experimental analysis of thermal mixing at reactor...

Experimental analysis of thermal mixing at reactorconditions

MATTIA BERGAGIO

Licentiate ThesisStockholm, Sweden 2016

TRITA-FYS 2016:74ISSN 0280-316XISRN KTH/FYS/16:74SEISBN 978-91-7729-190-9

AlbaNovaRoslagstullsbacken 21

10691 StockholmSweden

Akademisk avhandling som med tillstnd av Kungl Tekniska hgskolan framlggestill offentlig granskning fr avlggande av teknologie licentiatexamen i reaktortek-nologi fredagen den 16 december 2016 kl 10.00 i FB55, AlbaNova, Stockholm.

Mattia Bergagio, November 2016

Tryck: Universitetsservice US AB

Abstract

High-cycle thermal fatigue arising from turbulent mixing of non-isothermal flowsis a key issue associated with the life management and extension of nuclear powerplants. The induced thermal loads and damage are not fully understood yet.With the aim of acquiring extensive data sets for the validation of codes modelingthermal mixing at reactor conditions, thermocouples recorded temperature timeseries at the inner surface of a vertical annular volume where turbulent mixing oc-curred. There, a stream at either 333 K or 423 K flowed upwards and mixed withtwo streams at 549 K. Pressure was set at 72 105 Pa. The annular volume wasformed between two coaxial stainless-steel tubes. Since the thermocouples couldonly cover limited areas of the mixing region, the inner tube to which they weresoldered was lifted, lowered, and rotated around its axis, to extend the measure-ment region both axially and azimuthally.Trends, which stemmed from the variation of the experimental boundary conditionsover time, were subtracted from the inner-surface temperature time series collected.An estimator assessing intensity and inhomogeneity of the mixing process in theannulus was also computed. In addition, a frequency analysis of the detrendedinner-surface temperature time series was performed. In the cases examined, fre-quencies between 0.03 Hz and 0.10 Hz were detected in the subregion where mixinginhomogeneity peaked.The uncertainty affecting such measurements was then estimated.Furthermore, a preliminary assessment of the radial heat flux at the inner surfacewas conducted.

Keywords: Mixing estimator, empirical mode decomposition, Hilbert-Huangtransform, uncertainty assessment, radial heat flux

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Licentiatuppsatssammanfattning

Termisk hgcykelutmattning innebr en betydande risk fr krnkraftverk och lik-nande processanlggningar. Fenomenet har bring p bde skerhet, underhll ochlivstidsfrlngning av krnkraftverk i drift. Termisk hgcykelutmattning kan or-sakas av turbulent blandning av vattenflden av olika temperatur. I detta fall rdet en utmaning att frutsga och bedma den d vr nuvarande frstelse avturbulens, vrmeverfring och olika materials reaktioner p termisk belastning rbegrnsad.

I detta arbete redovisas en experimentell underskning av turbulenta blandning-ar med potential att leda till termisk utmattning. Under frsken, som genomfrtsi en testsektion installerad i HWAT-kretsen (High-pressure WAter Test) vid KTHi Stockholm, studerades blandningar vid frhllanden liknande dem i Oskarshamn3 och Forsmark 3, vid vilka ett antal styrstavsfrlngare uppvisade sprickbildningtill fljd av termisk utmattning.Testsektionen bestod av tv vertikala och koaxiala rr i rostfritt stl. I det annulraomrdet mellan rren leddes ett kallare vattenflde uppt och tv varmare vatten-flden nedt.Temperatursignaler registrerades vid det inre rrets radie och benmns hr inne-ryttemperaturer. Dessa data r avsedda fr att kunna anvndas vid validering avkoder som kombinerar numerisk strmningsdynamik (CFD) med finit elementana-lys (FEM) fr frutsgelse av termisk hgcykelutmattning.Tio fall undersktes, vart och ett med olika randvillkor: i sex av fallen var samp-lingsfrekvensen satt till 100 Hz och hjd till 1000 Hz i de terstende fyra. Inne-ryttemperaturen loggades normalt sett vid tta azimutala och fem axiella positionerfr varje fall. Denna temperatur mttes med hjlp av sex stycken 0.5-millimeters,K-typ, ojordade termoelement, fastldda vid tv diskar som monterats in i test-sektionens inre rr. Fr att kunna mta temperaturen i hela blandningsomrdetkunde detta rr roteras frn 0 till 360 och frflyttas vertikalt ver en strcka av387 mm med hjlp av en fjrrkontrollerad stegmotor. I fem av fallen var, fr attverensstmma med driftfrhllanden i tidigare nmnda reaktorer, trycket satt till7.2 MPa vid 333 respektive 549 K fr de kalla och varma vattenfldena. Dessa fr-hllanden motsvarar en densitetsskillnad p 227 kg m3 och 2.1 i Prandtltal.De inlopp dr de varma fldena nr det annulra omrdet benmns hdaneftervarma inlopp.

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Kvalitativt r det ett tecken p omfattande blandning om inneryttemperatursig-nalen uppvisar frekventa fluktuationer av jmfrbar amplitud, tillfllig blandningom den uppvisar pltsliga toppar och ingen blandning om den r mer eller mind-re konstant. Inneryttemperaturerna p var och en av diskarna fanns vara starktkorrelerade. Vidare karakteriseras inneryttemperaturer vid samma mtposition avliknande medelvrden, omfng och dominerande frekvenser.Mtningar vid 1000 Hz r att fredra framfr mtningar vid 100 Hz om en precisuppskattning av variansen nskas.

Efter avtrendning jmfrdes normaliserade inneryttemperaturer frn de tio fal-len med avseende p medelvrde, minimum och maximum samt maximum vid axiellniv z. Vidare berknades en estimator fr blandningsinhomogenitet genom att slihop standardavvikelser fr samtliga temperatursignaler vid samma mtpositionfr ett givet fall, och drigenom erhlls en enskild skalr deskriptor. Detta gjordeseftersom termisk utmattning r mera troligt i omrden med liten omblandning.Denna estimator utvrderar blandningsinhomogenitet p ett tillfredsstllande stt,tminstone fr konstant azimutal vinkel och tillrcklig samplingsfrekvens. Inne-ryttemperaturerna vid den axiella niv dr blandningen r som minst homogen rlgre n motsvarande adiabatiska blandningstemperaturer.Den axiella niv dr blandningen r som minst homogen ser ut att bestmmas avtv mekanismer: dels penetrationen av varma flden in i det annulra omrdet,dels termisk stratifiering. Den frra mekanismen frstrks av hga massflden ge-nom de varma inloppen och frflyttar det kritiska omrdet, d.v.s. det omrde drblandningen r som minst homogen, nedt medan den senare mekanismen frstrksav hga temperaturskillnader mellan kalla och varma flden. Den ser ut att kadet strsta blandningsestimatet, antagligen p grund av att stora axiella tempe-raturgradienter ger en betydande frlust av lokal temperaturuniformitet. Vidareser termisk stratifiering ut att minska den normaliserade temperaturens spridningvid den axiella niv dr denna spridning nr sitt hgsta vrde, givet att de var-ma massfldena understiger ett trskelvrde. P samma stt antas det att termiskstratifiering smetar ut mindre blandade omrden ver flera axiella niver samtidigtsom massfldet i de varma strmmarna trycker ihop dessa omrden till frre niver.Fr en given axiell niv, givet att de varma massfldena verstiger ett trskelvrde,verkar blandningsinhomogeniteten vara strre vid 360 n vid 180. Detta kan beroantingen p geometrisk asymmetri eller olika stora massflden genom de varmainloppen.

Temperatursignalerna i blandningsomrdet r icke-stationra och mycket hac-kiga, varfr konventionella spektrala metoder riskerar att misslyckas med att iden-tifiera de dominerande frekvenserna. En Hilbert-Huang transform, vilken kombine-rar empirisk moddekomposition (EMD) med Hilberts spektralanalys, appliceradesistllet p dessa data vid de positioner dr blandningsinhomogeniteten hade sinahgsta vrden. Vid dessa positioner, visar marginalspektra baserade p Hilbert-Huang transformen att dominerande frekvenser terfinns mellan 0.03 och 0.10 Hz.Detta frekvensband ser ut att ka i omfng med hgre varma massflden och rsmalare n vad som erhlls frn Fourierspektra.

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EMD:n delar upp varje signal i flera inre modfunktioner (IMF:er). De domineran-de topparna i Hilbert-Huang marginalspektrat hrstammar ofta frn de IMF:ersom har de lngsta tidsskalorna. Dessa IMF:er behlls i signalen efter avtrendningtack vare en parameter som spelar en stor roll fr bestmmandet av dominantafrekvenser.

Fr att validera frutsgelserna frn de numeriska modellerna krvs hg nog-grannhet i temperaturmtningarna vid den inre ytan. Drfr genomfrdes en os-kerhetsuppskattning. Oskerheten i ovan nmnda mtningar r 1.58 respektive3.78 K vid 1000 respektive 100 Hz. Den strsta kllan till oskerhet vid 100 Hztogs frn tabellerade vrden frn tillverkaren, medan motsvarande oskerhetskllavid 1000 Hz bestmdes med hjlp av kalibreringsdata fr hela datainsamlingssyste-met. Det r rimligt att anta att sdana kalibreringsdata vid 100 Hz skulle kunnaminska ven denna oskerhet. Dock menar vi att den acceptabla noggrannheten vid1000 Hz och testsektionens relativt enkla geometri gr att de numeriska modellernakan anses validerade.

Det transienta vrmefldet ver den inre annulra ytan berknades frn upp-mtta data. I brist p datapunkter i radiell riktning fr givna (, z), uppskattadesdetta vrmeflde med hjlp av en Crank-Nicolson diskretisering av den endimen-sionella vrmeledningsekvationen fr varje termoelement. Denna metod verifieradesdelvis med en analytisk lsning, delvis med en kombinerad analytisk-numerisk ls-ning.

List of publications

Journal articles

M. Bergagio and H. Anglart. 2016. Experimental investigation of mixing ofnon-isothermal water streams at BWR operating conditions. Submitted toNuclear Engineering and Design.In this experimental investigation, wall surface temperatures have been measured duringmixing of three water streams in the annular gap between two coaxial stainless-steel tubes.The inner tube, with an outer diameter of 35 mm and a thickness of 5 mm, holds six 0.5-mm K-type, ungrounded thermocouples, which measured surface temperatures with asampling rate of either 100 Hz or 1000 Hz. The tube was rotated from 0 to 360 and movedin a range of 387 mm in the axial direction to allow measurements of surface temperaturesin the whole mixing region. The outer tube has an inner diameter of 80 mm and a thick-ness of 10 mm to withstand a water pressure of 9 MPa. A water stream at a temperatureof either 333 K or 423 K and a Reynolds number between 1.27 104 and 3.23 104 rosevertically in the annular gap and mixed with two water streams at a temperature of 549 Kand a Reynolds number between 3.56 105 and 7.11 105. These two water streams en-tered the annulus radially on the same axial level, 180 apart. Water pressure was kept at7.2 MPa. Temperature recordings were performed at five axial and eight azimuthal loca-tions, for each set of boundary conditions. Each recording lasted 120 s to provide reliabledata on the variance, intermittency and frequency of the surface temperature time seriesat hand. Thorough calculations indicate that the uncertainty in the measured temperatureis of 1.58 K. Due to the high accuracy of measurements and a relatively simple geometry,the present experimental data can be used to validate computational methods for predict-ing thermal mixing. Furthermore, these data can provide new insight into compressible,turbulent mixing at BWR operating conditions and, more generally, into mixing coupledto the dynamics, also termed level-2 mixing.

M. Bergagio, R. Thiele, and H. Anglart. 2017. Analysis of temperature fluctu-ations caused by mixing of non-isothermal water streams at elevated pressure.International Journal of Heat and Mass Transfer, 104:979 992.Temperatures were measured at the inner surface of an annulus between two coaxial tubes,where three water streams mixed. These temperatures were sampled at either 100 Hz or1000 Hz. The acquisition time was set to 120 s. Two water streams at 549 K, with a

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Reynolds number between 3.56 105 and 7.11 105, descended in the annular gap andmixed with a water stream at 333 K or 423 K, with a Reynolds number ranging from1.27 104 to 3.23 104. Water pressure was kept at 7.2 MPa. Inner-surface temperatureswere collected at eight azimuthal and five axial positions, for each combination of boundaryconditions. To better analyze these temperatures and mixing in the vicinity of the wall,scalars estimating the mixing intensity at each measurement position were computed fromdetrended temperature time series. Fourier and Hilbert-Huang marginal spectra were cal-culated for the time series giving rise to the highest values of a mixing estimator of choice.The relationship between temperature and velocity was explored by examining the resultsof an LES simulation using the same boundary conditions as in one of the experimentalcases.

R. Thiele, M. Bergagio, and H. Anglart. 2015. Large Eddy Simulation ofthermal mixing in an annulus with conjugate heat transfer. Submitted toNuclear Engineering and Design.Thermal fatigue is present in most metals under varying heat loads and can become aproblem for the structural integrity of metal parts. Detailed knowledge of these loads is ofutter importance in order to avoid these kind of problems. This study uses Large-Eddy-Simulation in conjunction with the WALE sub-grid turbulence model and conjugate heattransfer to investigate thermal mixing in an annulus with a pair of opposing cold inletsand a pair of opposing hot inlet at different axial levels rotated by 90 to one another.The geometry represents a simplified model of a control rod guide tube of a nuclear powerplant. The numerical results are compared to experimental data at reactor conditions fromthe experimental facility. The comparison shows a good agreement of the wall temperaturefluctuation magnitude and the frequency spectrum.

Conference papers

M. Bergagio, S. Hedberg, S. Rydstrm, and H. Anglart. 2015. Instrumentationfor temperature and heat flux measurement on a solid surface under BWRoperating conditions. In Proceedings of the 16th International Topical Meetingon Nuclear Reactor Thermal Hydraulics, volume 7, pages 59625975.A new experimental facility has been developed at KTH Royal Institute of Technologyto measure temperature and heat flux propagations in solid walls due to mixing of non-isothermal water streams in their vicinity. The main purpose of the measurements has beento obtain a high-precision experimental database suitable for validation of ComputationalFluid Dynamics (CFD) codes. Consequently, a set of experiments have been performed ina test section simulating the annular region in the BWR control-rod guide tubes. Sincepreliminary CFD results implied that 0.1-1 Hz temperature oscillations were to be expected,this experimental research intends to assess the magnitude of temperature fluctuationswithin the abovementioned frequency range. To this end, water and wall temperatureshave been measured in the innermost part of the test-section annulus, with a variety ofboundary conditions. As thermocouples would otherwise be available at few axial andazimuthal coordinates only, the tube they are installed on has been lifted, lowered and

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rotated by a software-controlled motor to record temperature fluctuations in the wholemixing region. At each measurement point, data have been collected over a time longenough to detect the existence of the aforesaid fluctuations. Moreover, an uncertaintyanalysis has been carried out concerning water temperatures. Thermocouples meant tomonitor these temperatures have been modelled with a finite-element method for this verypurpose. The wall heat flux has also been estimated using experimental data, thanks to acorrected finite-difference Crank-Nicolson scheme.

H. Anglart, M. Bergagio, S. Hedberg, S. Rydstrm, and W. Frid. 2015. Mea-surement of wall temperature fluctuations during thermal mixing of non-isothermal water streams. In Proceedings of the 16th International TopicalMeeting on Nuclear Reactor Thermal Hydraulics, volume 1, pages 807818.This paper is dealing with measurement of temperature fluctuations during mixing of twowater streams in an annular test section at BWR operational conditions. The experimentsare simulating conditions existing in a guide tube of BWR control rods, where relativelycold water at about 333 K is mixing with hot water at 550 K. It is shown that the mixingis causing high amplitude temperature fluctuations in the solid walls of the control rod ex-tender. Using new movable multi-sensors it became possible to obtain a large experimentaldatabase, containing wall temperature measurements at 8 azimuthal and 5 axial positions,with 13 thermocouples at each position. In total 520 temperature readings were performed,each lasting about 2 minutes and recording transient temperature with frequency of at least100 samples per second and with estimated non-calibrated uncertainty less than 3.9 K. Thepresent experimental results can be used to analyze the governing phenomena during ther-mal mixing and also to validate CFD conjugate heat transfer models of thermal mixingapplied to actual reactor geometries.

Both conference papers are peer-reviewed.

Acknowledgments

I would like to thank my colleagues for creating a nice working environment andproviding thoughtful feedback. Among them, I am most grateful to Prof. HenrykAnglart for his valuable guidance. I would also like to express my gratitude toStellan Hedberg and Stefan Rydstrm for their help in the lab.Credits go to Anders Riber Marklund for the translation of the sammanfattning.I acknowledge the financial support of the Swedish Radiation Safety Authority(SSM), the Swedish Center for Nuclear Technology (SKC), and Berkningsgruppen(a panel of representatives from Forsmarks Kraftgrupp AB, Oskarshamns Kraft-grupp AB, Ringhals AB, and Teollisuuden Voima Oyj).

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Contents

Contents xv

List of Figures xvii

List of Tables xix

1 Introduction 1

2 Background 32.1 Thermal mixing and thermal fatigue . . . . . . . . . . . . . . . . . . 3

2.1.1 Thermal mixing . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Thermal fatigue . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Connection between thermal mixing and thermal fatigue . . . 72.1.4 Experiments and simulations of thermal mixing . . . . . . . . 92.1.5 Estimators of thermal mixing . . . . . . . . . . . . . . . . . . 9

2.2 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Heat flux assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Methods 133.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Overview of the facility . . . . . . . . . . . . . . . . . . . . . 133.1.2 Test-section thermocouples . . . . . . . . . . . . . . . . . . . 143.1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.1 Data acquisition tasks . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Inner-tube movement pattern . . . . . . . . . . . . . . . . . . 24

3.3 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Inverse- and low-pass filtering . . . . . . . . . . . . . . . . . . 263.3.2 Empirical Mode Decomposition . . . . . . . . . . . . . . . . . 263.3.3 Windowing, DFT, HHT, and Hilbert-Huang marginal spectrum 28

3.4 Mixing intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Uncertainty analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6 Heat flux assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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xvi CONTENTS

3.7 Heat flux verification . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Results 414.1 Results of inverse- and low-pass filtering . . . . . . . . . . . . . . . . 414.2 Results of mixing intensity . . . . . . . . . . . . . . . . . . . . . . . . 484.3 Results of DFT, HHT, and Hilbert-Huang marginal spectrum . . . . 564.4 Results of uncertainty analysis . . . . . . . . . . . . . . . . . . . . . 644.5 Results of heat flux assessment and verification . . . . . . . . . . . . 66

5 Conclusions 69

Bibliography 71

List of Figures

2.1 Example of level-2 mixing (Kuschewski et al. (2013)). . . . . . . . . . . 52.2 Temperature time series in solid bodies. . . . . . . . . . . . . . . . . . . 8

3.1 Key components of the HWAT loop. . . . . . . . . . . . . . . . . . . . . 153.2 A picture of the test section. . . . . . . . . . . . . . . . . . . . . . . . . 163.3 A picture of one of the thermocouple discs. . . . . . . . . . . . . . . . . 163.4 A picture of the motor shaft. . . . . . . . . . . . . . . . . . . . . . . . . 163.5 Coordinate system attached to the test section. . . . . . . . . . . . . . . 173.6 Cut view of the test section. . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 Exploded-view drawing of the inner tube. . . . . . . . . . . . . . . . . . 193.8 Sketch of the left and right thermocouple discs. . . . . . . . . . . . . . . 203.9 Longitudinal section of an inner-tube thermocouple in a thermocouple

disc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.10 The inner-tube movement pattern for Case 9 . . . . . . . . . . . . . . . 253.11 Illustrative temperature time series T (t) and its spectra. . . . . . . . . . 323.12 Coordinate systems on a plane z = constant. . . . . . . . . . . . . . . . 36

4.1 Inner-surface temperatures for Case 1 at (45, 0.65 m). . . . . . . . . . . 434.2 Inner-surface temperatures at 0.65 m for Case 1, from thermocouple H2. 444.3 Inner-surface temperatures at 360 for Case 1, from thermocouple H2. . 454.4 Inner-surface temperatures at 0.65 m for Cases 6, 8, and 10, from ther-

mocouple H2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.5 Inner-surface temperatures at 0.60 m for Case 6 from thermocouple H2. 484.6 Axial distribution of the highest, lowest, and average values of the nor-

malized inner-surface temperatures T f, d for Cases 1, 2, 5, 6, 8, and 9. . 504.7 Axial distribution of the highest, lowest, and average values of the nor-

malized inner-surface temperatures T f, d for Case 10. . . . . . . . . . . . 514.8 Mixing estimator for Case 1. . . . . . . . . . . . . . . . . . . . . . . . . 514.9 Mixing estimator for Case 5. . . . . . . . . . . . . . . . . . . . . . . . . 524.10 Mixing estimator for Case 2. . . . . . . . . . . . . . . . . . . . . . . . . 524.11 Mixing estimator for Case 6. . . . . . . . . . . . . . . . . . . . . . . . . 534.12 Mixing estimator for Case 8. . . . . . . . . . . . . . . . . . . . . . . . . 53

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xviii List of Figures

4.13 Mixing estimator for Case 9. . . . . . . . . . . . . . . . . . . . . . . . . 544.14 Mixing estimator for Case 10. . . . . . . . . . . . . . . . . . . . . . . . . 544.15 Low-pass filtered and detrended inner-surface temperatures for Case 1

at 45, from thermocouple H1. . . . . . . . . . . . . . . . . . . . . . . . 584.16 Some of the low-pass filtered and detrended inner-surface temperatures

for Case 1 at 225. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.17 Some of the low-pass filtered and detrended inner-surface temperatures

for Case 2 at 315. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.18 DFTs for Case 1 at 45, from thermocouple H1. . . . . . . . . . . . . . 604.19 Hilbert-Huang marginal spectrum for Case 1, at (45, 0.65 m), from ther-

mocouple H1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.20 Hilbert-Huang marginal spectrum for Case 1, at (45, 0.65 m), from ther-

mocouple V 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.21 DFTs for Case 1 at 225. . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.22 Hilbert-Huang marginal spectrum for Case 1, at (225, 0.67 m). . . . . . 624.23 DFTs for Case 2 at 315. . . . . . . . . . . . . . . . . . . . . . . . . . . 634.24 Hilbert-Huang marginal spectrum for Case 2, at (315, 0.68 m). . . . . . 634.25 Peaks in the Hilbert-Huang marginal spectra of the inner-surface tem-

peratures where the mixing estimator is at its highest. . . . . . . . . . . 654.26 Evaluation of normalized errors err0 and err1. . . . . . . . . . . . . . . 674.27 Radial heat flux at r = Rio and measurement position (360, 0.65 m) for

Case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.28 Highest weighted RMS of azimuthal correction qr, 1, for every measure-

ment position in Case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

List of Tables

2.1 T-junction experiments on mixing of water streams at different temper-atures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Geometry, dimensions, temperatures, and pressure in the test section ofthe HWAT loop and in BWRs. . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Positions of the inner-tube thermocouple tips. . . . . . . . . . . . . . . . 213.3 Experimental matrix for the measurement of test-section temperatures

at a sampling rate of 1000 Hz. . . . . . . . . . . . . . . . . . . . . . . . 213.4 Experimental matrix for the measurement of test-section temperatures

at a sampling rate of 100 Hz. . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Overview of the key data acquisition parameters. . . . . . . . . . . . . . 23

4.1 Largest gain G and highest, lowest, and mean differential spread at asampling rate of 1000 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Largest mixing estimators for each experimental case. . . . . . . . . . . 554.3 Values of arrays B+, B, and S with a sampling rate of 1000 Hz. . . . . 664.4 Values of arrays B+, B, and S with a sampling rate of 100 Hz. . . . . 66

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List of Symbols

AcronymsSymbol Description UnitsCN Crank-Nicolson scheme -DFT discrete Fourier transform -DI Duhamels integral -EMD empirical mode decomposition -FIR finite impulse response -HHT Hilbert-Huang transform -IMF intrinsic mode function -NC number of channels -OI orthogonality index -

Greek SymbolsSymbol Description Units thermal diffusivity m2 s1t time discretization step sx space discretization step m dimensionless axial coordinate - angle Q[l;m] m-th position of Q in terms of for case l density kg m3 standard deviation K mixing estimator - instantaneous frequency Hz

Roman SymbolsSymbol Description UnitsA length of each Tf,DAS after being low-pass filtered S it1 ch1B negative-side systematic uncertainty K

Continued on next page

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xxii List of Tables

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Roman SymbolsSymbol Description UnitsB+ positive-side systematic uncertainty Kc0 number of the first mode in the trend -ch channel -Co Courant number -f frequency HzfS sampling rate Hzg IMF -G number of IMFs for Tf, lf -it iteration of a given task -k thermal conductivity W m1 K1l case identifier -

m progressive entry number in the movement patternfor case l

-

m mass flow rate kg s1m number of entries in the movement pattern for case l -

muprogressive entry number in the deduplicatedmovement pattern for case l

-

nnumber of temperature arrays at a certain positionfor case l

-

Nt number of time intervals -Nx number of spatial nodes reduced by 1 -NSj number of samples per iteration j per channel S it1 ch1qr, 1 azimuthal correction of the heat flux W

Qcenter of the circular base of the mid thermocoupledisc

-

(r, , z) cylindrical coordinate system attached to the innertube (m, ,m)R1 azimuthal region defined as R1 = {135 225} -

R2azimuthal region defined asR2 = {315 360} {0 45}

-

Re Reynolds number -S samples -S random uncertainty Kt time sT temperature KT normalized temperature -v velocity m s1

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List of Tables xxiii

Continued from previous page

Roman SymbolsSymbol Description UnitsZQ[l;m] m-th position of Q in terms of z for case l m

SubscriptsSymbol Description Unitsb branch pipe -bc before calibration -C cold inlets -d detrended -DAS acquisition of temperatures from the test section -f any of thermocouples H1, H2, H3, H4, V 1, and V 4 -H hot inlets -if inverse-filtered -lf low-pass filtered -m main pipe -q spatial node q, q = 1, ..., Nx 1 -TC thermocouple -w windowed -

SuperscriptsSymbol Description UnitsI lower sampling rate -s time interval s, s = 0, ..., Nt 1 -

Chapter 1

Introduction

This thesis presents an experimental investigation of the turbulent mixing of onecold and two hot streams in a vertical annular volume. A partial post-processingof the temperature time series measured at the inner surface of this annulus isincluded as well.

The aforesaid measurements are necessary for proper validation of codes com-bining computational fluid dynamics (CFD) with finite element analysis (FEA) topredict thermal fatigue damage caused by turbulent, non-isothermal mixing. Thisdegradation mechanism poses a potential threat to safety, management, and lifeextension of nuclear power and process plants. In particular, it is challenging tomonitor high-cycle thermal fatigue (defined in Subsection 2.1.1) with todays plantinstrumentation. In addition, this kind of fatigue is hard to predict with existingtools, in that the current understanding of turbulence, heat transfer, and materialresponse to thermal loading is rather inadequate.

In the experiments performed as part of this research, mixing was investigatedat conditions similar to those in Oskarshamn-3 and Forsmark-3 reactors, in bothof which some control-rod stems experienced thermal fatigue damage cracking (seeSubsection 2.1.1). Namely, in accordance with the operation of the aforesaid reac-tors, the cold and hot streams were respectively kept at temperatures of 333 K and549 K, whilst pressure was set at 7.2 MPa. These settings translate into a differ-ence of 227 kg m3 in density, and of 2.1 in Prandtl number. Any analysis of thetemperature time series measured at the inner radius of the annulus, as well as anycomparison of them with results from simulations, cannot overlook such significantchanges in water properties.

The suitability of these time series for validation of CFD/FEA codes must besupported by a low uncertainty level. Thus, an accuracy assessment is essential.

We can also remark that the positions of the thermocouples at the inner surfaceallow the calculation of the transient radial heat flux there.

Concerning the structure of this thesis, Chapter 2 roughs out thermal mixing,thermal fatigue, and the relationship between them. Furthermore, the tasks of

1

2 CHAPTER 1. INTRODUCTION

assessing the uncertainty in the aforesaid temperature time series and the transientradial heat flux from them are introduced.

Chapter 3 depicts the test facility and the data acquisition system (DAS). Italso delves into the post-processing of the above time series, including filtering andspectral analysis, and into the assessment of mixing intensity and inhomogeneityby means of a simple scalar descriptor, inasmuch as thermal fatigue is more likelyto occur in areas exhibiting higher unmixedness. This chapter also outlines howuncertainty and heat flux were evaluated.

Some of the temperature time series recorded at the inner radius of the annulusare given in Chapter 4, which then discusses the variation of mean temperatures,temperature ranges, and mixing estimator in the measurement region. Inferencesare drawn from the most relevant figures. Highlights from a spectral analysis con-ducted in critical subregions are then reported. In this chapter, an estimate of theuncertainty in temperature measurements is also provided, while our method forthe evaluation of the transient radial heat flux at the inner radius of the annulus ispartially verified.

Chapter 2

Background

2.1 Thermal mixing and thermal fatigue

2.1.1 Thermal mixingMixing of streams at different temperatures exemplified in Fig. 2.1 can generatecyclic thermal stresses in the adjoining walls, which can then cause fatigue damage.

Concerning the kind of non-isothermal mixing (thermal mixing) at hand, aclassification is attempted on the basis of the boundary conditions in Tables 3.3and 3.4: since water density can vary between 759 kg m3 and 986 kg m3, and thehot jets are expected to flow downwards in the test-section annulus by gravity, atleast in the area closest to the inner tube (Pegonen (2012); Thiele et al. (2015)),significant differential accelerations are conjectured to be generated here.

Moreover, since the Reynolds number Re exceeds 104 at the four inlets (seeTables 3.3 and 3.4) and the hot streams flowing downwards are expected to keepvelocities analogous to those at the inlets (Thiele et al. (2015)), the mixing in thearea closest to the inner tube can be described as turbulent, of level 2 (Dimotakis(2005)).

Turbulent mixing is a multi-scale process developing through three differentstages: entrainment or injection, which is driven by large-scale dynamics; stirringor dispersion, which happens at large and intermediate scales; and diffusion, whichoccurs at small scales (Eckart (1948)). For liquids, where kinematic viscosity ismuch larger than mass diffusivity, diffusion happens in two steps: in the former,kinematic viscosity leads to small-scale vorticity, whereas in the latter mass diffusionoccurs, if mass fractions can be defined.

Three levels of mixing can be identified: 1, 2, and 3. Level-1 mixing is sometimestermed passive mixing: here, the distribution of a scalar quantity modeling themixing process is determined by fluid advection and molecular diffusion, but it hasno dynamic effect on fluid dynamics. Generally speaking, in incompressible flowstemperature can be treated as a passive scalar (Sakowitz (2013)), provided thattemperature gradients are small.

3

4 CHAPTER 2. BACKGROUND

Conversely, level-2 mixing couples back on the flow dynamics. The mismatch indensity and (hydrostatic) pressure gradients p generates baroclinic vorticity~, which amplifies instability. Instability will then create more surfaces of constantpressure (isobaric) and density (isopycnal); in other terms, it will smear density andpressure gradients, which will modify the production of baroclinic vorticity (Dimo-takis (2005)). Level-2 mixing may occur when large density gradients are found inacceleration/gravitational fields, similarly to those encountered in the present re-search. Thus, Rayleigh-Taylor and Richtmyer-Meshkov instabilities are often citedas an example of level-2 mixing. The same can be said about the Kelvin-Helmholtzinstability.

Level-3 mixing is strongly coupled to fluid dynamics, to the point that fluid-intensive properties, such as density and composition, are changed. Combustion isa common example of such mixing.

Level-2 and level-3 mixing are termed active mixing. Until now, level-2 andlevel-3 mixing have not been properly investigated. One of the major reasons is that,in these cases, turbulence is anisotropic at certain scales. Anisotropy stems from theasymmetry caused by large-scale features, such as acceleration/gravitational fields.Therefore, the classical Kolmogorov-Obukhov-Corrsin (KOC) theory, based on thelocal isotropy of passive scalars at high Re, cannot be directly applied (Movahedand Johnsen (2015)). Because of this, and since understanding level-1 mixing hasbeen researchers main interest, level-2 and level-3 mixing can be regarded as openresearch topics. Furthermore, in the matter of flows at high Re, researchers havemostly investigated canonical flows (such as flow in pipes, jets, and free shear layers)and built their results on empirical data (Dimotakis (2005)).

In industrial applications (such as stirred vessels and multifunctional heat ex-changer-reactors), a distinction can also be made between macromixing, mesomix-ing, and micromixing. The first can be defined as convective heat transfer involvinglarge-scale motions connected with the macroscale circulation time and the size ofthe mixer. The second is governed by turbulent diffusion occurring at intermediatescales associated with inertial-convective disintegration and turbulent dispersion.The third is dominated by kinematic viscosity and molecular diffusion, and relatedto small-scale motions (Torbacke and Rasmuson (2004)). Macromixing is usuallymuch slower than micromixing (Bird et al. (2007)).

2.1.2 Thermal fatigueBefore attaining homogeneity, uniformity, or a high degree of mixedness, mixingnon-isothermal water streams produces fluctuations in the temperature field. Thesefluctuations are transmitted to adjoining solid walls, where they propagate accord-ing to their frequency content and generate cyclic thermal stresses. These stresses,even if lower than the engineering yield stress, could induce thermal fatigue; thatis, a damage mechanism results from such stresses, which randomly creates shortcracks on the surface. After that, a crack network might form and propagate. Itspropagation through the wall results in failure of the component. Given the large

2.1. THERMAL MIXING AND THERMAL FATIGUE 5

Figure 2.1: Left: temperature distribution at five hydraulic diameters downstreamof a T-junction where turbulent mixing of level 2 occurs (Kuschewski et al. (2013)).Non-isothermal streams at a pressure of 7.5 MPa and temperatures of 298 K and404 K were simulated by injecting fluids at different densities and 293 K. Associatedtemperatures were computed by dispersing a dye in each stream. The NWLED-IFtechnique helped to detect the dye fraction, which was conjectured to be propor-tional to density. Density was then converted into temperature. The viscosity ratioof the streams at 7.5 MPa was not preserved. Right: temperature deviation fromlocal mean at five hydraulic diameters downstream of the T-junction (Kuschewskiet al. (2013)).

stress gradients, cracks resulting from thermal fatigue can be often described aslong defects (elephant skin), with a very large aspect ratio, or length-to-depth-ratio (Gosselin et al. (2007)). Thermal loadings, crack interaction, and aspect ratioseem to control the crack growth.

Thermal fatigue is one of the key safety-related issues connected with agingmanagement and lifetime extension of existing nuclear power plants (Walker et al.(2009)), as well as with the design of new reactors.

The probability of thermal fatigue does not gradually increase with time, assuggested by several failures imputable to this kind of damage, which occurredin less than a year (Dahlberg et al. (2007)). To worsen the picture, thermal fa-tigue seems to be more detrimental than uniaxial isothermal fatigue (Fissolo et al.(2009)). We recall here that thermal fatigue is commonly associated with biaxialstresses and strains (Dahlberg et al. (2007); that is, with one-dimensional cyclicthermal stresses, concurrent with tensile hoop or axial stresses. Hence, thermalfatigue damage needs to be carefully considered.

We can distinguish between two regimes: low-cycle (LCTF) and high-cycle ther-mal fatigue (HCTF). HCTF might occur if the number of stress cycles to failureexceeds 104 105. Unlike LCTF, HCTF cannot be detected by common plant in-strumentation systems, such as thermocouples mounted on the outer surface of thestructures to be monitored, because of delays in response and frequency attenuation


in the wall (Bergholz and Bruckmueller (2012)). Moreover, lack of suitable dataon HCTF precludes a deeper understanding of its initiation and growth. Becauseof this, and since HCTF assessment methods may not encompass all the loadingconditions and responses of materials, they are often either too conservative or notconservative enough (Metzner and Wilke (2005)). Likewise, identifying thresholdsfor fatigue crack growth, in terms of temperature amplitudes and frequencies, is akey challenge, as is discussed below.

Up to now, thermal fatigue failures have been observed in light-water reactors,such as Farley-2 in 1987, Tihange-1 in 1988, Loviisa-2 in 1994 and 1997, Civaux-1in 1998, Tsuruga-2 in 1999, and Tomari-2 in 2003 (Farley (1987); Hytnen (1998);Shah et al. (1999); Faidy et al. (2000); Sugano et al. (2000)); sodium-cooled fastreactors (such as PHENIX in 1991); refineries; petrochemical and liquefied naturalgas facilities (Maegawa (2006); Qian et al. (2015)), especially in close proximityto T-junctions. The thermal fatigue failures of major interest in this study wereobserved in Oskarshamn-3 and Forsmark-3 boiling water reactors (BWRs) in 2008.In these BWRs, the stems of many control rods were detected to be either broken oraffected by cracks, particularly in regions of high stress concentration such as welds,holes, and sudden changes of geometric shape (Tinoco and Lindqvist (2009)).

Listed below are several projects centered on assessing thermal fatigue risk:

The Materials Reliability Project (MRP), set up by the Electric PowerResearch Institute (EPRI) and aiming at the definition of guidelines for as-sessing, alleviating, and monitoring thermal fatigue (Keller et al. (2004)).

The THERmal FATigue evaluation of piping system tee-connections (THER-FAT), initiated by the European Commission (EC). Among other objectives,it aimed at finding parameters responsible for fatigue in T-junctions, deter-mining lower thresholds for fatigue crack growth, and developing methodolo-gies for predicting thermal fatigue life (Metzner and Wilke (2005)).

The Thermal Fatigue Project, set up by the Network for Evaluation of Struc-tural Components (NESC) and aiming at devising a common methodologyfor the evaluation of HCTF, with a focus on turbulent mixing in T-junctionsof light-water coolant systems (Dahlberg et al. (2007)).

The Thermal Fatigue - Basics of the system-, outflow- and material-char-acteristics of piping under thermal fatigue, funded by the German FederalMinistry of Education and Research (BMBF) and aiming at developing andvalidating material models and procedures predicting damage growth andlifetime under cyclic thermal stresses (Schuler et al. (2012)).

Furthermore, guidelines such as that issued by JSME (JSME (2003)), codessuch as the ASME Section III code for design or the French RCC-M and RCC-MR codes, and standards such as the German safety standard KTA, can be usedfor assessing thermal fatigue. Since a comprehensive review of the above projects,


guidelines, codes, and standards falls outside the scope of this analysis, here weonly stress that no full international agreement has yet been reached on thermalfatigue assessment.

2.1.3 Connection between thermal mixing and thermal fatigue

From the analysis in Subsection 2.1.2, it follows that thermal fatigue is an interdisci-plinary topic, spanning thermal-hydraulics; heat transfer; mechanics; and materialsscience. Thermal-hydraulics is essential to properly assess load types such as tur-bulent mixing, turbulent penetration and thermal cycling, thermal stratification,and thermal striping. Knowledge of heat transfer is of significant importance tomodel the heat exchange between fluid and wall, which may differ from the sta-tionary case. A good comprehension of mechanics is paramount to predict stressesdue to thermal loading, while further progress in materials science could help tounderstand the resistance of the wall material to thermal loading and cracking(Chapuliot et al. (2005)), as well as sequence effects; viz., whether the damagecaused by low-amplitude loading (i.e., high-cycle fatigue) cycles followed by high-amplitude loading (i.e., low-cycle fatigue) cycles is more pronounced than that inthe opposite case (Taheri et al. (2013)).

Here, only a few remarks on the first topic viz., thermal-hydraulics areprovided. With respect to temperature fluctuations in the adjoining solid walls, wecan consider the amplitude of the quasi-steady temperature in a half-infinite wall,which varies with coordinate x as T exp

(xf/

)(Taler and Duda (2006)). In

this formula, T denotes the amplitude of the temperature fluctuations at x = 0;in other words, at the wall surface. f represents the frequency of such fluctuations,and indicates the thermal diffusivity of the material the wall is made of. Theabove formula can be applied if certain conditions are met: the temperature at thewall surface must be given by T cos(2ft) that is, it must be sinusoidal over timet ; the wall temperature at x and that at time t = 0 must be the same as themean temperature at x = 0. Other sinusoidal components are left out of the aboveformula. Thus, in the case of stainless steel with diffusivity 4 106 m2 s1, iffrequency f amounts to 0.1 Hz, the amplitude of temperature fluctuations at 5 mmfrom the wall surface is approximately one-quarter of that at the surface. When freaches 10 Hz, this amplitude reduces to less than 106 times that at the surface.Hence, high frequencies are meaningful only in the vicinity of the wall surface (seeFig. 2.2).

Likewise, thermal fatigue cracks were predicted to be initiated by surface tem-peratures fluctuating at frequencies between 0.01 Hz (Tinoco et al. (2009)) and0.5 Hz (Angele et al. (2011)) in annular volumes, and between 0.1 Hz (Chapuliot etal. (2005)) and 3-5 Hz (Ayhan and Skmen (2012)) in T-junctions. These valueswere found in sections where turbulent mixing of non-isothermal streams occurred.Expanding on this subject, Kasahara et al. (2002) estimated fatigue damage ina half-infinite wall when fluid temperature is a sinusoidal function of time, with


0 1 2 3 4 5 6

t (s)400

420

440

460

480

500

T(x

,f,

t)(-

)x = 0.005 m, f = 0.1 Hzx = 0.005 m, f = 1.0 Hz

x = 0, f = 10.0 Hzx = 0.005 m, f = 10.0 Hz

(a) Temperatures at 5 mm from the surface of a half-infinite wall made of 316L(N) stainless steel.The initial temperature is T0 everywhere. The temperature at x is kept equal to T0, whilethe inner-surface temperature is given as T cos(2ft), where T = 50 K. Thermal diffusivity atT0 = 450 K. Formulas from Taler and Duda (2006).

0 1 2 3 4 5 6

t (s)400

420

440

460

480

500

T(r

,f,

t)(-

)

r = 0.042 m, f = 0.1 Hzr = 0.042 m, f = 1.0 Hz

r = 0.04 m, f = 10.0 Hzr = 0.042 m, f = 10.0 Hz

(b) Temperatures at 2 mm from the inner surface of a hollow cylinder made of 316L(N) stainlesssteel. The initial temperature is T0 everywhere. The outer-surface temperature is kept equal to T0,while the inner-surface temperature is given as T sin(2ft), where T = 50 K. Inner and outersurfaces at r = 0.04 m and at r = 0.05 m, respectively. Thermal diffusivity at T0 = 450 K. Formulasfrom Radu et al. (2009).

Figure 2.2: Temperature time series in solid bodies.


frequency f . In this case, a heat transfer coefficient was introduced, so that thefrequency response function, estimating the wall stresses induced by fluid tempera-tures, could be expressed as the product of an effective heat transfer function, gagingthe reduction in temperature from fluid to surface, and of an effective thermal stressfunction, estimating the aforesaid wall stresses due to surface temperatures. Withincreasing frequency f , heat transfer loss decreases the gain of the former effectivefunction, whereas the less efficient thermal homogenization increases that of thelatter. Thus, it is clear that the highest thermal stresses, which might cause fatiguedamage, occur at intermediate frequencies; namely, from 0.1 to 10 Hz.

Even though sinusoidal methods are known to yield overly conservative esti-mates of the fatigue lifetime (Hannink and Blom (2011)), it is evident that per-forming a proper spectral analysis of surface temperatures can help to predict therisk of thermal fatigue cracking. Anyway, given the complexity of such loads inthe case of turbulent mixing, 3-D coupled finite volume/finite element analyses orfactors accounting for plasticity at geometric discontinuities are usually taken intoaccount (Dahlberg et al. (2007)).

2.1.4 Experiments and simulations of thermal mixingBecause of the lack of accurate prediction methods for assessing HCTF induced bymixing of non-isothermal streams, a number of studies attempted to define suchmethods. Namely, this kind of mixing has been widely studied using computa-tions (see Hu and Kazimi (2006), Naik-Nimbalkar et al. (2010), Ayhan and Skmen(2012), and Qian et al. (2015)) and experimental testing. Table 2.1 reports manyexperiments performed in T-junctions, since most of the current work focuses onpost-processing of experimental data. From this table it can be noted that, un-til now, no experiments have accurately replicated the key phenomena leading tothermal fatigue cracking at BWR conditions. This holds particularly true for fa-tigue cracks in control rods: the T-junction geometry cannot correctly representthat around the control-rod stems, which is mostly annular. To address this issue,experiments described in Angele et al. (2011) and Tinoco et al. (2009) were carriedout in a test section reproducing the annular volume around the stems. Water tem-peratures were sampled at 50 Hz using 0.13-mm thermocouples. These probeswere positioned 1 mm from the surfaces of the inner and outer tubes that is, inthe water domain , at many azimuthal and axial measurement positions. How-ever, these experiments were conducted at low temperatures and pressures, far fromthose encountered in BWRs. The same can be concluded about most experimentsin Table 2.1. Thus, more experimental data about mixing of water streams at BWRconditions are required.

2.1.5 Estimators of thermal mixingWhen intensity, inhomogeneity, and efficiency of mixing of non-isothermal waterstreams have to be inferred from large datasets, regardless of whether they contain


Table 2.1: T-junction experiments on mixing of water streams at different temper-atures. Only sensors measuring water temperature are listed here.

SourceTm

(K)

Tb

(K)

vm

(m s1)

vb

(m s1)

p

(MPa)Sensors

fS

(Hz)

y 1

(-)

Fukushima et al. (2003)

296.98,

...,

343.63

296.93,

...,

343.36

0.02,

0.15

0.04,

0.3 0.1

0.5-mm ,

ungrounded

thermocouples

50

0,

...,

0.5

Kawamura et al. (2003),

Hu and Kazimi (2006)

290.85,

...,

297.95

324.95,

...,

329.65

0.27,

...,

2.54

0.21,

...,

2.52

0.1 Thermocouples 25 0.03

Westin et al. (2006)

297.15,

...,

300.45

332.95,

...,

339.05

1.69,

...,

3.97

1.68 0.1 Thermocouples 90 1/190

Kamide et al. (2009) 321 306

0.11,

...,

2.9

0.5,

...,

1.5

0.10.25-mm

thermocouples100

1/150,

...,

0.5

Braillard and Edelin (2009),

Kuhn et al. (2010)356 281 2.55 0.85 0.1

0.5-mm

K-type

thermocouples

5 2/54, 5/54

Naik-Nimbalkar et al. (2010) 303 318

0.33,

...,

1

0.5,

...,

1.32

0.1

Constant-current,

hot-wire

anemometer

1000

0.1,

...,

0.5 2

Kuschewski et al. (2013),

Selvam et al. (2014)

415,

421298

0.11,

0.160.08 3

1-mm

K-type

thermocouples

100 2/71.8

Chen et al. (2014) 363 293

0.05,

...,

0.2

0.96,

...,

3.37

0.49 Thermocouples NA

0.112,

...,

0.5

1Scalar quantity y is defined as the gap between wall and measurement points, over the hydraulic diameter of the conduit.

2For the data analyzed.

2.2. UNCERTAINTY 11

experimental or simulation data, deriving significant indicators and developing al-gorithms to correctly interpret such data seem to be intricate tasks. As an example,in Angele et al. (2011) these datasets resulted from experiments, Reynolds-averagedNavier-Stokes (U-RANS) and scale-adaptive simulations (SAS) of non-isothermalwater streams mixing in an annulus. The temperatures in these datasets were nor-malized. Their average and RMS values were also determined, at many axial andazimuthal measurement positions. Power spectral densities (PSD) of experimentaland SAS temperatures were then derived, to demonstrate that the most prominentspectral components emerge at low frequencies (f < 0.5 Hz), typical of thermalfatigue.

In Sakowitz et al. (2014), the mixing quality in a T-junction was assessed bythree estimators, all found from the passive scalar modeling the mixing process.First, a uniformity index UI was calculated as the weighted difference between thetime-averaged concentration of the passive scalar and its mean value over a crosssection of the computational domain. The RMS value of the passive scalar wasthen computed, to account for the variation of this scalar with time. After that,the integral time scale of the fluctuations of the passive scalar was evaluated, toestimate the longest time over which they are correlated.

In El Omari and Le Guer (2010), where thermal mixing and heat transfer ina two-rod mixer were investigated, an estimator referred to as composite mixingindicator was calculated as the integral mean value in time of the cell-averagedimensionless fluid temperature over its standard deviation; that is, over its levelof homogenization inside the mixer. The higher the composite mixing indicator,the better the thermal mixing. A quantity termed temperature scalar dissipationindicator was then introduced to measure the production and destruction of thetemperature gradient.

Other researchers explored mixing parameters gaging micro- and macromixing.As an example, in Koop and Browand (1979), a parameter called mixedness(Konrad (1977)) was computed to assess the amount of micromixing.

2.2 Uncertainty

One of our main objectives was to estimate uncertainty in thermocouple measure-ments. Procedures devised for evaluating the effect of thermocouple design and lo-cation on uncertainty, such as that proposed in Ould-Lahoucine and Khellaf (2005),could not be applied here, since they assume that water temperatures can be sim-plified to analytical expressions.

This deficiency is overcome in Dusarlapudi et al. by developing a finite-elementmodel of the thermocouple. Nonetheless, this technique alone is not suitable forappraising the relationship between uncertainty and thermocouple design whenhandling an extensive database of time series such as ours.

Methods based on end-to-end calibration data for the data acquisition system(DAS) (Nakos (2004)) were only partially viable here, because of time constraints.


2.3 Heat flux assessment

Heat flux has been recently calculated using different methods, such as

bare (uninsulated) micro-thermocouples measuring temperature at variousdepths inside the wall (Braillard and Edelin (2009), similarly to Bouvier etal. (2005)), so as to estimate the heat flux by using inverse heat conductionmethods;

gradient-type heat-flux sensors (Sapozhnikov et al. (2006)), exploiting theanisotropy of monocrystalline bismuth, where a thermo-electromotive forcecan be measured when the heat flux is not aligned with the principal crystalaxes; and

thermocouples or thin platinum resistance thermometers monitoring surfacetemperatures (Reichelt et al. (2002)), from which the heat flux can be assessedby solving the heat conduction equation in a one-dimensional body with eithersemi-infinite or finite thickness, provided that the boundary condition, eitherat infinity or on the other side, is known.

Only the last method could be applied here, because nothing but surface tempera-tures were available in the current experimental analysis. However, this method ei-ther requires significant approximations when being implemented (see, e.g., DAleoand Prasser (2013)), or is limited to cases where water temperatures are expressedanalytically (Taler and Duda (2006)). Moreover, since we intended to estimate theradial heat flux at the outer radius of a cylinder subjected to non-axisymmetrictransient thermal loading, determining the heat flux in the aforementioned one-dimensional body can be regarded as only the first step towards our goal. Unfortu-nately, since this derivation is highly dependent on the thermocouple arrangementat each axial level z, our literature research on the matter at hand has not revealedany convincing similarities between our case and others.

Chapter 3

Methods

3.1 Experimental setup

3.1.1 Overview of the facility

The HWAT (High-pressure WAter Test) loop used in the current experiments andthe test section are shown in Figs. 3.1, 3.2, 3.5, 3.6, and 3.7. The test section iscomposed of two coaxial vertical tubes: an inner and an outer one. The inner tube(see Fig. 3.6) is 2000 mm long, is formed of 316L(N) stainless steel, and has aninner radius, Rii, of 12.5 mm, whereas its outer radius, Rio, reaches 17.5 mm. Theouter tube (see Fig. 3.6), designed to sustain a pressure of 9 MPa, has an innerradius, Roi, of 40 mm, whereas its outer radius reaches 50 mm.

Water enters the annulus between the inner and the outer tube through two coldand two hot inlets. After mixing has occurred, water leaves the annular volumethrough two outlets. The inner diameters of the inlet tubes are 7.5 mm. Bothhot inlets are located at z = 800 mm (see Fig. 3.6) i.e., at the same height, at azimuthal angles = 180 and = 360. Both cold inlets are located atz = 150 mm (see Fig. 3.6) i.e., at the same height , at azimuthal angles = 90and = 270. Consequently, they are offset by 90 from the hot inlets. This offsetwas necessary to uniformly distribute residual stresses associated with welding andto keep the test section from bending. Two cold inlets, instead of one, were providedto avoid an uneven flow distribution of the cold streams reaching the mixing region.Furthermore, these inlets lie so far from the mixing region that their influence onphenomena occurring there is deemed negligible (Pegonen (2012)). Thus, the coldstreams entering the mixing region could be treated as one.

With reference to the outlets, their inner diameters are 14 mm. They are locatedat z = 1000 mm, not to interfere with the hot inlets.

The water flow in the loop is ensured by a circulating pump, which feeds waterto a preheater (see Fig. 3.1). This heat exchanger comprises 18 heating elements,each with a capacity of 8 kW. By contrast, the cold stream bypasses the preheaterand is supplied to the primary coolers, so as to attain the desired temperature at

13

14 CHAPTER 3. METHODS

Table 3.1: Geometry, dimensions, temperatures, and pressure in the test section ofthe HWAT loop and in BWRs.

Parameter HWAT loop BWR (Tinoco et al. (2009))

Number of hot inlets 28 (upper bypass inlets)& 4 (lower bypass inlets)

Number of cold inlets 2 1

Diameters of the hot inlets 7.5 mm14.6 mm (upper bypass inlets)& 8 mm (lower bypass inlets)

Diameters of the cold inlets 7.5 mm 38 - 43 mm (hydraulic)Outer diameterof the inner tube

35 mm 65 - 70 mm

Outer diameterof the outer tube

100 mm 140 mm

Water temperatureat the hot inlets

549 K 549 K

Water temperatureat the cold inlets

333 - 423 K 333 K

Pressure 7.2 MPa 7.2 MPa

the cold inlets. A pressurizer vessel downstream of the preheater dampens possiblepressure oscillations (see Fig. 3.1).

Table 3.1 contrasts the key dimensions and boundary conditions in the experi-mental facility at hand with those in reactors Oskarshamn-3 and Forsmark-3.

3.1.2 Test-section thermocouples19 K-type thermocouples monitor temperatures at the inner and outer tube inthe test section. The thermocouple placement was influenced by Pegonen (2012).Here, we examine only six of the test-section thermocouples, all with diametersof 0.5 mm. They are labeled H1, H2, H3, H4, V 1, and V 4 and are attached tothe wet surface of the inner tube. Henceforth, H1, H2, H3, H4, V 1, and V 4 aretermed inner-tube thermocouples, whereas we refer to the wet surface of the innertube as inner surface. The inner-tube thermocouples can collect inner-surfacetemperatures since their tips are flush with the inner surface. Fig. 3.7 depictsthree so-called thermocouple discs; that is, casings recessed into the inner tube tokeep the inner-tube thermocouples in position. These discs, whose caps were madecoincident with the inner surface by TIG welding, are positioned 90 of azimuthfrom each other. The center of the mid thermocouple disc, to be called point Q,

3.1. EXPERIMENTAL SETUP 15

Preheater

Pressurizer

Main cooler

TS hot inlets

TS outlets

TS cold inlets

Pump cooling circuit

Pump

Auxiliary cooler

Test section (TS)

Makeup water inlet

Figure 3.1: Key components of the HWAT loop. The flowmeters are sketched asrectangular cuboids.


Figure 3.2: A picture of the test section.

Figure 3.3: A picture of oneof the thermocouple discs.

Figure 3.4: A picture of themotor shaft.


z

0

Figure 3.5: Coordinate system attached to the test section.

serves as a reference point for determining the positions of the thermocouple discsand inner-tube thermocouples see Table 3.2. The thermocouple discs are arrangedin such a way that, if the inner tube were kept still, the inner-tube thermocouplessoldered to them could measure inner-surface temperatures in only two narrowsubregions within the mixing region, at roughly the same axial level and 180from each other. Furthermore, the inner tube could be equipped with only a fewthermocouples, because of technical limitations. To overcome these issues, theinner tube was turned around its axis, lifted, and lowered during the experimentalsessions. This was achieved by remotely controlling a step motor, whose shaft wassecured to the inner-tube base.

In order to measure inner-surface temperatures, in the thermocouple attachmentmethod adopted here six 0.7-mm through holes were drilled in the aforesaidthermocouple discs. Since thermocouples H1, H2, H3, and H4 are soldered to theright disc, four holes were made there. Moreover, since V 1 and V 4 are solderedto the left disc, two holes were made there. Then, casings were used to achievewatertight seal and position the inner-tube thermocouples appropriately (see Fig.3.9). All these casings are hollow cylinders featuring the same outer diameter of0.7 mm. Thus, each of them was partially pushed into one of the above-mentionedholes and silver-soldered to the base of the respective thermocouple disc. Casingscame in a range of lengths to ease soldering. Afterwards, as already mentioned,


GG

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3C

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MATERIAL:

TITLE:

DWG NO.

SCALE:1:20 SHEET 1 OF 1

A4

WEIGHT:

Outer tube without holes and inner tube_lic_thesis

z

z = 0

Figure 3.6: Cut view of the test section. 1 (red): inner tube. 2 (gold): outer tube.3 (green): thermocouple discs. Dimensions are in mm. Axis of cold inlet 1 at(90, 150 mm); axis of cold inlet 2 at (270, 150 mm); axis of hot inlet 1 at (360,800 mm); axis of hot inlet 2 at (180, 800 mm); axis of outlet 1 at (360, 1000 mm);and axis of outlet 2 at (180, 1000 mm).


A

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disc

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FINISH:UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN MILLIMETERSSURFACE FINISH:TOLERANCES: LINEAR: ANGULAR:

Q.A

MFG

APPV'D

CHK'D

DRAWN

Figure 3.7: Exploded-view drawing of the inner tube, together with the thermo-couple discs welded to it.

each inner-tube thermocouple was inserted into one casing until its tip was alignedwith the inner surface; viz., until its tip reached r = Rio. Given that all holes weredrilled at the same distance from the center of the circular base of the respectivethermocouple disc, and given that all tips of the inner-tube thermocouples lie atr = Rio, angle is identical for H1, H2, H3, H4, V 1, and V 4 and is definedby = arcsin (ri cos(45)/Rio). is the angle on plane z = constant between theaxis of a thermocouple disc and the line joining the inner-tube axis and the tip ofa thermocouple on the same disc. ri represents the distance between the centerof the circular base of a thermocouple disc and a through hole on the same disc.This distance was measured in the plane of the base (see Fig. 3.8). Lastly, a high-temperature solder (at 870970 K) was melted onto the outer end of each casing,to fasten the thermocouple inside it in place.


0.70 2.70

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FINISH:UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN MILLIMETERSSURFACE FINISH:TOLERANCES: LINEAR: ANGULAR:

Q.A

MFG

APPV'D

CHK'D

DRAWN

cap

Figure 3.8: Sketch of the left and right thermocouple discs. If not otherwise stated,all dimensions are in mm. 6 designates 2 ri.

xTC

L

1

2

34 4

Figure 3.9: Longitudinal section of an inner-tube thermocouple in a thermocoupledisc. 1 (green): thermocouple disc; 2 (yellow): thermocouple; 3 (gray): casing;4 (red): solder. The cylinder in Fig. 3.8 has not been included for simplicity.L 4.5 mm.


Table 3.2: Positions of the inner-tube thermocouple tips. Scalars Q[l;m] andZQ[l;m] are discussed in Subsection 3.2.2. H1, H2, H3, and H4 are attached tothe right thermocouple disc, while V 1 and V 4 are attached to the left one.

Label rf (mm) f () zf (mm)

H1 Rio Q[l;m] + 90 + ZQ[l;m] ri sin(45)H2 Rio Q[l;m] + 90 ZQ[l;m] ri sin(45)H3 Rio Q[l;m] + 90 + ZQ[l;m] + ri sin(45)H4 Rio Q[l;m] + 90 ZQ[l;m] + ri sin(45)

V 1 Rio Q[l;m] 90 + ZQ[l;m] ri sin(45)V 4 Rio Q[l;m] 90 ZQ[l;m] + ri sin(45)

Table 3.3: Experimental matrix for the measurement of test-section temperaturesat a sampling rate of 1000 Hz.

Caseno., or l

TH(K)

TC(K)

mH(kg s1)

mC(kg s1)

ReH(-)

ReC(-)

T mix(-)

1 549 333 0.8 0.07 711,367 12,696 0.9292 549 333 0.6 0.07 533,525 12,696 0.9083 549 423 0.6 0.14 533,525 32,265 0.8274 549 348 0.6 0.08 533,525 17,890 0.895

3.1.3 Boundary conditions

Ten experimental cases were considered. Their boundary conditions are listed inTables 3.3 and 3.4. Mass flow rates and temperatures at the test-section inletswere varied from case to case, while pressure was kept equal to p = 7.2 MPa.Test-section temperatures were monitored with a sampling rate of 1000 Hz in allcases from Table 3.3, and with a sampling rate of 100 Hz in all cases from Table3.4. To assess experimental repeatability and the effect of sampling rate on thetest-section temperatures acquired, Cases 1 and 2 were evaluated with the sameboundary conditions as Cases 5 and 7, respectively. The mixing temperature T mixin Tables 3.3 and 3.4 was evaluated as T mix = (Tmix TC)/(TH TC). As in Bergagioand Anglart (2016), the adiabatic mixing temperature Tmix was computed frompressure p and from adiabatic mixing enthalpy hmix. This enthalpy is given ashmix = (mChC + mHhH)/(mC + mH), where hH and hC represent the enthalpies of thehot and cold streams, respectively.


Table 3.4: Experimental matrix for the measurement of test-section temperaturesat a sampling rate of 100 Hz.

Caseno., or l

TH(K)

TC(K)

mH(kg s1)

mC(kg s1)

ReH(-)

ReC(-)

T mix(-)

5 549 333 0.8 0.07 711,367 12,696 0.9296 549 423 0.8 0.07 711,367 32,265 0.9277 549 333 0.6 0.07 533,525 12,696 0.9088 549 423 0.6 0.07 533,525 32,265 0.9059 549 333 0.4 0.07 355,684 12,696 0.86710 549 423 0.4 0.07 355,684 32,265 0.864

3.2 Data acquisition

For each case in Tables 3.3 and 3.4, water was circulated long enough to reachsteady-state boundary conditions. Once this precondition had been met, the innertube was driven to preset positions. There, the inner-tube thermocouples solderedto it measured inner-surface temperatures.

3.2.1 Data acquisition tasks

The experimental data were acquired on two laptops: a laptop A, controllingthe inner-tube movement and collecting temperature readings from the test-sectionthermocouples; and a laptop B, recording temperatures, pressure, and pressuredrops from the rest of the HWAT loop. As evident from Table 3.5, two devicescommunicate with laptop A: a National Instruments (NI) SCXI-1000 chassis; anda Measurement Computing (MC) 1608FS device. The former houses a SCXI-1102thermocouple input module, to whose front connector a SCXI-1303 terminal blockis attached. The extension cables of the test-section thermocouples are connected tothis terminal block. The SCXI-1000 is cabled to a DAQ-6024 card, which is pluggedinto a slot on laptop As side in order to link the SCXI-1000 chassis to laptop A.Concerning the MC 1608FS device, it returns the readings of two potentiometers,from which the position of point Q can be determined. It also allows to send digitalon/off signals to the circuit board on the step motor, thus seeing to the inner-tubemovement. For this purpose, three channels were configured as digital outputs.Unlike an analog input channel, which reads voltages produced by thermocouples,pressure transducers, and potentiometers, a digital output channel provides a digitalon/off signal.

Other two devices communicate with laptop B: a second MC 1608FS device,which collects pressure readings from the flowmeters in Fig. 3.1 and from the pres-sure transducer; and an Agilent 34980A data acquisition platform, which recordstemperature readings from the thermocouples in other components of the HWAT

3.2. DATA ACQUISITION 23

Table 3.5: Overview of the key data acquisition parameters. Underlined devicesare attached to laptop A. Pressure drops help to determine mass flow rates.

Device Data acquired fS (Hz) NS0 (S it1 ch1) NC

NI SCXI-1000 Test-section temp.s 1,000 or 100 120,000 or 12,000 19MC 1608FS 1 Position of point Q 1,000 10 2MC 1608FS 2 Pressure drops 100 10 4MC 1608FS 2 Pressure 10,000 3,000 1Agilent 34980A Loop temp.s 48.74 10 13

loop. The MC 1608FS devices and the Agilent 34980A communicate with thelaptops via USB.

The mass flow rates at the test-section inlets were determined from empiricalcorrelations based on pressure, pressure drops, and some loop temperatures. Thesemass flow rates, pressure, and several loop temperatures were displayed on the userinterface of laptop B and updated at a frequency of 1 Hz. During every experi-ment, mass flow rates and temperatures at the test-section inlets, as well as pressure,were compared with the values in the corresponding row of either Table 3.3 or 3.4.If the aforesaid boundary conditions matched the values in these tables within someconvenient tolerance, the inner tube was rotated around its axis and lifted up anddown until point Q viz., the center of the circular base of the mid thermocoupledisc reached the target position in the inner-tube movement pattern (see Sub-section 3.2.2). There, test-section temperatures were measured and stored in anarray of NS0, DAS samples for each thermocouple, NS0, DAS = 120 000 S it1 ch1.After these data had been collected, we checked whether the variances of the de-trended inner-tube temperatures were lower than a given value. If not, test-sectiontemperatures were measured again and stored in an array of NS1, DAS samples foreach thermocouple, where NS1, DAS > NS0, DAS . The PyDAQmx package (Clad(2010)) was essential in order to perform an effortless data acquisition in agreementwith these needs.

Two computers were required to improve the accuracy of the inner-tube move-ment, the quality of the inner-surface temperatures measured, and the consistencyof the experimental boundary conditions with the respective values in Tables 3.3and 3.4. NTP (Network Time Protocol) ensured synchronization between the twocomputers. Table 3.5 shows the key data acquisition parameters for the exper-iments at hand. As detailed in Table 3.5, test-section temperatures were mon-itored at fS,DAS = 1000 Hz: despite the fact that temperature fluctuations upto 10 Hz could be detected with a sampling rate of 100 Hz so as to comply withthe Nyquist-Shannon theorem and compensate for aliasing artifacts (National In-struments (2016)), a sampling rate of 1000 Hz is believed to provide more accuratevariances for the inverse-filtered inner-surface temperatures, as confirmed in Section


4.1. The reason behind inverse-filtering is elucidated in Section 3.3.Nevertheless, as already reported in Table 3.4, test-section temperatures for

boundary conditions matching those in Cases 1 and 2 were also sampled at f IS,DAS =100 Hz, with NSI0, DAS = 12 000 S it1 ch1. Thus, test-section temperatures wereaways monitored over 120 s it1 in the first iteration at a given position in theinner-tube movement pattern. This acquisition time was selected as it allowed totravel the whole inner-tube movement pattern in a feasible amount of time, withoutcompromising the statistical significance of the data acquired.

The temperature arrays of length NS0, DAS (that is, those yielded by the inner-tube thermocouples at the end of the first iteration at a given position) are calledTf,DAS and are of foremost relevance for our present study.

3.2.2 Inner-tube movement pattern

Among other objectives, these experiments were designed for obtaining a thoroughknowledge of the temperature distribution at the wet surface of the inner tube. Asalready pointed out, this objective was attained by translating and rotating theinner tube, because only a few thermocouples could be soldered to this tube andtheir measurement area would be too narrow otherwise. As illustrated in Fig. 3.5,a coordinate system was attached to the outer tube. Consequently, the position ofpoint Q could be easily expressed in terms of and z. Throughout the experiments,this position was returned by two potentiometers (see Table 3.5 for details). Byoperating the step motor whose shaft was fastened to the inner tube , pointQ could be moved up from z = 550 mm to z = 937 mm and then back downto z = 550 mm, whereas the inner tube was fully rotated around its axis, bothclockwise and counterclockwise. As a matter of fact, the inner tube was usuallyrotated counterclockwise (if seen from above) in steps of 45; that is, the distancebetween two neighboring azimuthal positions on the same axial level is 45. Theinner tube was rotated clockwise only once per axial level, when point Q had tobe moved from = 360 back to = 0 to stay below the voltage threshold on theboard controlling the motor.

The acquisition of test-section temperatures was initiated 60 s after the innertube had reached the desired position, to attenuate perturbations in the temper-ature field due to rotation and translation. The sequence of positions occupiedby point Q is the so-called inner-tube movement pattern (Q[l; 1, ...,m], ZQ[l;1, ...,m]) for case l. Fig. 3.10 illustrates this pattern for Case 9. Each position(Q[l;m], ZQ[l;m]) in a given inner-tube movement pattern was turned into mea-surement position (, z) by calculating f and zf from the expressions in Table 3.2,where and ri sin(45) were disregarded. Angle was computed after roundingthe resulting angle. Moreover, 360 were added to or subtracted from it if needed,in such a way that = 45 k, 1 k 8. It can be demonstrated that z differs nomore than (0.99 cm + ri sin(45)) 1.20 cm from zf , while differs no morethan ( + 1) 7.96 from f , if differences of 360 are neglected. On the

3.3. POST-PROCESSING 25

0 45 90 135 180 225 270 315 360 ()

0.55

0.60

0.65

0.70

0.75

0.80

z(m

)

1 2 3 4 5 6 7 8 9

10, 1811 12 13 14 15 16 17

19, 27, 46 20 21 22 23 24 25 26

2829 30 31 32 33 34 3536

37, 45 38 39 40 41 42 43 44

Figure 3.10: The inner-tube movement pattern for Case 9; that is,(Q[9; 1, ...,m], ZQ[9; 1, ...,m]). Numbers in boxes indicate the chronological or-der in which the respective points were reached; that is, they show m, 1 m 46.

whole, the mismatch is so small that, in Subsection 3.3.1 and Section 4.1, f andzf are approximated by and z.

3.3 Post-processing

As already stressed, we mostly examined inner-surface temperature time series mea-sured during the first acquisition of test-section temperatures at a target position.Data were acquired for 120 s at a sampling frequency of 100 or 1000 Hz. These timeseries were saved on laptop A after being conditioned by the 2 Hz low-pass filter onthe SCXI-1102 module. Each of the aforesaid time series was stored in an arrayTf,DAS of length NS0, DAS (see Section 3.2), if sampled at 1000 Hz. Four stepswere required to post-process array Tf,DAS :

1. Each array Tf,DAS was inverse-filtered; that is, a new array Tf, if was com-puted, virtually unaffected by the 2 Hz low-pass filter on the SCXI-1102 mod-ule (National Instruments (2004)).

2. Tf, if was cleared of its high-frequency spectral components by an appropriatelow-pass filter. The filtered array was termed Tf, lf and contained A samples,A < NS0, DAS . Generally speaking, plots of Tf, lf evidence non-stationarytemperature fluctuations. Furthermore, underlying trends can be occasionallyobserved, most likely due to some unsteadiness in boundary conditions.

3. A trend was identified and removed from each array Tf, lf . The detrendedarray was termed Tf, d.

4. Each array Tf, d was windowed. The windowed array was termed Tf, w andFourier-transformed. Array Tf, d was also Hilbert-Huang transformed if the


temperature data stored in it revealed strong, inhomogeneous mixing. In thatcase, its Hilbert-Huang marginal spectrum was calculated.

Data acquired at 100 Hz were post-processed in an analogous way.

3.3.1 Inverse- and low-pass filteringWith reference to steps 1 and 2, each Tf,DAS was discrete Fourier-transformed. Thespectrum thus calculated was multiplied by the inverse-filter response function andthen inverse Fourier-transformed, to retrieve the actual temperatures in the mixingregion. The filter response was estimated accurately enough up to 10 Hz, on thebasis of several experiments. Given that this research mainly centers on tempera-ture spectral components at frequencies lower than 4 Hz, a finite impulse response(FIR) filter was implemented to produce a substantial damping of amplitudes atf > 4 Hz and to contain ringing artifacts around the new cutoff frequency. Let usdesignate the filter order as C. C samples were then discarded after filtering inthat they were corrupted by initial conditions. Thus, the length A of each low-passfiltered array was found as A = NS0, DAS C.

To assess how the inverse-filtered temperatures were altered by sampling rates,a gain parameter G was developed, together with a differential spread variable .The gain parameter was computed as G = T2T1 (T4T3). T1 and T2 represent,respectively, the lowest and highest values in all the inverse-filtered temperaturesat measurement position (f , zf ) for a given case. Measurement position (f , zf )was defined in Subsection 3.2.2. Deriving T3 was more involved: first, each of theinverse-filtered temperatures at measurement position (f , zf ) for a given case wassliced with a step size of 10 S it1 ch1, which resulted in ten sub-arrays. The arith-metic average of the minima of these arrays was then included in a set. Applyingthis procedure to each inverse-filtered temperature at (f , zf ) for a given case re-sulted in a set of length L1, 6 L1 14. Variable T3 was found as the minimumof this set. Variable T4 was defined in a similar way, by replacing minima withmaxima.

The differential spread variable was computed as =(2

2 32)/22. Variable

22 is the mean variance of the inverse-filtered temperatures at (f , zf ) for a given

case. Variable 32 represents the mean variance of the ten aforesaid sub-arrays at(f , zf ) for a given case.

3.3.2 Empirical Mode DecompositionTrend extraction and detrending are required since boundary conditions exhibitedsome unsteadiness throughout the data acquisition process. In addition, inasmuchas trend is a low-frequency signal, detrending allows to analyze Fourier and Hilbert-Huang transforms at low frequencies. Furthermore, detrending helps to betterestimate the thermocouple mounting error, since trends alter the spectra of inverse-filtered temperatures at low frequencies. Disregarding detrending would impact on


the estimate of thermocouple responses, which depend on both amplitudes andfrequencies of water temperatures in the vicinity of thermocouples (see, e.g., Ould-Lahoucine (2004) for evidence).

The detrending method cannot but be adaptive, based on the nature of thesignals to be detrended. Furthermore, inasmuch as most time series exhibit oscil-lations of various amplitudes, the detrending method must handle non-stationarysignals well enough.

Several approaches to trend extraction for one-dimensional time series are avail-able, such as nonparametric linear filtering, the Model-Based Approach, SingularSpectrum Analysis (Alexandrov et al. (2012)), the Smoothness Prior Approach(Tarvainen et al. (2002)), wavelets, and Empirical Mode Decomposition (EMD).The algorithm implementing the last approach can be outlined as follows:

1. Array Tf, lf is renamed as r0. Index c is set equal to 1.

2. All local extrema of rc1 are found.

3. An envelope emax, c1 is built by interpolating all local maxima with a spline.

4. Another envelope emin, c1 is created from local minima in the same fashion.

5. The envelope mean emean, c1 is computed as the arithmetic average of emax, c1and emin, c1.

6. A detail dc is extracted as dc = rc1 emean, c1.

7. The same procedure is applied to residual emean, c1 iteratively.

As a matter of fact, steps 2 to 6, termed sifts", are repeated after rc1 has beenreplaced by dc, until dc can be regarded as zero-mean according to some stoppingcriterion (Huang et al. (2003)). Once this criterion is met, dc is considered anintrinsic mode function (IMF) gc, while residual rc is computed as rc = rc1 gc.Here, as already mentioned, c = 1. After setting c = 2, the same procedure isapplied to r1 and all other residuals after it, until a monotonic residual is found orthe maximum number of IMFs is reached. Once either condition has been satisfied,Tf, lf can be written as

Tf, lf =g1 + r1=g1 + g2 + r2=...

=Gc=1

gc + rG , (3.1)

where rG is the final residual and gc denominates the c-th IMF. For a detailed de-scription of the EMD algorithm we refer the interested reader to Huang et al. (2003),Rilling et al. (2003), and Wu and Huang (2009). Cubic or rational splines can be


used in steps 3 and 4. To build envelopes interpolating all local maxima and min-ima, we selected rational splines with end-condition and tension parameters equalto 1 and 5 (Peel et al. (2009)), respectively. Rational splines were implemented asthey are expected to reduce the over- and undershooting often observed when localmaxima and minima are interpolated using cubic splines. Details were regardedas IMFs if the number of zero-crossings and that of extrema differed at most byone, and if such numbers stayed the same for five consecutive sifts (Huang etal. (2003)). The algorithm stopped working on residuals when either the residualbecame a monotonic function or the maximum number of IMFs K = blog2(A)cwas extracted. The floor function attempted to conciliate suggestions from Wu andHuang (2009) with Song et al. (2012). Trend was found as

=G

c=c0

gc + rG , (3.2)

where gc indicates the c-th IMF and rG designates the final residual, whereas c0was selected according to Yang et al. (2013). Namely, c0 is the index i starting fromwhich the correlation coefficient between the Hilbert-Huang marginal spectrum ofIMF gi and that of IMF gi+1 becomes larger than a preset threshold , 0 < < 1.

3.3.3 Windowing, DFT, HHT, and Hilbert-Huang marginalspectrum

DFTs, HHTs, and Hilbert-Huang marginal spectra need to be computed in orderto identify which frequencies are prominent at the measurement positions wherethermal fatigue is most likely to occur. When mixing of non-isothermal streams isinvestigated, the spectral analysis of experimental data and computational resultsis commonly performed using the DFT (this is the case, e.g., in Kamide et al. (2009)and Pasutto et al. (2005)). This approach is very common because most studiesaim at assessing whether the results obtained agree with Kolmogorovs power-lawscaling of turbulence in the inertial subrange that is, E() 5/3, where E isthe energy spectrum function and is the wavenumber and at finding a numberof frequencies for the evaluation of thermal stresses in the adjacent walls. The for-mer purpose was pursued in Ayhan and Skmen (2012) and Timperi (2014), whilethe latter was explored in Radu et al. (2009) and Hannink and Blom (2011). Inthe latter case, either the bulk or the inner-surface temperature is assumed to varysinusoidally with time in order to evaluate stresses in the wall, consistently with theso-called sinusoidal method. Thus, any intermittency and non-periodicity in thesetemperatures are discarded so as to derive a range of critical frequencies that couldyield the minimum crack initiation life. From the above purposes, it follows thattime-frequency transformations able to analyze intermittent, non-periodic temper-ature time series are seldom examined. Accordingly, such transformations are onlyoccasionally applied to turbulence data.


In our case, since the inner-surface temperature time series acquired had to bedetrended in any case, a set of IMFs was extracted from each of them. Given thatthe Hilbert spectral analysis studies how the frequencies of each IMF evolve overtime, the Hilbert-Huang transform (HHT), combining the EMD approach with theHilbert spectral analysis, came as a natural choice.

This method was adopted in a certain number of studies on turbulence, suchas Meng et al. (2011), Huang et al. (2013), and Konsoer and Rhoads (2014). InMeng et al. (2011), the pressure oscillations measured in a Kenics static mixerwere examined with the aid of the HHT, under several flow regimes (that is, withdifferent Reynolds numbers). Huang et al. (2013) examined the velocity time seriesfrom direct numerical simulations (DNS) of a three-dimensional homogeneous andisotropic turbulent flow at Re = 400, where Re is a Reynolds number based onTaylor microscale . is an intermediate scale between the Kolmogorov length scale, characteristic of small eddies, and the outer spatial scale l, associated with largeeddies. Specifically, the so-called second-order Hilbert-based statistical moments ofthe above velocity time series were computed from their Hilbert spectra.

In Konsoer and Rhoads (2014), time series of temperature, streamwise andlateral velocity, and bac

experimental analysis of thermal mixing at reactor...

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