal geometry and comparison with analytical solutionsrdquoNuclear Science and Engineering vol 175 no 3 pp 340ndash3512013

[14] H N Tran F Zylbersztejn C Demaziere C Jammes andP Filliatre ldquoA multi-group neutron noise simulator for fastreactorsrdquo Annals of Nuclear Energy vol 62 pp 158ndash169 2013

[15] F Zylbersztejn H N Tran I Pazsit P Filliatre and C JammesldquoCalculation of the neutron noise induced by periodic deforma-tions of a large sodium-cooled fast reactor corerdquoNuclear Scienceand Engineering vol 177 no 2 pp 203ndash218 2014

[16] G L Fiorini and A Vasile ldquoEuropean commissionmdash7thframework programme the collaborative project on Europeansodium fast reactor (CP ESFR)rdquo Nuclear Engineering andDesign vol 241 no 9 pp 3461ndash3469 2011

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

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