Wavelet and Spectral Analysis of Some Selected Problems in Reactor Diagnostics

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2 CTH-RF-83 Wavelet and Spetal Analysis of Some Seleted Poblems in Reato Diagnostis Cal Sunde Akademisk uppsats fö avläggande av teknologie lientiatexamen i Reaktofysik vid Chalmes tekniska högskola. Examinato oh handledae: Pofesso Ime Pázsit Ganskae: Doent Ninos Gais, SKI Depatment of Reato Physis Chalmes Univesity of Tehnology Götebog 004

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4 Abstat Both spetal and wavelet analysis wee suessfully used in vaious diagnosti poblems involving non-stationay oe poesses in nulea powe eatos. Thee diffeent poblems wee teated: two-phase flow identifiation, deteto tube impating and oe-bael vibations. The fist two poblems ae of non-stationay natue, wheeas the last one is not. In the fist poblem, neuton adiogaphi and visible light images of fou diffeent vetial two-phase flow egimes, bubbly, slug, hun and annula flow, wee analysed and lassified with a neuo-wavelet algoithm. The algoithm onsists of a wavelet pat, using the -D disete wavelet tansfom and of an atifiial neual netwok. It lassifies the diffeent flow egimes with up to 99% effiieny. Deteto tubes in a Boiling Wate Reato may exeute vibations and may also impat on neaby fuel-assemblies. Signals fom in-oe neuton detetos in Ringhals- wee analysed, fo detetion of impating, with both a lassial spetal method and wavelet-based methods. The wavelet methods inlude both the disete and the ontinuous -D wavelet tansfom. It was found that thee is ageement between the diffeent methods as well as with visual inspetions made duing the outage at the plant. Howeve, the wavelet tehnique has the advantage that it does not equie expet judgement fo the intepetation of the analysis. In the last pat two analytial alulations of the neuton noise, indued by shell-mode oe-bael vibations, wee aied out. The esults ae in good ageement with alulations fom a numeial simulato. An out-of-phase behaviou between in-oe and ex-oe positions was found, whih is in ageement with ealie measuements fom the Pessuised Wate Reato Ringhals-3. The esults fom these alulations ae planned to be used when diagnosing the shell-mode oe-bael vibations in an opeating plant. Keywods: noise diagnostis, wavelet analysis, deteto tube impating, two-phase flow, oe-bael vibations

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6 This thesis onsists of an intodution and the following papes: I. C. Sunde, S. Avdi and I. Pázsit (Septembe 004), Classifiation of two-phase flow egimes via image analysis by a neuo-wavelet appoah, Applied Computational Intelligene, Po. of the 6th Intenational FLINS Confeene, Septembe - 3, Blankenbege, Belgium II. III. IV. C. Sunde, S. Avdi and I. Pázsit (004), Classifiation of two-phase flow egimes via image analysis and a neuo-wavelet appoah, submitted to Pogess in Nulea Enegy C. Sunde and I. Pázsit (004), Investigation of deteto tube impating in the BWR Ringhals-, submitted to Intenational Jounal of Nulea Enegy Siene and Tehnology, IJNEST C. Sunde and V. Azhanov (Septembe 003), Calulation of the neuton noise indued by shell-mode oe-bael vibations in a -D -goup -egion slab eato, CTH-RF-73, Chalmes Univesity of Tehnology, Götebog, Sweden. V. C. Sunde, C. Demazièe and I. Pázsit (004), Calulation of the neuton noise indued by shell-mode oe-bael vibations in a -D -goup -egion slab eato, submitted to Nulea Tehnology

7 Sientifi publiations elated to the lientiate topi but not inluded in this thesis: I. C. Demaziée, C. Sunde, V. Azhanov and I. Pázsit (Deembe 003), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 8, CTH-RF- 77/RR-0, Chalmes Univesity of Tehnology, Götebog, Sweden. II. C. Demaziée, I. Pázsit, C. Sunde and J. Wight (004), Reseah and Development Pogam in Reato Diagnostis and Monitoing with Neuton Noise Methods, Stage 0. Final epot, SKI Repot, Statens Känkaftsinspektion, Swedish Nulea Powe Inspetoate, Stokholm Sweden, to be published

8 Contents. Intodution. Wavelets 3. Time and fequeny 3. One-dimensional wavelet tansfom 4.3 Two-dimensional wavelet tansfom 7 3. Two-phase flow identifiation 0 3. Flow images 0 3. Wavelet pe-poessing 3.3 The lassifiation algoithm 3.4 Results fom the lassifiation 4. Deteto tube impating 4 4. Physial model 4 4. Analysing the measuements Measuement Measuement Continuous wavelet tansfom Conlusions of the deteto tube impating 0 5. Coe-bael vibations 5. Desiption of the model 5. The adiabati appoximation 5.3 The full spae-fequeny dependent model Compaison of numeial and analytial solutions 5 6. Conluding emaks 7 7. Aknowledgments 7 8. Refeenes 8 9. Abbeviations 3 0. The Papes I-V

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10 . Intodution. Intodution Noise diagnostis has been used in Nulea Powe Plants (NPPs) sine the ealiest days of eato physis [-4], beginning fifty yeas ago with the stat of the fist eato. Diffeent methods have been developed duing the deades up till now and still thee is a lot of development going on. This thesis is a ontinuation of the noise diagnosti wok at the Depatment of Reato Physis, Chalmes Univesity of Tehnology. Then, what is noise diagnostis? In odinay life, noise is onsideed to be something unwanted o distubing, the noise in adio boadasting o distubing noise whih an ou when speaking in a ell-phone. In ode to get a lea, noise-fee signal, the noise is often filteed away. Howeve, in noise diagnostis it is the othe way aound, the stati signal is filteed out and the noise is onsideed as the impotant pat of the signal. Thus, the noise is extated athe than filteed away. What is the use of the noise? Take, again, the ell-phone as an example. When alling to someone, thee may be some distubanes (noise) in the voie tansmitted ove the phone. Based on the noise it ould be possible to daw the onlusion that the peson you ae alling fo example is diving a a, whih is an example of a simple fom of noise diagnostis. Hene, the noise is used to daw onlusions about a system, whih an be vey useful. In eato noise diagnostis, the noise in signals fom detetos plaed inside and outside the eato oe ae used to diagnose the eato. Compaed to the simple ell-phone example the situation is somewhat diffeent when diagnosing a eato. Fist, the magnitude of the noise an be a ouple of powes of ten lowe ompaed to the signal itself. Seond, it is possible to have moe than one noise-soue. Thid, it is not always easy to get a simple esponse; you an not simply ask a eato whethe it is diving a a o not! Rathe, ontinuing with the same example, it is like finding out in whih of New Yok s thousands of steets the a is in, what speed it has et. The pupose of eato diagnostis is mainly to have ontol ove the eato and make sue eveything is woking popely. Howeve, if something stats deteioating, the on-line diagnostis of the eato should detet the eo and if neessay alet the opeato, who in etun may initiate a shut-down. The off-line diagnostis of the eato means mainly to investigate the behaviou of the eato and to give infomation about the eato status, e.g. find omponents that may need maintenane o undestand tends, unexpeted phenomena et. Hene, the diagnostis must be as eliable as possible. One does not want to stop the eato by false alam o maintain a woking omponent, sine, it is time onsuming to estat the eato and, of ouse, thee is a loss of money if the eato is not opeating. Hene, it is impotant to undestand and develop noise diagnosti methods, whih ae eliable. The methods developed in this thesis ae not fo on-line use in powe poduing nulea plants, but moe of an undestanding level of the poesses in a eato and fo off-line use. At the Depatment of Reato Physis, Chalmes Univesity of Tehnology, eato noise diagnostis have been used fo diffeent diagnosti tasks [5-6], suh as detetion of deteto tube impating, identifiation of oe bounday (bael) vibations and Modeato Tempeatue Coeffiient (MTC) detemination. Thee ae two ongoing eseah pojets on noise diagnostis, one in oopeation with the Swedish NPP Ringhals [0-7], and one in oopeation with the Swedish Nulea Powe Inspetoate (SKI) [8-6].

11 . Intodution In this thesis noise diagnostis is applied within the eseah aeas of oe-bael vibations, two-phase flow identifiation and deteto tube impating. One of the majo goals of this thesis is to inlude wavelet analysis in the noise diagnosti wok. Wavelets ae espeially suited fo analysing non-stationay poesses, e.g. intemittent signals. Hene, the fous of the noise diagnostis pefomed in this thesis is on non-stationay poesses. Sine wavelets ae a elatively new tool in eato physis, a shot intodution to wavelet theoy is given in Setion. The outline fo the est of the thesis basially follows the ode of the appended papes. The wok done on two-phase flow identifiation, whih is desibed in papes I and II, is summaised in Setion 3. Setion 4 is a summay of the wok with deteto tube impating found in pape III. Setion 5 is a summay of the wok with oe-bael vibations desibed in papes IV and V.

12 . Wavelets 3. Wavelets One goal of this thesis and the eseah within the pojet is to use wavelet tehniques in eato noise diagnostis. Wavelets ae still onsideed to be a new field in signal poessing, even though they have been in use fo almost two deades sine they wee fist intodued in the mid-80s. The eal development stated in the ealy 90s [7], and has ontinued eve sine by the use of wavelets in diffeent sientifi aeas suh as fluid dynamis, mediine, finane, physis and geophysis [8]. This Setion gives a vey shot intodution to wavelets and thei appliation. A moe mathematial detailed explanation an be found in [7-3].. Time and fequeny The lassial Fouie tansfom an be used to map a time signal into the fequeny domain, as illustated in Fig.. The Fouie tansfom an only be applied on stationay signals, whee thee ae no hanges in fequeny ove the time inteval of inteest. Howeve, if the signal is non-stationay, the Fouie tansfom annot be used. In that ase a windowing of the signal an be done, using the so-alled windowed Fouie tansfom o Shot Time Fouie Tansfom (STFT). The STFT maps a time signal into a two-dimensional signal of both time and fequeny. Hene, it is possible to get infomation about both when and at what fequeny a etain event ous. Howeve, thee is one dawbak with the STFT, namely that the fequeny esolution is the same fo all fequenies. Often thee is a need fo bette esolution in time at highe fequenies. The next step is to onstut a tool whih an map a time signal into time and fequeny but with diffeent esolution fo diffeent fequenies. The wavelet tansfom is able to ope with this equiement, even though it is mapping the signal into timesale o time-level athe than time-fequeny, but thee is a onnetion between sale, level and fequeny. Howeve, the bette esolution at high fequenies in time is ahieved by eduing the esolution in fequeny. The esolution in time beomes pooe at lowe fequenies as an be seen in Fig.. Fouie tansfom Shot time Fouie tansfom Wavelet tansfom Fequeny Fequeny Sale Fequeny Amplitude Time Time Fig.. Shemati pitue of the Fouie, shot time Fouie and wavelet tansfoms. The Fouie tansfom uses an infinitely long sinusodial funtion as the analysing tool, hene it has no time esolution. On the othe hand the wavelet tansfom uses a small, loalized wave funtion (see Fig. ) as the analysing tool, hene the name wavelet. The use of loalised funtions makes the wavelet tansfom well suited fo analysing non-stationay signals suh as tansients and intemittent signals. The wavelet displayed in Fig. is the so-alled Mexian hat, whih has the haateisti featues of a wavelet. Howeve, this wavelet will not be used in the est of the Setion sine it does not have ompat suppot and is not othogonal. Without going into details, this means that it an not be used when pefoming the disete wavelet tansfom.

13 . Wavelets 4 In the est the Daubehies 4 (db4) wavelet will be used instead, sine it has ompat suppot and is othogonal [7]. mexian hat wavelet Fig.. The Mexian hat wavelet has the fom of a small loalized wave, hene the name wavelet. The dotted uve is a osine funtion with the same fequeny as the ente fequeny of the Mexian hat wavelet.. One-dimensional wavelet tansfom It is possible to use the wavelet tansfom on one- o two-dimensional data. In this subsetion the one-dimensional tansfom is desibed and in the next a shot desiption of the two-dimensional wavelet tansfom is given. Thee ae two ways of pefoming the one-dimensional tansfom: ontinuous with ontinuous sales o levels (fequenies), o disete with disete sales o levels (fequenies). In this thesis the fous is on the disete tansfom, hene this will be desibed in moe details. Although pat of this thesis touhes upon the ontinuous wavelet tansfom, it will not be desibed in this Setion. The wavelet tansfom is based on a so-alled mothe-wavelet, Ψ, whih is dilated and tanslated (see Fig. 3), with the paametes a and b. ψ ab, () t ψ t b () a a At eah sale a thee is a haateisti fequeny, F a, whih an be alulated though the soalled ente-fequeny of the mothe-wavelet, F, and the sampling peiod, t, of the analysed signal: F a F () a t The ente-fequeny is the haateisti fequeny of the mothe-wavelet, see Fig.. One m m way of hoosing the paametes in () ae, a a 0 and b nb 0 a 0 whee m is alled the

14 . Wavelets 5 level. The most ommon hoie of a 0 and b 0 ae and espetively. This gives the disete one-dimensional wavelet tansfom of the signal x(t) as: T mn, xt () ψ( m t n) dt (3) m Hee T is alled the detailed wavelet oeffiients. They ontain infomation about the details of x. The oeffiients T m,n ae alulated by tanslating and dilating the mothe wavelet along the signal, illustated in Fig. 3, and pefoming the integation at eah step. The tansfom is pefomed at eah disete level m. Fist sale mothe wavelet Tanslated wavelet tanslation Dilated Wavelet dilation x Fig. 3. The db4 mothe wavelet with tanslated and dilated vesion, analysing a signal. With eah wavelet thee is an assoiated saling-funtion, φ. The wavelet and the saling funtions ae othogonal and have the following elation: ψ() t ( ) k k φ( t k) (4) k Hee is a saling oeffiient. The saling funtions an be used to alulate appoximation oeffiients of the signal, x(t), in the same way as the alulation of the detail oeffiients: S mn, xt () φ( m t n) dt (5) m

15 . Wavelets 6 The appoximation oeffiients ontain infomation about the mean behaviou of the signal. With the use of both the detail oeffiients and the appoximation oeffiients the signal an be eonstuted as: m 0 xt () x m0 () t + d m () t (6) m whee the detail of the signal, at level m, is defined as: d m () t T m n ψ( m t n) (7) n, and the appoximation of the signal at level, m 0 is: m x m0 () t S m0, n φ m 0 ( t n) (8) n If the input signal, x 0, is disete and of finite length, e.g N M, as it nomally is when dealing with measuements, m and n ae also finite and it is possible to ewite the above fomulae as follows: m 0 M x 0 () t x M () t + d m () t m x M () t S M n φ( M t n) M m, M d m () t T m n ψ( m t n) n 0, m (9) Hee x M is the mean of the signal. Fom this it is possible to get a elation between appoximation and detail at one level and the appoximation at the next level: x m () t x m () t + d m () t (0) With this hoie of a and b the deomposition of the signal into details and appoximation is alled the wavelet multiesolution analysis. Fom the multiesolution it is lea that the wavelets, used in the disete tansfom, onsists of an othogonal set of basis funtions in whih an abitay funtion an be expanded. As an example of the multiesolution analysis take a hip signal, shown at the top in Fig. 4 a), with some white noise added. It is deomposed into details and appoximations at diffeent sales using the db4 wavelet in the MatLab Wavelet Toolbox [3]. The details at eah level epesent the ontent of the signal at that level (fequeny). Hene it is possible to extat

16 . Wavelets 7 infomation about both when and at what fequeny a etain event happens. As expeted, the noise is pesent in the lowest level (highest fequeny) details -3, see Fig. 4 b). The high fequeny omponents of the egula hip signal ae visible at the middle levels, 5-6, to the ight in Fig. 4 b). The low fequeny omponents of the egula hip signal ae visible to the left at highe levels, 7-8, in the same figue. An almost noise fee hip signal is visible at appoximation level 4, x 4 (t) in Fig. 4 a). a) Appoximation of the signal b) Details of the signal x 0 (t) x (t) x (t) x 3 (t) x (t) 4 x 5 (t) x 6 (t) x (t) 7 x 8 (t) x 9 (t) x (t) 0 x 0 (t) d (t) d (t) d 3 (t) d 4 (t) d 5 (t) d 6 (t) d 7 (t) d 8 (t) d 9 (t) d (t) 0 x 0 (t) t t Fig. 4. Appoximation x m, a), and details d m, b), at ten diffeent levels of the wavelet tansfom using the db4 wavelet. The oiginal signal x 0 is at the top in both plots. The detail oeffiients T m,n ontain the same infomation as the eonstuted details, d m, and the same is valid fo the appoximation oeffiients S m,n and the eonstuted appoximation, x m. Hene it is possible to use eithe the oeffiients o the eonstuted appoximation and details when analysing the signal. The advantage with the oeffiients is that the size of them ae deeasing with a fato of two at eah level, ompaed with the eonstuted signals whih have the same size as the oiginal signal. Fom Fig. 4 it is possible to daw the onlusion that wavelets an be used to denoise a noisy signal. By taking the appoximation at level 4 of the signal, an almost noise-fee hip signal an be obtained. Howeve, all infomation fom detail level -4 is negleted, when using this appoximation. One way of pefoming a bette denoising is to use some infomation fom the lowest levels (highest fequenies). This an be done by thesholding the detail oeffiients, in this example details at level -4. When eonstuting the signal, the appoximation and the thesholded detail oeffiients ae used to alulate a denoised epesentation of the signal. Also it is seen that the noise an be extated though the details. By using the thee fist levels of details in Fig. 4 b) it is possible to almost ompletely extat the noise. An appliation whee this ould be of use is the possibility to extat shot intemittent pats of a signal, so-alled spikes. The one-dimensional wavelet tansfom (multiesolution) will be used in Setion 4..3 Two-dimensional wavelet tansfom If the input data ae two-dimensional, i.e. an image, it is possible to use a two-dimensional wavelet tansfom. The piniples of the tansfomation ae the same as fo the onedimensional tansfom, but instead of having one detail at eah level thee ae thee. The

17 . Wavelets 8 infomation in eah of the thee details ae fom thee dietions of the two-dimensional input data, hoizontal, vetial and diagonal details. The fist equation in (9) an then be expessed as M h v d x 0 ( x, y) x M ( x, y) + d m( x, y) + dm ( x, y) + d m( x, y) () m This is the eonstution fo the two-dimensional disete wavelet tansfom. The details, the detail oeffiients, the appoximations and the appoximation oeffiients ae alulated in the same way as fo the one-dimensional tansfom, exept that diffeent two-dimensional wavelets ae used fo the diffeent dietions and a two-dimensional saling funtion is used. All the two-dimensional wavelet and saling funtions an be alulated fom the onedimensional ones. ψ h ( x, y) φ( x)ψ( y) ψ v ( x, y) ψ( x)φ( y) ψ d ( x, y) ψ( x)ψ( y) φ( x, y) φ( x)φ( y) () Depited in Fig. 5 b)-e) ae the eonstuted appoximation and details at the fist level (highest fequeny) of a two-dimensional wavelet tansfom, using the db4 wavelet, on the image in Fig. 5 a). In the hoizontal detail ), the hoizontal high fequeny pats (shap tansitions) ae lealy visible, e.g. the line in the middle. In the same way the vetial high fequeny pats (the tees) ae lealy visible in d), the vetial details, as expeted. Thee ae no lea featues in the diagonal details e), sine the oiginal images have no shap diagonal tansitions. The two-dimensional tansfom an be used to haateise an image by identifying diffeent featues of the image at diffeent details levels (sales) and/o diffeent dietions. This is used in Setion 3. Anothe featue of the wavelet tansfom, whih is not used in this thesis, is the possibility to ompess images. The diffeenes between the oiginal image, a), and the fist level appoximation, b), ae hadly notieable. Howeve, the appoximation ontains only 5% the amount of data as the oiginal image. Hene, it an be used as a ompessed vesion. Fo futhe infomation on wavelet theoy and appliations the inteested eade is efeed to [7-3]. Thee all the mathematial details, whih ae left out in this thesis, an be found.

18 . Wavelets 9 a) Oiginal Image -D wavelet tansfom b) Appoximation ) Hoizontal detail Vetial detail d) e) Diagonal detail Fig. 5. Appoximation, b), and details, )-e), fom a two-dimensional wavelet tansfom, using db4 wavelet, at the fist level. The oiginal image is depited in a).

19 3. Two-phase flow identifiation 0 3. Two-phase flow identifiation The fist appliation of wavelet tehniques in this thesis is in the aea of two-phase flow identifiation. It is vey impotant to lassify the diffeent flow egimes in a eato, sine they have quite diffeent flow popeties. Befoe using a flow equation, the egime must be detemined in ode fo the ight expession to be hosen fo e.g. the intefaial shea oeffiient o some othe oeffiients like the heat tansfe oeffiient. The task at hand is to lassify two-phase flow egimes with image analysis. The appoah in this Setion is to use wavelets to pe-poess flow images and then extat statistial featues to use as inputs to an Atifiial Neual Netwok (ANN), see Fig. 6. The use of dynami images as the signal fom the flow means that the method is nonintusive. Both non-intusive methods, wavelets and ANN have been used peviously when lassifying two-phase flow egimes [3-37]. Howeve, they have not been used in ombination befoe. A moe detailed desiption of the two-phase flow identifiation is found in pape I and II. Flow image wavelet pe-poessing featue extating atifiial neual netwok flow egime Fig. 6. Outline of the two-phase flow lassifiation algoithm In nulea tehnology the two-phase flow is wate and vapou whih flow within the oe of a boiling wate eato. In vetial two-phase flow thee ae fou main egimes, bubbly, slug, hun and annula flow. Bubbly flow is the flow of dispesed vapou in ontinuous liquid, with small bubbles of vapou in the wate. In slug flow the bubbles of vapou have fomed lage egions, with a size of appoximately the size of the pipe diamete. If even moe vapou is pesent in the pipe the bubbles beak and thee is an unstable egime of liquid mix with vapou, hun flow. In the last type, annula flow, the pipe is almost filled with vapou, only a thin pat, lose to the wall, ontains liquid. The geometial stutue of the egimes is vey diffeent, but at the same time, it is diffiult to expess this diffeene in quantitative tems. This is why twophase flow identifiation is diffiult to pefom with algoithmi methods. 3. Flow images Two diffeent sets of flow images have been analysed. The fist set inluded dynami neuton adiogaphi images of two-phase flow within a metalli wate loop, see Fig. 7 a). This expeiment was pefomed at the Kyoto Univesity Reato Reseah Institute (KURRI) [38]. By ontinuously ineasing the heating of the wate in the loop, all fou flow egions wee eated in sequene. One dawbak with these images is that they had to be onveted into digital fomat sine the eoding was available on an analogue VHS in NTSC fomat. This onvesion somewhat ineases the noise in the images. The seond set of images, eoded at ou depatment with the use of an odinay digital amode, used visible light instead of neuton adiogaphy. In the expeiment the two-phase flow was simulated by injeting ai into a thin etangula pipe filled with oloued wate, see Fig. 7 b). Due to the simple set-up it was only possible to simulate bubbly and slug flow. On the

20 3. Two-phase flow identifiation othe hand, a muh bette image quality was ahieved as ompaed to the neuton adiogaphi images. a) b) Fig. 7. a) Images of the fou diffeent flow egimes using neuton adiogaphy, flike noisy images. b) Images of bubbly and slug flow egimes using visible light and oloued wate, almost noise-fee images 3. Wavelet pe-poessing Befoe extating inputs fo the ANN, the images (-D matix with gay sale pixel intensity) wee pe-poessed with the two-dimensional wavelet tansfom. Wavelets ae suitable to highlight featues of diffeent length sales (fequenies) in an image. It is also possible to extat infomation about featues in diffeent dietions in an image, see Fig. 5 (Setion.3). The diffeent flow egimes ae assumed to have diffeent featues at diffeent length sales and in diffeent dietions. The idea is to use some featues of the tansfomed wavelet oeffiients whih ae haateisti fo eah egime. One possible featue is the enegy ontent of the detail oeffiients. In this ase the fist level of detail oeffiients ae onsideed and one way of expessing the enegy ontent is given in [8] as: E x i x T, i (3) Hee x stands fo the diffeent dietions (hoizontal, vetial and diagonal) in the two-dimensional tansfom, see Setion.3 fo details. The enegy featue was used to haateise the visible light images. Unfotunately, using the enegy featue did not wok in the ase of the adiogaphi images, pobably due to the flike noise. Instead, the mean value of the fist level appoximation oeffiients was used. Anothe possible featue, whih ould be used, is the vaiane of the vey same oeffiients. Hene, one value fo the neuton adiogaphi images, the vaiane of the appoximation oeffiients, and thee values fo the visible light images, one fo the details oeffiients in eah dietion ae also used. This gives a total of two featues extated fo the adiogaphi images and six featues fo the visible light images. These featues ae, in the next step of the lassifiation poess, used as inputs fo an ANN. 3.3 The lassifiation algoithm The lassifiation task is solved by using an ANN with the wavelet pe-poessed featues as inputs. The piniple of an ANN is to have a set of inputs with known output values. The ANN is fed with these inputs whih popagate though the nodes of the netwok. The output is

21 3. Two-phase flow identifiation ompaed with the known output values and some eo paametes ae alulated whih ae used to hange the nodes. Thee ae diffeent ways in hanging the nodes, i.e. diffeent taining poedues. The taining is epeated until some pe-defined minimal iteion of the eo is eahed. Eah epetition is alled an epoh. Afte the taining it is possible to feed the ANN with inputs whose outputs ae to be detemined. A popely tained ANN will geneate the oet output values. Fo the two-phase flow lassifiation the ight flow egime type is seahed. The ANN used in this thesis is onstuted fom the Neual Netwok Toolbox in MatLab. Diffeent types of netwoks and taining algoithms wee tested and by tial and eo the best ombination was seleted fo the task at hand. A feed-fowad netwok with an input laye, an output laye and one hidden laye tained with the esilient bakpopagation (BP) algoithm was found to be the most effetive one. The numbe of input nodes used depends on whih type of flow images is used. Two input nodes wee used fo the adiogaphi images and six fo the visible light images. The numbe of nodes in the output laye also depends on the images, one output fo eah flow egime that is lassified. Hene, fou output nodes wee used fo the adiogaphi ones and two fo the visible light images. The log-sigmoid tansfe funtion was used fo the output laye, giving values between 0 and. Fo both types of images 40 hidden nodes with tan-sigmoid tansfe funtions wee used. The taining taget values wee set to 0.9 fo the oet egime and 0. fo the othe outputs. The input data onsist of 00 images fom eah of the flow egimes in the ase with neuton adiogaphy and 75 images fom eah of the two egimes in ase of the visible light. A 5-fold oss-validation ove the taining data was used, i.e. /5 of the input data was used as a test set to veify the lassifiation suess of the netwok tained with the emaining 4/5 of the inputs. This was epeated five times by using diffeent images in the test set. All images wee used only one time in the test set. When lassifying the test set, the outputs wee thesholded in ode to get eithe fo the ight flow o 0 fo the wong ones. If moe than one output is o none is the image is lassified as unknown flow egime. 3.4 Results fom the lassifiation To investigate the advantage with the wavelet pe-poessing, the mean value and the vaiane fom the aw image pixel intensity wee also fed in to the netwok, and the esults wee ompaed to those with the wavelet pe-poessed input data. Six diffeent disete wavelets wee used fo the pe-poessing, Haa, Daubehies 8, Coiflet 4, Symmlet 6 and Biothogonal 3. all available in the Wavelet Toolbox in MatLab. In the ase of the noisy adiogaphi images, the suess atio of the lassifiation was aound 95% fo the test set of images fo all the diffeent wavelets and the same fo the aw data input. In Fig. 8 the esult fo the input pe-poessed with the Daubehies 8 wavelet in shown. Hene, thee is no advantage, fom the point of view of suess atio, in pe-poessing the images. When using the visible light images the suess atio was even highe, aound 99%, fo both the pe-poessed and the aw inputs. Howeve, the numbe of epohs used duing the taining poedue is edued with a fato of 00 when using wavelet pe-poessing. Fo the aw data input the maximum numbe of epohs, set to , was always eahed befoe the taget value of the Mean Squae Eo (MSE), set to 0-3 was eahed. In the pe-poessed ase the MSE taget was eahed within appoximately 300 taining epohs. Hene, the use of wavelet pe-poessing has lage advantages fom the patial point of view.

22 3. Two-phase flow identifiation 3 Also woth mentioning is that 00% of the annula flow images wee lassified oetly in the neuton adiogaphi ase. As expeted, the slug flow and the hun flow wee the ones most likely to be mixed up, i.e. lassified wong o as unknown. The next step in the pojet will be to analyse neuton adiogaphi images, eoded in digital fomat. Hopefully, this will give images with the same quality as the visible light images, but with all fou flow egimes available. It is expeted that in the end this will lead to bette pefomane of the wavelet pe-poessing when used on all fou egimes. # of images Fig. 8. Classifiation atio of the neuton adiogaphi images using a theshold of 0.5 afte the ANN. A total of 00 images fom eah egime wee lassified and the aveage suess atio was 95%.

23 4. Deteto tube impating 4 4. Deteto tube impating Methods fo detetion of deteto tube impating in Boiling Wate Reatos (BWRs) with noise diagnosti methods, applied to signals fom neuton detetos, have been used fo a long time. The fist methods used wee based on lassial Fouie analysis [3, 39]. In these methods the Auto Powe Spetal Density (APSD), oheene and phase uves ae used to identify impating tubes. Typially, a boadening of the eigenfequeny peak in the APSD and/o a distoted phase uve give infomation about an impating tube. Moe about the spetal methods will be mentioned below. At ou Depatment a new method, based on wavelet analysis, has been developed. Fist, simulations and measuements fom the Swedish nulea powe plant Basebäk- wee analysed with both spetal and wavelet methods [5-6]. Late, measuements taken fom Oskashamn- wee analysed and the esult was ompaed with visual inspetions made befoe the analysis [4-5, 40]. Between the two diffeent analyses the wavelet algoithm was modified. In this Setion a ontinuation of these analyses is summaised. Fo a detailed desiption see pape III. This time, measuements fom Ringhals- ae analysed both with the taditional spetal methods and the new wavelet based method. Reently the ontinuous wavelet tansfom has also been applied to the same poblem, and it is also desibed biefly in the thesis. 4. Physial model The task is to identify deteto tubes whih not only vibate but also impat on the neighbouing fuel assemblies. If a deteto tube hits a fuel assembly it an damage the fuel box whih may ause also damage to the fuel ladding. Any suh event must be avoided in an opeating plant. Deteto tube F~-3Hz L~4m Neuton detetos Fuel box F~0-0Hz Coe suppot plate Fig. 9. Illustation of a deteto tube in a BWR oe with suounding fuel assemblies, thee out of fou shown. Some typial data of inteest ae also shown. Fig. 9 shows a geneal outline of the physial setup of a deteto tube togethe with the suounding fuel assemblies. The vibations aise fom the stong flow of oolant wate in the eato and the fat that the deteto tubes, whih ae oughly fou metes long, ae fixed only in thei ends. The eigenfequeny is aound Hz, [3]. If the vibation is lage enough, the tube

24 4. Deteto tube impating 5 may impat on the neaby fuel assemblies, whih in etun will exeute a shot, damped osillation afte eah hit, with an eigenfequeny of 0 to 0 Hz, [3]. Detetos ae plaed at fou diffeent axial levels in the tube. Fom the Ringhals- measuements, analysed hee, signals fom two of the fou detetos in eah tube wee available. Thee ae 36 deteto tubes evenly distibuted in the oe giving a total of 7 signals in the Ringhals- ase. The signals fom vibating and impating deteto tubes ae assumed to ontain thee pats. Fist a global pat with some boad-band noise, N(t). The seond pat is a egulaly osillating pat fom the deteto tube vibation, S(t), and thidly an intemittent pat due to the vibation of the fuel-assembly, T(t). The intemittent stutue is due to the damped, andomly ouing vibation of the fuel-assembly, see Fig. 0. In geneal, the amplitude o the oot mean squae value of T(t) is muh smalle than that of N(t)+S(t). Theefoe, in a spetal analysis, the effet of T(t) is not visible. Hene, the lassial spetal method fouses on the vibation of the deteto tubes, wheeas the wavelet method ties to identify the intemittent signal fom the fuel assembly vibations. Based on this fat the sampling fequeny used in the wavelet method needs to be highe, sine the eigenfequeny of the fuel assemblies is highe than fo the deteto tubes. Flux gadient N(t): Global neuton noise S(t): T(t): Deteto sting noise (Stationay) Fuel box noise (Tansient) F(t) N(t) + S(t)]+T(t): Total signal Fig. 0. Shemati view of the signal, S(t), fom a deteto vibating in a flux gadient with bakgound noise, N(t), and fuel assembly impating, T(t). 4. Analysing the measuements As mentioned above, measuements fom the BWR Ringhals- wee analysed. In oopeation with plant pesonnel two diffeent measuements wee made. Both measuements wee taken duing full powe (09%) and full oe flow (~ 000 kg/s). The fist measuement was aied out on 6th of Septembe, 00. In the est of this thesis this measuement is efeed to as measuement. The seond measuement was made on 7th of Otobe, 003, efeed to as measuement. In measuement the sampling fequeny was too low,.5 Hz, fo the wavelet method to be pefomed and the measuement time was appoximately min. Fo measuement, on the othe hand, the sampling fequeny was aised to 64 Hz to bette use the infomation fom

25 4. Deteto tube impating 6 the fuel assembly vibations (0-0Hz). This time the duation of the measuement was 5 min. At the moment a thid measuement is planned with even highe sampling fequeny, 00 Hz. 4.. Measuement Sine measuement was not suited fo wavelet analysis, the judgment on whih tubes that may impat on the fuel assemblies is based mainly on the spetal method. The spetal method is based on the featues of the Auto Powe Spetal Density, APSD, the oheene, and the phase between two detetos in the same tube. The following fou iteia ae used to lassify a deteto tube as an impating one [5]. Fist, boadening of the eigenfequeny peak in the APSD ompaed to a vibating but non-impating tube. Seond, multiple peaks in the APSD, espeially at the double eigenfequeny. Thid, high oheene at the eigenfequeny, and finally, distoted (zeo) phase ove a lage fequeny ange. The deision of whethe a tube is impating o not is made by qualitatively taking all fou iteions into aount. This is moe o less a elative method. The idea with the wavelet method is to alulate a so-alled Impat Rate index (IR-index). The IR-index gives the numbe of intemittent signals due to the fuel assembly vibation. A high value means sevee impating. By using the IR-index, thee is no need fo an expet judgment whih is the ase fo the spetal method. Hene, the wavelet method is an absolute method. As mentioned above the deteto signals ae assumed to onsist of thee pats: φ() t N() t + S() t + T() t (4) whee S(t) is analysed in the spetal ase, desibed above, and T(t) is extated in the wavelet method whih will be desibed below. To edue the high fequeny noise pesent in the signal, a wavelet denoising is pefomed. The denoising is done by applying a level-dependent theshold to eah of the detailed oeffiients, T m,n as desibed in Setion, in the multiesolution analysis. The wavelet deomposition is made down to a level, M, oesponding to the eigenfequeny of the fuel assemblies (0-0 Hz). The denoised signal is then eonstuted by using the thesholded detailed oeffiients in (9), giving Den(φ(t)). Then the appoximation, X M (t), whih onsists of the low fequeny pat of the signal, S(t), is emoved fom the denoised signal. The esult is the intemittent signal fom the fuel assembly vibation. V() t Den( φ() t ) X M () t T() t (5) The IR-index is alulated as the numbe of peaks in V(t). Eah peak oesponds to the stat of an intemittent fuel assembly vibation. In ode to use the wavelet method, a value of the eigenfequeny of the fuel assemblies has always to be given to pefom the denoising to the oet level. Sine the sampling fequeny of measuement only allows investigation up to fequenies of 6.5 Hz, the assumed eigenfequeny of the fuel assemblies is set to 5 Hz. Even though this value is fa fom ealisti, it is the best that an be used when pefoming the wavelet analysis on this measuement. As is ommonly known when using wavelets, it is not easy to know whih kind of wavelet to use fo a etain analysing task. Using the Meye wavelet in the MatLab Wavelet Toolbox

26 4. Deteto tube impating 7 gave best ageement with the esult fom the spetal method. Howeve, due to the low sampling fequeny the judgment of whih tubes that was most likely to impat is mainly based on the spetal analysis in measuement. Fig. shows the esult fom the spetal analysis of detetos at position, LPRM. and LPRM.4. LPRM stands fo Loal Powe Range Monito and is the notation of the neuton detetos used. LPRM.4 is the uppe deteto and LPRM. is the lowe one. Clealy thee is a boad peak in the APSD aound the expeted eigenfequeny of the deteto tubes (- Hz). Multiple peaks ae also visible in the APSD. The oheene is high at the vey same fequeny and the haateisti distotion of the phase is pesent (non-linea and almost zeo). Hene, this deteto position was lassified as one of the most likely to impat. Fig.. Autospeta (APSD), oheene and phase fo the detetos at LPRM position, measuement. Afte the analysis was pefomed, the plant pesonnel made visual inspetions of some fuel assemblies, whih wee pointed out by the analysis, duing the outage of the plant in August 003. They inspeted fuel assemblies aound deteto (LPRM) positions, 9, 0 and out of whih had been lassified by us as non-vibating in the analysis and the thee othes wee the ones that we judged most likely to impat. The esult of the analysis and the inspetion is pesented in Table. As an be seen, LPRM 0 and wee pointed out by the visual inspetion. They showed some wea maks on the one of the fuel boxes. This is in good ageement with the pedition fom the analysis.

27 4. Deteto tube impating 8 Table : Results of measuement Impating status LPRM by analysis LPRM by inspetion most likely impating 9, 0 and 0 and pobably impating 4, 8, 6 and small hane of impating,, 30 and Measuement The analysis of measuement was pefomed in the same way as that of measuement. Howeve, this time the sampling fequeny, 64 Hz, was bette suited fo the wavelet method. The eigenfequeny of the fuel assemblies, was set to a moe ealisti value of 0 Hz in this ase. In Fig. a) the APSD, oheene and the phase of LPRM 6. and 6.4 is depited and in b) V(t) fom equation (5) fo LPRM 6.4 is shown. As an be seen, all the iteia fo the spetal method ae fulfilled and seveal spikes ae visible in V(t). Hene, this deteto tube was pointed out as most likely to impat. a) 0. LPRM b) Time, [s] Fig.. a) Auto speta (APSD), oheene and phase of detetos at LPRM position 6 fom measuement. In b) the intemittent signal afte the wavelet analysis of deteto 6.4, measuement. The spikes indiating impating ae lealy visible.

28 4. Deteto tube impating 9 Unfotunately, thee was no time fo visual inspetions duing the outage in 004, sine evey seond yea thee is a shot outage. The esults of the spetal and wavelet analysis ae pesented in Table. In the two goups of detetos with highest pobability of impating fou out of six ae pointed out by both methods, LPRM 6 4, 34 and 35. Thus, thee is a good ageement between the two methods. Table : Results of measuement Impating status By spetal analysis By wavelet analysis most likely impating 5, 6, 4 and 35 6, 3 and 4 pobably impating and 34 4, 34 and 35 small hane of impating 7 and Continuous wavelet tansfom Thee is also a possibility to use the ontinuous wavelet tansfom in the analysis. In the ase with deteto tube impating it ould be of inteest to alulate the so-alled wavelet oheene, [8], between detetos in the same tube. Fig. 3 shows the wavelet oheene and the spetal oheene fo two diffeent deteto positions. a) LPRM LPRM 6 oheene Fequeny Fequeny b) Fig. 3. Spetal, a), and ontinous wavelet, b), oheene fo detetos in position and 6. Clealy, thee is a muh lage diffeene between the wavelet oheene in the fequeny band between 0 to 0 Hz than it is in the spetal ase. The pesene of the lage values in the wavelet oheene aound 0-0 Hz is not fully undestood. One guess is that it ould be the

29 4. Deteto tube impating 0 fuel assembly vibations that ae seen. Deteto positions pointed out by the wavelet IR-index also display lage values in the fequeny band between 0 and 0 Hz in the wavelet oheene. Even though the esult fom the ontinuous wavelet tansfom is not fully undestood, this fist esult implies that futhe use may tun of to be of inteest when investigating deteto tube impating. 4.4 Conlusions of the deteto tube impating The spetal method is in good ageement with visual inspetions. Fo the measuements with a sampling fequeny that is suitable fo the new wavelet based method, it is in ageement with the spetal one. One may wonde if thee is need fo a new method if a woking one aleady exists. Howeve, the advantage with this new wavelet method is that thee is no need fo ompaison with a non-vibating sample. It is an absolute method ompaed to the moe elative spetal method, and its use equies no expetise, in ontast to the spetal based method, whih annot be pefomed e.g. by the ontol oom pesonnel. It is expeted that a new measuement, with a highe sampling fequeny that will be pefomed duing the yea of 004/005, and a visual inspetion duing the outage in late summe 005 will onfim the ageement between the two methods and the eal damage in the plant. The ontinuous wavelet oheene will be futhe investigated and hopefully bette undestood with the new measuement. Up to now the wavelet method has been tested on signals fom thee Swedish plants, Basebäk, Oskashamn and Ringhals. It woked satisfatoily in all thee ases, even though it had to be modified between its use in diffeent plants.

30 5. Coe-bael vibations 5. Coe-bael vibations Coe-bael vibations ae, to a lage extent, stationay poesses, and have been suessfully teated with spetal methods. The -D pendula vibations, howeve, show sometimes intemittent popeties (altenating isotopi - unilateal vibations). Theefoe, oe-bael vibations will also be analysed with wavelet methods in the futue. Aodingly, pepaations ae made by getting involved in the modelling of the in-oe neuton noise indued by oe-bael vibations. Howeve, in this thesis, only lassial, i.e. spetal analysis based, appliations will be inluded. This analysis, still, has led to inteesting new esults in the undestanding of etain expeimental esults. Shell mode Beam mode Fig. 4. Illustation of the beam- and shell-mode oe-bael vibations Analyses of ex-oe neuton noise of both beam-mode, ~8 Hz, and shell-mode, ~0 Hz, oe-bael vibations have been used a long time, see Fig. 4, [3, 9-7]. It has also been notied that the vibations may lead to in-oe noise whih also an be used to analyse the vibations. When analysing the shell-mode vibations the use of in-oe noise is espeially impotant in Westinghouse eatos, sine the ex-oe detetos ay the same infomation, due to the 90 o spaing. Theefoe, the amplitude and the dietion of the vibations an not be detemined at the same time. Fo this eason, in-oe detetos have been inluded in the analysis of the shellmode vibations within a eseah pojet in oopeation with the Swedish NPP Ringhals [7]. Howeve, the small numbe of in-oe detetos hindeed onfimation of the theoy as well as use of the esult. In ode to have onsistent intepetations it was neessay to assume that noise fom in-oe and ex-oe detetos lying on the same azimuthal position have opposite phase. To onfim this out-of-phase behaviou, two diffeent -dimensional analytial alulations of the noise indued shell-mode vibations have been pefomed: an adiabati appoximation and a full spae-fequeny dependent solution. A ompaison between the analytial esults and the esults fom a numeial simulato, developed at the depatment [4], was also made. These alulations ae desibed in detail in pape IV and V and a summay is given in this Setion. 5. Desiption of the model A -dimensional -goup -egion model is used with the bounday of the oe set to b6.5 m and the bounday of the efleto set to a79.5 m. The eato paametes, e.g. oss-setions, ae alulated fom the oe of the Pessuised Wate Reato (PWR)

31 5. Coe-bael vibations Ringhals-4, using the in-oe fuel management ode SIMULATE-3 (homogenization fom 3- D to -D). The hoie of whih eato to use is abitay. One only needs some ealisti paametes to visualize numeial esults in the end. The fluxes ae assumed to be symmetial aound the ente of the oe. Sine a -D -goup -egion diffusion model is used, thee ae two diffusion equations in the oe egion,, and two diffusion equations in the efleto egion,, fo the stati fluxes. One in eah egion fo the fast flux,, and one fo the themal flux,. D D d φ( x) Σa dx (, + Σ R )φ ( x) + -- ( νσ k f, φ ( x) + νσf, φ ( x) ) 0 d φ( x) Σa dx, φ ( x) + ΣRφ( x) 0 oe egion (6) D D d d d φ x d φ x ( x) ( Σa, + Σ R )φ ( x) 0 ( x) Σa, ( x) + ΣRφ( x) 0 φ efleto egion (7) The oss-setion notations ae standad, i.e. a stands fo absoption, f fo fission and R fo emoval (satteing) fom the fast goup to the themal. All the oss-setions and the diffusion oeffiients ae assumed to be onstant within eah egion. The fist step in alulating the phase behaviou of the noise, indued by shell-mode vibations, is to solve the stati equations (k-eigenvalue equations). That is easily done by using the symmety, the intefae (ontinuous fluxes and uents) and bounday (zeo flux) onditions. One a nomalised and itial solution of the stati fluxes is obtained the noise an be alulated. Citiality is ahieved by adjusting the fission oss-setions. 5. The adiabati appoximation In ode to alulate the noise fom shell-mode oe-bael vibations, equations fo a time-dependent system have to be used. One aveage goup of delayed neuton peusos, C, is assumed. The time-dependent diffusion equations now ead v ---- φ φ D ( Σ t x x a, + Σ s, )φ + ( β) ( νσ f, φ + νσ f, φ ) + λc φ φ D Σ v t x x a, φ + Σ s, φ C λc + βνσ ( t f, φ + νσ f, φ ) (8) The aguments ae left out, but eveything exept β, ν and λ is both time- and spae-dependent. The equations ae valid in both the oe and the efleto egions. The shell-mode vibation is modelled by letting the bounday between the oe and efleto osillate aound the stati positions, b and -b, in a symmetial way, b(t)b+ε(t) and -b(t)-b-ε(t).

32 5. Coe-bael vibations 3 The equations in (8) ae solved by an adiabati appoah, i.e. splitting the fluxes into a time-dependent pat, amplitude fato P(t), and a spae-time dependent pat, shape funtion ψ(x,t), as follows. φ ( x, t) Pt () ψ ( x, t ) P( 0) φ ( x, t) Pt () ψ ( x, t) ψ, ( x, 0) φ, stati x ( ) (9) When using the adiabati appoximation a new nomalization is needed. The shape funtion an be nomalized by using the adjoint funtion, [4]. Using (9) in (8) and multiplying with the adjoint flux, subtating the stati equations and integating ove the eato volume one ends up with the following equations afte using the nomalization ondition. dp() t dt d() t dt ρ() t β Pt () + λt () Λ() t P β () Λ() t t λt () (0) Lineaising by splitting all time-dependent quantities into a stati and a small deviation pat, negleting seond ode tems, δp an be expessed in the fequeny domain as: δp( ω) G 0 ( ω) δρ( ω) () Hee G 0 (ω) is a tansfe funtion, whih is simila to the odinay zeo powe eato tansfe funtion. Sine the fequeny of inteest fo the shell-mode vibations, 0 Hz, is within the soalled plateau egion the tansfe funtion is onstant and equal to: G 0 ( ω) -- () β Using this it is possible to expess δp in the time-domain and by inseting this in (9) and lineaising, splitting into stati and small deviation pats, negleting seond ode tems. The expession of the spae-time dependent neuton noise is: δφ ( xt, ) δφ ( xt, ) δρ() t φ β δρ() t φ β stati x stati x ad ( ) + δψ ( x, t) ad ( ) + δψ ( x, t) (3) Hee the petubed adiabati shape funtion is used. The adiabati appoximation is valid if the petubation is assumed to be small. Then the vaiation in time of the shape funtion is elatively small. It is then possible to alulate the petubed adiabati shape funtion at any time, as δψ ad (x,t)ψ ad (x,t)-φ stati (x), whee ψ ad (x,t) is a stati flux alulated fom new k-eigenvalue equations, i.e. (6) and (7) with a lage oe. δρ(t) is alulated, using the new k, as: δρ() t (4) k() t

33 5. Coe-bael vibations 4 All this gives the neuton noise indued by shell-mode oe-bael vibations in a -D -goup -egion eato in the adiabati appoximation. 5.3 The full spae-fequeny dependent model The othe way of alulating the noise indued by the shell-mode vibation in a -D model is to make a full spae solution in the fequeny domain. Fist, the stati oss-setions, in both the oe and efleto egion, ae witten as: Σ( x) { H( x b) }Σ + H( x b)σ (5) Hee H(x) is the unit step funtion. With a vibating bounday bb+ε(t), the time-dependent oss-setions ae witten as Σ( x, t) { H( x b ε() t )}Σ + H( x b ε() t )Σ (6) Splitting the oss-setions into a stati and a time-dependent pat and making a Taylo expansion, assuming a small vibation, gives: Σ( xt, ) Σ( x) + ε()δx t ( b)δσ, δσ Σ Σ (7) Using this togethe with a splitting into a stati and a small deviation pat of the fluxes and delayed neuton peusos in (8) and lineaising by negleting seond ode tems, equations in the fequeny domain ae obtained fo the noise in the oe and the efleto, whih an be witten in a ondensed fom as ˆ L ˆ L ( x, ω)δφ εω ( x, ω)δφ εω ( )δ( x b)ŝ( x)φ stati ( )δ( x b)ŝ( x)φ stati (8) Hee the veto fluxes in espetive egions epesent both the fast and themal fluxes. The maties S and L ae given in pape V. On the ight hand side in (8) it is not indiated if the stati flux is taken fom the oe o the efleto. But sine they ae equal at the intefae, xb, it does not matte whih one is used. The neuton noise an be omplex and the solution to (8) is given as: ( ) Ã 3 δφ x, ω δφ x, ω C κ( ω) ( ) Ã C µ ( ω) sinh( κ ( x a) ) + sinh( κ ( b a) ) Ã 0 4 os( µ x) + Ã C η( ω) sinh( κ ( x a) ) sinh( κ ( b a) ) osh( η x) osh( η b) (9) The bounday onditions of zeo fluxes at the extapolated boundaies x ± b have been used aleady in (9). The oeffiients, Ã, ae detemined fom the following matix equation:

34 5. Coe-bael vibations 5 A εω ( )M ˆ 0 0 S φ ( b) + S φ ( b) (30) S φ ( b) + S φ ( b) The matix M onsists of the intefae onditions and is given in pape V. Now, the neuton noise fom the shell-mode vibations in a -D eato is alulated in two diffeent ways. The analytial solutions fo the two ases ae found in equations (3) and (9). 5.4 Compaison of numeial and analytial solutions The themal noise, denoted with subsipt, is the one of inteest, sine the themal neutons ae the ones deteted by neuton detetos in the powe plant. The phase fom the seond analytial solution is easily alulated fom the omplex neuton noise in (9). Sine the noise is omplex, it is possible to have all values between 0 and -π. On the othe hand, in the fist ase, with (3), the phase is alulated as eithe in-phase, 0, o out-of-phase, -π, i.e. only two disete values ae possible. In Fig. 5 the themal noise fom both analytial solutions is shown. The most inteesting featue is the lage loal out-of-phase behaviou lose to the bounday, whee the noise also has a lage absolute value. The out-of-phase behaviou means that the themal noise lose to the intefae has an opposite value as in the est of the system, i.e. a deease of the flux at the intefae and an inease in the est of the eato. The global inease is due to the inease in eativity, δρ > 0. The deease at the intefae aises fom the fat that fission mateial is added into the efleto egion giving moe absoption of themal neutons, and modeato mateial is emoved, ausing less slowing down of fast neutons to themal ones. Hene, the loal deease of the themal flux in the alulations an be explained. The themal noise fom the full spae-fequeny dependent analytial solution and the numeial simulato is shown in Fig. 6. The phase behaviou is vey muh the same in both ases, but thee is some deviation in the absolute values. The diffeene in the absolute value is pobably due to the fat that the numeial simulato uses nodes instead of ontinuous spae. Hene, the petubation at the bounday is spead out ove a whole node athe than existing in a point (Dia-delta funtion). The onlusions of these alulations ae that thee is a lage loal omponent of the noise lose to the intefae between the efleto and the oe and that it is indeed possible to have an out-of-phase behaviou between in-oe, lose to the efleto, and ex-oe detetos. Howeve, it is not lea if this out-of-phase behaviou is pesent in a -D model o not. Hene, the next step would be to expand the alulations into -D. If thee is a lage loal out-of-phase behaviou lose to the intefae between the oe and the efleto, it ould be possible to measue it with detetos plaed in the outemost fuel-assemblies. If this tuns out to wok it ould be used as a diagnose tool fo the shell-mode oe-bael vibations.

35 5. Coe-bael vibations 6 a) 6 x 0 Absolute value of themal noise 5 b) δφ (x,ω)/ε(ω) Phase size [m] full spae dependene, 0 Hz adiabati appoximation Fig. 5. The absolute value, a), and the phase, b), of the themal noise alulated with the two diffeent analytial appoahes. a) 6 x 0 Absolute value of the themal noise 5 b) δφ (x,ω)/ε(ω) analyti numeial simulato Phase size [m] Fig. 6. Absolute value, a), and phase, b), of the themal nosie fom shell-mode oe-bael vibations. The dotted line is alulated with the numeial simulato and the full line is the full spae-fequeny dependent analytial solution.

36 6. Conluding emaks 7 6. Conluding emaks In this thesis thee diffeent poblems wee investigated, two of them within the aea of lassial eato noise diagnostis, oe-bael vibations and deteto tube impating. The elatively new mathematial tool, wavelet analysis, is used in two of the poblems, deteto tube impating and two-phase flow identifiation. Espeially the use of the ontinuous tansfom and the two-dimensional tansfom ae new onepts in the eseah going on at the depatment. Both of the new wavelet methods seem pomising fo futhe development and appliation within the two pojets and pobably in othe pojets as well. Wheneve thee is non-stationay poess, wavelet analysis ould be of use. Hopefully, thee will be seveal diffeent new as well as old poblems to takle with the wavelets in the ontinuing wok onneted to the thesis. As mentioned above in Setions 3 to 5, thee is still some wok to be done in eah of the pojets. As egads the oe-bael vibations, the next step would be to extend the model into two dimensions and veify the esults with measuements. Conening the deteto tube impating, thee is aleady a new measuement unde way. This new measuement will be pefomed duing the ongoing fuel yle of Ringhals- and hopefully thee will be time fo visual inspetions duing the next outage, whih an veify the esult. Moeove, thee is need to exploe the ontinuous wavelet tansfom to get a bette undestanding and knowledge of its featues and possible appliations. A physial intepetation of the esults is of majo impotane fo futhe use of the ontinuous tansfom. The next step in the two-phase flow identifiation pojet would be to apply the enegy featue extation to noise-fee neuton adiogaphi images. Howeve, it is not lea whethe it is possible to get suh images o not. The fous of using wavelets in the noise diagnostis will ontinue in both pojets mentioned above and in othe pojets as well, e.g. estimation of MTC whee a so-alled wavelet spetogam ould be of use. 7. Aknowledgments Fist of all I would like to thank my olleagues at the Depatment of Reato Physis. A speial thank goes to my supeviso Pofesso Ime Pázsit, who always seems to have answes to my many questions. I would also like to give speial thanks to the following thee people fo the o-opeation on the wok onneted with this thesis, D. Senada Avdi, D. Chistophe Demazièe and D. Vasily Azhanov. Pofesso emeitus Nils Göan Sjöstand and Håkan Mattsson ae thanked fo eading the poof and oeting my English. The NPP Ringhals is aknowledged fo poviding neuton measuement data fo us. Kyoto Univesity Reato Reseah Institute is aknowledged fo the neuton adiogaphi images of the two-phase flow. Svenskt Käntekniskt Centum (SKC) is aknowledged fo the finanial suppot.

37 8. Refeenes 8 8. Refeenes [] J. A. Thie (963), Reato Noise, ANS, Rowman and Littlefield, In., New Yok [] M. M. R. Williams (974), Random Poesses in Nulea Reatos, Pegamon Pess, Oxfod [3] J. A. Thie (979), Coe motion monitoing, Nulea Tehnology, Vol 45, 5-4 [4] J. A. Thie (98), Powe Reato Noise, ANS, La Gange Pak [5] I. Pázsit and O. Glökle (993), BWR instument tube vibations: Intepetations of measuments and simulations, Ann. Nul. Enegy, Vol, [6] A. Ráz and I. Pázsit (997), Diagnostis of deteto tube impating with wavelet tehniques, Ann. Nul. Enegy, Vol 5, [7] V. Azhanov (00), Powe Reato Noise Studies and Appliations, PhD Thesis, Depatment of Reato Physis, Chalmes Univesity of Tehnology, CTH-RF-6 [8] C. Demazièe (00), Development of a Noise-Based Method fo Detemination of the Modeato Tempeatue Coeffiient of Reativity (MTC) in Pessuized Wate Reatos (PWRs), PhD Thesis, Depatment of Reato Physis, Chalmes Univesity of Tehnology, CTH-RF-66 [9] I. Pázsit and V. Azhanov (000), Linea Reato Kinetis and Neuton Noise in Systems with Flutuating Boundaies, Ann. Nul. Enegy, Vol 7, [0] I. Pázsit (Edito) (996), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage. CTH-RF-/RR3, Chalmes Univesity of Tehnology, Götebog, Sweden. [] I. Pázsit, N. S. Gais and J. K.-H. Kalsson,(997), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage. CTH-RF-3/RR4, Chalmes Univesity of Tehnology, Götebog, Sweden. [] J. K.-H. Kalsson and I. Pázsit (998), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 3:Analysis of oe bael vibations in Ringhals, 3 and 4 fo seveal fuel yles. CTH-RF-35/RR5, Chalmes Univesity of Tehnology, Götebog, Sweden. [3] C. Demazièe, V. Azhanov, J. K.-H. Kalsson and I. Pázsit (999), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 4. CTH-RF-45/RR6, Chalmes Univesity of Tehnology, Götebog, Sweden. [4] C. Demazièe, V. Azhanov and I. Pázsit (000), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 5. CTH-RF-56/RR7, Chalmes Univesity of Tehnology, Götebog, Sweden. [5] C. Demazièe, V. Azhanov and I. Pázsit (00), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 6. CTH-RF-6/RR8, Chalmes Univesity of Tehnology, Götebog, Sweden. [6] C. Demazièe, V. Azhanov and I. Pázsit (00), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 7. CTH-RF-67/RR9, Chalmes Univesity of Tehnology, Götebog, Sweden

38 8. Refeenes 9 [7] C. Demazièe, C. Sunde, V. Azhanov and I. Pázsit (003), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 8. CTH-RF-77/RR0, Chalmes Univesity of Tehnology, Götebog, Sweden. [8] I. Pázsit, N. S. Gais (995), Foskningspogam angående häddiagnostik med neutonbusmetode. Etapp. Slutappot, SKI Rappot 95:4, Statens Känkaftsinspetion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. [9] I. Pázsit, N. S. Gais, O. Thomson (996), Foskningspogam angående häddiagnostik med neutonbusmetode. Etapp. Slutappot, SKI Rappot 96:50 Statens Känkaftsinspetion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. [0] I. Pázsit, N. S. Gais, J. Kalsson, A. Ráz (997), Foskningspogam angående häddiagnostik med neutonbusmetode. Etapp 3. Slutappot, SKI Rappot 97:3, Statens Känkaftsinspetion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. [] I. Pázsit, J. K-H. Kalsson (998), Reseah and Development Pogam in Reato Diagnostis and Monitoing with Neuton Noise Methods. Stage 4. Final Repot, SKI Repot 98:5, Statens Känkaftsinspetion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. [] I. Pázsit, J. K-H. Kalsson, P. Lindén, V. Ajanov (999), Reseah and Development Pogam in Reato Diagnostis and Monitoing with Neuton Noise Methods. Stage 5. Final Repot, SKI Repot 99:33, Statens Känkaftsinspetion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. [3] I. Pázsit, C. Demazièe, S. Avdi, B. Dahl (000), Reseah and Development Pogam in Reato Diagnostis and Monitoing with Neuton Noise Methods. Stage 6. Final Repot, SKI Repot 00:8, Statens Känkaftsinspetion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. [4] I. Pázsit, C. Demazièe, V. Azhanov, and N. S. Gais (00), Reseah and Development Pogam in Reato Diagnostis and Monitoing with Neuton Noise Methods. Stage 7. Final Repot, SKI Repot 0:7, Statens Känkaftsinspetion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. [5] I. Pázsit, C. Demazièe, and V. Azhanov (003), Reseah and Development Pogam in Reato Diagnostis and Monitoing with Neuton Noise Methods. Stage 8. Final Repot, SKI Repot 003:08, Statens Känkaftsinspetion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. [6] I. Pázsit, V. Azhanov, A. Nodlund, D. Olsson (004), Reseah and Development Pogam in Reato Diagnostis and Monitoing with Neuton Noise Methods. Stage 9. Final Repot, SKI Repot 003:30, Statens Känkaftsinspetion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. [7] I. Daubehies (99), Ten letues on wavelets, CBMS 6, Soiety fo Industial and Applied Mathematis, Philadelphia, US [8] P. S. Addison (00), The Illustated Wavelet Tansfom Handbook, Institute of Physis Publishing, Bistol and Philadelphia, US [9] S. Mallat (998), A Wavelet Tou of Signal Poessing, Aademi Pess, London, UK

39 8. Refeenes 30 [30] J. Begh, F. Ekstedt and M. Lindbeg (999) Wavelets, Studentlitteatu, Lund, Sweden [3] Wavelet toolbox, Uses guide, The Math Woks In. [3] M. A. Vine and R. T. Lahey (98), Development of an optial digital intefeomete, Intenational Jounal of Multiphase Flow, Vol 8, 93-4 [33] A.M.C. Chan and D. Bzovey (990), Measuement of mass flux in high tempeatue pessue steam-wate two-phase flow using a ombination of Pitot tubes and a gamma densitomete, Nulea Engineeing and Design, Vol, [34] M. Balasko, E. Svab, L. Cse (987), Simultaneous Dynami Neuton And Gammaadiogaphy, NDT Intenational, Vol 0, [35] K. Mishima, T. Hibiki, Y. Saito, H. Nakamua and M. Matsubayashi (999), The eview of the appliation of neuton adiogaphy to themal hydauli eseah, Nul. Inst. Methods A, Vol 44, 66-7 [36] H. Wu, F. Zhou and Y. Wu (00), Intelligent identifiation system of flow egime of oilgas-wate multiphase flow, Int. J. Multiphase Flow, Vol 7, [37] R. Ulbih, M. Kotkiewiz, N. Szmolke, S. Anweile, M. Masiukiewiz and D. Zaja (00), Reognition of two-phase flow pattens with the use of dynami image analysis, Po. Inst. Mehanial Enginees Pat E-Jounal of Po. Meh. Engineeing, Vol 6, 7-33 [38] K. Mishima, T. Hibiki, Y. Saito, J. Sugimoto and K. Moiyama (999), Visualization study of molten metal-wate inteation by using neuton adiogaphy, Nul. Eng. Design, Vol 89, [39] J. A. Thie (98), Powe Reato Noise, ANS, La Gange Paks, Illinois, US [40] V. Azhanov and I. Pázsit (00), Deteting Impating of BWR Instument Tubes with Wavelet Tehniques, Powe Plant Suveillanes and Diagnostis - Applied Reseah with Atifiial Intelligene, Editos Da Ruan and Paolo F. Fantoni, Spinge, Physia Velag XIV, pp [4] C. Demazièe (004), Development of a -D -goup neuton noise simulato, Ann. Nul. Enegy, Vol 3, [4] G. I. Bell and S. Glassstone (970), Nulea Reato Theoy, Van Nostand-Reinhold, New Yok

40 9. Abbeviations 3 9. Abbeviations ANN APSD BWR IR-index LPRM MSE MTC NPP PWR STFT SKC SKI Atifiial Neual Netwok Auto Powe Spetal Density Boiling Wate Reato Impat Rate index Loal Powe Range Monito Mean Squae Eo Modeato Tempeatue Coeffiient Nulea Powe Plant Pessuised Wate Reato Shot Time Fouie Tansfom Swedish Cente of Nulea Tehnology Svenskt Käntekniskt Centum Swedish Nulea Powe Inspetoate Statens känkaftsinspektion

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49 Classifiation of Two-Phase Flow egimes via Image Analysis and a Neuo-Wavelet Appoah CARL SUNDE,*, SENADA AVDIC AND IMRE PÁZSIT,# Depatment of Reato Physis, Chalmes Univesity of Tehnology, SE Götebog, Sweden Faulty of Sienes, Depatment of Physis, Univesity of Tuzla Tuzla, Bosnia-Hezegovina * kalle@nephy.halmes.se # ime@nephy.halmes.se Numbe of pages: 3 Numbe of tables: 0 Numbe of figues: 9

50 Classifiation of Two-Phase Flow egimes via Image Analysis and a Neuo-Wavelet Appoah CARL SUNDE,*, SENADA AVDIC AND IMRE PÁZSIT Depatment of Reato Physis, Chalmes Univesity of Tehnology, SE Götebog, Sweden Faulty of Sienes, Depatment of Physis, Univesity of Tuzla Tuzla, Bosnia-Hezegovina * kalle@nephy.halmes.se ABSTRACT A non-intusive method of two-phase flow identifiation is investigated in this pape. It is based on image poessing of data obtained patly fom dynami neuton adiogaphy eodings of eal two-phase flow in a heated metal hannel, and patly by visible light fom a two-omponent mixtue of wate and ai. Classifiation of the flow egime types is pefomed by an atifiial neual netwok (ANN) algoithm. The input data to the ANN ae some statistial moments of the wavelet pe-poessed pixel intensity data of the images. The pe-poessing used in this pape onsists of a one-step multiesolution analysis of the -D image data. The investigations of the neuton adiogaphy images, whee all fou flow egimes ae epesented, show that bubbly and annula flows an be identified with a high onfidene, but slug and hun-tubulent flows ae moe often mixed up in between themselves. The eason fo the faulty identifiations, at least patially, lies in the insuffiient quality of these images. In the measuements with ai-wate two-omponent mixtue, only bubbly and slug flow egimes wee available, and these wee identified with nealy 00% suess atio. The use of the wavelet pepoessing, as ompaed to using the aw data without pe-poessing, did not have an influene on the suess atio of the identifiation; howeve, it deeased the taining time (numbe of epohs) with about a fato 00. KEYWORDS: Two-phase flow lassifiation; Image analysis; Neual netwoks; Wavelet analysis.. INTRODUCTION Two-phase flow pattens ae usually lassified into fou lassial so-alled flow egime types. These ae ) bubbly, ) slug, 3) hun-tubulent, and 4) annula flow egimes (see Fig..). Reognition and, possibly ontol, of the flow egime types is essential in numeous - -

51 Fig.. Shemati dawings of the diffeent phases of vetial two-phase flow. patial appliations. Fo instane, in two-phase flow alulations, in ode to use the oet algoithm in a fluid dynamis ode, the egime type in question must be known in advane, so that the ight expession an be hosen fo e.g. the intefaial shea oeffiient o some othe oeffiients like the heat tansfe oeffiient. Although the flow lassifiation an be done eliably in fully instumented hannels in whih themoouples, pessue tansdues, flow-metes et. ae available, a moe hallenging altenative would be to use non-intusive methods. In this field, the availability of the methods is muh moe limited. Non-intusive methods so fa have been based on adiation attenuation measuements, suh as X-ays (Vine and Lahey, 98) o gamma-ays (Chan and Banejee, 98; Chan and Bzovey, 990; Kok et al., 00). These methods ae usually based on the detetion of ollimated ays penetating the flow, and the poessing of the vaiation of the intensity, modulated by the flow, by vaious statistial methods (pobability distibutions, auto- and oss-oelations and speta). A qualitatively diffeent appoah, whih will be pusued in this pape, is to use image analysis, in ombination with a neual netwok based identifiation method with a wavelet pe-poesso. Afte all, the onept of flow egime aises fom an intuitive judgement of the topology of the flow, based on visual obsevation. Coespondingly, image analysis, although not ombined with so-alled intelligent omputing methods, has quite eently been eognised as one possible avenue of flow identifiation (Ulbih et al., 00; Kashdan et al. 004a, b). Images of two-phase flow ae though not easy to podue fo eal high-pessue high-tempeatue flows. Ulbih et al. (00) used a wate-ai two-omponent flow with visible light, and this is one of the altenatives that will also be investigated in this pape. Kashdan et al. (004b) investigated patile beam in ai. Images of a flow an be podued in tanspaent pipes easily with visible light, but in metal pipes, needed fo eal two-phase flow, neithe X-ays o gamma-ays ae appliable. X-ays do not penetate the wall, and - -

52 gamma-ays an not, in geneal, be podued with desied intensity suh that an image with good ontast and dynamis an be ahieved. Howeve, in the past one and a half deade, dynami neuton adiogaphy has been developed to the extent that it an podue dynami images of two-phase flow in metal pipes. Suh measuements wee made at the Kyoto Univesity Reseah Reato Institute (Mishima et al., 98). Some of these measuements wee made available fo us, and wee used in the pesent analysis. Howeve, the quality of these eodings tuned out to be insuffiient fo the pupose of effetive image analysis. Hene we have set up model expeiments with oloued wate - ai loop in a plexiglas slab pipe. In these expeiments visible light and an odinay digital video amode was used. This way images with muh bette ontast and shapness ould be obtained. The pupose of these latte expeiments was to hek the algoithms, elaboated fist in onnetions with the adiogaphy images, and see if the impovement in the input data quality led to impovement in the flow egime lassifiation. Moe details on the data aquisition and some image examples will be shown in the next setion. The image analysis, leading to the flow egime identifiation, is based on wavelet tansfom, i.e. multiesolution analysis, and neual identifiation. Suh methods have been used fo two-phase flow identifiation (Wu et al., 00), but the input signals wee simple poess signals and not images. Multiesolution analysis is apable of extating infomation fom a one- o two-dimensional stutue, oesponding to diffeent sales. Suh a poedue leads also to the edution of the dimensionality of the infomation epesented by the image, and an be used as input data to an Atifiial Neual Netwok (ANN) fo the lassifiation. This equies taining samples (images) being available to tain the netwok, whee the flow egime type is known. The tained netwok an be fed by simila, but unknown type, images as input, and used to lassify the type o egime. It is this pogam that has been pefomed in the pesent pape and whih will be desibed moe in detail below.. THE FLOW IMAGES Two diffeent types of images wee used. The fist one was obtained with neuton adiogaphy, and it ompises all fou flow egimes (Fig. ). The seond one was obtained with visible light of only bubbly and slug flow (Fig. 3). The way of poduing dynami neuton adiogaphi images, and in patiula fo twophase flow, an be found in the liteatue (Mishima et al. 988, 999a; Balasko et al., 987, 999). The essene is to use a ollimated beam of themal neutons fom a eato to illuminate the objet, in this ase the two-phase flow in the pipe. Afte penetating the pipe and the flow, whee some neutons ae emoved by satteing and absoption in the sample, the beam is dieted to a neuton onvete, whih onvets the neuton beam into visible light. This light, afte a efletion by a mio (in ode to filte out gamma-ays that aompany the neutons fo most neuton soues) ae eoded with an odinay CCD video amea. The images used in this wok wee eoded at the Kyoto Univesity Reato Reseah Institute (KURRI), by the Division of Nulea Engineeing (Mishima et al., 998). Duing the expeiment, the wate was iulated in a loop with an oveflow tank, and was heated up. The expeiment stated with pue wate in the loop, and the heating was ineased. With - 3 -

53 ineasing heating, all fou flow egimes oued in sequene. The measuement esults wee available fo us in fom on an analogue VHS eoding in NTSC fomat. Fom the whole sequene, fou setions, oesponding to the fou diffeent egimes, wee seleted, and fo eah egime a numbe of individual fames wee seleted. Eah fame was then saved as an individual jpg file. Although, by a visual inspetion of the video, the vaious flow egimes an be lealy seen, the ontast and shapness of the stati images obtained by the above poedue was elatively poo. Some examples ae shown in Fig.. One eason fo the poo quality is the digitization of the oiginal analogue eoding; the fat that the pitues wee not eoded with digital tehnology leads to a high level of bakgound noise. Fig.. Images of two-phase flow using neuton adiogaphy. To obtain bette quality images, a simple expeiment was set up at ou depatment. In this ase a thin tanspaent plasti pipe, filled with oloued wate, was used to geneate the images. The images wee eoded with a digital video amea. The two-phase flow was simulated by injeting ai in the bottom of the pipe. By this way only two-omponent flow ould be eated, with only bubbly and slug flow egimes. The same way as with the adiogaphy eodings, the a numbe of fames oesponding to the (now only two) available egime types wee seleted, and onveted as individual jpg files. Some examples ae shown in Fig. 3. It is seen that due to the digital signal handling thoughout, and the bette ontast and shapness of the oiginal image to be eoded, the image quality of these files is muh supeio to the adiogaphy ones. It has to be noted that the diffiulty we enounteed with the neuton adiogaphy images is not due to the tehnique itself. Vey high quality images of two-phase flow, and even steam explosions, have been podued lately with pulsed eatos and/o high speed digital ameas (Mishima et al., 999b). It is the intention of the pesent authos to epeat the analysis pesented hee also on bette quality adiogaphy images when and if they will beome available fo us

54 Fig. 3. Images of two-phase flow using visible light and oloued wate. In the pesent study, fom the neuton adiogaphi images a total of 00 fames fom eah of the fou egimes wee used fo the identifiation and lassifiation poess, wheeas 75 fames fom eah of the two egimes wee used fom the expeiment with visible light. 3. WAVELET DATA PROCESSING The infomation in eah image in the digital (jpg) files is ontained in appoximately pixels. Natually, this infomation ontent needs to be edued in dimension befoe using it as input to the neual netwok. Wavelet tansfom is known to often be an effetive tool to ahieve this goal (Addison, 00; Begh et al., 999). Apat fom data ompession, wavelet tansfom oeffiients ae also often bette (moe sensitive) featues in patten eognition tasks than the oiginal data. What egads two-phase flow, the fou flow egimes have stutues that show up spatial vaiations at diffeent sales. Hene, wavelet oeffiients fom a multi-esolution analysis (o the oiginal input data, but edued by a one-step wavelet multiesolution analysis) seem to be suitable input data. In ode to impove the lassifiation poess of the flows, it is advisable to pe-poess the images using wavelet tehniques befoe extating the input data fo the Atifiial Neual Netwoks (ANNs), (Hazaakia et al., 997; Fadhel and Bhattahayya, 999; Vema et al., 00; Kandaswamy et al., 003). The advantages using wavelet tansfom inlude some noise edution and featue extating at diffeent sales and dietions of the images. The one-dimensional wavelet tansfom maps a time signal into a time and fequeny signal at diffeent fequeny levels, N, (Addison, 00; Begh et al., 999; Mallat, 999). At eah level the signal is deomposed into an appoximation and a detail. It is possible to do both a disete (with disete fequeny levels), and a ontinuous (with ontinuous fequeny levels), tansfom. In the ase with an image, one uses the two-dimensional wavelet - 5 -

55 tansfom, whih maps a two-dimensional signal (image), in spatial athe than time oodinates, into the two oodinates at diffeent fequeny (wave numbe) levels. The twodimensional tansfom an also be done both disete and ontinuous in the fequeny oodinate o level. At the fist level of a -D disete wavelet tansfom, the oasest level of appoximation oeffiients, A ontains 5% of the infomation of the oiginal image, S 0. The appoximation oeffiients, A, at level, an be used to make a eonstution of an h v appoximation, S, of the oiginal image. In the same way the detail oeffiients, T, T and, d T (Fig. 4) ontaining the high fequeny infomation of the image, at level an be used to h v d eonstut hoizontal, D, vetial, D, and diagonal, D, details of the oiginal image. Adding the details to the appoximation one an eonstut the oiginal image ompletely without any loss of infomation. h v d S 0 ( x, y) S ( x, y) + D ( x, y) + D( x, y) + D( x, y) () It is possible to do the tansfomation into lowe levels wee the details at eah level ontain the infomation of the signal oesponding to the fequeny of that level. But in this wok, only the fist level of tansfomation was found to be useful to apply. The oeffiients and the oesponding eonstuted images ontain the same infomation, hene it is possible to use the oeffiients when extating input data fo the ANNs. The eonstuted images ae useful when displaying the tansfom. Fig. 4. Fist level of the D wavelet tansfom. S 0 is oiginal image o data, A is the set of the fist level appoximation oeffiients and T ae the fist level detail oeffiients in eah of the thee dietions, vetial, hoizontal and diagonal. 4. IDENTIFICATION WITH ANNS As mentioned above, it is possible to impove the lassifiation if the input data ae pe-poessed with wavelets befoe using them in the ANN. With the wavelets it is possible to extat featues whih ae not visible in the oiginal data. One set of featues, mentioned in Hazaika et al. (997), is the mean and vaiane of the fist level appoximation oeffiients. Anothe possibility would be to use the enegy of the diffeent wavelet details. The enegy of eah detail is defined as the sum of the squae of the absolute value of the detail oeffiients, (Addison, 00)

56 E d d T, i () In the fist un, the neuton adiogaphy images wee analysed. Input data fom the aw images wee extated and ompaed with the wavelet pe-poessed input data. In this ase the mean and vaiane fom both the aw images and the fist level appoximation oeffiients wee used fist, as efeene input data. It tuned out that, due to the poo image quality, no othe featues ould be suessfully used in the lassifiation poedue. In the seond ase the images podued with visible light, that had muh bette quality, wee used. Again the mean and vaiane of the aw data wee used as efeene. Hee, howeve, wavelet tansfom pe-poessing did lead to impovement. With the wavelet pepoessing, the enegy of the fist level of detail oeffiients and thei vaianes wee used, giving a total of 6 inputs fo the ANN (see Fig. 5). Fig. 5. Input data of the images made with visible light afte wavelet pe-poessing. As usual with wavelet analysis, the hoie of the ight wavelet fo the task at hand is not self-obvious. One guideline of hoosing a suitable wavelet is to selet one, whih has the same featues as the data analysed. Fo this lassifiation task the following wavelets wee tested: Daubehies of ode 8 (db8), Symmlet of ode 6 (sym6), Coiflet of ode 4 (oif4), - 7 -

57 Daubehies of ode (Haa) and biothonomal (bio3.). These ae all available in the Wavelet Toolbox in MatLab (Anon, 000). 4. Classifiation using atifiial neual netwoks ANNs ae apable to takle vey ompliated tasks, inluding non-linea lassifiation poblems. The bakpopagation (BP) algoithm is the most fequently used algoithm fo the taining of suh netwoks (Pazsit and Kitamua, 997). We have used the multi-layeed peepton (o simply the feed-fowad netwok) onsisting of an input laye, an output laye and one hidden laye. The netwok eeives input though the nodes in the input laye, fom whih the signals popagate fowad to the nodes of the onseutive laye and output signals ae podued in the output laye. In the bakwad phase, eo signals ae popagated bakwad though the netwok and some paametes ae adjusted in efeene to the eo signals. The pefomane of ANNs depends on the pope hoie of the input paametes. We have investigated the pefomane of the ANN fo vaious input paamete sets. The numbe of the input and output nodes is defined by the poblem itself. Fo the adiogaphi images, the numbe of input nodes was and fo the visible light images it was 6, as mentioned befoe. All the input featue vetos wee nomalized so that they fall in the ange [-,]. Fo eah type of flow, a oesponding output lass is assoiated. The ANN has 4 output nodes fo the adiogaphi ase and output nodes fo the visible light ase, oesponding to the 4 o diffeent flow types (Fig. 6). The taget value, duing taining, fo eah lass ontains value of 0.9 fo the oet ategoy and thee o one dummy vaiables with value of 0.. ANN Input Tansig 40 nodes hidden laye Logsig 4 nodes Thesholding Output Bubbly Slug Chun Annula Fig. 6. The Atifiial Neual Netwok used in the lassifiation poess of the neuton adiogapi images. The thesholding is only used duing testing. Sine the output laye hosen fo this lassifiation task has a log-sigmoid tansfe funtion, the output values ange fom 0 to, and thesholding has to be pefomed on the output data, to get 0 o, when lassifying. Two diffeent theshold levels wee used, 0.5 and 0.7. All output values lage than the theshold ae set to unity and all othe to zeo. If the output data, afte the thesholding, ae all fou zeos, the oesponding image is lassified as unknown o unlassified flow egime. The same applies fo the ase of moe than one nonzeo value. Lowe theshold makes the lassifiation less etain but a too high theshold will lassify many images as unknown

58 The optimal numbe of nodes in the hidden laye, the taining algoithm and the ativation funtions wee detemined by tial and eo. Tan-sigmoid funtion was used fo the hidden laye and log-sigmoid fo the output laye. A few of the modified bakpopagation (BP) algoithms suh as adaptive leaning ate, esilient BP, saled onjugate gadient and gadient desent algoithm with momentum wee examined fo taining the ANN. Cossvalidation was used to estimate whih leaning ANN model will pefom the best on the poblem at hand. Fo eah of the models a 5-fold oss-validation ove the taining set was used, whih means that /5th of the taining data was used as a validation set and the poess was epeated with non-ovelapping otations. In the ase of the neuton adiogaphi images, eah leaning model was tained with 640 samples fo the fou vaious types of flow, and then they wee tested on one subset of 60 samples whih was not used duing taining. Fo the visible light images, the taining set onsisted of 0 samples and the test subset of 30 samples (Fig. 5). The esilient bakpopagation (RP) algoithm was found to have the highest aveage test set soe. Namely, the lassifiation effiieny with RP algoithm was 00% (the peentage of the flows that wee oetly lassified) when the eall test was pefomed on the taining set, and 95% when the eall test was made on the subset not used duing taining. The best pefomane was obtained fo the taining and validation test set with an ANN stutue onsisting of one hidden laye having 40 nodes. The default pefomane funtion fo feedfowad netwoks was the mean squae eo, i.e. the aveage squaed eo between the netwok outputs and the taget outputs. The ANN taining was pefomed until the maximum numbe of epohs, set to 30000, was eahed o the mean squae eo, MSE, taget value of 0-3 was ahieved. All alulations wee aied out by using the toolboxes available with the tehnial omputing softwae MATLAB (Anon, 000). 5. Neuton Radiogaphi images 5. RESULTS AND DISCUSSION The pefomane of diffeent wavelets, db8, sym6, oif4, haa and bio3., on the lassifiation effiieny has been investigated. The lassifiation effiieny is defined as the peentage atio of the numbe of flow pitues oetly lassified to the total numbe of pitues oesponding to one type of flow. Aveage effiieny of the flow lassifiation fo eah type of the hosen wavelets by using ANN with mean values and vaiane as input and fo theshold levels of 0.5 and of 0.7, is depited in Fig. 7. The aveage effiieny is shown with eo bas (standad deviation). Almost the same esult is ahieved independent of whih wavelet type that is used. Though, oif4 has a slightly lowe peent of oet lassified egimes and Haa has the best lassifiation effiieny. Clealy a theshold of 0.5, with auay of about 95%, is bette than a theshold of 0.7, whih only has about 87% oetly lassified images. But even by using the mean and vaiane of the oiginal image values, the effiieny is the same. So thee is not muh impovement, in lassifiation, using the wavelet tansfom in this ase. All of the diffeent wavelets lassified the annula flow with 00% effiieny, whih is also the ase fo the oiginal data. In Fig. 8 eah ba epesents the 00 images fom eah flow and the diffeent olous show whih egime the images wee identified with. White is fo unknown o unlassified images. As it is seen, a vey few images tun out to be unlassified fo a theshold of 0.5. The most diffiult egimes to lassify fo both the wavelet - 9 -

59 Fig. 7. Classifiation effiieny fo the diffeent wavelets and the oiginal data, theshold 0.5 to the left and 0.7 to the ight. and the oiginal data ae the slug and hun flow. This is of ouse due to the fat that these flows have simila featues NN with db8 and th0.5 bubbly slug hun annula unknown NN with db8 and th Bubbly Slug Chun Annula 0 Bubbly Slug Chun Annula Fig. 8. Result of the lassifiation of the neuton adiogapi two-phase flow images using input data pe-poessed with the db8 wavelet, with a theshold of 0.5 to the left and 0.7 to the ight. The maximum numbe of epohs, , wee used in the taining poess, meaning that the MSE taget value of 0-3 was not eahed. 5. Visible light images The same five wavelets as above wee used also in the ase with visible light images. The esults ae patly simila to the adiogaphy images. Namely, thee is not muh inease of the lassifiation effiieny when using wavelets ompaed to the featues of the aw images. Howeve, in this ase thee is a lage impovement in the numbe of epohs (iteations) used when taining the ANN. When using the wavelet pe-poessed data, the numbe of epohs is edued with a fato of 00, ompaed to the aw data input (Fig. 9). With the aw data, the numbe of epohs always exeed the maximum numbe of , befoe eahing the MSE taget values of 0-3. The lassifiation effiieny is also slightly - 0 -

60 bette in this ase, ompaed to the neuton adiogaphi ase. This is pesumably also attibuted to the bette quality of the images. Fo the wavelet pe-poessed data one slug flow image was lassified as bubbly and the est wee oetly lassified. That is, 49/ % fo wavelet pe-poessing, wheeas suess atio of lassifiation with the aw image inputs was 46/ %. The esult was the same fo all wavelets with a theshold of 0.7. Fig. 9. Depited is the numbe of epohs used when taining the netwok. The input data ae wavelet pe-poessed visible light images. 6. CONCLUSIONS The methodology used in this pape appeas to be suitable to lassify two-phase flow egimes by image analysis, in a non-intusive way. The oiginal idea was to use data fom dynami neuton adiogaphy of two-phase flow in a heated metal pipe. Suh measuements have been pefomed and by now good quality images an be obtained. Unfotunately no suh measuements wee available to us, only old analogue eodings. Howeve, a poof-ofpiniple investigation was made of the appliability of the image analysis method, with positive outome. The potential of the method wee then investigated by using expeiments with visible light on a wate-ai mixtue. In this expeiment only a few basi flow stutues ould be ahieved, but the images had bette quality. Hene it was possible to pove the extended effiieny of the wavelet pe-poessing of the data befoe the neual identifiation. The two diffeent measuements togethe show that on-line, non-intusive identifiation of two-phase flow in a metal pipe an be ahieved with poessing dynami neuton adiogaphy images with a neuo-wavelet appoah. 7. ACKNOWLEDGEMENT This wok was suppoted by the Swedish Cente fo Nulea Tehnology, and the Swedish Nulea Powe Inspetoate. 8. REFERENCES Addison P.S. (00) The Illustated Wavelet Tansfom Handbook, Institute of Physis Publishing, Bistol and Philadelphia. - -

61 Anon (000) Uses manual MatLab. The Math Woks In., Natik, MA US. Begh J., Ekstedt F. and Lindbeg M. (999) Wavelets, Studentlitteatu, Lund, Sweden. Balasko M, Svab E, Cse L (987) Simultaneous Dynami Neuton And Gammaadiogaphy. NDT Intenational 0, Balasko M, Koosi F, Svab E, Eodogh I (999) A novel type epithemal neuton adiogaphy deteting and imaging system. Nul. Inst. Methods A 44, Chan A.M.C. and Banejee S (98) Design aspets of gamma densitometes fo void fation measuements in small sale two-phase flows, Nulea Instuments & Methods 90, 35. Chan A.M.C. and Bzovey D. (990), Measuement of mass flux in high tempeatue pessue steam-wate two-phase flow using a ombination of Pitot tubes and a gamma densitomete, Nulea Engineeing and Design, 95. Fadhel E.A. and Bhattahayya, P. (999), Appliation of a Steeable Wavelet Tansfom using Neual Netwok fo Signatue Veifiation, Patten Analysis & Appliations, 84. Hazaika N., Chen J. Z., Tsoi A. C. and Segejew, A. (997), Classifiation of EEG signals using the wavelet tansfom, Signal Poessing 59, 6. Kandaswamy A., Sathish Kuma C., Ramanathan R. P., Jayaaman S. and Malmuugan N. (003), Neual Classifiation of Lung Sounds using wavelet oeffiients, Computes in Biology and Mediine 6, 53. Kashdan JT, Shimpton JS, Whybew A (004a) Two-phase flow haateization by automated digital image analysis. Pat : Fundamental piniples and alibation of the tehnique. Patile & Patile Systems Chaateization 0, Kashdan JT, Shimpton JS, Whybew A (004b) Two-phase flow haateization by automated digital image analysis. Pat : Appliation of PDIA fo sizing spays. Patile & Patile Systems Chaateization, 5-3 Kok H.V., van de Hagen T.H.J.J., Mudde R.F. (00) Subhannel void-fation measuements in a 6 x 6 od bundle using a simple gamma-tansmission method. Intenational Jounal Of Multiphase Flow 7, Mallat S. (999), A wavelet tou of signal poessing, Aademi Pess, USA. Mishima K., Fujine S., Yoneda K., Yonebayashi K., Kanda K.and Nishihaa H. (988) Po. 3d Japan-US Semina on Two-Phase Flow Dynamis, Ohtsu, Japan. Mishima K, Hibiki T, Saito Y, Nakamua H, Matsubayashi M (999a) The eview of the appliation of neuton adiogaphy to themal hydauli eseah. Nul. Inst. Methods A 44, 66-7 Mishima K, Hibiki T, Saito Y, Sugimoto J, Moiyama K (999b) Visualization study of molten metal-wate inteation by using neuton adiogaphy. Nul. Eng. Design Pazsit, I. and Kitamua, M. (997), The Role of Nulea Netwoks in Reato Diagnostis and Contol, Advanes in Nulea Siene and Tehnology 4, 95. Vema M. S., Patt L., Ganesh C. and Medina C. (00), Hai-MAP: a pototype automated system fo foensi hai ompaison and analysis, Foensi Siene Intenational 9, 68. Vine M. A. and Lahey R. T. (98) Intenational Jounal of Multiphase Flow 8,

62 Ulbih R, Kotkiewiz M, Szmolke N, Anweile S, Masiukiewiz M, Zaja D (00) Reognition of two-phase flow pattens with the use of dynami image analysis. Po. Inst. Mehanial Enginees Pat E-Jounal of Po. Meh. Engng Wu H., Zhou F and Wu Y (00) Intelligent identifiation system f flow egime of oil-gaswate multiphase flow. Int. J. Multiphase Flow 7,

63

64 Investigation of deteto tube impating in the Ringhals- BWR Cal Sunde and Ime Pázsit Chalmes Univesity of Tehnology Depatment of Reato Physis SE-4 96 Götebog Sweden Tel: Fax: Abstat: Neuton noise measuements wee made in two onseutive fuel yles in the Swedish BWR Ringhals- with the pupose of diagnostis of vibations and impating of deteto stings. The analysis was based on two diffeent methods. The fist was the taditional spetal analysis, whih uses autospeta and the oheene and phase between detetos in the same sting, fo a qualitative judgement. The seond method was based on alulating the so-alled impating ate by a wavelet-based poedue, developed at the Depatment of Reato Physis at Chalmes eently, and whih was peviously tested on data fom the Oskashamn BWR. Also a new method, wavelet-based oheene, was tested with suess. The fist measuement seies was not suitable fo a wavelet analysis, beause of the low sampling fequeny. The taditional method is moe obust; howeve, it only gives a qualitative esult and equies the subjetive deision of a noise expet. Based on the esults of the analysis, with emphasis on the taditional method, the deteto tubes wee divided into thee goups with espet to the seveity and likelihood of impating. Fo the fist seies of measuements, these onlusions ould be heked against visual inspetion of the fuel assemblies duing efuelling afte the yle, in ode to find impating damage. A good oelation between the pedition of the analysis and the inspetion esults was found. Keywods: noise analysis, neuton noise, deteto impating, spetal analysis, APSD, oheene, phase, wavelet analysis. Biogaphial notes: Cal Sunde eeived his MS in engineeing physis fom Chalmes Univesity of Tehnology, Sweden, in 00 and is woking as PhD student at the Depatment of Reato Physis of Chalmes. His eseah inteests ae noise diagnostis, diagnostis of two-phase flow and intelligent omputing methods inluding atifiial neual netwoks and wavelet analysis. Ime Pázsit is pofesso at the Depatment of Reato Physis, Chalmes Univesity of Tehnology, Götebog, Sweden. He got his PhD at the Univesity Loand Eötvös in Budapest, Hungay in 975 and woked in the Cental Reseah Institute fo Physis, Budapest, until 983. Afte that he spent a peiod in Studsvik, Sweden, befoe he joined Chalmes in 99. His eseah inteests ae flutuations in neuton tanspot and atomi ollision asades; eato diagnostis based on noise analysis; elaboation of invese methods in neuton noise diagnostis; intelligent omputing methods suh as atifiial neual netwoks and wavelet analysis; diagnostis of two-phase flow and fusion plasma; and eently, positon annihilation spetosopy and positon tanspot. He is the dieto of the Chalmes Cente fo Nulea Tehnology, and Head of the Setion fo Mathematial Physis of the Swedish Physial Soiety. Intodution

65 Exessive vibations and impating of deteto stings in BWRs have been a known poblem fo a long time [-6]. The situation is illustated in Figue. High flow tubulene aound the deteto fom the bypass flow at the lowe oe suppot plate an ause stong vibations of the deteto tube (also alle instument tube, o deteto/lprm sting). Vaious methods of deteting impating fom the signals of the vibating neuton detetos have also been elaboated and tested duing the past few deades [3-9]. These methods an be divided into two ategoies: taditional o spetal analysis based methods, and wavelet analysis methods. The spetal analysis methods ae based on assumptions onening how the spetal popeties of the APSD, and in patiula the popeties of the vibation peak(s) and the phase of the oss-speta hange ompaed to vibating but not impating, o not vibating, ases. These methods assume stationaity of the signal, and they ae mostly qualitative. They have nevetheless been found to wok in seveal ases. As is lea fom the desiption, these methods mostly ely on the knowledge of a efeene base, i.e. vibation haateistis of vibating but not impating stings. In othe wods these methods ae in need of some fom of alibation. The wavelet-based methods ae moe absolute, i.e. have less need of alibation, and supply a quantitative measue of the seveity of impating. They do not assume a stationay behaviou, athe they an handle intemittent signals as well. They ae, on the othe hand, based on a hypothesis, whih, although being vey plausible, has not been possible so fa to pove o dispove in patie. The hypothesis is that eah impat of the deteto sting against the fuel box indues a shot, damped vibation of the assembly, with a fequeny whih is highe than the eigenfequeny of the deteto sting (fo an illustation, see Figue ). The neuton noise indued by this shot lived vibation is sensed by the neuton deteto as a spike in a boad-band noise, whih an be extated effiiently by wavelet filteing methods. This latte method has also been veified on measuements taken in seveal Swedish BWRs [9-]. In addition, ontinuous wavelet tansfom methods wee also tested in this wok as a novel method fo deteting deteto impating. A time-fequeny dependent wavelet oheene was alulated, whih shows high values in the fequeny ange of the impating deteto stings aound the eigenfequeny of the damped vibations of the fuel assemblies. When assessing the pefomane of the vaious methods, it should be noted that thee is no geneal assessment available about the pefomane of the vaious methods at diffeent eatos. In most ases, one kind of method was used by one diagnosti goup at one type of oe and oesponding instumentation. Thee ae also some indiations that a method, whih was used suessfully at one eato unit, did not pefom suessfully when tansfeed to anothe oe. In Sweden, both spetal and wavelet based methods, that woked well at the Basebäk eato, did not pefom so well when tansfeed to measuements made on the Oskashamn eato [5-6, 9-]. The wavelet based method, fo instane, equied futhe development befoe it woked just as effetively fo the Oskashamn measuements as fo the Basebäk ones. In othe wods, the expeiene with the tansfe of methods fom one oe to anothe is elatively spase. Theefoe, it appeas inteesting to epot on the appliability of some, aleady tested, methods in a new oe. In Sweden, vibation analysis has been pefomed in the Basebäk and Oskashamn eatos, but this is the fist ase when the two methods epoted on in this ommuniation wee tested on the Ringhals- BWR. Unfotunately, only one (the seond) out of the two measuement ampaigns was suitable fo wavelet analysis, wheeas post-yle heking of the aused damage of impating, whih an be used fo ating the pefomane of the diagnostis, is uently available only fo the fist measuement seies. (As of this witing, the fuel yle in whih the seond measuement seies was made, is still on-going). The wavelet-based analysis indeed showed that a diet tansfe of the method that woked quite effetively fo the Oskashamn measuements, is not possible, and futhe modifiations of the method beame neessay. On the othe hand, the taditional spetal analysis ould be pefomed fo both yles, and the pefomane of the diagnostis in the fist yle ould be heked duing fuel inspetion duing the evision. A good oelation

66 was found between the pedition of the diagnostis and the atual damages found. A postyle onfimation of the seond analysis will be made duing efuelling in late summe 004. Piniples of analysis. The spetal analysis method This method was elaboated and fist used duing the analysis of the ealy Basebäk measuements in 986 by the Studsvik noise goup [5]. Its piniples ae desibed thooughly in [6]. The essene is that in the absene of vibations, the APSD (auto-spetum) of the LPRMs, i.e. the individual detetos, is a smooth funtion of the fequeny, and the oheene and phase between any two detetos follows a patten haateisti fo popagating petubations in the pesene of a loal noise omponent (in-oe BWR noise). The phase is linea up to 0 Hz, and the oheene shows a peiodi peak-sink stutue, whose maxima and minima oinide with the zeo- and π -ossings of the phase. The pesene of weak vibations an be notied in the appeaane of a elatively naow peak in the APSD at the fundamental fequeny (eigenmode) of the deteto sting, usually in the ange of -5 Hz. With an ineasing amplitude of the vibations, the fundamental peak ineases; also, new peaks ou at fequenies of the highe hamonis (double and tiple of the fundamental fequeny). When impating stats, the vibation peak gets boadened. Conuently, the linea phase, and to a lesse extent also the sink stutue of the oheene, gets distoted with ineasing vibations. The most visible is the distotion of the phase aound the vibation fequeny; with ineasing stength of the vibations, the phase tends to be zeo ove an ineasing fequeny egion aound the vibation fequeny, owing to the simultaneous movement of the two detetos. It is impotant that out of the above quantities, only the peak boadening is a lea indiato of impating. The highe hamonis and phase distotion ae, stitly speaking, only an indiato of ineasing vibation amplitudes. Nevetheless, a lage vibation amplitude is assoiated with a highe pobability of impating. Expeiene so fa shows that the set of above indiatos an be used to a quantitative lassifiation of the pobability and seveity of impating into thee basi goups as follows: No vibations: smooth stutue of the APSD and the oheene, no peaks linea phase between the two detetos in the same sting, showing the time delay of the signal (bubble tansit time). Vibations but no impating: a single naow peak in the APSD and the oheene linea phase distoted in the naow fequeny ange of vibations, the phase is zeo thee. Indiations of impating: a boad peak in the APSD seveal peaks, mainly a seond peak at the double fequeny of the fundamental mode phase uve distoted and zeo ove a lage fequeny ange The seveity of the impating an be judged by whethe o not all thee points ae fulfilled simultaneously, and also on the quantitative value of the indiatos (bodening, distotion et).. The wavelet-based method The fist method that has been used in seveal appliations befoe, is based on disete wavelet tansfom and wavelet filteing. The piniples of this method ae desibed in [8-]. An illustation of the assumptions is shown in Figue. In ase of impat-fee vibations, the deteto signal onsists of a boadband bakgound noise, and a naow fequeny band

67 omponent, both omponents being stationay. The impating indues a shot, damped vibation of the fuel assembly, whih is onveted into a shot, deaying tansient in the deteto signal. It is assumed that the eigenfequeny of the tansient vibations is notieably highe than that of the deteto stings, and is in the ange 0-5 Hz. These tansients o spikes ou in an intemittent manne, and thei ontibution to the signal APSD is negligible ompaed to the othe two omponents, due to thei small amplitude and spase ouene. The task of the wavelet analysis is to detet the pesene of suh spikes and give a measue of the fequeny of thei ouene. This quantity is named the impating ate, and is alulated in an algoithmi way by wavelet filteing. The essene of the method is a wavelet filteing, based on the disete wavelet tansfom (DWT) and a thesholding of the wavelet tansfom oeffiients befoe invese tansfom. A usual poblem in suh an appliation is the detemination of the filte theshold. Usually, the vaiane of the bakgound noise is used to set the theshold in an algoithmi way. The bakgound noise level, on its tun, is detemined fom the high fequeny tail of the APSD, fo fequenies above the fuel assembly eigenfequeny. In the fist appliation [8] Haa wavelets wee used, and the beak fequeny of the high-pass filte of the bakgound extation method was set in an ad ho way at a fequeny that was assumed above the fuel eigenfequeny. Late this method was futhe developed by using sale-dependent thesholding, i.e. the thesholds ae diffeent fo eah of the diffeent sales in the wavelet multisale esolution [9-0], when analysing measuements taken in Oskashamn-. Fist a global theshold is alulated [3-5]: thg * log ( N ) () whee N is the length of the signal. Then a sale dependent theshold is alulated as the median absolute deviation: ths ( a) median( ( a) ) / () whee (a) ae the wavelet oeffiients at sale a. The final theshold at sale a is then: th( a) thg * ths( a) (3) The multisale esolution is pefomed down to sales oesponding to the fequeny of the fuel-box, 0 Hz. The onnetion between sale and fequeny is given by the so-alled ente fequeny of the wavelet [3]. When analysing the Oskashamn measuements it was also notied that, unlike in the Basebäk ase, the hoie of the mothe wavelet did play a ole. Moeove, the algoithm had to be impoved to be ompetitive with othe, non-wavelet based tehniques. The impovement onsists of a final thesholding of the diffeene V between the wavelet denoised signal Den(S) and the appoximation at the lagest sale level A in the multisale esolution V Den( S) A (4) using the vaiane of the bakgound noise. In analysing the measuements fom Ringhals- the disete Meye wavelet was used without the final thesholding of V. The esult is patly a visual display of the emaining spikes that ae supposed to epesent the impating effets and thus give an intuitive measue of the impating fequeny o seveity, and patly a numbe alled Impat Rate, whih is the total numbe of spikes (impats) in the signal. In addition, a new method was tested in the pesent measuements the fist time, based on wavelet oheene. The inspiation fo the test of this method was taken fom a eent wok by Pokol and Po [6]. Similaly to the ase of the taditional spetal analysis method, the wavelet oheene was alulated between the signals of two detetos in the same instument tube. The wavelet oheene between two signals f and g is detemined as follows [6]. Fist a shot-tem aveaging integal of the wavelet tansfoms is alulated as: t+ T/ w * f, g(,) f(, ) g(, ) t T/ C f t W f τ W f τ dτ (5)

68 Hee W ( f, τ ) is the ontinuous wavelet tansfom of signal f and g, espetively and T is hosen to be 00 times the sampling fequeny in ode to get a good aveaging. Note hee that, aoding to the haate of the appliation, the wavelet tansfom is given in tems of the fequeny athe then as a funtion of the sale. The two vaiables ae invesely elated, but the oespondene is unambiguous. The oheene is then alulated as w Cf, g( f, t) w γ f, g( f, t) (6) w w C f, f( f,) t Cg, g( f,) t This way a oheene funtion is obtained, whih, similaly to the odinay CWT tansfom, is dependent on both time and fequeny. One an hoose the aveaging window in (5) to be of diffeent length; if the integal is extended to the whole time inteval of the measuement, one obtains a oheene funtion whih, similaly to the FFT-based oheene, is only dependent on fequeny. Even in suh a ase, as a ule, the infomation ontent of the wavelet oheene diffes fom that of the spetal oheene, as is demonstated in [4]. Hee the expetation is, as was shown in [6], that the wavelet oheene an identify haateistis at etain fequenies that annot be seen in the spetal funtions (APSD) o the wavelet-based spetogams of single signals. 3 The measuements Two diffeent measuements ae analysed, the fist measuement, #, was taken duing statup of Ringhals- on Septembe 3-6, 00 and the seond measuement, #, one yea late on Otobe 7, 003, also duing stat-up. The fist measuements onsist of fou diffeent measuements, thee at edued oe flow and edued powe, to investigate BWR stability, and one measuement at full flow and full powe. The late one is the most inteesting one egading deteto tube vibations, hene solely this measuement was used in this analysis. Signals fom a total of 36 oe positions wee eoded, eah position ontaining two detetos, one at a lowe axial elevation (4) and anothe one at a highe axial elevation (). This gives a total of 7 diffeent signals, but the signals fom one position wee defet. The measuement was aound min long with a sampling fequeny of.5 Hz. Afte the analysis of the fist measuement it was ealised that the sampling fequeny was too low fo the wavelet-based method. A new measuement was then taken duing the stat-up of the following yle in 003. This measuement also onsists of 7 signals, but with a highe sampling fequeny of 60 Hz and a length of 5 min. The layout of the oe, with the positions of the deteto stings is shown in Figue 3. The positions of the LPRM stings 9, 0 and that wee judged in the analysis to have impating in the fist seies of measuements (see Setion 4), as well as sting whih did not even indiate vibations, ae maked in the figue. 4 Results of the analysis 4. The measuements made in 00 (seies #) Expeiene fom ealie studies shows that lassifiation and anking of the pobability and seveity of impating of the vaious stings is most effetive if all ases ae investigated and an intenal ompaison between the diffeent stings is made. This is beause the methods, at least fo the taditional spetal analysis method, ae not absolute, and the iteia fo judgement ae diffeing fom oe to oe. Fo the fist seies of measuements, analysed in this setion, the wavelet based analysis was not effetive, due to the low sampling fequeny, hene in this setion we shall onentate on the spetal analysis. Fo all stings inluded in the measuements, the APSD of both detetos within a sting, as well as the oheene and the phase between the two detetos wee made. A omplete list of these indiatos, as well as the aw time signals and the APD funtions (amplitude

69 pobability distibutions) wee listed in the intenal epot []. Based on these data, the stings wee lassified as follows. Some stings did not show signs of vibations and hene had no impating at all. Some stings showed signs of mild vibations whee the pobability of impating was judged to be negligible. The next ategoy is that of stings with definite vibations and a non-negligible likelihood of impating, suh that the impating was not judged to be sevee. Finally some stings wee judged to have sevee impating with a lage onfidene in the judgement, based on the expeimental evidene. Examples of some of the ases, and in patiula fo the heavily impating stings, will be shown below. LPRM (Figue 4) shows no signs of vibations, hene this instument tube has no impating. As Figue 4 shows, the APSD has a smooth dependene on fequeny, without any appaent peaks. The most onvining indiato of negligible vibations is supplied by the phase uve. It has a lea linea shape up to 4.5 Hz, whih is the maximum fequeny ange fo Ringhals- whee linea phase an be obseved. Fo ompaison it is inteesting to notie that in Basebäk- the linea phase extended well ove 0 Hz, showing that the signatues ae diffeent even fo non-vibating stings fo diffeent oes. The oheene also has a stutue that is assoiated with pue two-phase flow indued noise, with minima and maxima oesponding to the π - and zeo-ossings of the phase, espetively. The stings with the lagest likelihood of heavy impating wee found to be LPRMs 9, 0 and. The oesponding auto-speta, oheene and phase uves ae shown in Figues 5-7. These all show one o moe peaks in the auto-spetum, and a signifiantly distoted phase and oheene stutue. LPRMs 9. and 9.4 (Figue 5) both show a boad peak at aound.5 Hz. Hee, LPRM 9. denotes deteto in sting 9 et. In addition, peaks ae pesent even at highe fequenies; in LPRM 9. the fist highe hamonis at 3 Hz is seen, wheeas 9.4 shows a peak at aound 4.5 Hz, whih oesponds to the seond hamonis. The phase is nealy zeo eveywhee between 0 and 4 Hz, and the sink stutue of the oheene is diffeent fom that of LPRM, i.e. the non-vibating ase. Thee is a peak in the oheene at.5 Hz, whih is the fundamental fequeny of the instument tube vibations. The analysis of the othe two stings that wee judged to have impating goes in a simila manne. In LPRM 0 thee is a boad peak at the fundamental fequeny, and taes of peaks at highe fequenies. The phase stats out in a linea manne, but above Hz it dops to zeo. In LPRM thee is a lage peak at.5 Hz, whih is not as boad as in the othe two impating stings, but seveal highe hamonis ae seen with boad peaks. The phase and oheene ae distoted, and the peak stutue of the oheene is ompletely ditated by the positions of the peaks of the APSD, i.e. the vibation eigenfequenies. Finally, one ase is shown, LPRM 5, with some possibility of impating, whose likelihood and/o seveity is nevetheless judged to be lowe than fo the above thee ases (Figue 8). The same haateisti featues ae seen as fo LPRMs 9, 0 and, but to a smalle extent. The peak at.5 Hz is smalle and is naow, and the phase shows a linea dependene on fequeny ove a lage pat of the egion 0-4 Hz, exept at the vibation fequeny. This ase also illustates the fat that thee is no lea-ut sepaation between the ases of heavy and light impating, and that the poedue is based on an expet evaluation of the alulated quantities. As was mentioned ealie, fo this measuement, the wavelet based impat detetion algoithm was not appliable. The eason is the low sampling fequeny of the measuement (whih was pimaily made fo investigating BWR stability, whose haateisti fequeny is about 0.5 Hz). The sampling fequeny of.5 Hz means that the spetal quantities an only be alulated up to 6.5 Hz. This is suffiient to see the deteto tube vibation peaks and the lowest highe hamonis. Howeve, the damped vibations of the fuel assemblies, indued by the impating, ae expeted to lie in the ange 0-5 Hz. The low sampling fequeny auses two poblems. Fist, the shot tansients ( spikes ) in the signal, oesponding to the effet of the fuel assembly vibations, an go undeteted fully o patially. Seond, the vaiane of the bakgound noise, whih is neessay to know in ode to set the wavelet filte theshold, annot be detemined sine this equies aess to the high fequeny tail of the APSD, above 0 Hz. Fo this easons, no wavelet-based wavelet analysis of measuement was pefomed.

70 4. The measuements made in 003 (seies #) This measuement was pefomed with a sampling fequeny of 64 Hz, whih makes it suitable also fo wavelet analysis. Howeve, the analysis will be fist pefomed with the taditional spetal analysis, similaly as in the pevious ase. The piniples ae the same, theefoe hee only a summay of the findings will be given hee. In these measuements, fou stings wee found to show the stongest vibations, and these ae LPRMs 5, 6, 4 and 35. As an illustation, the ase of LPRM 6 is shown in Figue 9. It shows the same featues as those in measuement. Thee ae seveal boad peaks at aound, 4 and 6 Hz, i.e. at the fundamental fequeny and oesponding hamonis. One an note that these fequenies ae somewhat highe than those in the pevious measuement. The phase is lose to zeo at low fequenies up to 4 Hz and the peak stutue of the oheene follows the peaks in the APSD losely. The othe LPRMs judged to exeute impating show simila featues. They ae not shown hee fo bevity. A summay of all the findings with lassifiations is given in the next Setion. Fo measuement #, the wavelet based impat diagnostis was also possible to pefom. Hee, howeve it was notied that the method, based on wavelet filteing and thesholding, whih was also used in the evaluation of the Oskashamn- measuements, had to be slightly modified. This is in line with the fat that when applying the method, oiginally used fo Basebäk, to the Oskashamn measuements, also a modifiation was neessay befoe a good disimination powe of the method was ahieved as follows. The theshold given in Eqn (3) was used on the wavelets oeffiients of the two smallest sales, oesponding to fequenies highe than 0 Hz, of the multisale esolution using the disete Meye wavelet [3]. Afte the thesholding the signal was eonstuted and the diffeene V (Eqn 4), was examined fo the possible ouene of spikes, whih ae supposed to epesent the intemittent fuel-box vibations. In this measuement the best disimination was obtained at oe level 4, i.e. the lowest axial elevation. The quantity V (Eqn 4) is shown in Figue 0a fo LPRM 0.4 whih is assumed to be non-vibating, and on Fig. 0b fo LPRM 6.4 whih is assumed to be impating aoding to the spetal analysis. The diffeene between the two signals, what egads the numbe of impating spikes found, is lealy visible. The IR index fo LPRM 6.4 is 4.6 impats/min wheeas fo LPRM 0.4 it is only 0. impats/min, whih quantifies the impating status of the two stings. Out of the 36 LPRM signals at level 4, the following ones have IR index highe than impats/min: LPRMs 4, 6, 3, 4, 34 and 35. Out of these, LPRMs 6, 4 and 35 ae idential with thee of the fou LPRMs, pointed out by the spetal method to be the most pobable impating ones. LPRM 4 is in the seond highest ategoy of impating pobability based on the spetal method. Thus the esult fom the wavelet based analysis is in good ageement with the lassial spetal analysis. As was mentioned ealie, the new method of wavelet oheene was also tested on these measuements. The wavelet oheene was alulated by using a Meye wavelet via Eqns (5) and (6) fo eah sting, using the two deteto signals available pe sting. It was found that fo etain stings a high value of the oheene was found between 0 and 0 Hz, whih is the fequeny ange whee the fuel box vibations ae expeted to take plae. Fo the othe stings no suh omponent was obseved, so this analysis divided the LPRM tubes into two ategoies. A ompaison with the esults fom the spetal and wavelet filteing analysis showed that the LPRM stings showing a lage wavelet oheene aound 5 Hz oinide with those that wee pointed out as most likely impating. An illustation is given in Figue, whih shows fou LPRM stings: two with high wavelet oheene at 5 Hz (LPRM 6 and 35) and two othes that have an aveage (low) oheene at these fequenies (LPRM and 7). A ompaison with Table II onfims that these stings ae suspeted fo stong impating by the othe two methods.

71 It is inteesting to note that the odinay FFT-based oheene has a lage, shap peak at aound 5 Hz fo all deteto signals, with vey little vaiation in the amplitude (Figue ). The oigin of this vey naow peak in the spetal based oheene is not undestood. Although it is in the middle of the fequeny ange found by wavelet oheene method, whih is attibuted to the fequeny of the impating indued damped vibations of the fuel assemblies, yet this appeas to be given ise by a diffeent physial phenomenon. Namely, stongly damped vibations lead to a boad peak (peak boadening is in fat one of the taditional methods to detet impating); the peak in the spetal based oheene is too naow (monohomati) to be due to impating. The naow peak in the spetal based oheene, on the othe hand, does not show up in the wavelet oheene, due to the fat that the fequeny esolution of wavelet methods ae muh oase (espeially fo highe fequenies) than the FFT-based methods. Hene it seems that, despite the quantitative oinidene of the fequenies whee both the spetal and the wavelet oheene has peaks o high values, espetively, these oespond to physially diffeent phenomena. It is also seen that the wavelet oheene has a muh highe disimination powe fo deteting impating than the odinay oheene. The pefomane of the wavelet oheene method has though to be onfimed by futhe tests. 5 Disussion In measuement, only the taditional diagnosti poedue was appliable, so a ompaison between the two methods was not possible. On the othe hand, the peditions of the taditional spetal method ould be ompaed to the esults of visual inspetion of seveal, but not all, fuel assemblies. The eason fo the patial obsevation is that duing an aveage efuelling, time onstaints do not pemit to inspet all fou fuel assemblies aound all 36 LPRM positions, only a few seleted ones. One use of the vibation diagnostis is to estit the numbe of inspeted fuel assemblies into a limited set of suspeted positions. The esult of the taditional diagnostis of measuement an be summaized in the Table. In the left side the lassifiation of the diffeent LPRMs is given with espet to the pobability o seveity of impats. This infomation was made available to the powe plant as a eommendation on whih LPRM positions should be inspeted. In patiula, the impating LPRMs 9, 0, and the non-vibating LPRM (as efeene) wee eommended to be heked. Duing the efuelling in Otobe 003, a total of ten fuel assemblies aound the fou positions wee inspeted. Aound LPRMs and 9 no wea damages was obseved, but two assemblies aound LPRM 0 and one aound LPRM showed maks of wea, see Table to the ight. It is seen that a good oelation exists between the LPRMs pedited to have impating and the atual damage. What egads measuement, no post-yle inspetions ae available yet (the yle is still on-going as of this witing). On the othe hand, both the spetal and the wavelet-based analyses wee possible to pefom. The analysis esults ae summaised in Table fo the two methods. The Table shows that the LPRMs pointed out as the most seveely vibating ae not ompletely idential fo the two methods, but they have a lage ovelap between the two goups. The LPRMs 6, 4 ae the ones pointed out by both methods to have the highest pobability of impating, hene they ae the pimay suspets fo impating. LPRM 35 is in the highest goup of the spetal method and in the seond highest in the wavelet method. Finally LPRM 34 is in the goup with the seond highest pobability in both of the methods. Also the new method of using wavelet oheene was also tested fo this measuements, and supplied esults ompletely onsistent with the findings of the spetal and wavelet filteing methods. The appliability of this latte method needs, howeve, onfimed in futhe measuements. It is also impotant to get feedbak fom the inspetions duing efuelling to onfim o deny the appliability of the methods. The inspetion duing a late evision will bing a vey useful and impotant laifiation egading the pefomane of the two methods. 6 Conlusions

72 Diagnostis of impating of deteto stings was shown to be possible by both spetal and wavelet based methods. The spetal method was possible to apply to the Ringhals- ase without any modifiations ompaed to ealie appliations. Howeve, this method is not suitable fo on-line monitoing by the opeatos; athe it equies expet judgement and woks best off-line. The wavelet based method had to be modified ompaed to ealie appliations befoe it tuned out to be effetive. This shows that the wavelet method, at least when fist applied to a new oe, does not fulfil the expetations of being suitable fo an algoithmi, absolute (without alibation) and quantitative method that an be used fo on-line monitoing by the opeatos. Nevetheless, it an be tuned to a speifi oe suh that afte tuning it an be used fo on-line monitoing by non-expets. The tuning means finding the suitable mothe wavelet fom and the oesponding theshold values. It equies an analysis of all signals, and the optimum paametes ae found by assuming that thee ae both non-impating and impating stings. In othe wods, the tuning of the method is just as subjetive and based on expet knowledge as the taditional method. Howeve, afte having optimized fo a given oe, it fulfils the expetations of being an absolute, algoithmi method. This latte statement will be followed up in applying the wavelet method, optimized in this study, fo late yles in the Ringhals- oe. In addition, a new method of wavelet-based oheene was tested whih, although at this point appeas to be moe like empiial without deep theoetial justifiation, seems to be dietly appliable fo a new oe without tuning. Its appliability has though to be tested in moe appliations. Aknowledgements This wok was suppoted by Ringhals AB, ontat Nos and Refeenes J. A. Thie (979) Coe Motion Monitoing, Nulea Tehnology, Vol. 45, pp J. A. Thie (98) Powe Reato Noise, ANS, La Gange Pak, Illinois. 3 J. E. Mott, J. C. Robinsson, D. N. Fy and M. P. Bakin (976) Detetion of Impats of Instument Tubes Against Channel Boxes in BWR-4s Using Neuton Noise Analysis, Tansations of Ameian Nulea Soiety Vol. 3, p D. N. Fy et al (977) Summay of ORNL Investigations of In-Coe Instument Tube Vibations in BWR-4, ORNL/NUREG/TM-0, Oak Ridge National Laboatoy. 5 I. Pázsit, F Åkehielm, B-G. Begdahl, and R. Oguma (988) BWR Instument Tube Vibations: Theoy of In-oe Deteto Speta and Intepetation of Measuements, Pogess in Nulea Enegy, Vol., pp I. Pázsit and O. Glökle (994) BWR Instument Tube Vibations: Intepetation of Measuements and Simulations, Annals of Nulea Enegy, Vol., No., pp O. Thomson, N. S. Gais and I. Pazsit (997) Quantitative Indiatos of Deteto Sting Impating, Nulea Tehnology, Vol. 0, pp A. Ráz and I. Pázsit (998) Diagnostis of Deteto Tube Impating with Wavelet Tehniques, Annals of Nulea Enegy, Vol. 5, No. 6, pp V. Azhanov and I. Pázsit (00) Deteting Impating of BWR Instument Tubes by Wavelet Analysis. Powe Plant Suveillane and Diagnostis-Applied Reseah with Atifiial Intelligene, Editos: Da Ruan and Paolo F. Fantoni, Spinge, Physia Velag XIV, pp I. Pázsit, C. Demazièe, V. Azhanov and N. S. Gais (00) Reseah and Development Pogam in Reato Diagnostis and Monitoing with Neuton Noise Methods, Stage 7. Final Repot, SKI Repot 0:7, Statens Känkaftsinspektion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. I. Pázsit, C. Demazièe and V. Azhanov (003) Reseah and Development Pogam in Reato Diagnostis and Monitoing with Neuton Noise Methods, Stage 8. Final Repot,

73 SKI Repot 003:08, Statens Känkaftsinspektion (Swedish Nulea Powe Inspetoate), Stokholm, Sweden. C. Demazièe, C. Sunde, V. Azhanov and I. Pázsit (003) Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 8, Chalmes epot CTH-RF- 77/RR-0, Chalmes Univesity of Tehnology, Götebog, Sweden. 3 Wavelet Toolbox Use s Guide (000) The MathWoks, In. 4 P. S. Addison (00) The Illustated Wavelet Tansfom Handbook, Institute of Physis Publishing, Bistol and Philadelphia. 5 S. Mallat (999) A Wavelet Tou of Signal Poessing, Aademi Pess. 6 G. Pokol and G. Po (004) ALPS Loose Pat Monitoing System and Wavelet Analysis.IMORN-9, Budapest, Hungay, 7-9 May 004. Submitted fo publiation to IJNEST.

74 Table Results of measuement # Impating status LPRM by analysis LPRM by inspetion most likely impating 9, 0 and 0 and pobably impating 4, 8, 6 and Vey small hane of impats,, 30 and Table Results of measuement # Impating status By spetal analysis By wavelet analysis most likely impating 5, 6, 4, 35 6, 3 and 4 pobably impating and 34 4, 34 and 35 Vey small hane of impats 7 and

75 Figue Illustation of the position of the deteto sting in a BWR oe between fou fuel elements, with indiation of some typial data of inteest. Figue Shemati view of the signal of a deteto vibating in a flux gadient with bakgound noise and impating. Fo explanation see text in the pape.

76 Figue 3 The layout of the oe of Ringhals- with the loations of the deteto stings. Sting # did not show vibations, wheeas stings 9 0 and wee found to have impating in the measuement seies #.

77 Figue 4 Autospeta, oheene and phase of the detetos in LPRM, measuement #. Figue 5 Autospeta, oheene and phase of the detetos in LPRM 9, measuement #

78 Figue 6 Autospeta, oheene and phase of the detetos in LPRM 0, measuement # Figue 7 Autospeta, oheene and phase of the detetos in LPRM, measuement #

79 Figue 8 Autospeta, oheene and phase of the detetos in LPRM 5, measuement # Figue 9 Autospeta, oheene and phase of the detetos in LPRM 6, measuement #

80 Figue 0 Diffeene V of the detetos at level 4 in LPRM 0 and LRPM 6 showing the spikes oesponding to the fuel-box vibations, measuement #.

81 Figue Wavelet-based oheene, as a funtion of time, fo fou diffeent LPRM stings. Figues on the left: LPRMs and 7. Low oheene eveywhee. Figues on the ight: LPRMs 6 and 35. High oheene in the fequeny band 0-0 Hz, indiating impating.

82 Figue Spetal based oheenes fo the fou stings shown in Figue. In all fou stings a naow peak aound 5 Hz is shown in the oheene, with somewhat lowe magnitude fo the stings that wee not suspeted as impating by the othe methods.

83

84 CTH-RF-73 Septembe 003 Calulation of the neuton noise indued by shell-mode oe-bael vibations in a -D -goup -egion slab eato C. Sunde and V. Azhanov Depatment of Reato Physis Chalmes Univesity of Tehnology SE-4 96 Götebog, Sweden ISSN

85

86 Coe-bael vibations CTH-RF-73 Calulation of the neuton noise indued by shell-mode oe-bael vibations in a -D -goup -egion slab eato C. Sunde and V. Azhanov Depatment of Reato Physis, Chalmes Univesity of Tehnology SE-4 96 Götebog, Sweden Abstat The subjet of this wok is the alulation of the in-oe neuton noise, indued by the shell-mode vibation of the oe-bael. The oiginal motivation was to investigate whethe an out-of-phase behaviou an exist between the in-oe and ex-oe detetos lying on the same azimuthal position. To this ode a two-egion two-goup diffusion model was used in one dimension. The noise was alulated in the adiabati appoximation. It was found that an out-ofphase behaviou an exist between in-oe detetos, lose to the oe bounday, and the exvessel detetos, fo lage systems. In addition, due to the use of two-goup theoy, the stong loal noise omponent at the bounday of the vibating oe was also found. The pape gives details of the alulations and the esults. -3-

87

88 Coe-bael vibations CTH-RF-73. Intodution The ex-oe neuton noise, indued by oe-bael vibations have long been used to diagnose both beam-mode and shell-mode vibations. In onnetion with the study of in-oe neuton noise indued by flutuating system boundaies, [], it has been ealized that oe-bael vibations might lead also to in-oe noise. This is espeially useful fo the diagnostis of shellmode vibations in Westinghouse eatos beause all ex-oe detetos ay the same infomation, due to the 90 o spaing. Hene it is not possible to detemine both the vibation amplitude and the dietion fom the deteto signals. In patiula it is not possible to find out if the eason fo a hange in the signal amplitude is due to the hange in the vibation amplitude o the dietion of the vibations. Fo this eason we have stated inluding the in-oe deteto signals in the analysis of oe-bael vibations in the ollaboative eseah pojet between Ringhals and the Depatment of Reato Physis at Chalmes Univesity of Tehnology ([] - [8]). Howeve, onfimation of the theoy as well as use of the esults fo diagnostis was hindeed by the low numbe of in-oe detetos. In patiula, duing evaluation of the only available measuement in Ringhals-3 when both in-oe and ex-oe detetos wee available, in ode to have a onsistent intepetation, it was neessay to assume that ex-oe and in-oe detetos lying on the same azimuthal position have opposite phase. This out-of-phase behaviou was ontaditing the simple theoy that was used so fa. Howeve, in that theoy the in-oe noise was only alulated in a non-efleted system. One annot exlude the possibility that the oe-bael vibations, if teated in a efleted system, will lead to anothe stutue of the neuton noise than in a bae system. Fo instane, the outbound movement of the bounday in the unefleted system means some multipliative mateial added outside the stati bounday, whih then would lead to both an inease of eativity and, in addition, to a loal inease of the neuton flux. Both omponents would be in phase with the signal of the ex-oe detetos. Howeve, in a efleted system, an outbound movement of the oe bounday means a deeasing efleto thikness, and, unde etain onditions, also possibly a deease of eativity. Hene, it is not obvious whethe suh a movement will lead to the inease of the in-oe flux o not. The pupose of this epot is to investigate the in-oe noise indued by oe-bael vibations in efleted systems by the extension of the model used ealie. In ode to be able to handle a efleted system, it was neessay to use two-goup theoy. To ompensate fo the ompliations that this extension inus, the desiption was edued to one spatial dimension. Density vaiations, indued by the -D shell mode vibations, wee also aounted fo. The noise was alulated in the adiabati appoximation. This made it possible to handle the poblem analytially thoughout, with a numeial evaluation of the final fomulas.. Desiption of the eato model A one-dimensional model of a efleted eato is seleted fo this study with a ental oe and oute efletos plaed symmetially aound the oe (Fig. ). Two-goup diffusion theoy will be used to alulate the noise, with oesponding oss setions and diffusion onstants in the fast,, and themal,, goup, espetively. One aveaged goup of delayed neutons was used in the dynami alulations. The shell-mode vibations will be modelled by simultaneous, -5-

89 Coe-bael vibations CTH-RF-73 symmeti vibations of the oe bounday aound the stati position x ± b. Hene both the stati and the dynami ase will be symmeti aound the oigin, and this simplifies the solution of the poblem. φ, (x) φ, (x) φ, (x) Coe Refleto -a -b 0 b a Fig.. -D Coe with efleto x Sine the oiginal idea was to see if thee an, at least unde etain iumstanes, exist an out-of-phase behaviou between in-oe and ex-oe (ex-vessel) detetos, the bounday ondition at the oute bounday of the system was hosen to be the no inoming uent ondition, whih an be expessed by the logaithmi flux deivative and the extapolation length, see below. This way the sala flux does not vanish at the system bounday, and the exoe deteto signals wee hosen to be equal to the flux at the bounday. 3.. Bounday and intefae onditions 3. The stati ase To stat we define two funtions in the oe: φ, ( x), -b x b, and two funtions in the efleto: φ, ( x), -a x b, b x a. The subsipt indiates if it is the fast,, o the themal,, enegy goup. The supesipt stands fo the efleto and fo the oe. Beause of the symmety of the poblem we have fist of all: φ, φ, ( x) φ, ( x), -b x b ( x) φ, ( x), -a x b, b x a This gives us the possibility to set up the poblem only fo x 0. The bounday and intefae onditions used ae desibed below in equation () and (3) () D x D x φ φ φ φ ( x) x b+ ( x) x b+ ( x) x b+ ( x) x b+ φ ( x) x b- φ ( x) x b- D φ( x) x x b- D φ( x) x x b- Intefae onditions () -6-

90 Coe-bael vibations CTH-RF-73 x x φ φ ( x) x a ( x) x a d d ---- φ ( x) x a ---- φ ( x) x a Bounday onditions (3) Hee d and d ae the extapolation lengths. They ae defined though the oss-setions and diffusion onstants in the efleto. d 0.7 Σ D t, Extapolation lengths (4) d 0.7 Σ D t, These ae all the onditions we need to solve the poblem and we ae fee to nomalize the flux. 3.. Equations in the efleto The diffusion equations fo the fast and themal fluxes in the efleto egion, wee no fission ous ae as follows: D D d d d φ x d φ x ( x) ( Σa, + Σ R )φ ( x) 0 ( x) Σa, ( x) + ΣRφ( x) 0 φ (5) Now, the subsipt fo the oss setions displays whih type of eation it is, a fo absoption, R (emoval) fo satteing fom the fast to the themal goup and f fo fission in the oe egion. The supesipt stands fo the efleto and late fo the oe. The geneal solutions of these oupled diffeential equations ae easily deived and simplified by using (). The onstants will be detemined late when solving the oe equation an using the bounday and intefae onditions. Although, thee ae six onstants and six onditions one of the onstants will be undetemined sine an equation fo the itiality appea fom the six onditions. Though, we have to deal with the lage exponents due to the lage size of a eal efleto (a~300 m and e 0 ). By multiplying the solution with e - b and use new integation onstants, the numeial values beome smalle: φ φ ( x) α3 e κ x b ( ) ( x) α5 e κ x b ( ) α 4 e κ x b + ( ) α 6 e κ ( x b ) φ ( x) D ( κ κ ) Σ R a b whee b (6) -7-

91 Coe-bael vibations CTH-RF-73 Hee the following notations ae used: 3.3. Equations in the oe κ , κ L Σ Σ D Σ a, + Σ R L Σ a, D (7) The equations in the oe ae a little bit moe ompliated due to the fat that they involve fission tems that ae not pesent in the efleto egion. This means that the fast and themal equations ae moe oupled than in the efleto egion. We assume that all fission neutons ae fast but one ould also assume that some pat of them ae themal. Then one would have to add a fission tem in the equation fo the themal ones, using the spetum of the fission neutons. D D d φ( x) Σa dx (, + Σ s, )φ ( x) + -- ( νσ k f, φ ( x) + νσf, φ ( x) ) 0 d φ( x) Σa dx, φ ( x) Σs, + φ ( x) 0 (8) When solving this equation system we ty with the following fom of the fluxes. φ ( x) φ0, φ ( x) φ0, e λx e λx (9) With this inseted into (8), we aive with a matix equation: D λ 0 0 D λ φ 0, φ 0, + Σ Σ R --νσ k f, Σ a, φ 0, φ 0, 0 0 (0) with Σ Σ a, + Σ R --νσ k f, () and if det Σ D λ Σ R --νσ k f, Σ a, D λ 0 () thee exist a solution to the equations. And the eigenvalues, λ, an be detemined and expessed -8-

92 Coe-bael vibations CTH-RF-73 in tems of the oss setions and diffusion onstants of the two enegy goups as: ( λ ) D Σa,, ( + D Σ ) ( D Σa, D Σ ) 4 ± + --νσ k f, Σ RDD D D (3) This gives fou eigenvalues whih appea in two pais: λ, ± η ± λ 34, ± iµ ± ( D Σa, + D Σ ) ( D Σa, D Σ ) νσ k f, Σ RDD ( D Σa, D Σ ) 4 D D + --νσ k f, Σ RDD ( D Σa, + D Σ ) φ 0, φ 0, D D We also have a elation between and fom equation 0 eading as (4) φ 0, Σ a, Σ R φ 0, D λ (5) This gives us two onstants: C C Σ a, Σ R D η Σ a, Σ R D µ Togethe with the eigenvalues this gives us the solution of the fast and themal flux in the oe although we have to detemine the itiality of the system. (6) φ ( x) α e ηx + α ηx e + α e iµx + α iµx e φ ( x) C ( α e ηx + α ηx e ) + C ( α e iµx + α iµx e ) (7) Due to the symmety onditions expessed by (), the above an be simplified to: φ ( x) α oshηx + α osµ φ ( x) α C oshηx + α C osµx with α α α and α α α (8) -9-

93 Coe-bael vibations CTH-RF-73 Now (6) and (8) ae the geneal solutions in the efleto and the oe, and to detemine the onstants and the itiality the bounday and intefae onditions have to be used Citial equation Two of the onstants ae aleady detemined with the symmety so six onstants and the eigenvalue (itiality) ae to be detemined. To fully detemine the eigenvalue and the onstants we need a nomalization ondition, whih will be done by setting the fast flux to unity, in the ente of the oe. Using () and (3) we aive at a matix equation A*α 0whee the matix A is: osh ηb osµb ε ε C oshηx C osµx ε κ κ ---- ε ε ---- ε 0 0 κ κδ κ ---- κ ---- κ δ ---- κ ---- d 0 0 κ δ κ d d d δ δ d d δ (9) ηsinh ηb µ sinµb C ηsinhηb C µ sinµb κ D ε D κ D ε κ D κ D D ε κ D κ D ε κ D ε D κ D D ε Hee the onstants used to simplify the matix ae ( ε e b b )B, e b b ε ( )B ( δ e a b )B (, δ e a b )B, κ Σ R ( κ κ )D (0) n and u wee defined in (4). The itial equation fo detemining k eff is: f ( k) det( A) 0 () Using Mathematia one aives at an equation fo k eff : -0-

94 Coe-bael vibations CTH-RF-73 f ( k eff ) Σ a, D η * ( ηtanhbη + h )* µ bµ µ Σ a, tan *h + h D () µ Σ a, * ( h µ tan bµ )* ηtanhbη η Σ a, + *h + h 0 D D whee η η( k eff ), µ µ ( k eff ) and h, h and h ae onstants whih onsists of some tan and tanh funtions and must be solved numeially. The onstants in this equation ae just some expessions involving othe known tems as the extapolation lengths and some exponential tems whih ae defined in (0). The solution, whih is physial, is the on with the lagest k eff. See Fig.. By using that value in (4) the eigenvalues, η and µ ae detemined. The stutue of the itial equation Citial Equation ( λ eff.096; K eff /λ eff 0.90) 30 χ χ λ min λ λ λ Axis λ /k λ Fig.. Plot of the k-funtion plotted as a funtion of the invese vaiable λ k is displayed in Fig.. It shows that the k- funtion onsists of infinitely many banhes sepaated by the asymptotes: ( D µn + ( Σ a, + Σ R ))( D µn + Σ a, ) λ n ; n 0,,, (3) νσ f, ( D µn + Σ a, ) + νσ f, Σ R A minimal bound, λ min, may be defined by µ(λ min ) 0, whih gives --

95 Coe-bael vibations CTH-RF-73 ( Σ λ a, + Σ R )Σ a, min ( + Σ R ) νσ f, Σ a, νσ f, (4) Then inseting them into the matix equation and solving it gives the integations onstants. Sine this gives us six onstants that ae linealy dependent we use the nomalization ondition ( φ ( x0) ) to fully detemine all six onstants. The fast and themal fluxes ae now fully detemined and ae displayed in Fig. 3 with a oe adius of 6.5 m and a efleto adius of 79.5 m. The flux fom a numeial simulato [9], developed at ou depatment is also plotted and as one sees the analytial solution agee vey well with the solution fom the simulato. This togethe with the efleto peaks of the themal flux indiates the oetness of the solution. The size of the eato is supposed to epesent a eal eato. All the oss-setions ae taken fom SIMULATE-3. Although SIMULATE-3 stated with a 3-D itial system the -D system is supe-itial sine it is less leakage in a -D system. The value fo k eff is.0046 fom the simulato and the same fo the analytial alulation done in this epot. Although, thee is a mismath in the sixth deimal. Note that the fast flux is about ten times lage than the themal. Fast flux analytial solution numeial simulato 0.8 flux Coe Refleto size [m] Themal flux 0.5 analytial solution numeial simulato flux 0. Coe Refleto size [m] Fig. 3. Fast and Themal flux fom semi-analyti alulations and fom SIMULATE-3. k.0046 in both ases. To eally visualise the efleto peaks and the atio between the fast and themal fluxes they ae plotted togethe in Fig. 4. Only the egion lose to the bounday between the oe and efleto --

96 Coe-bael vibations CTH-RF-73 is displayed. The peak of the themal flux in the efleto aises fom the slowing down of the fast neutons to themal ones that ous in the efleto. Notie also that the system is itial in this figue ompaed to Fig. 3, this is done by esaling the fission oss-setions and using the same paametes as in the supe itial system. Dim Goup, K eff 0. Fast flux Themal flux flux size [m] Fig. 4. Bounday between oe and efleto 3.5. Adjoint flux What we want to investigate is how a vibation of the oe-bael effets the deteto signal, i.e. petub the themal flux. To alulate this noise we need to alulate the adiabati flux, and to nomalize that we need the stati adjoint flux. The adjoint flux and the diet flux have the same eigenvalue, same k, so we do not need to detemine the itiality one moe. Also the intefae and bounday onditions ae the same. The adjoint equations ae simila to the diet ones, so the idea of solving is the same. The adjoint equations in the efleto ae: D D d + φ ( x) Σa dx, Σ + + ( + s, )φ ( x) + Σs, φ ( x) 0 d + + φ ( x) Σa dx, φ ( x) 0 (5) and in the oe: D D d φ ( x) Σa dx (, + Σ s, )φ ( x) + --νσ k f, φ ( x) + Σs, φ ( x) 0 d φ ( x) Σa dx, φ ( x) + --νσ k f, φ ( x) 0 (6) The solutions fo the adjoint fluxes in the efleto ae: -3-

97 Coe-bael vibations CTH-RF κ φ ( x) α5 e x b + + κ φ ( x) α3 e x b ( ) ( ) + κ α 6 e x b + + κ + α 4 e ( ) ( x b ) Σ R φ ( x) D ( κ κ ) (7) Fo the oe the solution looks like φ ( x) α oshηx + α osµ φ ( x) α C oshηx + α C osµx C + Σ R + Σ , C R ( Σ D η ) ( Σ + D µ ) (8) Compaed with the diet solution, (6) and (8), we see that the fast and themal flux in some sense have hanged plaes. Though, the onstants ae somewhat diffeent. To see the diffeene between the diet and adjoint fluxes they ae plotted in Fig. 5. The system has a oe adius of 0 m and a efleto adius of 40 m, but it is still itial. Showing this smalle system makes the diffeene between the diet and adjoint fluxes leae. Flux and adjoint flux Fast diet Themal diet Fast adjoint Themal adjoint 0.8 flux 0.6 Coe Refleto size [m] Fig. 5. Diet and Adjoint flux -4-

98 Coe-bael vibations CTH-RF The dynami ase We now poeed with a dynami model that desibes the esponse of the oiginally stati system to time-dependent petubation. Assuming one aveage goup of delayed neuton peusos, C, in addition to two pompt neuton goups, we have fo the slab eato ---- φ φ D ( Σ v t x x a, + Σ s, )φ + ( β) ( νσ f, φ + νσ f, φ ) + λc φ φ D Σ v t x x a, φ + Σ s, φ C λc + βνσ ( t f, φ + νσ f, φ ) In ontast to the stati system (5) and (8) now the unknown funtions and oss setions ae spae and time dependent φ φ ( xt, ); D D ( x, t) ; Σ a φ φ ( xt, ); D D ( x, t) ; Σ a C C( x, t), Σ a, Σ a, ( xt, );, ( xt, ); We model the shell mode vibations by letting the oe bounday, b(t), osillate aound the stati position, b, at both sides of the oe in a symmetial manne: bt () b + δb(); t δb() t 0 (3) In addition, we assume the value of the oss setions in the oe to be popotionally modified b Σ x( xt, ) Σx, stati D b ( x, t) D (3) b + δb() t stati b + δb() t wheeas the value of the oss setions in the efleto is assumed to be unaffeted by the vibations. It should be noted hee that beause of the moving oe bounday all the oss setions must be egaded as time and spae dependent even in the efleto. Instead of solving equations (9) dietly we ae going to deive popeties of the petubed system by using the adiabati appoximation. To this ode we stat by fatoising the unknown fluxes into an amplitude fato, P(t), and a shape funtion, ψ(x,t), as follows φ ( x, t) Pt () ψ ( xt, ) P( 0) stati (33) φ ( x, t) Pt () ψ ( xt, ) ψ, ( x, 0) φ, ( x) Sine we have intodued thee new quantities instead of the two unknown funtions, φ and φ, we need to impose an additional onstaint that nomally eads as [0] + φ 0 i a a φ v 0, ( x)ψ ( xt, ) φ v 0, ( x)ψ ( x, t) dx onst Hee,, denotes stati adjoint fluxes. Usually, a deivation of the kineti equations, details of whih one an find in [0], goes as follows. One puts the fatoization (33) into (9), multiplies (9) (30) (34) -5-

99 Coe-bael vibations CTH-RF-73 eah equation by the oesponding stati adjoint flux, then subtats the stati equations, and finally, integates ove the eato volume. This yields in the end fo the amplitude fato, P(t), the equations: dp() t dt d() t dt ρ() t β Pt () + λt () Λ() t β P() t λt () Λ() t (35) The new quantities that appea in (35) involve an abitay fato, F(t), that is usually defined as Then the pompt neuton geneation time eads as The eativity tem beomes ρ() t a + F() t φ 0, ( x) [ νσ f, ψ ( xt, ) + νσ f, ψ ( x, t) ] dx Λ() t a a a F() t a a φ v 0, ( x)ψ ( x, t) φ v 0, ( x)ψ ( x, t) dx F() t φ + 0, [ νσ ( f, )ψ + νσ ( f, )ψ ] dx + a a Hee, denotes the flutuation in a oesponding oss setion, fo example a Σ ( ) [ + + φ 0, φ ]ψ dx , Σ ( [ a, )φ 0, ψ + Σ ( a, )φ 0, ψ ] dx νσ f, a ( ) νσ f, ( xt, ) νσ f, ( x) (36) (37) (38) (39) Finally, the peuso density, (t), eads as t () a Λ ()F t () t φ + 0, ( x)c( x, t) dx a It should be noted hee that β, whih is involved in the kineti equations (35), depends on time t in the most geneal ase. But ou model assumes that both pompt fission and delayed neutons ae bon in the fast goup only. It follows fom the exat definition, [0], that β is time independent and equals to the stati delayed neuton fation unde this assumption. Ou next step is to lineaize the kineti equations (35) in ode to obtain the zeo eato tansfe funtion fo ou two-goup model. The standad lineaization tehnique stats with splitting quantities into a steady pat plus a small deviation: ρ() t ρ 0 + δρ() t ; F() t F 0 + δf() t ; Λt () Λ 0 + δλ() t (4) (40) -6-

100 Coe-bael vibations CTH-RF-73 One then puts this into the oiginal equations (35), and neglets seond-ode tems. Hee a + F 0 φ 0, ( x) [ νσ f, φ 0, ( x) + νσ f, φ 0, ( x) ] dx a a + + Λ φ F 0 v 0, ( x)φ 0, ( x) φ v 0, ( x)φ 0, ( x) dx a In the end one aives at the following fomula in the fequeny domain: δp( ω) G 0 ( ω) δρ( ω) (4) (43) Hee, G 0 (ω) is a tansfe funtion that is simila to the odinay (one-goup) zeo eato tansfe funtion: G 0 ( ω) β iω Λ + 0 λ iω with an enegy and spae aveaged onstant, Λ 0, defined in (4). In the plateau egion, λν<νων<νβ/λ 0 (see Fig. 6), an appoximate equality holds: G 0 ( ω) Equation (43) in onjuntion with (45) allows us to deive both a vey impotant and useful elationship in the plateau egion, namely: δp( ω) The impotane of the above fomula stems fom the fat that δρ(t) may be elatively easily evaluated by alulating a oesponding eigenvalue poblem: Hee k(t) depends on time t paametially though the time-dependene of the oesponding oss setions. A staightfowad lineaization of (33) gives us -- β --δρ( ω) δpt () β δρ() t k() t --δρ() t β (44) (45) (46) (47) δφ ( x, t) δpt () φ δφ ( x, t) δpt () φ 0, x 0, x ( ) + δψ ( x, t) ( ) + δψ ( x, t) (48) The fist tem on the ight-hand-side is alled the point eato tem wheeas the seond one is efeed to as the spae-dependent tem. The spae-dependent tems may be elatively easily evaluated though the adiabati appoximation: -7-

101 Coe-bael vibations CTH-RF-73 ad ψ ad δψ ( x, t) δψ ( x, t) ad δψ ( x, t) δψ ( x, t) ad ψ ( x, t) φ0 ( ) Hee,, ( x, t) ae the positive eigenfuntions fo the eigenvalue poblem (8) with the oss setions oesponding to the time instane t (i. e. the momentay oe bounday b(t)). Eq. (48) ombined with (49) and (46) gives, finally, the following appoximation: δφ ( xt, ) δφ ( xt, ), x ad ψ ( x, t) φ0 ( ), x ad --δρ() t φ β 0, ( x) + δψ ( x, t) ad --δρ() t φ β 0, ( x) + δψ ( x, t) (49) (50) In ou numeial alulations we use the following paametes β ; λ ; Λ (5) that oespond to a plateau egion of 0 - < ω <0. The shell mode vibations ae expeted to be aound 5 ad/s, whih is within the plateau egion. G 0 (ω) Amplitude λ /β β/λ angle(g 0 (ω)) 0. Phase (adian) λ β/λ Fequeny [ad/s] Fig. 6. Zeo powe tansfe funtion -8-

102 Coe-bael vibations CTH-RF Numeial wok Now we have eveything we need to alulate the effet on the flux due to a vibation of the oe-bael. The vibations ae assumed to hange the volume of the oe but not its mass. Hene, the density of the mateial pesent, hanges due to the lage (o smalle) size of the oe and this will affet all oss setions as well as the diffusion onstants. δb δb Coe Refleto -a -b 0 b a Fig. 7. Coe with moved bounday The new oss setions and diffusion onstants in the oe ae: b Σ x( x, t) Σx, stati b + δb D b ( xt, ) D stati b + δb (5) Hee δb is the hange of the bounday between the oe and the efleto. In the efleto on the othe hand, the density is assumed to be the same so all oss setions and diffusion onstants ae the same. By this we assume that some wate is emoved (added) fom the efleto when its size is smalle (lage). In othe wods we assume that it is impossible to ompess the wate. Now, it is possible to alulate a new k eff and a new flux φ adiabati ( x). And by using the new k eff, ρ(t) an be alulated. Then one an use (50) to alulate the new flux with its time-dependent petubation. Out of this it is possible to detemine the phasebehaviou between in-oe and ex-oe positions (detetos), due to this petubation. 6. Results and Disussion When investigating the phase behaviou one has to estimate the size of the oe bael vibation. We assume that it is in the ode of mm if the oe has a size of seveal metes. The oe used in SIMULATE-3, fo alulation of the eato paametes, is Ringhals 4 and it has a oe adius of 5.0 m and the oute adius of the efleto is 66.3 m (the efleto is 4.3 m thik). The small thikness of the efleto is due to the fat that SIMULATE-3 only uses one node fo modelling the efleto. But on the othe hand we an extend the efleto to a adius of 79.5 m and the oe adius is atually 6.5 m in the -D model used by ou simulato. Using this we alulated k eff.0046 and in ou analytial model desibed above we also eeived k eff.0046, as one an see they math pefetly. Howeve, it is possible to hoose any size of the system. One just has to adjust the fission oss setions so that the system beomes itial. To display the featues of the system we hoose a small system and a lage unphysial movement of the oe bounday. By using a oe adius of 0 m, a efleto adius of 40 m and a displaement of the oe of m we see that thee is an out-of-phase behaviou (shaded aea, Fig. 8) between the whole oe and the exoe deteto. Sine this hange of the bounday, is quite lage, the fato ρ/β is also lage -9-

103 Coe-bael vibations CTH-RF-73 meaning that the point eato tem, i.e. the fist tem on ight hand side in (50), will ontibute to the petubed flux togethe with the spae-dependent tem.. Dim Goup, ρ dx m a) b) stati themal flux petubed themal flux stati fast flux petubed fast flux Dim Goup, ρ dx m stati themal flux petubed themal flux stati fast flux petubed fast flux New oe bounday New oe bounday flux 0.6 flux 0.6 Coe Refleto Coe Refleto size [m] size [m] Fig. 8. Stati and Petubed flux in an unphysially small eato, 0 m oe and 40 m oute adius of the efleto. In a) the bounday is moved m outwads and in b) it is moved m inwads. The shaded aeas in both figues display the pat of the eato that is out-of-phase ompaed to ex-oe positions (detetos). Fo the ex-oe value we us the themal flux at the oute edge of the efleto, wheeas the inoe values ae of ouse the themal flux in the oe. What one also an see is that the system beomes less itial, i.e. ρ is negative, fo ineasing size. On the othe hand a deease of the oe ineases the eativity and the system is supe-itial. The out-of-phase behaviou is still pesent in the whole oe in both ases. When we inease the system size and use a moe ealisti displaement of mm it is still possible to notie the out-of-phase behaviou but it is not appeaing as distintly as fo the smalle system. The eativity hange is eally small, ~0-5, this means that the point eato tem is almost negligible, as expeted, beause a small hange to a lage system is not affeting -0-

104 Coe-bael vibations CTH-RF-73 the eativity that muh. As one an see in Fig. 9 the out-of-phase (shaded aeas) behaviou appeas in the middle fo both mm lage and mm smalle oe. Dim Goup, ρ.333e 5 dx 0. m a). stati themal flux petubed themal flux stati fast flux petubed fast flux b) Dim Goup, ρ 6.663e 6 dx 0. m stati themal flux petubed themal flux stati fast flux petubed fast flux New oe bounday New oe bounday flux 0.6 flux 0.6 Coe Refleto Coe Refleto size [m] size [m] Fig. 9. Stati and Petubed flux in a eal eato. Figue a) is with a mm outwad displaement of the bounday and b) is with mm inwad displaement. The shaded aeas ae the out-of-phase pats of the oe and the efleto. Thus, by ompaing the ex-oe position (deteto) with in-oe positions (detetos) lose to the ente of the eato it should be possible to detet shell mode vibations. In fat it should be possible by using only in-oe positions (detetos), one lose to the ente and one somewhee between the ente and the oe-bael. The petubed themal flux, δφ (x), in Fig. 0, lealy illustates the phase behaviou. One an also see that the ex-oe flux is ineasing with ineasing size and deeasing with deeasing size. x 0 4 δφ(x) with dx 0. m x a) b) 4 δφ(x) with dx 0. m δφ(x) 6 δφ(x) size [m] size [m] Fig. 0. The themal noise, δφ (x). In a) with mm outwad movement of the oe bounday and in b) mm inwad movement. a) and b) ae almost the efletion oh eah othe, i.e. -*a) ~ b). Fo a lose look at the inteesting aea aound the bounday between the oe and the efleto Fig. displays this pat of Fig. 9. Hee it is possible to see that a position lose to the oebael also shows an out-of-phase behaviou with espet to an ex-oe position. The advantage with this position ompaed to the one in the ente of the eato is that hee the hange of the --

105 Coe-bael vibations CTH-RF-73 flux is muh geate (see also Fig. 0) so it would be easie to detet vibations. But on the othe hand it is impossible to have a deteto at this position in a eal oe Dim Goup, ρ.333e 5 dx 0. m stati themal flux a) petubed themal flux stati fast flux petubed fast flux b) Dim Goup, ρ 6.663e 6 dx 0. m stati themal flux petubed themal flux stati fast flux petubed fast flux flux flux size [m] size [m] Fig.. This figue is a zoom in, of the oe bounday egion, of Fig. 9. The dashed vetial line is the oe bounday befoe the movement and the dotted one is afte the movement. One othe way of desibing the vibation would be to not hange the oss setions and diffusion onstants, instead just hange the size of the oe. Then the eativity hange is muh lage, meaning a ontibution fom the point eato tem and that emoves the out-of-phase behaviou fom the ente of the oe, Fig. a) Dim Goup, ρ e 06 dx 0. m stati themal flux petubed themal flux stati fast flux petubed fast flux 4 x 0 4 b) δφ (x) with dx 0. m New oe bounday flux 0.6 δφ (x) 4 Coe Refleto size [m] size [m] Fig.. a) displays the stati and petubed flux fo a mm outwad movement of the oe bounday without any hange of the oss setions and the diffusion onstants. b) displays the themal noise φ (x). --

106 Coe-bael vibations CTH-RF Conlusion We have solved the -goup diffusion equation in a -egion -D system and simulated a oe bael vibation by assuming a lage (smalle) oe with highe (lowe) density, hange of the oss setions and diffusion onstants. Then we applied eato dynamis to see if it is possible to get an out-of-phase behaviou between ex-oe and in-oe positions. The esults show that thee is an out-of-phase behaviou between ex-oe positions (detetos) and positions aound the ente of the oe and also positions (detetos) lose to the bounday between the oe and the efleto. The out of phase behaviou in the ente if the oe disappeas if the density is kept onstant. It is also shown that the eativity hange of the system is almost negligible meaning that the point eato tem of the petubation is small. Thus, it is the loal pat of the noise that is dominating. 8. Aknowledgements We would like to thank D. Chistophe Demazièe fo poviding us with oss setion data and othe eato paametes. We ae also thankful fo his alulations of the neuton noise with his neuton noise simulato, it gave us some numeial values to ompae ou semi-analytial solution with [9]. -3-

107 Coe-bael vibations CTH-RF-73 Refeenes [] I. Pázsit and V. Azhanov (000), Linea Reato Kinetis and Neuton Noise in Systems with Flutuating Boundaies, Ann. Nul. Enegy, 7, [] I. Pázsit (Edito) (996), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage. CTH-RF-/RR3, Chalmes Univesity of Tehnology, Gothenbug, Sweden [3] I. Pázsit, J. Kalsson, and N. S. Gais (997), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage. CTH-RF-3/RR4, Chalmes Univesity of Tehnology, Gothenbug, Sweden [4] J. K.-H. Kalsson, and I. Pázsit (998), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 3: Analysis of oe bael vibations in Ringhals, 3 and 4 fo seveal fuel yles. CTH-RF-35/RR5, Chalmes Univesity of Tehnology, Gothenbug, Sweden [5] C. Demazièe, V. Azhanov, J. K.-H. Kalsson, and I. Pázsit (999), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 4. CTH-RF-45/RR6, Chalmes Univesity of Tehnology, Gothenbug, Sweden [6] C. Demazièe, V. Azhanov and I. Pázsit (000), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 5. CTH-RF-56/RR7, Chalmes Univesity of Tehnology, Gothenbug, Sweden [7] C. Demazièe, V. Azhanov and I. Pázsit (00), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 6. CTH-RF-6/RR8, Chalmes Univesity of Tehnology, Gothenbug, Sweden [8] C. Demazièe, V. Azhanov and I. Pázsit (00), Final Repot on the Reseah Pojet Ringhals Diagnostis and Monitoing, Stage 7. CTH-RF-67/RR9, Chalmes Univesity of Tehnology, Gothenbug, Sweden [9] C. Demazièe (003), Development of a -D -goup neuton noise simulato, aepted fo publiation in Ann. Nul. Enegy. [0] G. I. Bell and S. Glassstone (970), Nulea Reato Theoy, Van Nostand-Reinhold, New Yok [] I. Pázsit (00), Tanspot theoy and Stoasti poesses, Letue notes, Chalmes Univesity of Tehnology, Gothenbug, Sweden [] Use manual fo Matlab on the web -4-

108 Coe-bael vibations CTH-RF-73 Appendix The following values wee alulated fo oss setions and diffusion onstants in -D with SIMULATE-3. Note that the absoption oss setion fo the fast goup in the oe is negative, this is due do the homogenization fom 3-D to -D. Σ a, Σ a, Σ R νσ f, νσ f, Σ a, Σ a, Σ R D D D D 0.05 m 0.09 m 0.05 m m 0.45 m.4376 m m m m m.36 m 0.64 m fo the oe (53) fo the efleto (54) -5-

109

110 Calulation of the neuton noise indued by shell-mode oe-bael vibations in a -D -goup -egion slab eato model C. Sunde, C. Demazièe and I. Pázsit Depatment of Reato Physis, Chalmes Univesity of Tehnology SE-4 96 Götebog, Sweden Numbe of pages: 9 Numbe of tables: 0 Numbe of figues: 6

111 Calulation of the neuton noise indued by shell-mode oe-bael vibations in a -D -goup -egion slab eato model C. Sunde, C. Demazièe and I. Pázsit Depatment of Reato Physis, Chalmes Univesity of Tehnology SE-4 96 Götebog, Sweden Abstat The subjet of this wok is the alulation of the in-oe neuton noise, indued by the shellmode vibations of the oe-bael. The oiginal motivation was to investigate whethe an out-of-phase behaviou an exist between the in-oe and ex-oe (ex-vessel) detetos lying at the same azimuthal position. To this ode a two-egion two-goup diffusion model was used in one dimension. The noise was alulated by epesenting the vibations of the oe-bael by a model developed ealie to desibe ontol od vibations. It was found that suh an out-of-phase behaviou indeed exists, although only fo in-oe deteto positions lose to the oe bounday. This behaviou is due to the loal omponent of the noise, whih is aounted fo in a two-goup teatment. The finding is in aodane with the expeiment whose esult pompted the pesent wok. In addition to its effet on the phase, the loal omponent also manifests itself by a lage amplitude of the noise aound the vibating oe bounday, i.e both in the oe and the efleto. The appeaane and the popeties of the loal omponent of the neuton noise fo oe-bael vibations is the main finding of this wok. The esults suggest that the effiieny of oe-bael vibations an be enhaned if, in addition to the ex-oe detetos, in-oe detetos in the outemost fuel assemblies ae used.. Intodution The ex-oe neuton noise, indued by oe-bael vibations have long been used to diagnose both beam-mode and shell-mode vibations []-[]. The oesponding methods have undegone a quite long development stage, and beame quite effetive fo the diagnostis of beam-mode (pendula) vibations. By stating out with undelying models estited to eithe unidietional o isotopi vibations, they have suessively advaned to a stage of being able to teat abitay anisotopi -D andom motions. This way it beame possible to monito a hange in the pefeed dietions and the amplitude of the vibations simultaneously. Even eonstution of the -D andom motion of the oe bael has been pefomed with suess in some ases, although this latte has less diet diagnosti value. The quantitative diagnostis of shell-mode vibations, howeve, has not eahed a simila state, at least not in Westinghouse-type eatos whee thee ae fou ex-oe detetos with an equal 90 o spaing. Due to the fat that the shell-mode vibations, and hene also the neuton noise indued by suh vibations, exhibit the same symmety against otations with 90 o as the detetos, the infomation ontent in all detetos is equivalent, i.e. all ex-oe detetos ay the same infomation. (the situation is diffeent at eato onstutions, suh as the East- Euopean VVER-440 eatos, whee 3 ex-oe detetos ae used with a 0 o spaing, and in --

112 some Japanese eatos with 5 ex-oe detetos [9]). Hene it is not possible to detemine both the vibation amplitude and the dietion fom the deteto signals. In patiula it is not possible to find out if the eason fo a hange in the signal amplitude is due to a hange in the vibation amplitude o to a hange in the dietion of the vibations. In onnetion with the study of in-oe neuton noise indued by flutuating system boundaies [3], oiginally onsideed fo the desiption of the neuton noise indued by vibating ontol ods, it was ealized that oe-bael vibations might lead also to in-oe noise. Fo this eason the pesent authos stated using the in-oe deteto signals fo the analysis of oe-bael vibations as pat of a ollaboative eseah pojet between the Ringhals powe plant and the Depatment of Reato Physis at Chalmes Univesity of Tehnology ([4] - [7]). A ompat solution fo the adial and angula dependene fo the inoe noise, indued by oe-bael vibations, was given in [8]. The full onfimation of the theoy, howeve, was hindeed by the low numbe of in-oe detetos (a maximum of five movable in-oe detetos at a time). In patiula, duing the evaluation of the only measuement in Ringhals-3 when both in-oe and ex-oe detetos wee available, in ode to have a onsistent intepetation, it was neessay to assume that ex-oe and in-oe detetos lying on the same azimuthal position exhibited opposite phase [8]. This latte statement, and the appaent ontadition that it implies, will be expounded hee in some detail beause it was this obsevation that pompted the pesent wok. The theoetial esults and expeimental ompaison ae summaized in Fig. below. It was shown in [8] that a b Out of phase deteto with lage APSD deteto with vey low APSD Fig.. The stutue of in-oe noise indued by shell-mode vibations (a) and evaluation of measuement fom Ringhals-3 (b) the angula and adial dependene of the in-oe noise, indued by shell-mode vibations at the vibation eigenfequeny ω, an be given in a -D pola o-odinate system (, ϕ) as δφ(, ϕ) AJ ( B( ω)) os( ϕ χ) () whee B( ω) is the dynami bukling ([3]), J a Bessel funtion and χ is the angle of the vibation axis. As () shows, and as an be expeted also by simple onsideations, the indued noise has a otational symmety by 90. In patiula, the APSD, given fom () as --