Pulsed Eddy Current Sensing for Critical Pipe Condition Assessment

25 Pages·2017·15.45 MB·English

sensors
Article
Pulsed Eddy Current Sensing for Critical Pipe
Condition Assessment
NalikaUlapane* ID,AlenAlempijevic ID,TeresaVidalCalleja ID andJaimeVallsMiro ID
CentreforAutonomousSystems(CB11.09.300),FacultyofEngineeringandInformationTechnology,
UniversityofTechnologySydney,15Broadway,Ultimo,NSW2007,Australia;
[email protected](A.A.);[email protected](T.V.C.);
[email protected](J.V.M.)
* Correspondence:[email protected]@gmail.com;
Tel.:[phone]or+61-4-0451-2441
Received:1August2017;Accepted:20September2017;Published:26September2017
Abstract: PulsedEddyCurrent(PEC)sensingisusedforNon-DestructiveEvaluation(NDE)ofthe
structuralintegrityofmetallicstructuresintheaircraft,railway,oilandgassectors. Urbanwater
utilities also have extensive large ferromagnetic structures in the form of critical pressure pipe
systemsmadeofgreycastiron,ductilecastironandmildsteel. Theassociatedmaterialproperties
renderNDEofthesepipesbymeansofelectromagneticsensinganecessity. InrecentyearsPEC
sensinghasestablisheditselfasastate-of-the-artNDEtechniqueinthecriticalwaterpipesector.
ThispaperpresentsadvancementstoPECinspectioninviewofthespecificinformationdemanded
fromwaterutilitiesalongwiththechallengesencounteredinthissector. Operatingprinciplesofthe
sensorarchitecturesuitableforapplicationoncriticalpipesarepresentedwiththeassociatedsensor
designandcalibrationstrategy. AGaussianprocess-basedapproachisappliedtomodelafunctional
relationshipbetweenaPECsignalfeatureandcriticalpipewallthickness. Acasestudydemonstrates
thesensor’sbehaviouronagreycastironpipeanddiscussestheimplicationsoftheobservedresults
andchallengesrelatingtothisapplication.
Keywords: criticalpipes;eddycurrents;Gaussianprocess;NDE;NDT;PEC;sensors
1. Introduction
PulsedEddyCurrent(PEC)sensingisconsideredasthemostversatilememberofthefamily
of Eddy Current (EC) Non-Destructive Evaluation (NDE) techniques. PEC signals are known to
possessabroadfrequencyspectrumenablingthemtopenetrateintodifferentdepthsandtoprovide
informationaboutthegeometryofthetestpiecebeingevaluated[1]. ThiscapabilityhasmadePEC
sensingapopularNDEtechniqueforferromagneticmaterialinspectioninresearchandcommercial
domains[2–5]. Criticalpipesarelargediameter(usually≥300mm)pressurizedpipesownedand
managedbywaterutilitiesthroughouttheworldforthepurposeofdistributingconsumablefresh
watertocustomers. Globally,mosturbanwaterutilitieshaveextensivelarge,criticalpressurepipe
systems,partsofwhichhavebeeninserviceforoveracentury[6–8]. Sincetheseagedpipesarefound
intheformofthreeferromagneticmaterials,namelygreycastiron,ductilecastironandmildsteel,
NDEofthesepipesbymeansofelectromagneticsensingisanecessity. WaterutilitiesundertakeNDE
toascertainthestructuralintegrityoftheirassetsandmakecrucialdecisionsonpipemaintenance
andrenewaltopreventcostlyandcatastrophicpipefailures. Motivatedbythisneedanapproachto
effectivelyemployPECsensingforNDEofcriticalpipesispresentedandtheunderlyingchallenges
relatedtotheapplicationarediscussed.
The work in [9] provides a comprehensive review of state-of-the-art inspection technologies
usedforconditionassessmentofpipes. Providedthatwaterutilitiesdrivetowardsestimatingstress
Sensors2017,17,2208;doi:10.3390/s17102208 www.mdpi.com/journal/sensors
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concentrationonpipewallsinordertopredictfailures,thecriticalpieceofinputinformationrequired
forthatisadensemapoftheremainingpipewallthickness[10]. PECsensingthereforehashigher
preferenceforthisapplicationasitiscapableofdirectlymeasuringtheremainingthicknessofhealthy
materialandproducingdensemapswhileexhibitinglowsensitivitytotuberculationformedonaged
criticalpipesintheformofcorrosionandgraphitisation. Inaddition,itisnotsensitivetoinsulated
protectionpresentintheformofinternalcementliningshouldsensingbeundertakenfrominsidethe
pipe[11–13]. Sincecriticalpipematerialsareferromagnetic,recentworksinvestigatingPECsensing
offerromagneticmaterialthickness[2–5]closelyrelatetotheworkofthispaper. Acommonalityin
thoseworksisusingtheconcentrically-woundexciter-detectorcoil-basedPECsensorarchitecture,
whichhasbeendeemedtohavebettersensitivityforferromagneticmaterialthicknessquantification.
Althoughthereareothersensorarchitectureshavingmagneticsensorsforsignalreceptionasin[14–18],
theyhavetypicallybeenusedonnon-ferromagneticmaterials. Recentworkin[19,20]hasreported
preliminaryworkexplicitlyonferromagneticpipeinspection. Theworkin[19]introducesanovel
sensor design integrated with a distributed EC inspection system, while [20] presents a study of
applyingpulsedremotefieldeddycurrentsforinternalinspectionofpipes. Thosetechniquesfocuson
defectdetectionandhaveproducedpromisingresults. However,theyrequirefurtherworktobeable
todeliverquantifiedremainingwallthicknessintheformofdensemaps,whichisthecriticalpieceof
informationdemandedbywaterutilities.
Derivingfromtheworkof[2–5],thispaperexploitstheexciter-detectorcoil-basedPECsensor
architecturetoinspectcriticalpipesbyestimatingremainingwallthicknessusingatimedomainsignal
decayrate-basedfeature. Thesignalfeatureisderivedfromtheanalyticalmodelproposedin[3]for
PECsignalsresultingfromthisarchitecture. Although[2–5]haveanalysedPECsensorperformance
onstandardsteelssuchasQ235andA3undercontrolledexperiments, theirworkonapplications
relatingtocriticalpipeinspectionislacking.
Work specifically related to critical pipe inspection can be found in [11,21]. A study of the
designofaPECsensorthatachievessignalsensitivitytogreycastironthicknessispresentedin[11].
However,in[11],thesensor’sperformancewasnotevaluatedonactualpipes. Ontheotherhand,[21]
studiestheuseoftheGaussianProcess(GP)[22]toinfergreycastironpipewallthicknessbyexploiting
multiplefeaturesextractedbyaveragingregionsofthesignalsproducedbyacommercialPECsensor.
Theapplicabilityoftheresultsin[21]islimitedtothespecificcommercialsensor. Althoughthese
tworeferencesformthebasisforthispaper,themaincontributionhereisthein-depthstudyofgeneral
PECsensingappliedtocriticalpipeassessment. Morespecifically,itstudiestheelectricalandmagnetic
propertiesofacriticalpipematerial,andbasedonthequantificationofthesepropertiesitproposesthe
sensordesign,sensorexcitation,signalacquisitionandcalibrationstrategiessuitableforsuchmaterials.
Finally,itproposesaGP-basedapproachtomodelthefunctionalrelationshipbetweenaPECsignal
featureandmaterialthickness,whichissuitableforcriticalpipewallthicknessprediction.
The outline of this paper is as follows: Section 2 contains the theoretical formulation, which
presents the PEC sensor operating principles with respect to ferromagnetic material thickness
estimation,andtheGP-basedapproachtomodelthefunctionalrelationshipbetweenasignalfeature
andpipewallthickness. Section3presentsPECsensordesignprincipleswithparticularreference
to grey cast iron pipe assessment. Section 4 presents the experimental evaluation of the designed
sensor’sperformance. Acasestudydepictingthesensor’sperformancewhenusingGPtoestimate
wallthicknessofagreycastironpipeisalsopresented. Section5concludesthispaperbydiscussing
theimplicationsofobservedresultsandsummarisingviablepracticesandchallengesforexploiting
PECsensingforstate-of-the-artconditionassessmentofcriticalpipes.
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2. TheoreticalFormulation
2.1. Exciter-DetectorCoil-BasedPECSensorOperatingPrinciple
The typical exciter-detector sensor architecture sensitive to ferromagnetic material thickness,
asexploitedinthispaperandintheferromagneticmaterialthicknessquantification-relatedworks[2–5],
iscomposedoftwoconcentrically-wound,aircored,conductivecircularcoils,asshowninFigure1.
Rarer, though also practical, is the use of concentrically-wound rectangular coils [12,21]. In both
configurations,onecoilbehavesastheexciter,whiletheotheractsasthedetector,whichcapturesthe
signal. TheexcitercoilisexcitedwithavoltagepulsethatcantheoreticallybemodelledasaHeaviside
stepfunction. Thepulsedexcitationcausesarapidchangeinthesurroundingmagneticfield;thisin
turninduceseddycurrentsinthetestpiecebeingassessed. Theneteffectofinducededdycurrents
andtheexcitationpulseinducesauniquetime-varyingvoltageinthedetectorcoil. Itisthisdetector,
thecoilvoltage,thatisidentifiedasthePECsignal,whichcarriesinformationaboutthetestpiece.
ThetypicalshapeofsuchaPECsignalisanexponentialdecayasshowninFigure2.
Figure1.Cross-sectionalviewofatypicalexciter-detectorcoil-basedPECsensor.
Figure2.ThetypicalshapeofaPECsignal:inducedvoltageinthedetectorcoil.
As done in [3], the decaying part of the time (t) domain PEC signal V(t) can be modelled as
aninfinitesummationofexponentialtermsasshowninEquation(1)wheretheb andc termsare
i i
constantscontaininginformationaboutthepropertiesofthesensorandtheferromagnetictestpiece.
∞
∑
V(t) = b exp(−c t) (1)
i i
i=1
Allc ≥0,c (cid:54)= c wheni (cid:54)= jfori,j ∈N.
i i j
Toderivethesignalfeatureusedinthispaper,weexpressEquation(1)initslogarithmicformas
showninEquation(2).
(cid:34) ∞ (cid:35)
∑
ln[V(t)] =ln b exp(−c t) (2)
i i
i=1
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Sinceallc ≥0,Equation(1)becomesasumofexponentialdecays. Therefore,weconsiderthe
i
laterstageofthesignal(i.e.,t >>0)andapplythepropertiesofthelogsumofexponentials[23]to
Equation(2). Fortheregiont >>0,Equation(2)canhencebereducedtothedominantexponentialas:
(cid:12)
(cid:12)
ln[V(t)](cid:12) ≈ln[b1exp(−c1t)] (3)
(cid:12)
t>>0
wherec isthedominanttimeconstantandb isthecorrespondingcoefficientoftheexponentialterm.
1 1
Expandingtheright-handsideofEquation(3)resultsin:
(cid:12)
(cid:12)
ln[V(t)](cid:12) ≈ −c1t+ln[b1]. (4)
(cid:12)
t>>0
WenowdefinethesignalfeatureβbytakingthederivativeofEquation(4).
(cid:12)
dln[V(t)](cid:12)
(cid:12) ≈ −c1 (5)
dt (cid:12)
t>>0
1
β = (6)
c
1
Equation(4)suggeststhatthebehaviourofthelaterstageofaPECsignaltakenonaferromagnetic
materialshouldapproximateastraightlinewithanegativegradientwhenexpressedinitslogarithmic
form. Numerical simulations and experimental results in the subsequent sections validate this
approximation. The feature β is the reciprocal of the absolute gradient of the logarithmic signal
regionbehavingasastraightline. Inreturn,βextractsthedominanttimeconstantc ,whichdictates
1
the later stage of the signal V(t) expressed in Equation (1). Previous work [4] has noted that this
dominanttimeconstantbehavesasc ∝1/(µσd2)whereµ,σandd,whicharemagneticpermeability,
1
electricalconductivityandthicknessoftheinspectedferromagneticmaterialunderthedomainof
influenceofthesensor, respectively. Sincethisbehaviourhasbeenreportedonflatplatesandthe
focusofthisworkispipewalls,whicharecurved,thebehavioursforlargediameterpipesandthe
correspondingsensorgeometryareverifiedthroughFiniteElementAnalysis(FEA)in[12],andithas
beenobservedthatthecriticalpipesurfacesbehavethesamewayasflatplatesduetothecurvature
beinglowwithrespecttosensordimensions. Thus,βbehavesasβ ∝ µσd2forlargediameterpipes.
Therefore,whencalibratedtoaparticularmaterial,βreducestoafunctiondependentonthickness
alone,makingitpossibletousetheinverseofthisfunctionforthicknessquantification. Furthermore,
simulated and experimental results in this paper suggest that β has a desirable quality of having
lowsensitivitytolift-off,making βasuitableoptionforcriticalpipeinspectionapplicationswhere
unknownlift-offisaprevalentchallenge. Thenon-linearityofβisparticularlyprevalentinlowerand
higherendsofthicknessduetolimitationsinhowhighorhowlowtheexcitationstrengthdeliveredby
sensordrivingelectronicscanbe,suggestingthatanon-linear,potentiallynon-parametricmodelling
techniquewouldbesuitablefortheapplication.
2.2. GaussianProcessFormulation
As established in Section 2.1, once calibrated for a material (i.e., for a particular material
having properties µ and σ), thickness d reduces to a non-linear function dependent on β alone.
Therefore,estimatingthicknessfromPECsensorsignalscanbeformulatedasanon-linearregression
problem. Gaussianprocessmodelsareapowerfultooltosolvesuchregressionproblems.
Given the set of β values extracted from PEC signals B = [β ,β ,...,β ]T, where each β is
1 2 m i
associated with a noisy value of thickness d in the set D = [d ,d ,...,d ]T, the aim is to find
i 1 2 m
the underlying function that maps β values to actual thickness. GP is used to learn the thickness
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distributionandtopredictthisdistributionforarbitrarypointsβ∗. Inthispaper,thetrainingdataset
[B,D]isproducedthroughnumericalsimulationasexplainedinSection4.4.
ToapplytheGPframeworktothisregressionproblem,akernelK(B,B)whoseelementsaregiven
byk = k(β ,β )hastobeselected. Afterevaluatinganumberofcommonlyusedkernels,thesquared
i,j i j
exponentialkernelwaschosenforthisworkasitwasfoundtobeeffective. Thiskernelisgivenby:
(cid:26) (cid:27)
1
k(β ,β ) = α2exp − (β −β )2 . (7)
i j 2η2 i j
whereαandηarehyper-parameters. Thesehyper-parameterstogetherwithnoisestandarddeviation
σ arelearnedfromthetrainingdata,byminimizingthenegativelogmarginallikelihood:
n
1 1 m
−logp(D|B,θ) = DTΣ−1D+ log|Σ|+ log(2π) (8)
2 2 2
wherethecovariancefunctionΣisgivenby:
Σ = K(B,B)+σ2I (9)
n
withrespecttoθ = {α,η,σ },where I isthecorrespondingidentitymatrix.
n
Oncehyper-parametersθ arelearnedfromthetrainingdata,arbitrarypoints β∗ areprovided
asinputtotheGPmodeltopredictthecorrespondingthicknessestimateasaGaussianprobability
distributionwhosemeanisµ∗ andstandarddeviationisσ∗. µ∗ andσ∗ arecalculatedasfollows:
d d d d
µ∗ = K(β∗,B)Σ−1D; (10)
d
(σ∗)2 = α2+σ2−K(β∗,B)Σ−1K(B,β∗). (11)
d n
3. ASensorDesignExample
Thissectiondetailstheprocedurefordesigninganexciter-detectorcoil-basedPECsensorsuitable
forcriticalpipeinspection. Thedesignexampletargetsgreycastironpipeassessment;themaximum
thicknessexpectedonpipeswas20mm. Theprocedureincludesthefollowingsteps: (1)identifying
electricalandmagneticpropertiesofthepipematerialtobeinspected; (2)numericallysimulating
asensortodeterminesuitablesensordimensions;and(3)sensorfabrication. Thefollowingsubsections
detailthethreesub-stepsofthedesignprocedure.
3.1. IdentificationofMaterialElectricalandMagneticProperties
Since β becomes a function in the form of β ≈ g(µ,σ,d) and predominantly depends on
the electromagnetic properties and material thickness, it is in fact heavily independent of sensor
dimensions. Practicallimitationsinsensorexcitationandsignalacquisitionelectronicsdictatethose
dimensions, and they should be decided upon before fabrication in order to achieve sufficient
penetration depth in the ferromagnetic material (grey cast iron in this case). Sensor dimensions
aredeterminedinthispaperthroughnumericalsimulation,andtoachievethat,knowingµandσ
beforehandisnecessary.
Measuring electrical and magnetic properties was done by extracting a coupon (hot
tapping [24–26] is a viable option for coupon extraction from on-site critical pipes), making
a specimen (dimensions = 3 mm × 2 mm × 2 mm) through Electric Discharge Machining (EDM)
wirecutting[27–29](usingcoolingliquid)andfeedingittoaPhysicalPropertyMeasurementSystem
(PPMS) [30–32]. An average representation of properties is derived by performing measurements
onmultiplespecimens. Atotalof27specimensmadefromcouponsextractedfromequally-spaced
locations along a 1 km-long grey cast iron pipeline, with details provided in Table 1, were tested
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bymeasuringtheirmagnetizationcurve(i.e., BHcurvewhereBismagneticfluxdensityandHis
magneticfiledintensity)andelectricalconductivity.
Table1.Specificationsofthegreycastironpipeusedfortheexperimentalworkofthispaper;adapted
from[7].
Location VeronaStreet,StrathfieldNSW2135,Australia
YearInstalled 1922
NominalPipeDiameter 600mm
InternalPipeDiameter 579mm–590mm(withcementlining)
ExternalPipeDiameter 662mm–666mm
NominalWallThickness 27mm
Material Pitcastiron
InternalLiner Cement(installedin1964)
CementLiningThickness 9.5mm–16.5mm
PipeLengths 3.6m
Jointing Leadrunjoints(withtar-soakedhempsealants)
TotallengthinUseforResearch Approximately1km
Sinceelectricalconductivityisknowntovarysignificantlywithtemperature,thedependencewas
capturedbymeasuringtheconductivityofeachspecimenacrossarange(220K–350K).Theaverage
representationforconductivity(i.e.,2.16×106S/m,approximately)wasobtainedbycomputingthe
meanovertemperature(between283Kand313Ktoresembleatmospherictemperaturevariation
in Sydney Australia), as well as specimens. Although the average value of σ was considered for
simulation,itwasnotablethatσ ofcastironisconsiderablyvariable. ThiscanbeseeninFigure3,
whichshowsahistogramofconductivitiesresultingfromallconductivityvalues(1097intotal)between
temperatures283Kand313Kcapturedfromall27specimens. Thestandarddeviation(std)ofthis
datasetwas0.261×106S/m,and94.8%ofthedatafellwithin±2standarddeviations.Suchavariation
in conductivity creates a unique difficulty in calibrating PEC sensors for critical pipe assessment.
ConductivitydatafromwhichthestatisticswerecalculatedareprovidedasSupplementaryMaterial.
Significantvariationinmagneticpropertieswithinatmospherictemperatureconditionsanda
correlation between electrical conductivity and magnetic permeability were not evident from the
availabledata;thereforeamagnetizationcurveperspecimenwasmeasuredwhilemagnetizingand
demagnetizing. Resultingcurveswereaveragedeventuallyacrosssamples. Figures4and5depict
a magnetization curve and conductivity measurements performed on a particular grey cast iron
specimen, respectively, and raw data are provided as Supplementary Material. A fine sampling
resolutiontomeasurethemagnetizationcurveregioncoveringlowmagneticfieldsisrecommended
in order to capture the typical non-linear behaviour present. A sampling interval of 10 A/m was
usedwhenmagneticfieldintensity≤100A/m. Therelativepermeabilityvalueµ = 63calculated
r
fromthelowmagneticfieldregionofthemagnetizationcurveinFigure4(consideringthesensor
excitationstrength,highmagneticfieldsarenotexpectedinsidethepipematerial)andtheaveraged
conductivityvalue2.16×106 S/mwereusedtonumericallysimulateaPECsensoranddetermine
suitabledimensionsfortheapplication,theprocessisdescribedinsubsequentSection3.2.
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Figure3.Histogramoftheelectricalconductivityofgreycastironfortemperaturesbetween283Kand
313Kcaptured.
Figure4.Ameasuredmagnetizationcurveofaspecimentakenfromagreycastironpipesegment.
Figure5.Temperaturevariationoftheelectricalconductivityofgreycastiron.
3.2. NumericalSimulationofthePECSensor
Duetothesimplicityofmodelling,andthecommonuseforferromagneticmaterialthickness
estimation[2,3,5],acircular-shapedPECsensorhavingconcentrically-woundaircoredcoilsasshown
in the cross-section in Figure 1 was selected for this work. For a fixed excitation, the sensor size
hasbeenobservedtobeadominantfactorinfluencingthesensor’spenetrationcapability(i.e.,the
maximumthicknessofaparticularmaterialtowhichthesensorwillbesensitive)[16].Therefore,before
fabrication,thesensorinteractionwithgreycastironwasnumericallysimulatedusingFEA.A2D
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axisymmetricmodelofthesensorplacedaboveagreycastiron(showninFigure6)wasdeveloped
usingCOMSOLMultiphysics®. TheinputparametersrequiredforsimulationaredefinedinTable2.
ThesimulationmodeloutputsthedetectorcoilvoltageV(t)asafunctionofmanyinputvariablesas
showninEquation(12). V(t)iscalculatedusingthemagneticvectorpotential,whichisdeterminedby
solvingthemagneticvectorpotentialequationshowninEquation(13)foranygivenlocationinthe
(cid:126) (cid:126)
model,where Aisthemagneticpotentialatanylocation,tistimeand J isthesourcecurrent.
s
V(t) = f(r ,r ,h ,lo ,n ,σ ,µ ,r ,r ,h ,lo ,n ,σ ,µ ,d,µ,σ,Z ,I ,t) (12)
di do d d d d d ei eo e e e e e dl e
(cid:126)
∂A
∇2A(cid:126) −µσ = −µ(cid:126)J . (13)
s
∂t
Figure 6. 2D axisymmetric model of the PEC sensor placed on a cast iron block (developed in
COMSOLMultiphysics®).
Table2.Parametersrequiredforsimulation.
Symbol Description
r Innerradiusofdetectorcoildomain
di
r Outerradiusofdetectorcoildomain
do
h Heightofdetectorcoildomain
d
lo Verticaloffsetofthedetectorcoil
d
n Numberofdetectorcoilturns
d
d Diameterofthedetectorcoilwire
d
σ Electricalconductivityofthedetectorcoil
d
µ Magneticpermeabilityofthedetectorcoil
d
r Innerradiusofexcitercoil
ei
reo Outerradiusofexcitercoil
he Heightofexcitercoildomain
loe Verticaloffsetoftheexcitercoil
ne Numberofexcitationcoilturns
de Diameteroftheexcitationcoilwire
Re Resistanceoftheexcitationcoilwire
σe Electricalconductivityoftheexcitercoil
µe Magneticpermeabilityoftheexcitercoil
d Platethickness
σ Electricalconductivityofpipematerial
µ Magneticpermeabilityofpipematerial
Ie Amplitudeortheexcitationcurrentpulse
Z Loadimpedanceconnectedtothedetectorcoil
dl
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Tonarrowdownthesuitablesetofsensordimensions,theheightsofexciteranddetectorcoils
(h andh )andverticaloffsetsofthetwocoils(lo andlo )arefixed. Copperwiresareusedtowind
e d e d
coils; thus, we use the standard permeability and conductivity of copper (µ , µ , σ and σ ) for
e d e d
simulation. Inaddition,roughestimatesofpermeability(µ)andelectricalconductivity(σ)ofgreycast
ironarerequired. Table3showsthefixedparametersforsimulation. Aspertheestimatedvaluein
Section3.1,theapproximatedelectricalconductivityusedforgreycastironwasσ =2.16×106S/m.
Since the magnetic properties of grey cast iron are non-linear, the relative permeability value
µ =63calculatedfromthelowmagneticfieldregionoftheexperimentally-measuredBHcurvein
r
Section3.1wasusedtorepresentµ. Theamplitudeoftheexcitationcurrentpulsewasalsoconsidered
tobefixedat200mA.Thesensorexcitationcircuitwasdesignedtoproduceavoltagepulsehaving
a10-Vamplitudeandcurrentamplitudeof200mA;thus,theexcitercoilresistancewasrequiredtobe
R =50Ω. Duetoavailability,standardcopperwireofd = d =0.315mmdiameter(AWG28wire
e e d
class)waschosentowindbothexciteranddetectorcoils. AsdiscussedinSection3.3,thedetectorcoil
outputisdirectlyconnectedtoaninstrumentationamplifierhavinghighinputimpedance. Thisresults
intheeffectiveimpedance(Z )feltastheloadbythedetectorcoiltobehigh(indicativeof∞).
dl
Table3.Fixedparameters(constants)usedforsimulation.
Symbol Value
he =hd 10mm
loe =lod 2mm
de =dd 0.315mm
σe =σd 5.998×107S/mforcoppercoils
µe =µd 4π×10−7H/mforcoppercoils
σ 2.16×106S/m
µ µ=µrµ0,µr =63,µ0 =4π×10−7H/m
Ie 200mA
Re 50Ω
Z ∞
dl
Giventheconstraints,theobjectivewastoselectsuitableinnerandouterradiiofbothexciter
anddetectorcoils(i.e.,r ,r ,r andr )alongwiththeirrespectivenumberofcoilturns(i.e.,n and
ei eo di do e
n )inordertohavecastironthicknesssensitivityfromabout5–20mm. Whileparameterselection
d
canbeformulatedasanoptimisationproblem,similartotheworkin[19]relatedtoadistributedEC
inspectionsystem,solvingitforthiscasewouldrequiretime-consumingstochasticoptimisationdue
toderivingclosed-formequationstoperformaquickerconvexoptimisationbeingdifficult. Therefore
theparametersinTable4wereselectedthroughsimulationandexperimentallyvalidatedtoyield
sufficientsensitivitytothicknesswithbothlowandhighlift-off,asshowninFigure7. Thelaterstage
ofallsignalsinFigure7behavesasastraightlinewithanegativegradient;thisbehaviourisexpected
aspertheformulationinEquation(4)andvalidatesthestraightlinebehaviourtheorizedinSection2.1.
Raw data plotted in Figure 7 and the relevant COMSOL simulation model are provided with the
SupplementaryMaterialforinterestedreaderstouseandwithwhichtoexperiment.
Table4.EstimatedparametervaluesusedtofabricatethePECsensor.
Symbol Value
r 25mm
di
r 28mm
do
n 300
d
r 50mm
ei
reo 57mm
ne 600
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Figure7. Numerically-simulatedsignalsbythe2Daxisymmetricmodelfordifferentgreycastiron
thickness with and without lift-off (visualized as ln[V(t)]); sharp rising edges of signals are not
discriminableinthemstimescaleandappeartobeoverlapping.
Due to operational practicalities, water utilities are interested in robotic tools, which can
autonomouslyinspectpipesinternally;adevelopmentrelatedtothepurposeispresentedin[13]. Asin
thecasewiththepipeinTable1,criticalpipesusuallyhaveaninsulatedinternalprotection,typically
madeofcement. Therefore, wheninspectinginternally, itisnecessaryforPECsensorstoassessa
pipewallwitha10–15mmlift-off. Motivatedbythisneed,andforthesensortobesuitableforboth
externalandinternalinspection,thedimensionsinTable4werechosenforthesensortobesufficiently
sensitivetothicknesswithalift-offrangingfrom2–14mm,asindicatedbythesignalsinFigure7.
3.3. SensorFabrication
Figure 8 shows the PEC sensor fabricated using the values provided in Table 4. Both exciter
anddetectorcoilswerewoundusingAWG28wireshavinganapproximatediameterof0.315mm.
ThesensorcorewasdesignedinSOLIDWORKS©softwareandwas3DprintedusingPolylactide(PLA)
biodegradablepolyester.