Capacitance sensor for void fraction measurement in water steam

FlowMeasurementandInstrumentation15(2004)317–324

http://www.77cn.com.cn/locate/ owmeasinst

Capacitancesensorforvoidfractionmeasurementinwater/steam

ows

A.JaworekÃ,A.Krupa,M.Trela

InstituteofFluidFlowMachinery,PolishAcademyofSciences,POBox621,Fiszera14,80-952Gdansk,Poland

Received17December2003;receivedinrevisedform25February2004;accepted16April2004

Abstract

AcapacitancesensoroperatingatRFrangeforvoidfractionmeasurementswasdeveloped.Twoelectrodesofthecapacitoraremountedontheoutersideofpipewalls.Thevariationsinthepercentageofphasesintwo-phase owcausechangesoftheequivalentpermittivityofthedielectricbetweentheelectrodes.Thecapacitorisconnectedinaresonantcircuitofanoscillatortunedtohighfrequencyof80MHz.Thechangesoffrequencygeneratedbytheoscillatorarethemeasureofthevoidfractioninthetwo-phase ow.Aneight-channelsystemwithcapacitancesensorsofthistypewasusedfordeterminationofthephasecon-versionalongasteaminjector.

#2004ElsevierLtd.Allrightsreserved.

Keywords:Two-phase ow;Voidfractionmeasurement;Steaminjector;Capacitancesensor

1.Introduction

Two-phase owsarefrequentlymetinmanytechni-calandenergyconversionprocesses.Avarietyofmeth-odshavebeendesignedforinsitudeterminingthegasvolumefractionintwo-phase owwithoutdistortingthe ow,ortheneedofusingaseparationtechnique.TheradiationmethodsarebasedonX-orc-rayextinc-tion,andallowmeasurementofareavoidfractioninaselectedcrosssectionofapipe.Electricalsensorsmea-surethepermittivityorresistanceoftheliquidphase,whichsigni cantlydi ersfromthatofvoids.Admit-tancesensorsmeasuretheconductivitybetweentwoparallelwiresstretchedacrossthepipe.Admittanceprobecanbeappliedforslugorbubble owswhenthesizeofbubbleislargerthanthewirespace.Theadvan-tageofsuchdevicesistheirhighsensitivity.Thistypeofprobe,however,distortsthe owandcanchangethe owpattern.

Electricalcapacitancesensorsweredevelopedfornon-invasivemonitoringofphasepercentageingas–liquidtwo-phase owsinpipelines.Thecapacitance

Correspondingauthor.Tel.:+48-58-346-0881;fax:+48-58-341-6144.

E-mailaddress:jaworek@imp.gda.pl(A.Jaworek).0955-5986/$-seefrontmatter#2004ElsevierLtd.Allrightsreserved.doi:10.1016/j. owmeasinst.2004.04.002

Ã

sensorsmeasurethephasepercentagedetermininganelectricalcapacitancebetweenoneormorepairsofelectrodesmountedinsideoroutsideofthepipewalls.Thecapacitancecanvaryintherangeof0.1–10pF,butinordertoachievehighmeasurementsensitivityandgoodsignal-to-noiseratiothee ectofstraycapacitancemustbeminimised.Theoutputsignalisnotproportionaltothephasevolumepercentage,andalsodependsonthe owpattern,andthereforethecalibrationofthesensorisneeded.Thistypeofdevicewasusedforgas–liquidsystemsormeasurementsofconcentrationofsolidparticlesingas[1–10].Thecapacitance-sensortechniqueswerereviewedbyHuangetal.[5].

Thecapacitancesensorsaresuccessfullyusedinthecapacitancetomography,atechniquethatisabletodeterminethe owpatternofgas–solid,gas–liquidortwo-liquidmixtureinrealtime[11–16].Thesinglecapacitancesensorallowsonlyroughdeterminationofthephasespercentagewithoutanyinformationonthe owpattern.However,incontrasttothecapacitancetomographythesinglecapacitancesensorissimpleindesignandinoperation.Itdoesnotrequireacomplexsoftwaresolvinganinverseproblem,andismuchcheaper.

318A.Jaworeketal./FlowMeasurementandInstrumentation15(2004)317–324

Thispaperpresentsacapacitancesensorusedforvoidfractionmeasurementsinwater/steamtwo-phase ow.Thisdevicedi ersfromthosepresentedintheliteratureinthatitisconnectedinaresonantcircuitofanelectronicoscillatortunedtoradiofrequency.Var-iationsinthecapacitanceofthesensorcausedbyvoidpercentagechangewithintheelectrodesleadtofre-quencydeviationsoftheoscillator.Thesedeviationsareusedfordeterminationofthevoidfractioninthe ow.Asetofsuchsensorswasusedfordeterminingthevoidfractiondistributionalonganozzleinthelab-scalesteaminjectorattheInstituteofFluidFlowMachinery.

2.Sensortheoryandcharacteristics

Thecapacitancesensorpresentedinthispaperdeter-minesameanvalueofphasepercentageintwo-phase ow.Themeasurementsofcapacitanceofacapacitor lledwithaconductingliquid,likewater,aredi cultbecauseequivalentresistanceoftheliquid,whichisusuallylow,isconnectedinparallelwithcapacitivecomponentoftheadmittance,providedthewatercomponentisthecontinuousphaseinthemixture.Forlowfrequencies,thisresistanceislikea‘short-circuit’tothecapacitance.Tocut-o thee ectoftheresist-ance,http://www.77cn.com.cnmerciallyavailablemeasuringdevicesareusuallyusedforcapacitancedeterminationofcapacitancesensors.Stottetal.[2]testedexternalandinternalcapacitanceofasensorbutonlyforlowfrequencyof1.6kHz.AbouelwafaandKendall[1]usedaradio-frequencybridgeoperatingatthefre-quencyof1MHz,however,itwasstilltoolowtoover-cometheliquidconductancecomponentofthesensorcapacitance.Huangetal.[4]excitedthecapacitancesensorwithfrequencyofupto5MHz.Thefrequencyof80MHzforexcitationofacapacitancesensorusedinlaboratorytestsofvoidfractionmeasurementwasproposedbyJaworek[7].Themethodofoscillationfrequencydeviationwasusedbytheauthorfordeter-minationofthesensorcapacitance.

Thefrequencyof80MHzwasalsousedfordetermi-nationofcapacitancevariationinthispaper.Atthisfre-quencythereciprocalofthetimeconstantofelectricalrelaxationprocessesintheliquid(tapwater)de nedasqe(qistheliquidresistivity,eitsabsolutepermittivity)isafewtimeslowerthantheexcitationfrequency.Thisreducesthee ectofliquidconductanceonthemeasur-ingresults.Forexample,whentheresistivityoftap

waterisq¼25Xmatthetemperatureof20v

C,anditselectricalpermittivitye¼7Á10À10C=Vm[17]thenthereciprocalofthetimeconstantis1=s¼8:9MHz.

Thecapacitancesensorusedintheexperimentscon-sistedofapairofbrasselectrodesmadeintheformof

stripsofwidthof10mm,whichweremountedaroundtheoutsidewallofapipemadeofpolycarbonate.Thecrosssectionofthepipeandthesensor,andalsoanelectricalschemeofthemeasuringcircuitareshowninFig.1.

Themeasurementsofvoidfractionwithcapacitancesensorarequasi-local,i.e.,thesensordeterminesthepercentageofbothphasesnotstrictlyinaselectedcrosssectionofthepipebutinacertainvolume,basedontheelectrodesheight.Theexactboundaryofthisvolumecannotbepreciselydrawnduetofringee ects.Tominimisethenon-locale ects,theheightoftheelectrodesmeasuredalongthepipeshouldbeasshortaspossible,butthee ectofthefringe eldcannotbeeliminated.Shortelectrodeshave,however,smallcapacitanceandlowsensitivity,andinthiscaseacompromiseisneeded.Thesensorwasshieldedtominimisethedistortione ectsduetoouterobjectsandelectromagnetic elds.Theshielddimensionsshouldbeaslargeaspossibleinordertominimisestraycapaci-tance.

Inasimpletheoreticalmodel,thecomplexdistri-butionofliquidandvoidfractionsinapipecanberepresentedbyalumpedcapacitanceCeofunknownrelativepermittivityeeofthemediuminthepipe.Theequivalentcapacitanceinthissystemisthe

imaginary

Fig.1.Electricalschemeofthemeasuringcircuit.

A.Jaworeketal./FlowMeasurementandInstrumentation15(2004)317–324319

partofthee ectiveimpedanceoftheliquidwithinthepipeinserieswithtwovirtualcapacitances2Cwbetweenwaterandelectrodes,withthepipewallsasdielectric(Fig.1).TheparallelresistanceReqofwaterwithinthepipewasassumedtobenegligible.Thecon-ductanceofthepolycarbonatetubecanalsobeneglec-tedbecauseitisaperfectdielectric.Thee ectivecapacitanceofthesensoris:

C¼

CeCw

Cð1Þ

eþCw

ThecapacitanceCeisanidealisedcapacitance,whichvariesduetophasepercentagechanges.OtherstraycapacitancesincludingthattoagroundedshieldandgroundedelementsarerepresentedbyCc.Alsothecapacitanceduetofringee ects.i.e.,thatgeneratedby eldlinesnotpenetratingthemixturecanbeincludedtoCc.Thecapacitancevariationscausedbythechan-gesinphasepercentagearemeasuredbyfrequencymethod.TheelectrodesareconnectedasacapacitorinaLC-resonantcircuitoperatinginacircuitofaradio-frequency(RF)oscillator.TheRFoscillatorisverysensitivetothecapacitancevariationscausingfre-quencydeviations.Thesedeviationscanbeeasilydeterminedbycomparisonoftheactualfrequencyoftheoscillatorwithareferencefrequency.Theoscillatorandreferencegeneratorwereplacedonaprintedcir-cuitboardclosetotheelectrodestominimisestraycapacitanceandreduceexternaldistortions.Miniaturedimensionsofthisdevicewereachievedbyusingsur-facemounteddevices.Thedi erencefrequencywasnexttransmittedtoamicroprocessorsystemmeasuringlowfrequencyinupto4MHzrange.

TheelectricalcapacitanceCedependsonthee ectivepermittivity,ee,ofthemediumbetweentheelectrodes.Whentheproportionbetweentheliquidphaseofhighrelativepermittivityandgasphaseofdielectriccon-stantequalto1changes,thee ectivecapacitancevar-iesprovidinginformationonthegas/liquidcontent.Theproblemofe ectivepermittivityordielectricconstantofamediumcomposedoftwoimmiscibledielectricsofdi erentelectricpropertieswasstudiedbymanyauthors.Fourmodelscanbeapplicabletothesituationoftwo-phase ows(cf.Bruggeman[18]):1.Platevoidsplacedperpendicularlytotheelectrodesinacontinuousmedium,whichcanbereducedtotwovirtualcapacitancesconnectedinparallel,oneofpermittivityofwaterandtheotherofgaseousphase,inwhiche ectiverelativepermittivitywasgivenbyWienerformula[18]:ee¼dgegþdlel

ð2Þ

wheredg,anddlaregasandliquidfractions,respect-ively(dgþdl¼1),egandelaretherelativepermit-tivitiesofthegasandliquid,respectively.

2.Platevoidsplacedparalleltotheelectrodesinacon-tinuousmedium,whichcanbereducedtotwovir-tualcapacitancesconnectedinseries,alsoproposedbyWiener[18]:ee¼

1

dð3Þ

g=egþdl=el

3.Acontinuousmedium(water)withcylindricalvoidsplacedparalleltotheelectrodes,whichcouldbeamodelofannular ow[18]:

eðdÞðeq

gÀdlgÀelÞþðdgÀdlÞ2ðegÀelÞ2þ4egel

e¼

2

ð4Þ4.Acontinuousmediumwithsphericalvoids,whichcouldbeamodelofbubble ow[18]:

2ðddÞþq ð2ðd

egegþlelÞÀðdgelþdleggegþdlelÞÀðdgelþdlegÞÞ2þ8egel

e¼

ð5Þ

Forun-orderedplatevoids,Bruggemanproposedaformulaforrelativepermittivityasageometricmeanof(2)and(3):

s

e¼dgegþdleel

d=eeð6Þ

ggþdl=lAllmodelsofe ectiverelativepermittivityforagas–watermixtureareshowninFig.2aasafunctionofvoidfraction.Themodelbasedontwoparallel-capaci-tancesgivespermittivityproportionaltothecompo-nentsfraction.Othermodelsarenon-linear,andthepermittivityislowerthanthatfortheparallel-capaci-tancemodel.

Theequivalentcapacitanceofthemixturebetweentheelectrodescanbepresentedasacertainfunctionoftherelativepermittivityeeofthemedium:Ce¼Ce0fðeeÞ

ð7Þ

whereCe0istheequivalentcapacitancefora¼1,i.e.,forthepipewithoutwaterinside.

Theangularresonancefrequencyoftheoscillator,ingeneral,is:x2¼

1LC

ð8Þ

whereListheinductanceoftheresonantcircuit,Cisthetotalcapacitanceintheresonantcircuitcomprisingthecapacitanceofthesensor.ThecapacitanceCcanbewrittenasC¼C1

e

1þCþCð9Þ

e=Cw

c

whereCcisadditional(forexampleatrimmerusedfor

320A.Jaworeketal./FlowMeasurementandInstrumentation15(2004)317–324

Fig.2.(a)E ectiverelativepermittivityoftwo-phasemixturedeterminedfrommodels(2–6)).(b)Relativefrequencydeviationsofanoscillatorfordi erente ectivepermittivitymodels.

tuningtheresonantcircuit)andthestraycapacitanceofthewholecircuitinparallel(cf.Fig.1).

Takingintoaccount(7)and(9)in(8)theresonancefrequencyis:x2¼

1LCc

1

e0fðeeÞ1þ

1þðCe0=CwÞfðeeÞCc

deviations

ð10Þareð11Þ

oftheoscillatoris:

x2¼

1

LCc

1

1þ

e0ðaþelð1ÀaÞÞ

e0wlc

ð13Þ

Therelativeangularfrequencydeterminedfromtheequation:xgÀx1Àx=xg

¼

xgÀxl1Àxl=xg

inwhichxgistheangularfrequencyforthepipefreeofwater(aironly)i.e.,fora¼1,andxlforthepipetotally lledwithwater,whena¼0.

Allmodelsofe ectivepermittivity,givenbyEqs.(2–6),weretested,andrelativefrequencydeviationsforthesamevaluesofothercapacitancesarepresentedinFig.2b.Itwillbeshowninnextsectionthatonlypar-allel-capacitancemodel tstheexperimentaldata,whileotherpermittivityapproximationsgivequiteunreasonableresults.Forlowvaluesofvoidfraction,uptoabouta¼20%,parallel-capacitances,spherical-andcylindrical-voidmodelsgivesimilarresults.Fortheserial-capacitancesmodel,thepermittivityandrelativefrequencydeviationschangetoofastinthelowvoidfractionrange.

Theparallel-capacitancesmodelwill,therefore,beusedinthefollowing,andthee ectiverelativepermit-tivityoftheliquid–gasmixturewillbeapproximatedbyWienerformula(2).Foreg¼1,theequivalentcapacitanceCeis:Ce¼Ce0ðaþelð1ÀaÞÞ

ð12Þ

whereaisthevoidfraction.

Thismodelisequivalenttotwolumpedcapacitancesinparallel,onecontaininggasasthedielectric,andthesecond lledwithwater.Theresonancefrequency(10)

Characteristicsofthecapacitancesensorarenon-lin-ear,andcalibrationisrequiredfordeterminationoftherelationbetweenthefrequencydeviationsandvoidfraction.Thecalibrationcurvewasobtainedforthebubble ow,andforannular ow.Thebubble- owvoidfractionwasobtainedonlyupto15%intheseexperimentsbyinjectinggasbubblesonalowerpartofthepipeline.Theaveragegascontentinsidethepipeforbubbles owingupwardswasdeterminedfromanincreaseinthewaterlevelinameasuringcylinderonwhichtheelectrodesweremounted.Cylindricallyshapedvoidssimulatingannular owwereobtainedbyplacingend-closedglasspipesco-axiallyintothepipe-line lledwithwater.FrequencydeviationsforthesemeasurementsareplottedinFig.3.Thereisalittledif-ferencebetweenbubble owandsimulatedannular ow.

Forcomputationalpurposes,thecalibrationdatacanbeapproximatedbythefollowingpolynomialof6thorder:

df¼À12:927a6þ26:636a5À19:361a4

þ5:0284a3À0:1129a2À0:2641aþ1

ð14Þ

wheredfistherelativefrequencydeviationde nedasdf¼

fgÀffgÀfl

ð15Þ

ThislineisshowninFig.3ascontinuousline.

Thetheoreticalcurvewasalsoplottedinthis gure.Allcapacitancesin(13)cannotbeexactlydetermined,andthefrequencydeviations(11)canbepredictedthe-oreticallyonlywithacertainerror.Inthesystem

used,

A.Jaworeketal./FlowMeasurementandInstrumentation15(2004)317–324321

Fig.3.Relativefrequencydeviationvs.voidfraction(continuouslineinthe6thorderpolynomialapproximation).

thepipewallcapacitanceCwwasestimatedtoCw¼5pF,theequivalentcapacitancefora¼1toCe0¼0:5pF,andtheadditionalcapacitanceCc¼10pF.ThecapacitanceCwwasmeasuredbetweenthesensorelectrodesafterplacingametalsheetontotheinnerpipewall.ThecapacitanceCewaseliminatedbythisway.ThecapacitanceCcwasestimatedbymeasuringfrequencydeviationsafterreplacingthesensorbylumpedcapacitancesofknownvalues.Therelativefre-quencydeviationsweredeterminedfromEq.(11)andobtainedcurveisshowninFig.3asdashedline.Thedi erencebetweentheexperimentalandtheoreticalresultsisnotverylarge,andcanbeexplainedbyane ectofstraycapacitanceandwaterconductanceonfrequencydeviations.

3.Experiments

Theareavoidfractioninacertaincrosssectionofapipeisde nedastheratioofthesurfaceoccupiedbyvoidstothesurfaceofthecrosssectionofthepipe:að2Þ¼

Svoid

ð16Þ

S

Thevolumevoidfractioninacertainsegmentofapipeistheratioofthevolumeoccupiedbyvoidstothetotalvolumeofthesegment:að3Þ¼

Vvoid

ð17Þ

V

Thecapacitancesensorunderconsiderationmea-suresonlythevolumevoidfractionbecauseofthefringee ectsinacapacitorwithwide-spacedelectro-des.Thevolumeofthesegmentcan,however,notbedeterminedunambiguously.Thecharacteristicsofthe

sensorweredeterminedbymeasurementsoffrequencyresponseoftheoscillatorfordi erentvoidpercentage.Thecapacitancesensorsofthistypewereusedformeasurementofvoidfractioninselectedcrosssectionsofasupercriticalsteaminjector.TheexperimentalstandisschematicallyshowninFig.4.Steaminjectorisadeviceinwhichkineticenergyofsteamisusedforwatersuction.Thesteaminjectorconsistsofasteamnozzle,mixingchamberanddi user[19].Twopipeinstallationscanbedistinguishedinthescheme:thewaterandthesteam.Superheatedsteam owingthroughtheLaval-typenozzleexpandstosupersonicvelocitythatcauseslowstaticpressureattheinlettothemixingchamber.Thepressurebelowatmosphericdrawsthewaterthroughtheannularslotsurroundingthesteamnozzleintothemixingchamber,wherethesteamtransfersitsmomentumandheattothewater.Duetotheshockwave,whichdevelopsinthedi userdownstreamofthethroat,thetwo-phase owofwaterandsteamiscompressed,andthesteamcondenses,sothatonlywaterleavestheinjector.Thesteaminjectorwasequippedwiththermocouples,pressuretransdu-cers,andvoid-fractioncapacitancesensorsfordetermi-nationofthermodynamicparametersofthephaseexchangeprocesses(cf.Fig.4).Thepipeinthisstandwasmadeofpolycarbonatetoeliminatescreeninge ectthatcouldbecausedbyametalpipe.Thepipewasplacedvertically,withthesteaminjectorfacingdownwardsfromtheupperpartofthemixingcham-ber.Aphotographofoneofthesensorswithelectro-nicsisshowninFig.5.

Thevoidfractioninthissystemwasdeterminedinfourcrosssectionsofthemixingchamberandfourindi user.Schematicdiagramofmicroprocessorcon-trolledmeasuringsystemforthecapacitancesensorispresentedinFig.6.SignalofhighfrequencyUisin(xt

)

322A.Jaworeketal./FlowMeasurementandInstrumentation15(2004)317–324

Fig.4.Schemeoftheexperimentalstandforinvestigationofsupercriticalsteaminjector.

Fig.5.Aphotographofcapacitancesensorwithelectronics.

fromeachsensorwasmixedwiththereferencefre-quencysignalU0sin(x0t):umðtÞ¼

UiU0

ðcosðx0ÀxÞtÀcosðx0þxÞtÞ2

ð18Þ

Fig.6.Schematicdiagramofthemeasuringsystemforthecapaci-tancesensors.

Theproductwas lteredbyalow-passelectronic ltertoobtainlowfrequencysignal:uoutðtÞ¼Usinðx0ÀxÞt

ð19Þ

Thefrequencydeviationsweremeasuredbyamicro-processorsystemandconvertedtodigitalsignal,whichisnextstoredandprocessedbyacomputer.Thevoidfractionineachcrosssectionofthepipewasdeterm-inedfromthefrequencydeviationsbythecomputer.

Originallydevelopedprogramwasusedformeasure-mentcontrolanddataprocessing.

Anexampleofvoidfractionmeasurementsduringa3-hexperimentwitheight-channelsystemispresentedinFig.7.Thevariationinsignallevelsisduetochan-gesinexperimentalconditionssuchas owrates,tem-peraturesandpressures.Apeakoccurringinthechannel-2wastheresultofunexpected oodingwithwateroneoftheelectrodepairoutsidethepipe.

Voidfractiondistributionalongthemixingchamber–

system,startingfromthemixingchamberinlet,

A.Jaworeketal./FlowMeasurementandInstrumentation15(2004)317–324323

Fig.7.Variationsofvoidfractionina3-hexperimentatthesteaminjectorstand.Changesinvoidfractionrefertodi erentexperimentalcon-ditionssuchas owrate,temperature,and

pressure.

wasdeterminedfromthedatarecordedinFig.7,andareshowninFig.8.TheexperimentalconditionsforthisplotarelistedinTable1.Theactualfrequencydeviationswerenormalisedtothemaximumfrequency

di erence,xgÀxl(cf.Eq.(11)).Thereferencefre-quencyxl,withwholesteaminjector oodedwithwater,wasdeterminedjustafterallmeasurementsintheserieswerecompleted,andthewatertemperaturewasasthatduringthemeasurements.Itwasnecessarybecausewaterchangesitsproperties(conductivity,permittivity)withtemperature[17].The ood-frequencymeasuredintheseconditionsismoreappropriatethanthatdeterm-inedbeforetheinjectorsystemwasrunning,andwatertemperaturewaslower.Themeasurementaccuracybythismethodisestimatedtobeoftheorderofmagnitudeof10%.

4.Conclusions

Amethodofvoidfractionmeasurementintwo-phase owbasedonmeasurementoffrequencydevia-tionsgeneratedbyahighfrequencyoscillatorwithacapacitancesensorinitscircuitwaspresentedinthispaper.ThecapacitancevariationsduetopercentagechangesoftwophasesweremeasuredinaLC-resonantcircuit,whichisverysensitivetosmallcapacitancechanges.Characteristicsofthecapacitancesensorarenon-linear,andcalibrationisneededfordeterminationoftherelationbetweenthefrequencydeviationsandvoidfraction.Thefrequencydeviationsweredeterm-inedbycontinuouscomparisonofgeneratedfrequencywithareferencefrequencyofaquartzgenerator.Thedi erencefrequencywasmeasuredbyamicroprocessorsystemandconvertedtodigitalsignal,whichwasnextstoredbyacomputer.Thesystemwasdesignedfordeterminationofsteamcontentinawater-vapourtwo-phase owalongasupercriticalsteaminjector.Thisisasimple,lowcostandnon-invasivemethod,whichallowsdeterminingvoidfractionandavoiding owdis-tortionbecausenomechanicalelementisputinto

the

Fig.8.Examplesofvoidfractiondistributionalongthesteaminjector.

Table1

MeasuringconditionstoFig.8Lineno.

Steam

temperaturev

(C)142142142

Steam owrate(kg/h)132134132

Water owrate(kg/h)340028403400

Throatpressure(kPa)13.964.1130.3

123

324A.Jaworeketal./FlowMeasurementandInstrumentation15(2004)317–324

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