tFTLREPORTRR25NASAC15926THEINTdACTIOIOFBABIOi8230132FItEUUEICEL - PDF document

1 t,FTLREPORTRR2--5--(NASA-C[_-1592_6)THEI
t,FTLREPORTRR2--5--(NASA-C[_-1592_6)THEINT_.dACTIOI_OFBABIO_i82-30_132FI:tEU_UE_IC¥ELECTROM_GNKTICF!_FI_,pSW!T._i•i_O_cATmOSPhERICWATCHDBOSLE_5ANDAP_LICA_ICBSTOAIRCRAFtiCEP_5¥E_IO_Thesislas.(MassachusettsInst.cfTecn.)193pG3/322862_THEINTERACTIONOFRADIOFREQUENCYELECTROMAGNETICFIELDSWITHATMOSPHERICWATERDROPLETSAN_APPLICATIONSTOAIRCRAFTICE_PREVENTION_-,__1;bRobertJohRHa;isman,Jr,_•_June19821982022556 DFTLREPORTR82-5THEINTERACTIONOFRADIOFREQUENCYELECTROHAGNETICFIELDSWITHATHOSPHERICWATERDROPLETSANDAPPLICATIONSTOAIRCRAFTICEPREVENTIONbyROBERTJOHNHANSHAN,JR.June19821982022556-002 ITHEINTERACTIONOFRADIOFREQUENCYELECTROMAGNETICFIELDS•WITHATMOSPHERICWATERDROPLETSANDAPPLICATIONSTO

2 AIRCRAFTICEPREVENTIONtbylWm|RobertJohnHa
AIRCRAFTICEPREVENTIONtbylWm|RobertJohnHansman,Jr.m¢ABSTRACT_Inthlsworkthephysicsofadvancedmicrowaveanti-icingsystems,'_whichpre-heatimpingingsupercooledwaterdropletspriortoImpact,Isstudiedbymeansofacomputersimulationandisfoundtobefeasible.-Inordertocreateaphysicallyrealisticsimulation,theoreticaland_|experimentalworkwasnecessaryandtheresultsarepresentedinthis"i-_thesIs.Thebehavioroftheabsorptioncross-sectionformeltingiceparticles•ismeasuredbyaresonantcavitytechniqueandIsfoundtoagreewiththeoretlcaIpredictions.Valuesofthedielectricparametersof|supercooledwateraremeasuredbyasimilartechniqueat;k-2.82cmdownF•to-17°C.ThehydrodynamicbehaviorofacceleratedwaterdropletsIsstudied

3 photographlcallyInawindtunnel.Dropletsar
photographlcallyInawindtunnel.Dropletsarefoundtoinitially_-"deformasoblatespheroidsandtoeventuallybecomeunstableandbreakup"inBesselfunctionmodesforlargevaluesofaccelerationordropletsize.ThisconfirmsthetheoryastothemaximumstableJropietsizeintheatmosphere.AcomputercodewhichpredictsdroplettrajectoriesinanarbitraryflowfieldIswrittenandconfirmedexperimentally.Finally,theaboveresultsareconsolidatedintoasimulationtostudytheheatingbyelectromagneticfleldsofdropletsimpingingontoanobjectsuchasanairfoil.ResultsIndicatethatthereissufficienttimetoheatdropletsprlortoImpactfortypicalparametervaluesanddesigncurvesforsuchasystemarepresentedInthestudy.Ib=1982022556-003 +'_L•ACKNOWLEDGEHE

4 NTS+'Iwishtoexpressmyappreciationtoallth
NTS+'Iwishtoexpressmyappreciationtoallthosewhohelpedmake,thisworkpossible,ParticularlyProfessorWalterHollister_who+provideddirectionandsupport,andtotherestofmycommittee,+,ProfessorRobertKyhl,ProfessorRichardPassarelli,andProfessor,,BernardBurke,fortheirhelpfuladvice,IwouldalsoliketothankProfessorRobertSimpson,whoprovi6edmewithahomeintheFlightTransportationLaboratoryandCharlesMillerforhisphotographicadvice,\IwishtothankAIShawpPaulBauerandDonWelnerfortheirtechnicalsupport,JohnPararasforhishelpwith+_mputergraphicsand;especiallyBobHcKillipforhis"quickProgrammingCourse".IthankiJ:alsoSteveKnowltonforhisdiscussionsandcompanionshipandthemembersoftheFlightTransportationLaborato

5 ry.:Theproductionofthisthesiswasmadeposs
ry.:TheproductionofthisthesiswasmadepossiblebythetypingandeditorialskillsofAbbyCrear_thedraftingskillsofLauraWernickandthesoftwareofBobHcKillip.MydeepestthanksareduetomyparentsttoLauraWernick_andtomymanyfriendswhomotivatedandsupportedmethroughthehighsandlowsoftheresearchdescribedinthisthesis.ThisworkwassupportedinpartbyNASAgrantsNAG-I-IO0andNGL-22-oog-640andsponsoredbytheLangleyResearchCenterandagiftInmemoryofStuartDreger.1982022556-004 TABLEOFCONTENTSJABSTRACT2!ACKNOWLEDGEhENTS3+•+iTABLEOFCONTENTS4CHAPTERI.INTRODUCTIONGI.ITheIcingProblem7:!.2ConceptsforMicrowaveIceProtectionil1.3ThesisStructure16:2.THEORYOFABSORPTIONANDSCATTERINGBYHYDROMETERS182.1ScatteringTerminology

6 18i+2.2DielectricPropertiesofWaterandIce
18i+2.2DielectricPropertiesofWaterandIce212.3AbsorptionandScatteringbyblelectricSpheres252.3.1MieThe¢,ry252.3.2Rayleigh292.4AbsorptionandScatteringbyEllipsoids302.5AbsorptionandScatteringbyaCollectionof34Scatterers•3.ABSORPTIONANDSCATTERINGTHROUGHTHEICE-WATERPHASE39TRANSITION3.1TheoryofScatteringbyMeltingandFreezing39:!Hydrometers3.1.1WaterCoatedIceSpheresandEllipsoidsitO3.1.2HodeloftheHeltlngofScatterersInthe43Atmosphere3.2ExperimentalMethodsforMeasuringAbsorptionDuring47PhaseTransition3.2.1PerturbationTechniquesinaResonantCavity483.2.2ExperimentalSet-up523.3ExperimentalResults573.3.1AbsorptionCrossSectionsforWarmDrops603.3.2DielectricParametersforSupercooledWater603.

7 3.3AbsorptionCrossSectionsDuringPhase65T
3.3AbsorptionCrossSectionsDuringPhase65Transition,.[1982022556-005 •+,\i.,54,HYDRODYNAHICSOFACCELERATEDDROPS76,4.1HydrodynamicTheoryofWaterDrops764,1,1DropDeformation77+4.1.2_ropOscillations784,1,3InstabilityandDropBreak-up824,2ExperimentalTechniquesforWindTunnelObservations90ofDropShapeandVelocity4.3ExperimentalResults964.3.1DropDeformation974,3,2InstabilityandDropBreak-up975,WATERDROPLETTRAJECTORIES109+5,1ComputerSimulationsofDropletTraJectories109i5,1,1DropletEquationofHotlonllO5,1,2IterationAlgorithm114+5,1,3DragCoefficients1165,2WindTunnelValidationofComputerTraJectories1225,2,1DropletsInJectedIntoaUniformFlow122._+5,2,2DropletTraJectoriesNearaCylinder1255,3Simula

8 tionResultsNeartheLeadingEdgeofan128Airf
tionResultsNeartheLeadingEdgeofan128Airfoil5.3.1TwoDimensionalImpingementTraJectories1335,3,2DropletKinematicsontheStagnation143Streamline5.3.3AdditionalSimulations148il6,CONPUTERSIHULATIONSOFDROPLETHEATING1526.1DescriptionoftheSimulation1526,2DropletEnergyBalance1586,3SimulationResults165'i7°CONCLUSIONS177LISTOFSYHBOLS181REFERENCES188dm'r01982022556-006 r•/CHAPTER1INTRODUCTIONAircrafticln9haslongbeenrecognizedasoneofthemost!_seriousmeteorologicalhazardstoflight.Iceformationcanresultinlossofaerodynamicefficiency,controlandvlslbIllty,aswellasanIncreaseinaircraftweightandthefailureofvitalcommunicationorInstrumentationsources.WhiloavarietyoftechniqueshavebeenappliedtotheI

9 cingproblem,therearestillareaswhereprese
cingproblem,therearestillareaswherepresentiLsystemfallshort.1'2TheworkdescribedinthisthesisIscenteredaroundunderstandingtheI_yslcsofadvancedconceptsInaircraftIceprotectionwhichemploymicrowaveelectromagneticradiation.Whileunderstandingthephysicsofadvancedconceptswasthefundamentalthreadwhichunltedthework,itwasoftennecessary,Inthecourseofthework,tocressIntootherfieldsInordertoobtainasatisfactoryunderstandingofthephysicsInvolved.SomaofthefieldsInwhichexperimentalortheoreclcalworkwasnecessaryareAtmosphericPhysics,ExperimentalandComiau-rationalFluidDynamicsandtheInteractionofElectromagnetlc!RadiationwithHatter.Asaresultoftheresearch,contributionshavebeenmadeIneachoftheabovef

10 ieldsenroutetotheoriginalobjectiveofunde
ieldsenroutetotheoriginalobjectiveofunderstandingthephysicsunderlyingmicrowaveice_preventionsystems.InSection1.ItheIcingproblemwillbedefinedandcurrent"techniquesusedtodealwlthit_tlllbediscussed.InSection1.2!e1982022556-007 7I°:lsomeoftheconceptsformicrowaveiceprotectionwillbedescribed.'i_!InSection!.3anoutllneofthefollowingchaptersandtheirrelation-shiptothethesiswillbepresented.I.!TheIcingProblemIceformsonaircraftstructureswhenflightisconductedthrough•areasofsupercooledcloudorprecipitationdroplets.Supercooled;waterdroplets,whichoccurcommonlyintheatmosphere,existina;metastablestate.Ifsomestructure,suchasanaircraft,comesincontactwithasupercooleddrop,thenitwillbegintohete

11 rogeneously•nucleateandformice.Therateof
rogeneously•nucleateandformice.TherateofnucleationandsubsequentfreezingI•dependsonthestructureandonthetemperatureofthedrop.This;!isthebasicmechanismforiceformationonaircraft.,_Forslightlysupercooleddrops(-5°CtoO°C),thedropsfreezeIslc_iyafterimpactandsmooth"clear"iceisformed.Forcolderitemperatures(-20°Cto-5°C),dropletsfreezequicklyandanopaque,=Irregular"rime"iceisformed.Below-20°C,supercooledwater:-'becomeslesscommonintheatmosphereashomogeneousnucleationbeginstooccurandattemperaturesbelow-40°CessentiallyallwaterisInitheicephase.3FrozenhydrometersdonotcontributetotheIcingJi1problem,asthoseparticleswhichdostrikethestructurebounceoff.Thevaluesofdropletandliquidwatercontent

12 applicabletotU.S,continentalcumuliforman
applicabletotU.S,continentalcumuliformandstratifomcloudsarepresentedinthedesigncriterionforiceprotectioncertificationinPart25ofthe"•U.S.FederalAviationRegulations(FAR's).4,5Howewr,theFAR'si_neglectIcingfromsupercooledprecipitationdroplets,suchas1982022556-008 8ifreezingrain.TableIsummarizestherangeofdropletparameters!r7_,:applicableforcloudandprecipitationicing.•Whileadescriptionofthemeteorologicalconditionswhichleadto,,!_:Icingisfairlyeasytoprovide,accurateforecastingofthese:conditionsismuchmoredifficult.Thisisduetodifficultiesin:_,predictingwhetheracloudwillbeinthesupercooled-waterorice:_-'phase,whichcanchangequicklywithtime.Forexample,Ifsomeicephaseparticlesareintro

13 ducedintoasupercooledcloud,through:heter
ducedintoasupercooledcloud,through:heterogeneousnucleationorsomeothermeans,thenthecloudwill:veryrapidlytransitiontotheicephase,duetothefactthatthe6saturationvaporpressureovericelslowerthanoverwater.,:iInaddition,Icingzoneshavebeenobservedtobefairlylocalized7evenwhenaccuratelypredicted.'TheuncertaintyInpredictingicingconditionscausesforecasterstobeconservativeandtoforecasticingconditionsanytimethepotentialexists.Theconservativenatureoficingforecastinghastwodetrimentaleffects.ThemostImportantIsthe"crywolP'syndrome,wherepilotsbecomeaccustomedtoflyinginforecastIcing:_TconditionswithnodifficultyandignoreIcingfo_castsInmoresevereconditions.ThisIsborneoutbythefactthatin92_oft

14 heIcing•accidentsbetween1973and1977,Icin
heIcing•accidentsbetween1973and1977,Icingconditionswerecorrectly,forecast.2iTheseconddetrlmntaleffsctofconservativeForecastingIs1economic.Hostgeneralaviationaircraftarenotcortlfledforflight-Into"known"icingconditions.Whilethedefinitionof"known"IclngIssomewhatanbiguous,manyoperatorschoosenottoflyIn1982022556-009 Table1.ParametersoftheIcingProblemiiCloudDropletsPrecipitationDropletsTemperatureO°C6°C-5°CLiquidWaterContentOgm/m3Ogm/m3MeanDropDiameterIO40micronsI+5mm1982022556-010 &,v,'-.i,-II0forecastIcingconditions.CancellationsduetoInvalidforecastsareadrainontheaircraftoperatorsandtheeconomyIngeneral.,_Thereasonfortheconservativeapproachtoaircrafticingaretheinherentdan

15 gers.Someoftheseare:iLossofaerodynamicef
gers.Someoftheseare:iLossofaerodynamicefficiencyIncreaseddragDecreasedIIftIncreasedstallspeedIncreasedwelghtLossofcontrolmow_entEnginefailureAeroelasticflutterresultingfromchangeInstructuralmassdistrlbutionLossofvisibilitythroughwindshieldLossofnavigationandcommunicationantennaLossofcockpitinstrumentationsources(pltot,static,etc.)LossofonboardradarTheaboveIcing-relatedphenomenacanoccursinglyormultiplywithvaryingdegreesofseverity,dependingontheicingconditions,theaircraftandtheIceprotectionequipmentonboard.TheavailableIceprotectiondevicescomeIntwobasicforms,anti-Ice(noiceIsallowedtoform),andde-Ice(someIceformsx,andissubsequentlyremoved).Thecurrenttechniquesallhaveadvanta

16 gesanddrawbacks.Electrothermalandhot-air
gesanddrawbacks.Electrothermalandhot-airantl-lcedevicesareveryeffective,butrequirelargeamountsofenergyinthattheyevaporateallimpingingwateratacostof600cal/gm(2.5xI0lOarg/gm).Freezing-pointdepressantssuchasglycolorchemical"'pastesareefficientbutaresubjecttoerosionontheleadingedge,andco.bustionproblems.Pne_,:tlcde-icingbootsareefficient,!buttendtobeunreliableandsubje_ctomisuse._t"1982022556-011 :Fromtheforegoing,thereIsclearlyaneedforadvancement,both":,_intheabilitytoaccuratelyfor_.casticingconditionsandInIce•protectiononceicingconditionsareencountered,inaddi¢ion,a!,!betterunderstandingoftheIcingproblemonthepartofpilots,as'_."wellasresearchers,willhelpincreasethesafetyand

17 productivityofflight.i1.2ConceptsforMicr
productivityofflight.i1.2ConceptsforMicrowaveIceProtection:Theuseoft,icrowaveelectromagneticenergyforiceprotectionhasbeenproposedbyavarietyofresearchersforbothIcedetectionandicepreventionroles.Remotedetectionofloca|izedIcingzones•wasproposedbyAtlasin1954.8Usingmeteorologicalradar,zones.nfmoderatetohighliquidwatercontentabovethefreezinglevel-_,.canbeobservedinrealtime.WiththeadvancesInground-basedandairborneradar,thisapprcachholdsgreatpromise.In1976,',Hagenheimproposedusingmicrowavestodetectthelocalaccumulationof|ceonairfoilsandhelicopterrotorsbymeasuringthechangein!impedance,asIceaccumulatesonthedielectriccoatingofasurfacetwaveguldelocatedontheleadingedgeoftheairfoll.9

18 ThetechniqueIsfairlysuccessfulbutsuffers
ThetechniqueIsfairlysuccessfulbutsuffersfromanomalousmeasurementswhenIIqulewaterismixedwiththeice.MagenheimalsoproposedusingmicrowaveenergyInade-lcingsystem.Inthisscheme,microwaveheatingattheicedielectricInterfaceofasurfacewavegulde,locatedontheleadlngedgeofthealrfoiI,isusedtobreaktheadhesionbondofice.Thlstechnique_I0wasdemonstratedin1976.However,ithasnorealadvantageoverelectrothermalde-icingtechniques,whichoperateontheIdentical1982022556-012 ...................iii"_12Lprinclp_e,andhassomedisadvantagesintermsofcomplexityandefficlency.:_In1980,HansmanandHollisterproposedu_ingmicrowaveheating"_11,12":_topreventtheformation(anti-ice)onaircraftstructuresTheconceptistoprehe

19 atthesupercooled_aterdropletstoabovefree
atthesupercooled_aterdropletstoabovefreezingpriortoImpactbyamicrowavefieldaheadoftheairfoila_dr,therebypreventIcefromforming.Thepotentialadvantagesseenforamicrowaveanti-iclngsystemarelowpowerconsumption,lowmmintenanceand_erodynamiccleanliness,asopposedtootheranti-icetechniques.±Lowpowerconsumptionisanticipatedduetothesavingofthelatentheat_efusion(80cal/gm,3.35x109erg/gm)bycircumventingthewater-to-Ica-to-waterphasetransitions,andtoth•abiIItyof,_selectivelyheatingsupercooledwaterdroplets.Theselectiveheatingofwaterdropletsisaresultofthestrongabsorptioncharacter'stlcofwaterinthemicrowaveregime,whereassnow,iceandmetalsurfacesarepoorabsorbers.Thisimpliesthatthewingneednotnec

20 essarilybeashotaswithothertechniques,whi
essarilybeashotaswithothertechniques,whichminimizesconvectiveandevaporativelosses.TheadvantageofkeepingtheairfoilascoolaspossiblecanbeclearlyseenInFigure1.1,wheretheheatlossfromawetsurfaceexposedtoatangentialflowisplottedasafunctionofsurface13temperature.Theambienttemperatureis-20°Candthetangentialve|ocityis60m/sac.Theheatfluxincreasesdramaticallyabove:,-xO°Cduetoconductiveandevaporati_losses•Inordertoruntheairfoilascoolaspossibleandstlilpreventrunbackfreezingproblems,theoptimallyefficientanti-icingsystemismo,.tlikely1982022556-013 ,o,I!13•/20"sFigure1.1Heatlossfromawettedsurface.//(calculatedfromNACATH279913)/Ambientten.20°C/J.:_'Surfacevelocity60m/s/;I_'15"E/3/;I/-"L

21 _I0/-E•8/:0I"/mi"5•"rI/,wlW,..._2-_o0I02
_I0/-E•8/:0I"/mi"5•"rI/,wlW,..._2-_o0I02030;Surfacetempeb-ature(°C)1982022556-014 14amicrowavehybrid.Inahybridsystemamicrowaveleadingedcle•systemwouldbecombinedwitheitherafreezing-pointdepressantoranelectrothermalsystemontheaftalrfollsections.Advanced14freezing-polntdepressantpasteshavebeensuccessfulatsuppressingrunbackrefreezlng,andcanoperateatan/)ienttemperatures,butaresubjecttorapiderosionontheairfoilleadingedge.Electrothermalsystemswhichoperateslightlyabovefreezingarealsoefficient,buthaveaproblemInitiallyheatingthedroplets.Thesetechniquesare,therefore,well-suitedtobeingcombinedwithmicrowavepre-heatinginordertoapproachoptimalefficiencyincaseswhereantl-iclngisrequire

22 d.Powerrequirementsfortypicalgeneralavia
d.Powerrequirementsfortypicalgeneralaviationparametersareestimatedforthemicrowavesystemtobeontheorderof100Wforpropelleranti-icingandIk;/forwinganti-Icing.AnadditionaladvantageofthemicrowavesystemIsthat,neglectingcircuitlosses,powerisonlyconsumedwhenliquidwaterispresent,andtherebyhasthecapabilitytoserveasItsowndetector.Anexampleofapossiblefirst-ordermicrowaveanti-iclngschemeemployingsurfacewaveguidesisshownInFigures1.2aand1.2b.InFigure1.2a,across-sectlo_oftheairfoilshowingthedle|ectricinsetforthesurfacewaveIspresented.Theelectromagneticwavepropagatesalongtheleadingedgeoftheairfollandisboundtothesurfacewavegulde.Theelectromagneticfieldstrengthcharacterlstl-:I•I(:allydeca

23 ysexponentiallyawayfromthewaveguide.15Th
ysexponentiallyawayfromthewaveguide.15Thewavesare!',launchedatthewingrootandpropagatetothetip,wheretheenergy!notabsorbedbythewaterdropletsiscollectedandrecycledbackI1982022556-015 t_t;ISORiGiNALPAGEISOFPOORQUALITY_IElectro 16Insidethewingtotherootina"racetrack"pattern.Itshouldbe•notedthatthelaunching-retrievingprocessisnotperfectlyJefficient.ThepresenceofliquidwatercanbemeasuredbyapowerIlossasobservedthroughadirectionalcoupleranddetectoronthereturEpathofthecircuit.Themicrowaveanti-icesystemhassomedistinctanduniqueadvantagesoverotheranti-icingsystems.Thereare,however,severalquestionstobeanswered.TheprimaryquestioniswhetherthereissufficienttimetoheatImpingingdropletsatve

24 locitiesofaeronauticalrelevance(80to200w
locitiesofaeronauticalrelevance(80to200w4)h).Whilethereareotherquestionssuchasrunback-refreeze,electromagneticfieldoptimizationiandcircuitefficiency,thesebecomemootifthereIsnotsufficient!Itlmetoheatthedroplets.Forthisreason,themajorthrustofthisthesiswillbetoendeavortodetermine,inasystematicmanner,whetherthereissufficienttimetoheatatmosphericwaterdropletstoabovefreezingpriortoimpact.I.3ThesisStructureTheprimarygoalofthisthesis,asdiscussedInSection1.2,istounderstandthephysicsofasupercooledwaterdropletbeingheatedbymicro_aveelectromagneticenergy,asItapproachesanairfoil.ThlsproblemisstudiedbymeansofacomputersimulationInChapter6,whichwasdesignedtobeasphysicallycompleteasposs

25 ible.Inordertoprovidethephysicalbackgrou
ible.Inordertoprovidethephysicalbackgroundforthesimulation,somepreliminaryworkofamorebasicnaturewasnecessary.InChapter2,thetheoryofabsorptionbyhydrometersIs1982022556-017 17reviewed.InChapter3,theabsorptionandscatteringofmixed-phaseicewaterparticlesisstudied,boththeoreticallyandexperimentally.•Inaddition,valuesofthedielectricparametersforsupercooled_.aterdropletsareexperimentallymeasured.InChapter4,thedeformationandstabilityofdropletssubjecttoanexternalacceleratlonarestudied,boththeoreticallyandinthewlndtunnel.ThisworkisIntendedtobecombinedwiththatofChapter2,wheredeviationsfromspherlcltyarefoundtohaveapronouncedeffectontheabsorptionpropertiesofdroplets,rInChapter5,aco

26 mputersimulationofwaterdropletsispresent
mputersimulationofwaterdropletsispresentedandexperimentallyverified.Finally,inChapter6the"."resultsofthepreviouschaptersarecombinedintoacomputercode,°whichfollowsthetrajectoriesandheatingofdropletsastheyapproachanairfoil.InChapter7,theresultsandcontributionsofthe.workarebrieflyreviewedbywayofaconclusion."I*iiI1982022556-018 18CHAPTER2THEORYOFABSORPTIONANDSCATTERINGBYHYDROMETERSAbriefsummaryofbasicscatteringtheoryapplicabletoatmospherichydrometersispresentedinthischapter.InSection2.1somescatteringterminolo_/isdefined.TheknowndielectricpropertiesofliquidwaterandicearepresentedinSection2.2.InSection2.3andSection2.4theapproachesofRie,RaylelghandGansforscatteringfromspheres

27 andellipsoidsarebrieflydiscussed.Finally
andellipsoidsarebrieflydiscussed.Finally,inSection2.5theproblemofacollectionofscatterersisconsidered.2.1.Sc.a.,_tterIn9Temi.nologyIntheclassicalformulationofthescatteringproblem,aplanewaveisIncidentonthescatteringobject.ShowninFigure2.1arethecasesofInteresthere.Thew_veiselectromagneticandpropagatesinthedirectionwithwavevectork0--_-no.ThewaveIspolarizedwiththeelectricfieldE!inthe_0direction.ThescatteringobjectisofarbitraryshapeandIsinbeddedina_backgroundofunifomdielectricmaterialcharacterizedbythecomplexdielectricconstantto.Formostatmosphericapplications,¢0Istakentobeunity,thevalueforfreespace.Thescatteringobjecthasacomplexdielectricconstant¢whichcanvaryspatially.Thebac

28 kgroundandthescatteringobjectareassumedt
kgroundandthescatteringobjectareassumedtobedielectricinnature.Therefore,themagneticpermeabilltlesofthebackgroundandtheobjectareunity.[1982022556-019 tt-IrolOFpOORQUALI'6Y•,?A-t_n0El_i.)_'i__o_oFtgure2.1Thescatteringproblemshowingthetnctdentandscatteredpolarizationandpropagationvectors.!m1982022556-020 _,_.m_=;:,r_i"_",_."I__'_20OFPOC_Qj.L',LIT¥TheenergyfluxoftheIncidentwaveischaracterizedbyitsPoyntingvectoreuC"slx;i(2.i),Nhichisalignedinthe_0direction.WhentheincidentwaveImpingesontheobject,energycanbelosteither*.oscatteringortoabsorptionbytheobject.Theselossesarecharacterizedbyvariouscross-sections.Cross-sectionshaveunitsofareaand,whenmultipliedbytheIncidentPoyntlngflu

29 x,yieldthepowerflowIntoabsorptionorscatt
x,yieldthepowerflowIntoabsorptionorscattering.Additionoftheabsorptioncros--sectfonGandthescatteringcross-section(Tequalstheastotalcross-section°t=°a+Üs(2.2)whichmeasuresthetotalenergyflowoutofthewaveduetotheobject.Inmanycases,thescatteredenergyisnotIsotroplc.Thedifferentialcross-section,do_._(n_,__).)isthenused.WhenmultipliedU_AbyISilityieldsthepowerfluxwithpolarization•throughadifferentialofsolidanglecenteredaboutthedirectiondefinedbyn.Thescatteringcross-sectionisthenfoundbyIntegratingoverthetotalsolidangleandallpolarizations.1982022556-021 21OFPO0;:IQ_ALI'_Y'•Forradarapplicationstheenergyscatteredbacktowardthesource"i,isofspecialimportance.Towardthisend,theback-scatt

30 ercross-,sectionobIsdefined.Itisthescatt
ercross-,sectionobIsdefined.Itisthescatteringcross-sectionofan,_.isotroplcsourcewithaconstantdifferentialcross-sectionequaltothatoftheobjectinthebackscatterornegati_nodirection._,"C:do(-)o'b"d_cl.qdi_(2.Ii)_,_._,_ItshouldbenotedthatIntheliterature,C_a,Os,atare-sometimesreplacedbyQa'Qs'I_t'andobIssometimessimplyo.Theaboveconventionhasbeenchosentoavoidconfusionwiththe"qualityLfactod'(_Inlatersections.•_2.2TheDlelectrlcPropertiesofWaterandIceThedielectricpropertiesofanon-magneticmaterialcanbe_'characterizedbytherealandImaginarypartsofitscomplexdielectricconstant._.c(f,T)-¢'(f,T)-l¢"(f,T)(2.5)TherealpartIsameasureofthepolarizationofthematerialsubjecttoanappliedelectricfiel

31 d.TheImaginarypartisaloss"_termmeasuring
d.TheImaginarypartisaloss"_termmeasuringtheenergytransferfromthefieldtothematerial.The%dielectricconstantIsafunctlenoffrequencyfandtemperatureT.ThefrequencyisrelatedtothetreespacewavelengthXbythespeediofII_tc.-It:Q,j1982022556-022 I___,,L,_-_,'''-Y++_1i!22f.)'OF'_"""_':c(2.6):_f-y:)))_Unlessotherwisenoted,cwillbeassumedtobeequaltounityforLbackgroundmaterialsofairorfreespace.,+.."ValuesofthedielectricconstantforwaterandIcehavebeen,:'extensivelymeasured.Severalgoodtabulationshavebeenpublished,LwiththoseofyonH|ppe116andRyde17beingespeciallyuseful.Thetemperaturedependenceofc'andE"forwateratseveralwavelengthsisplottedinFigures2.2and2.3from_nHlppel'sdata.Thedependence,ofc'an

32 dthelosstermc"onTIsclearlynotnegligible,
dthelosstermc"onTIsclearlynotnegligible,particularlyatthelowertemperatures.Theabsenceofidataforsupercooledwaterisdistressing,becauseofthestrongdependenceofEontemperature.ThelackofdataIsprimarilydueto:thedifficultyinmaintainingsupercooledwater,intheliquidstate,undercontrolleddlelect,'icmeasuringconditions.H_vertsupercooledwaterisImportantbecauseitoccurscommonlyInnature.ForsomeapplicationsItisconvenienttodefinethecomplexrefractiveIndexmasthesquarerootof¢:m--n-(2.7)'wherenistherefractiveIndexandKiStheabsorptioncoefficientofthematerl_ll.Itwillalsobeconvenienttodefine2m-lmT&)i1982022556-023 h238o),-lOom70Figure2.2Realpartofthe30dielectricconstantversustemperature.(fromvonHi

33 ppel16)010203050-,T(oc)|•1982022556-024
ppel16)010203050-,T(oc)|•1982022556-024 80,_70Figure2.3Imaginarypartofthedielectricconstantversustemperature60(fromvonHippel)_II"50)gO,23O20::I0X=lOcm.0102030405060T(°c).i!.l1982022556-025 25Valuesofn,K,IK21andtheimaginarypartof-KgivenbyRydelO;areshoqnforwaterandiceInTables2.1and2.2.72.3AbsorptionandScatteringbyDielectricSpheresThescatteringofelectromagneticwavesbyadielectricsphereI_oneofthefewscatteringproblemsforwhichthereexistsacompleteanalyticalsolution.Thegeneralsolutionwasfirstpubl|shedbyHiein1908.18Hie'sworkwasprecededbythatofLordRayleigh.In1871Raylelghsolvedthescatteringprobleminthelongwavelengthlimitaspartofhisexplanationforthebluecolorofthesky19,20Bothsolutio

34 nsarediscussedinthefollowing.!v.2.,3.1Hi
nsarediscussedinthefollowing.!v.2.,3.1HieTheory4,I_le'sanalysisofscatteringbyadielectricsphere18issimpleinconceptalthoughsomewhatcomplexindetail.Thegeneralapproachwillbeoutlinedherealongwiththeresults.ThereaderIsreferredtoStratton'stext21orH|e'soriginalwork18forthedetails.:Hie'sapproachwastowritethefieldoftheincidentplanewaveasthesumofvectorsphericalwavefunctionscenteredatthesphere.Theinciden:fieldgivesrisetooscillatin9chergesInthespherewhichproducesecondaryfieldsInsideandoutsideofthesphere.TheseexternalandInternalfieldsarealso_xpandedinsphericaJwavefunctlons.TheexpansioncoefficientsforthesecondaryfieldsarefoundbyImposingboundaryconditionsattheorib,nandthesurfaceofthes

35 phere.1982022556-026 26•OFPOORQUALITYTab
phere.1982022556-026 26•OFPOORQUALITYTable2.1Valuesofthedielectricparametersfo-water(takenfromRyde,17)i,i•,,m,|ii�.(cm)I03.211.240.62T(°C)208.888.146.154.44109.027.805.453.94-_.n08.997.144.753.4,5-86.::84.153.10II_oo.++,.oo,.,,._9,oo90,._,,,o,._o_,.4,,._,_.7_,.o_II200.9280.92750.91930.892610IKI20.93130.92820,9152,0.8726II00.93400.93000.90330.8312-80,89020.7921ZO0.004740.018830.06,710.091510Im(-K)0.006880.92_70.06130.114200.011020.03330.08070.1441"80,10360.1713i•iimiiiimi|1982022556-027 27ORIGINALPAGEISOFPOORQUALITY+rable2.2Valuesofthedielectricparametersforice(takenfromRyde,17))itn1.78Atalltemperatures024x10-4-I0k7.9xIO-4-205.5x10"4IKIz0.197Atalltemperatures.Thi

36 sisforiceofunitdensity,thevaluetobeusedw
sisforiceofunitdensity,thevaluetobeusedwhen0isdiameterofmeltediceparticle.i(HarshallandGunn,1952.)"!09.6x10"4i-10Im(-K)3.2x10-4-202.2x10-4:--iim•iii••i......ii*Refractiveindexoficeisindependentofwavelengthinthecentimeterband.i1982022556-028 28Inordertocalculatethecross-sectionsoncethesecondaryfieldsareknown,thePoyntingfluxisintegratedoverasphericalsurface,concentricwiththescatterertoyieldtheenergyf|ow.IntegratingthetotalPoyntingvector,whichincludestheincidentandsecondaryfields,willresultintheenergyflowintothesphereandthusoa.IntegratingthescatteredPoyntlngvectorwithonlythesecondaryfieldswillyieldthescatteredenergyflowmayfromthesphereoro.Thesproblemissimplifiedbyassuming

37 thatthesphereofintegrationislargei.:-i.e
thatthesphereofintegrationislargei.:-i.enoughthattheasymptoticvaluesofthesphericalwavefunctionscanbeused.Theforegoinganalysisresultsinthefollowingcross-sections)`2O0,'_0t-_-_(-Re)23(2n+l)(an+bn)(2.9)n=l)2®',---z(2nl)(lanl2
Ibnl2):as21Tn,,1(2.10)aa-ot-os(2.11)iOb1_(2n+I)(an-bn)2n--Iwhere),isthewavelengthinthebackgroundmedium.Theexpansion'coefficientsoftheexterlorsecondaryfieldanandbncanbe2thoughtofasthenthmagneticandelectricmultiplecoefficients.TheyaremadeupofsphericalBesselfunctionsanddependonpropertiesofthespheresuchasthecomplexrefractiveIndexmand(1where2waa"_andaistheradiusofthesphere.•iI!l,1982022556-029 I*)Iz9iiiTheHiecross-sectionshavebeencalculatedforiceandwater

38 spheres°:1tbyavarietyofinvestigatorsford
spheres°:1tbyavarietyofinvestigatorsfordifferentvaluesof(xand;itemperature,i7,22-252.3.2RayleI9hTheoryForsmallspheresor1on9wavelengthsi2_aQ="X-I12.131thesimplifyingassumptionsofRaylelghcanbemade.Physically,theaboveconstraintimpliesthattheincidentfieldlsspatiallyunlform:overthesphere.Inthissituationtheexteriorsecondaryfields-fromthespherecanbereplacedbythoseofanoscillatingdipole"p2.I.(c_._;__.)c-1a3_'i.(:_..Z.)a3-_Ei(2.14)m+2whereaistheradiusofthesphereand"1_iIstheIncidentelectricfield.Thecross-sectionsoandobarecalculatedusingthesesfields.Theabsorptioncross-sectionocanbecalculatedfromtheaohmiclossesoftheoscillatingchargesrequiredtocreatethedipole.Theprecedingcross-secti

39 onscanalsobecalculatedusingtheHieformall
onscanalsobecalculatedusingtheHieformallsmbyexpandingthecoefficientsanandbnintermsofa.Byneglectingtermsofhigherorderthane6,allmultipoleshigherthanthedipoledropout.Bothapproachesyieldthefollowingcross-sections231982022556-030 30,.ot-oa+os(2.15)i#•2)`2_6IKI2°s"_712.161+;)2_30---Im(-K)(2.17)a_)`2ob-_-a6IKI2(2.J8)whereValuesofIK21andIm(-K)_retabulatedInSection2.2.Noting2_ra+thatcc"Tthescatteringcross-s-_ctionandbackscattercross-sectionhavethecharacteristicl/_4d?.pendenceknownasRayleigh'sla_.TheRayleighcross-sectionsareofextremeimportanceinthestudyofscatteringfromatmospherichydrometers.TheyarevalidoverawiderangeofusefulparametersandarethPbasisofcomparisonfor-thoserefinement

40 swhichattempttoincludesuchadditionaleffe
swhichattempttoincludesuchadditionaleffectsasnon-sphericalshapeornon-homogeneousmaterial.+II12.4Absorption.andScatteringbyEllipsoidsTheproblemofscatteringbyanellipsoidofrevolutionwastreatedInthelongwavelengthlimitbyGansin1912.26Gans'approachisessentiallyanextensionofRaylelgh_swork.Heresolvestheincident"electricfieldIntocomponentsalongthresperpendicularaxes,onebeingtheaxisofrotationoftheellipsoid.GansthenassumesthatI,,•1982022556-031 .,LOR!C;NAL_''"'-31OFPOORQUALITYtheelectricfieldcomponentsexcitedipoleoscillationsalongtheIellipsoidaxes.ScatteringphenomenacanthenbecalculatedbyanalogytoRaylei9hscattering.Iftheellipsoidcoordinatesystemis_,rl,x,with/_the_axisofre_lution,th

41 enthedipolemomentsgivenbyGansarePF,"gEe(
enthedipolemomentsgivenbyGansarePF,"gEe(2.20),:"Prl"g'En12.21)!PX"g'Ex(2.22)_er'_°g=V(m2-I)(2.23)4_+(m2-I)Pg,.V(m2-1)(2.24)+(2.I)P'foroblatespheroids.P-4_-2P'-_.[I-sin'le](2.25)andforprolatespheroids.P-kw-2P'-kw('_'eZn_lI](2.26)ewhereVisthevolumeoftheellipsoidand•istheeccentricitydefinedasi•--4I-(B/A)212.2711982022556-032 oRIGSNALp_.G'E.|332OFpOORQUALI'[Y_:i4V,,,"_"AB2Prolate(2.28)V-_A2BOblate(2.29)wlthAandSbeingthemajorandminorradiioftheellipsoid.TheGansbackscattercross-sectionsforwater,iceandsnowhavebeencalculatedforrandomlyandpreferentiallyorientedellipsoidsby27Atlas,Kerker,andHirschfeld.TheirworkindicatesthatforwaterellipsoidsorientedwithmajoraxisalongEl,°bIncrease

42 sroughlylinearlywithaxesratioA/Buptovalu
sroughlylinearlywithaxesratioA/Buptovaluesoforder10.Thiseffectisgeometricalandhaslittledependenceonwavelength.Iceandsnowiareseentobeonlyweaklydependentongeometricalfactorsduetotheirgenerallysmalldielectricconstant.Theabsorptioncross-sectionfororientedellipsoidswillbeofparticularinterestinlatersectionsandwasnotcalculatedbyAtlas,etal.ThereforeowillbecalculatedhereusingtheGansapproach.aTheincldenLc_.lectrlcfieldisassumedtoprc_agateperpendiculartotheaxisofrotationoftheellipsoid.Theproblemthenbecomesessentiallytwo-dlmenslonal.InFigure2.4theorientationanglo0.db'_isdefinedastheanglebetweenEland_.Theabsorptioncross-sectionoiscalculatedbyassumingthata"the/_andndipoleshavesepara

43 tecross-sectionso_andonwhich?areexcitedb
tecross-sectionso_andonwhich?areexcitedbytheappropriatecomponentsoftheIncidentfield.Theabsorptioncross-sectionisthen0a-O_cos2a+Or1sln2a(2.30)1982022556-033 O_!G':t_'.F_'__OFPOORQUALITY]Ii]OblateEllipsnidil"i;i(,!'n_Ei_"1CProlateEllipsoidFigure2.4OrientationoftheincidentelectricfieldwlthrespecttotheelllpsO]dcoordinatesystem.{1982022556-034 ilOFPO0:tQU::_L_'(Thepartialcross-sectionso_and_rlare_flnedbyanalogytoRaylelghscatteringas8_2O_"_Im(-g)(2.31)8_2On"TIm(-9')(2.32)_hereXisthefreespacewavelengthandg,g'aretheGansfactors.:Theratioofo6andor1forwateratO°Ctooaofan_equivolumetrlcsphereisplottedversustheaxisratioA/BforoblateandprelatespheroidsinFigures2.5and2.6.OfInterestisth

44 estrongIncreaseinabsorptionastheellipsoi
estrongIncreaseinabsorptionastheellipsoidbecomesmoreeccentric.Thisappearstobevalidindependentofwavelength,subjecttothelongwavelengthlimit.4_2.5AbsorptionandScatteringbyaCollectionofScatterersInmanycasesofsigniflcance,thescatteringobjectIsactuallyacollectionofsmallerscatterers.Theabsorptloncross-sectionoafor,isuch•collectionis,tofirstorder,justthesumofalltheIndividualcross-sectionsoa-z:o12.33)ja,jaslongasthescatterersarenotinImmediateproximitytooneanother.ThiscanbeIllustratedbyconsideringtwoRayleighspheresofradiusaseparatedbyadistancer.Thescatteredelectricfieldfromsphere1i1982022556-035 35:_Figure2.5Ratioofo_andOr1tothesphericalvalue'"versusAIB.50OblateE!lipsoLds3O.st¢•

45 3.%-I,24and10cm,!59A/If O_Lr.,!,_Z:;-_,.
3.%-I,24and10cm,!59A/If O_Lr.,!,_Z:;-_,...-.-.:OFpOORqL;_L_'I'YIFigure2.6Ratioofo{andontotosphericalvalueI00.IversusA/B.IProlateEllipsoids_so4_.---",-'°lS"•///._i-5__'__)_-I.2llandlOcm•/I")"._011982022556-037 \}_':,ORIGINALPAGEISit37OFPOORQUALITY;i_:i_feltbysphere2Isoforder(r)3EiwhereEiistheincident"•_ielectricfield.ItIsclearthatforlnterparticlespacingsgreaterthan_._•:afewradii,theIncidentfielddominatesandtheparticlesabsorbi':independently.Collectiveeffectscanbeimportantforthescatteringcross-sections.InthecaseofIdenticalscatterersthedifferential:(scatteringcross-sectioncanbewrittenas2_^aiIonescatterersscattererwhereF(q)isthestructurefactorusedinx-raycrystallography.It,

46 dependsonthedistributionofscatterersandi
dependsonthedistributionofscatterersandisdefinedas=IzexpClq•xj)lz(z.3s)jo:Thevectorqisthechangeinwavevector_andxjisthepositionofthejthscatterer.ItisInstructivetoconsiderF(q-'L)forthebackscattercross-'-"section_-2_"O.ForauniformdistributionoffixedIdentical"-j"scatterers:•FizZ'0)-0(2.361"Thisisduetothephasecancellationofbackscattersignalfrom1982022556-038 38•onescattererbyanotherseparatedbyanintegernumberofhalfwave-lengthsalon9theIncidentdirection.If,however,thereissomere|ativemotionbetweenthescatterers,thenthephasecancellationiaveragesoutandr(2_O)-N(2.37):whereNIsthenumberofscatterers.Thebackscattercross-section|:formeteorologicalscattererscanthenbewrlttenas,ob-_%,j(2.3

47 8)jThiseffectallowstheback,scattercross-
8)jThiseffectallowstheback,scattercross-sectiontoberelatedtosuchmeteorologicallyrelevantparametersastheliquidwatercontent:andtherainfallrate._."1982022556-039 39o*CHAPTER3_:iABSORPTIONANDSCATTERINGTHROUGHTHEICE-WATERPHASETRANSITION•_Theproblemofscatterln_frommeltingandfreezinghydrometersIst+consideredinthischapter.InSection3.1thetheoryofscattering_'frommixed-phaseparticleslsdiscussed,alongwithamode]forscatteringbymeltingatmosphericparticles.InSection3.2,•{experimentaltechniquesusedtomeasureabsorptionduringthephase1.._itransltlonsarepresented.Finally,Section3.3containstheresults:oftheabsorptioncross-sectionmeasurements.•Ji"3.1TheoryofScatterln_byHeitln_andFreezin_Hydrom

48 etersThescatteringbywater-coatedicespher
etersThescatteringbywater-coatedicesphereshasbeenstudied:theoreticallybyseveralinvestigators.In1951AdenandKerker29extendedHle'stheorytothecaseoftwoconcentricspheres.In1952,30LanglebenandGunncalculatedcross-sectionsforwater-coatedicespheresusingtheAdenandKerkerresults.Addltlonalcross-sectionshavebeencalculatedbyB¢..en,HermanandBrowning31,32Experlmental•!measurementsofthebackscattercross-sectionsforlargemeltingicespheresweremadebyAtlasetalin1960.33Labrumstudiedtheproblemofscatteringbytwoconfocal!_ellipsoidsin195;234inanattempttoexplainenhancedreflectionfromthe"brightband".The"brightband"isobservedonmeteorologicalradarasahlghly-reflectivezonelocatedJustbelowthefreezinglev

49 el.Theenhancedreflectioninthe"brightband
el.Theenhancedreflectioninthe"brightband"canonlybepartially:explainedbythedynamicsofmaltingprecipitationparticles.LabrumappliedtheGansmethodologytocalculatebackscattercross-sectlons.pi!1982022556-040 _r_Healsomadesomeexperimentalmeasurementsofbackscatterfrommelting35:_nonspher!ca]iceparticles.Becauseofexper|mentaldiff|culties,,however,Labrumwasonlyabletogetgeneralqualit°:iveagreement,wlththetheoreticalbackscattercross-sections.*.InSection3.1.1,theabovetheoreticalresultsforspheresandellipsoidswillbesummarized.InSection3.1.2,amodelofthemeltingandscatteringprocesseswhichincludethetheoreticalcross-sectionswiI!bepresented,:--;3.1.iWater-CoatedIceSpheresandEllipsoidsAnexampl

50 eoftheabsorption,scattering,andbackscatt
eoftheabsorption,scattering,andbackscattercross-.sectionscalculatedbyLanglebenandGunn30isshowninFigure3.1.Inthiscasethewavelengthis3cmandtheequivalentmelteddiameter¢Dofthesphereis2./Imm.Thecrosssectionsarenormalizedbyeqthelrmeltedvalues_meltedandareplottedagainstthemassfractionofwater_1f=(3.1)wHw+hiwhereHwisthemassofwaterInthesphereandHiisthemassofice•!Thebehaviorofthecross-sectlonsinFigure3.1withfwIstypicalofthosewater-coatedicesphereswithradius_essthan),•,Thescatteringcross-sectionsIncreaserapidlywithfwtothevaluoforwaterspheres•Theabsorptioncross-sectionrisesveryrapidlyiiwithfwtomorethantwicethemeltedva!uebetweenfw"•1andfw".2andthengraduallyreducestothemeltedvalue.•t

51 r_1982022556-041 ,i-,,____c*,Ii41ORIC-It
r_1982022556-041 ,i-,,____c*,Ii41ORIC-It,!ALPAG_ISOFPOORQUAUTY.1982022556-042 C42+-TheIncreasedabsorptionofapartially-meltedicespherecanbeunderstoodphysicallybyconsideringthemeltedcase.Theabsorptionbyameltedspherecanbethoughtofasohmiclossesarisingfromcurrentswhichcreateoscillatingmul°ipolestomatchtheboundaryconditionsatthesphere'ssurface.Ifanobstruction,inthiscaseanicesphere,isplacedinthecenteroi:thewatersphere,thepathlengthforthecurrentsisIncreasedandtheohmiclosseswillsimilarlyincrease.IftheIcesphereisverysmalltherewillbelittleeffect,andIfthe•sphereispredominantlyice,thentheexcitedmultiplesareweakerandmuchofthecurrentwillflowInthelowlossIce,resultinginlittleabsorption

52 .Forscatteringfrommeltingellipsoidsofrev
.Forscatteringfrommeltingellipsoidsofrevolution,itwillbeassumedthatthemeltingeffectsofthissectionandtheGansshapeeffectsofSection2.4canbeincludedmultiplicatively.Therefore;oO=Ogans,o(3-2)OmeltedOsphe+.esphereHereoIsthecross-sectlontobecalculated,oisthe°reeltedratioofpartlally-tototally-meltedcross-sectionsforasphere.Theo:ratio_isthatoftheGanscross-sectiontoanequivolumetric°sphere,sphereandOsphereisthecross-sectionoftheequlvolumetrlcwater+*sphere.Backscattercross-sectionscalculatedasaboveagreewlththoseofLabrum31ttoatleastfirstorder.Theabovesimplificationisconsideredacceptableconsideringtha_,formostapplications,theassumptionofanellipsoidofrevolutionIsitselfonlyanapproxima

53 tionofsomemorecomplicatedshape.iI+119820
tionofsomemorecomplicatedshape.iI+11982022556-043 43_.1.2HodelofHeltingofScatterersIntheAtmosphereTheapproximationofatmosphericscatterersasbodiesofrevolutionworkswellforice,wherethelowdielectricconstantmakesshapeeffectunimportant,andforsmallwaterdropswheresurfacetensiondominates.Formeltingicecrystals,however,asomewhatmorecomplexmodelmustbeused.Inordertomodelthemeltingprocess,snowflakeswereobservedastheyfellontoaplateofwarmglass.Fivedistinctphasesofthemeltingprocesswereobserved,althoughnoteveryparticleexhibitedallfivephases.ThefivephasesareshownschematicallyinFigure3.2.Phaselistheicephasebeforemelting.InPhase2thecrystalbeginstomeltatitsextremities,wheretheheattransferis

54 thegreatest.Smallwaterspheresbegintoform
thegreatest.Smallwaterspheresbegintoformattheextremities.InPhase3enoughwaterhasmaltedtocoalesceintoawatershell.Thereis,however,stillsufficienticestructureinthisphasetomaintainanon-spherlcalshape.InPhase4theicehasmeltedtosuchapointthatthestructuralintegrityoftheiceisgoneandthesurfacetensionofthewatercausesthedroptocollapseintoasphericalshape.Phase5isthatofthetotally-meltedsphere.TheabsorptionandscatteringcharacteristicsofameltingIceparticlearedifferentIneachofthefivephases.AnexampleoftherscatteringbehaviorofameltingcrystalisshownIrFigure3.3,whereoaandc;bareplottedagainstthemassfractionfw"InPhase]theparticlescattersandabsorbslikeanequlvolumetricIcesphere.IInPhase2thepart

55 iclecanbeappr_xlmatedasacollectionofRayl
iclecanbeappr_xlmatedasacollectionofRaylelgh,wrscatterers.Theabsorptioncross-sectionis,therefore,alinearly-increasingfunctionoffw"Thebackscattercross-sectionis4_1982022556-044 tOFPOORQUALITYPHASE1PHASE2•_PHASE30PHASE_:PPHASE5QFigure3.2Schematicrepresentationofthefive_asesofmelting.¢1982022556-045 45_dependentontheforefunctionF(2_O)definedinSection2.5.Itwill,,Ingeneral,beacomplicatedfunctiondependingontheorientationof,theparticleandthelocationswheremeltingbegins.Inmanycases4thecrystalwillhavesomesymmetry,hexagonalorotherwise,andthemeltingshouldinitiatesymmetrically,whichsimplifiescalculationorF.Themaximumvalue,ofFIsN2whereNisthenumberofwaterspheres.inPhase3,shapeeffects

56 •asapproximatedbyGansscattering,are|:com
•asapproximatedbyGansscattering,are|:combinedwiththemeltingeffectstocalculatethecross-sectlonsasinSection3.1.1.intheexampleshowninFigure3.3,thecross-sectionsareenhancedbyshapeeffects,Implyingthattheelectricfieldisorientedatleastpartlyalongthemajoraxis.Itshouldbenotedthattheshapeeffectscouldalsodegradethecross-sectionstovalueslowerthanthoseofameltingsphere,iftheelectricfieldwereoriented'+ialongaminoraxis.InPhase4,theparticleactslikeawater-coatedIcesphereforoaandob.Finally,inPhase5whentheparticleistotallymelted,thecross-sectionsarethoseofawatersphere."Freezingwaterdropsintheatmosphereexhibitmuchsimplerbehaviorthanthemeltingcase.Thedropsaregenerallyinthemetastablesupercoo

57 ledstatepriortomelting.Whencrystallizati
ledstatepriortomelting.Whencrystallizationoccurs•itdoessorapidlyand_lativelyhomogeneously.The;-_catterlngbehaviorofafreezingdropcanbeapproximatedbythatof•sphereoranelIIpsoidwithahomogeneousdlelectricconstant:¢lcoHi+%aterI_,i¢""I+".(3.3);1982022556-046 46ORIG,t_.,'.-.P,',.gSr,:_OFPOORQUALITY,!I1982022556-047 !47whichIsthemassweightedaverageofthedlelectrlcconstantseice.andawareroftheconstituents.3.2ExperimentalMethodsforHeasurlngAbsorptionDuringPhaseTransitionRelativelyfewexperimentalmeasurementshavebeenmadetoverifythetheoreticalcross-sectionsofmeltingandfreezinghydrometers.Radarobservationsofphenomenasuchasthe"brightbancf'yieldsomeInslght,butaresubjecttomanyuncertaintie

58 s,duetofactorssuchasprecipitationdynamic
s,duetofactorssuchasprecipitationdynamics,coalescence,andgeneraluncertaintyastothe•exactnatureofthescatterers.Controlledexperimentswhichmeasurethecross-sectionsofasingle,scattereraredifficult,duetotheverysmallpowerchangesinvolved.Thosemeasurementswhichhavebeenmade,byLabrum35andiithosebyAtlaseta1,33measuredbackscattercross-sectionsunderless-than-Idealconditions.Labrummeasuredobformeltinghemispheresplacedinawavegulde.Atlasetalmeasuredc;bforverylargemeltingspheressuspendedbyaballooninthenearfieldofamteorologlcalradar.Atechniqueforaccuratelymeasuringtheabsorptioncross-sectionsofmeltingmeteorologicalscaledropsusingperturbationtechniquesin•aresonantmicrowavecavityhasbeendeve

59 loped.Section3.2.1doscribesthetechniquea
loped.Section3.2.1doscribesthetechniqueandtheparticularcavltyusedIntheexperiments.InSection3.2.2theexperimentalset-upisdiscussed.Section3.3containstheexperimentalresultsobtainedusingthistechnique.1982022556-048 48'3.2.1PerturbationTechniquesInaResonantCavityThetechniqueusedformeasuringtheabsorptioncross-sectionoaconsists,basically,ofmeasuringthechangeInqualityfactorQofahighQresonantcavitywhereQ=_energystoredincavlty=uU(3.4)powerlossincavityPandm=2_f(3.5)I•U-energystoredinthecavity(3.6)P=powerlossinthecavity(3.7)ThevalueofI_canberelatedto_aIfthedroplsnon-magneticandItsdimensionsaresmallcomparedwiththescalelengthoftheelectricfieldinsidethecavity.Undertheseassumptionsthes

60 cattererseesanoscillatingelectricfieldof
cattererseesanoscillatingelectricfieldofstrengthEO.ThepowerabsorbedbythedropPdcanbewrittenusingaaandthe:Poyntingvectorwhichwouldresultiftheelectricfieldwereinfreespace.c(3.8)Pd=°a_-_EO2TorelatePdtothechangeInQwhenadropIsIntroducedIntoacavity,thecaseoftheemptycavitymustfirstbeconsidered.FortheemptycavityU%-13.9)W1982022556-049 49whereQeistheemptyQandPwisthepowerdissipatedinthewallsofthecavity._#henthedropIsplacedInthecavity,assumingoUdoesnotchange,theperturbedvalueO.pbecomesu(3.10)QP'_P+PdIfthedropletQisdefinedasUQd=_F_d(3.11)thenI11(3.12)NotingEquations3.8through3.12,theabsorptioncross-sectloncanbewrlttenas%.1CEoZ_dd(3.13)cavityIssuchthatUisproportionaltoE02,thenIfthet

61 heelectricfieldstre.gthwilldropout.Theca
heelectricfieldstre.gthwilldropout.ThecavitychosenfordropmeasurementswasarightcircularcylinderoperatlngIntheTHO10mode.ThismodewaschosenbecauseofItsfairlyunlfomelectricfieldatthecenterofthecavityorientedalongthecavityaxis.TheTHOI0modealsohasvc_/smallsurfacecurrentsontheendplatesneartheaxis,allowingholestobedrilledintheendplateswithmlnlmaleffectonthecavityfields."1982022556-050 :50',Thecavitywasdesignedtoresonateat10.66GHz.ItsdiameterDwas2.153cmanditslengthLwas1.229cm.Figure3.4isthemode36"chartforrightcircularcylinders,shGwingtheoperatingpointforthecylinder.Thereisgood_odeseparation,withtheclosestmodebeingtheTEllIat14.4GHz.SincetheTHOI0isthelowestmode,frequencieslowertha

62 ntheTMOI0resonancearecutoff.Thecavityisc
ntheTMOI0resonancearecutoff.ThecavityisconstructedoutofcopperandtheemptyQIscalculatedtobe9000.'TheelectricandmagneticfieldsfortheTROI0modeare-4.81.-l_tEz(p,t)=EOJoiT)e(3.14)8_(p,t)=-iEoJi(I°'_lp-)e-i_t(3.15)wherep,¢.andzarethestandardcylindricalcoordinates.Withthefieldsknown,itispossib'!etocalculatethestoredenergyU.NotingtheperiodictimedependenceandassumingtheunperturbeddielectricconstanttobeunityII+IB_(p,t)dvol(3.16)U=I-_IEz(p,t)l212¢avityU"L[Jo2(_J.p),ji2(4._lP__)lpdp(3.17)_0U=0.0084(D2L)E02(3.18)2NotethatUhasthedesiredE0dependence.ForthevaluesofDandLdescribedabove|I1982022556-051 OF£OORQU:\LI'i'Vi_.0Ce-;mzI_°_2.0_.I°TH011._._"_-TEll1Eo1.5_l_'Closestcompeting-'__-'po

63 int1.0_Operatingpoint_o_................
int1.0_Operatingpoint_o_..................4.........THO100X_._'_'0.5az.0z'.53.03.'5A2_-(OIL)Figure3,itHodechartforrightcircularcylindershowingtheTH010operatingpointandtheclosestcompetingpoint.1982022556-052 52U=0.048E02(3.19)Therefore\0"Ok8_E02I=0.048f(cm2)(3.20)-°a_E02Theabsorptioncross-sectionissimplyproportionaltotheproductoffandI/Qd.3.2.2ExperimentalSet-UpThemicrowavecircuitusedtomeasurecavityQIsshownin1Figure3.5.PowerwasgeneratedbyanX-bands_eeposcillatorwhichvariedfrequencylinearlyoveraspecifiedrangeneartheresonanL(frequency.Powerwastransmittedthroughavariableattenuatorand:byacommercialwavemetertoprovideafrequencyreference.The,cavitywasconnectedtothecircuitviaasho

64 rtstubtoacoaxialtee!•iThecoaxcoupledmagn
rtstubtoacoaxialtee!•iThecoaxcoupledmagneticallyIntothecavityfieldsbyalooporientedinthe_directionlocatedononeoftheendplates.Theteewas:tconnectedtotherestofthecircuitbyflexiblecoaxialcable.ThisilallowedtheentirecavityassemblytobeplacedInacoldpox.Thei/cablewasterminatedatbothendsbymatchedattenuatorsinordertoIireducereflections.tAfterpassingthecavity,powerwasmeasuredbyacrystaldetector.1Thecrystalhadbeencallbratedpreviouslyagainstabolometer.Theoutputofthecrystalwenttoadigitalscope.ThescopewastitriggeredbythesweeperandhadthecapabilitytostoretracesonfloppyB1982022556-053 JI_•53OFPO'3RQUALITY1982022556-054 7.4;disksfor]ateranalysis.Thisfeaturewasusedwhenmeasuringtime-dependen

65 tphenomena,suchasmeltingeffects.InFigure
tphenomena,suchasmeltingeffects.InFigure3.6atypicalscopetraceofcrystalvoltageversustimeor,equivalently,frequency,isshown.Th_referencemarkerandthecavityresonancearevisible.Whenthefrequencyisoffresonance,nopowerisabsorbedbythecavityandthepowertothecrystalIsatamaximum.Onresonancethecavityisveryabsorptiveandthecrystalpowerisatitsminimum.TheloadedQofthecavity,whichincludescouplingeffects,is,ijfo"Qt=-_(3.21)_FwherefoistheresonantfrequencyandI"Isthefullwidthathalf"41"maximumoftheresonance.ThecavityQdiscussedpreviouslyIstheunloadedQwhichisrelatedtoQI.foracircuitsuchasthisby00my__'x0.LT"x'0"__q__(3.22)wherePmexandPminarethemaximumandtheminimumpowersreceivedbythecrystal.37Theact

66 ualcavityassemblyusedinthe_xperimentIssF
ualcavityassemblyusedinthe_xperimentIssF,owninFigure3.7.Itwasmachinedoutof1.5inchcopperrod.ThecavitydiameterDwas2.153cmandthecavitylengthLwas1.229can.Oneendplatewasremovableforcavitycleaningandpolishing.Therewereholesdrilledonaxisineachofthe].5-cm-thicke)dplateswitha2mmradlustoprovic_eaccessfordropInsertion.Thecavitywas1982022556-055 °,55_ri--C_RQ_ALITY3OV£_OAIY£S_31982022556-056 Figure3.7ResonantCavityAssembly.1982022556-057 57supportedhorlzontallyonaveeblockIndirectcontactwitha.thermometertomeasurecavitytemperature.Dropsofdisti_ledwaterweresupportedbysurfacetensiononaquartzfiberwitharadiuslessthan.!nun.AnexampleisshowninFigure3.8.Dropswereplacedonthefiberbyasyringean

67 ddimensionsweremeasuredInsitub'iadirect-
ddimensionsweremeasuredInsitub'iadirect-readmeas_rl,gmicroscope.Thefiberwasorientedalongthez-ax'_ofthecavityandranthroughtheholesIntheendplates.Itwasheldonaxisbyplexiglasssupportsoneitherendofthecavity.Theentirecavitya_semblywasportablesothatitcouldbemovedinandoutofa-20°Ccoldboxwithoutdisturbingthedroporthemicrowavecircuit.TypicalcoolingandwarmingcurvesareshowninFigure3.9.TheresonantfrequencyandQoftheemptycavitywerefaeasured.Theresonantfrequencywas10.656HzandtheQwas8990,whichareveryclosetothedesignvaluesof10.66GHzand9000.Thedifferencesareattributedtotheendplateholesandimperfectcavitysurfaces.ItshouldbenotedthattheemptyQvariedsomewhatduetooxidationontheinteriorcavitysur

68 faces,resultingfromexposuretomoistureand
faces,resultingfromexposuretomoistureandthethermalcyclingofthecavity.Inordertocorrectforthis,thecavitywaspolishedperiodicallyandtheemptyQwasmeasuredpriortoeachdroprun.3.3ExperimentaI_Resu!t_sTheexperimentalresultsofabsorptionmeasurementsforwaterdrops•arepre__entedInthissection.Themeasuredcross-sectionsforroom-'temperaturedropsarefoundtoagreewiththeRayleighvaluesini:.1982022556-058 58Oi"P_""Q"_ag¢"Figure3.8_laterdropletsupportedonaquartzfiber,:_i:1982022556-059 IO20304050time(rain)1982022556-060 J603Section3.3.1.InSection3.3.2somemeasurementsofthedielectric:iconstantforsupercooledwaterarepresented.Finally,Section3.3.3.containsabsorptiondataforsphericalandnon-sphericalhy

69 drometersduringmeltingandfreezing.4,3.3.
drometersduringmeltingandfreezing.4,3.3.1AbsorptionCross-SectionsforWarmDropsi:!Absorptioncross-sectionsweremeasuredfordistilledwateridropsataroomtemperatureof2O°Cusingthetechniquesdescribedin!:iSection3.2.Theresonantfrequencyofmeasurementwas10.64GHzi*.OIGHz.Dropswithdiametersvaryingfrom0.5mmto2.0mmweremeasured.Themeasuredabsorptioncross-sectionisplottedagainstt|,dropvolumeVinFigure3.10.ThestraightlineisaplotoftheiiRayleighabsorptioncross-sectionst6_VIm(-K)(3.23)";°a=T,2withIm(-K)takenfromRyde17at2O°Ctobe0.01883.ThedataagreeswellwithRayleightheoryforthesesmallessentially-sphericaldrops,ThescatterinthedataismalnlyattributedtouncertaintyinmeasurementofV.,iI3.3.2Dielectri

70 cParametersofSupercooledWaterWiththeconf
cParametersofSupercooledWaterWiththeconfidenceInthemeasurementsofRayleighabsorptionIcross-sectionsgainedfromtheresultsofSection3.3.1,measurementsweremadeofoaasafunctionofcavitytemperatureTinordertoInferthedlelectrlcparameterIm(-K)fromtheRayieighcross-sectionsInequation3.23.HeasurementsofIKI2,whichcanbe1982022556-061 IFPCCRQUAL!I'Yo'.¢1"o..29m_4.0ix(w:))"ow1982022556-062 :62;linearlyrelatedtotheshiftintheresonantfrequencyfoby,perturbationtheory,werealsoattempted.MeasurementsofJKI2,however,wereunsuccessfulbecausetheverysmallchangesinIKI2iwithtemperatureweremaskedbyrelativelylargechangesintheresonancefrequencyduetothermalexpansionofthecavity.LackofIKIzmeasurementisnotcons

71 ideredtooimportantas,inapplications,If;I
ideredtooimportantas,inapplications,If;I2isgenerallyassumedtobeconstantwithtemperature.i-•ReasurementsofthesupercooledtemperaturedependenceofIm(-K)weremadebyinsertin9distilledwaterdropsinawarmcavityandi{tplacingthecavityassemblyinthe-20°Ccoldbox.Thedropswereitassumedtoremaininthermalequilibriumwiththecavity.Microwave''.heatingofthedropwasneglectedbecauseofthelo_powerofthesweeper(lessthan1roW)andthelowdutycycleonresonance(lessthan0.0OI).Liquiddropswereobservedattemperaturesaslowas-17°Cbefore.crystallization.__.Valuesof1/(_dandcavitytemperatureTversustimefora:typicalcoolingrunareshowninFigure3.ll.Thedropcooledwithincreasingabsorptionto-17°Cwhereitnucleatedandtheabsorptio

72 ndroppedtothevalueforice.At16minutesInto
ndroppedtothevalueforice.At16minutesIntotheruntherewasasuddenjumpin1/Qdwhichlastedfor12minutes.Similarjumps:wereobservedonsubsequentruns,althoughoccurringatslightlydifferenttimesandtemperatures.Theseanomalously-highvaluesofJI/QdarenotthoughttobephysicallyrelatedtochangesinIm(-K),:butrathertosomethermaleffectinthemicrowavecircuit.Inthefollowing,therefore,suchvalueswillbeomitted.Figure3.12isaplotoftheobservedveluesofIm(-_;asi,m1982022556-063 63O;,L_,,+,4Q;JALriYI=l•i•ttI,l=L•IImi=Oo••_•-=".II••l,e_°oQ?I!-_.•.+•i•=Q¢••,_i•II•••,_ot•iiii|.ii.i.i-RoooA.*,-,_-•1:301=+"pat,.,=od==.l.'e°lOf=i%11"_tjII.................,+.,_...............1982022556-064 •464,.ORIGIi'IALPAGEISOFP

73 oORQUALITY_,J"\C&OQD_L.,-4•'4)•i*°u�
oORQUALITY_,J"\C&OQD_L.,-4•'4)•i*°u�u_e,_B_),t?8-_4I•inooo:••v-|0•",O000,*•...•,;,,,__l1982022556-065 65calculatedbyequation3.22,alongwiththevaluesofIm(-K)1617_.calculatedfromthedielectricdataofvonHippelandRyde.Themeasureddataisafairlysmoothfittothepreviousvaluesandextendswellintothesupercooledrange.TherealandimaginarypartsofthedielectricconstantcouldbecalculatedfromIm(oK)byassumingIKI2isconstantwithItsvalueforO°C.TheabovetechniquecouldbeusedtomeasureIm(-K)atevenlowertemperaturesifaverycleancoldboxenvironmentwasmaintained.ValuesatotherfrequenciescouldbemeasuredIfasuitablecavitywasc_nstructed.|5.3-3AbsorptionCross-SectionsDuringPhaseTransitionInFigure3,11ofthepre

74 vioussection,asupercooleddropisobservedt
vioussection,asupercooleddropisobservedtocrysta111ze.ThevalueofI/QddroppedlinearlyintimefromavalueforliquidwatertothatoficeinlessthanIminute,ThisIsthebehaviorpredictedinSection3.1forafreezingsupercooleddrop.ThemeltingbehavioroficeparticlesisexpectedtobemorecomplIcated.Observationsofthemeltingcross-sectionsweremadebyplaclnganiceparticleontothefiberandintothecavitywhllethecavltywasinthermalequilibriumwiththecoldboxat-20°C.Thecavityassemblywasthenremovedfromthecoldboxandallowedtowarm.AtypicalexampleofthebehaviorofI/QdwithtimeisshowninFigure3.13foranIcesohereofequivalentmelteddiameterOofeq1.15ram.TheabsorptionbehavedqualitativelyaswaspredictedInSection3.1.At12:30minutesthe

75 dropbegantomelt.ThevalueofI/Qdquiclyrose
dropbegantomelt.ThevalueofI/Qdquiclyrosetoamaximumat16minutes.Theabsorptionthenilk_11982022556-066 66•tOR!G'_?I:_LF;_.'/ISoFPOOP,QUALITY0C•._Q._i'-""aU•'w)E&""=DNU_.IF'_Nq.olxP_/iq,1982022556-067 -,qlo-]/67begantodecayandat20minutesreachedthemelteavalue,whichwaslessthanhalfthepeakvalue.Inordertoquantitativelycomparethemeasurementswiththeory,0aI/Qdmustbeconvertedtothenormalizedcross-sectionandOmeitedsomerelatlcnmustbeassumedbetweentimeandthemassfractionMw(3.24)fw' +MITorelatefwtotlmeT,assumethatthedroplettemperatureIszeroduringmelting.Alsoassumethatmeltingbeginsattimetoandendsattimetf.TheheatlossfromthedropIsdQd-T"const.T(t)(3.Z5)whereT(t)isthetime-dependentcavitytemper

76 atureandtheconstantisdependentonlyonthec
atureandtheconstantisdependentonlyonthecavitygeometry.Fromthewarmingcurvein_'Figure3.9,itisassumedthatthecavitytemperatureIncreaseslinearlywithtimenearzero.ThereforedQ_--const,t(3.26)ThechangeinwatermasscanberelatedtotheheatlossbythelatentheatoffusionLiw.dMwl.J.._tt().27)"_'"Liw1982022556-068 i-ii68........uq'SincethetotalmassH+Hiisconstantandnotln_thatHw-Oat,;:W"itimetOfW(t)Hw(t)t"t_+Mi=const.(t-tO)d(t-tO)(3.28)0:.f(t)-const.(t-t0)2(3.29)Notingthatfw"1attimet-tf,then(t-t0)2=(3.30).,-(tf'-to)2whichistherequiredrelationbetweenmassfractionandtime.Figures3.14,3.15and3.16showtheexperimentalandtheoreticalonormalizedcross-sectionsaoasfunctionsoffwforIceameltedspheresequlw,le

77 ntmelteddiametersDeqof1.15,1.6and2.0mm.T
ntmelteddiametersDeqof1.15,1.6and2.0mm.Thegeneralbehaviorisingoodagreementwiththeory,althoughthetheoryunderestimatestheabsorptioncross-sectionsbyasmuchQs25_.atpeakvalueinthe,_orstcase.ThediscrepancyloduetothefactthattheAdenandKerkertheoryassumesdielectricvaluesforwaterona+wavelengthof3cmandatemperatureof18°C.Thedielectricconstantatthesevaluesislessabsorptivethantheactualexperimentalcondi-tionsof2.8cmand0°C.Theincreasedabsorption,alongwithslightdeviationsfromsphericity,couldaccountfortheobserveddifference.:Tf,eabsorptioncross-sectionsformeltingnon-sphericalIce,._,particleswereobserved,tocheckforthen,ultlplepha_eb_havlorpredlc_edinSection3.1.2.ShapedIceparticleswereconst

78 ructedby1982022556-069 69?C........_JOF,
ructedby1982022556-069 69?C........_JOF,_;,/..-.:C_-:,!-'.1''/!CI'P!oI•I0|IIi"-I!I:J_a"II'_.Io.atI:-.I,'_I-Io,I:...-/It/:•I8III].-I•I/N•Ic);II•I\.•__,,_,.\•'ii11982022556-070 :70.__'.-'"QUALITY_'CFF...._1982022556-071 71c.'....i_"c'b•i°itl|,,._:IIIIZZ.I,,o.oI::_,._,bI:!,/•I•!Zh,,,m/tI//•///•//,_•/•/#°•!ii•Ii1982022556-072 ,',-,L-,.."i.Ti72l!placingasmallwaterdroponafiberandimmersingthedropInailiquidnitrogenbath.Afterthedropfroze,itwasremovedfromthebatllandanadditionaldropwasadded.Theprocedurewasrepeateduntiltherequiredshapewasachieved.Themajordisadvantageofthismethodwastheinabilitytocreatesharpedgesduetosurfacetensionsmoothingofthewaterdrops.Figures3.17and3.18showexamp

79 lesoficeparticleswhichw_.reroughlyoblate
lesoficeparticleswhichw_.reroughlyoblateandprolateellipsoidsorientedwiththeiraxisofrevolutionalongtheelectricfield.FortheoblatecaseinFigure3.17,fourofthefivemeltingphasesarevisible.Theabsorptioncross-sectionstartedinPhaseIlikethatofanicesphere.Oncemeltingbegan,thecross-sectlonskippedPhase2becauseoftherelativelysmoothsurfaceandabsorbedinPhase3likeanoblatewater-coatediceellipsoidwithaxesratioA/B-1.8.Atamassfractionof0.2,thedropbegantocollapseintoPhase4ofawater-coatedicesphereandatamassfractionofI.Othedropab_._edlikeawatersphereInPhase5.InFigure3.18anexampleofaprolateellipsoidisshown(notethechangeinscale).Inthiscasetherewassufficientsurfaceirregularitytoshowanon-uniformme

80 ltingPhase2uptoamassfractionofapproximat
ltingPhase2uptoamassfractionofapproximately0.2where_neverystrongabsorption,inPhase3,ofanorientedprolateellipsoidwlthA/B-2.5prevailed.Phase3lasteduptoamassfractionof0.6whichisartificiallyhighduetothestructuralsupportofthequartzfiber.Abovefw"0.6thedropcollapsedintothesphericalshapeofPhases11and5.1982022556-073 73•.,'/:.,,..::.t_r-,••"...._.,_ur_tII••II1IjI_.3IJoI"Io!Ii}'I!•_|_,_II_.r-I•I-/!,,DL,4/I•II:i_=//--II/III/I•/I4",I'd/,/II-I,i",I•//I/"I///'"*,IIp.+_t':[4l,lInlk•I•It41,1',,.Ion.+1982022556-074 +++71tORIGINALPAGEIS_p'_OUALITY+,\1982022556-075 'r,•75iInconclusion,theexperimentalmeasurementsoftheabsorptioncross-sectionsseemtosupportthetheoryforwater-coatedicespheresa

81 ndellipsoidsandthemodelforatmosphericmel
ndellipsoidsandthemodelforatmosphericmelting.Theobservedt!cross-sectionswerefoundLobegreaterthanorequaltothetheoreticalivaluesforwater-coatedicespheres.Thegreatestdifferencesoccurredatthepeakofthemeltingcurve,whereforonecasethetheoreticalcross-sectionunderestimatedthemeasuredvalueby25_.Thedifferenceisthoughttobetheresultofdiscrepanciesbetweenthe:theoreticalandexperimentalvaluesofthedielectricconstant.1982022556-076 f76+,,+-;+CHAPTER4+HYDRODYNAMICSOFACCELERATEDDROPS'+Theabsorptionandscattcringcross-sectionsofwaterdropswereIshown,inChapters2and3,tobestrongfunctionsofdropletshape.:+Inordertoaccuratelypredictcross-sections,therefore,itisnecessary,+tohavesomemodeloftheexpec

82 teddropletshape.TheCloudPhysics++communi
teddropletshape.TheCloudPhysics++communityhasdoneagreatdealoftheoreticalandexperimentalworkondropletshapeandbreak-upinthequasi-steady-statecaseofdrops•fallingintheearth'sgravitationalfield.38"46InSection4.1thisworkisbrieflyreviewedandthegeneralizationtoaccelerateddrops!Ismade._InSection1t.2experimentaltechniques,employinghigh-speed,_photography,formeasuringdropletshapesandvelocitiesinthewind;+tunnel,arediscussed.InSection11.3theresultsofexperimentsondropdeformationandstabilityarediscussed.4.1HydrodynamicTheoryofWaterDropsForverysmallwaterdroplets,wheresurfacetensionisthedominantsurfaceforce,thedropletsassumeasphericalshape.Section11.1.1discussesdeformationsfromspherica

83 lshapeasaresultof!+hydrodynamicaccelerat
lshapeasaresultof!+hydrodynamicaccelerationofthedropletsbyanotherfluid.Sectionc:__1.I.2presentstheoscillationswhichresultfromtherestoringnature'ofthesurfacetension.Finally,inSection4.1.3theinstabilitiesiwhicharisewhenthesurfacetensionisnolongersufficienttomain-tainthedroparepresented.I1982022556-077 r77.!jII4.1.1DropDeformation!Thesteady-stateshapeofawaterdropbeingacceleratedhydrodynamicallybyanotherfluid,inthiscaseair,can,inprinciple,befoundbybalancingallforcesactingonthesurface.Theseforcesare•SurfacetensionCentrifugalforcefrominternalcirculationofthewaterinsidethedropAerodynamicforcefromairflowingaroundthedropHydrostaticpressuregradientwithinthedropresulting,,fromacc

84 elerationInvestigatorshaveattemptedtomod
elerationInvestigatorshaveattemptedtomodeldropletsbyincludingsurface38-42tensionwithdifferentcombinationsoftheaboveforces.Themostcomplete,althoughreasonablycomplicated,methodwasthatofIPurppacherandPiterin197038:j.TheyIncludedeachoftheabove1forces,eitheranalyticallyorsemi-empirically,intoFourierexpansioncoefficientsinelevationangledetailingthechangeinradiusofthe:drop.Theresultsagreedwellforquiescentfreely-fallingdropswithequivalentdiameterDlessthan5mmwiththeexperimentaleqresultsofPruppacherandBeard,39Includingsuchsecond-ordereffectslasthedimpleobservedonthebot*.omoflargedrops.Fordropslargerthan5Bin,thePruppacherandPitermodelunderestimatesthedeforma-i}:tionsomewhat.1Ifit

85 issufficienttoapproximatethedeformeddrop
issufficienttoapproximatethedeformeddropasanoblateiellipsoid,whichiscertainlyadequateforcalculationsofGansabsorptionandscatteringcross-sections,thenthesimplerapproachof1982022556-078 •+:;78ORIO!P_.LP,.-'...CZISi+i_OFPOORQUALITY._DGreen46providesaccuracyequivalenttothatofPruppacherandPiter.:.Greenlsapproach,in1975,wastoneglectfloweffectsaltogetherand_Includeonlythesurfacetensionandhydrostaticforces.UsingthistapproachthediameterofanequivolumetrlcsphereDecanbewritten•_intermsofthemajor-to-minoraxisratioA/Boftheellipsoidandtheaccelerationaas,!,(_)1/2=(_)i.,2[(A/s)2-2(A/B)1/3,111/2:',Deq(A/B)"l/6(4.I)t_,,,wheregistheaccelerationduetogravity,disthesurface,i,tension(n.b.,_isa

86 lsousedtodenotecross-sections),Pwisthe:I
lsousedtodenotecross-sections),Pwisthe:Idensityofwateranda/gIstheaccelerationofthedropingunits.t;Greenoriginallyassumedthatthedropswerefallingatterminal'velocityandthereforea/gwasunity.Themoregeneralcaseforanyquasi-steady-stateaccelerationisincludedhere.Equation4.1isplottedinFigure4.1forwaterat20°C.,4.1.2DropOscillationsWhendropsareperturbedfromtheirequilibriumshape,surface'tensionactsasarestoringforce.ThisrestoringforceresultsIn1'_droposcillations.TheoscillationscanbecharacterizedbyadiscreteI",t".tsetofnormalmodes.ThesemodeswerefirstidentifiedbyRaylelgh,'+6,47whoidentifiedtheallowedfrequenciesforasphereast+fn"r2-n(n"l)(n+2)_11/2(4.2)"_2pwDeq3,:Ii_+1982022556-079 •:JI7

87 9it,_,_-I_'/_LpAGEIS-q•,!OFpOORQUALITY,i
9it,_,_-I_'/_LpAGEIS-q•,!OFpOORQUALITY,i:!i::'2.-ft,I'1iq'i|_J_,J1•••|1982022556-080 80OR;G:;;,,LV,.C-"ISOFPOORQUALITYwherenisthemodenumber.Thefundamental_odeisn=2withafrequency=[t6o]112(4.3)f2._2PwOeq3="z_efundamentalfrequencyisplottedagainstDinFigure4.2eqformIllimeter-sizedrops.Inthefundamentalmodethedropisflattenedalongoneaxisinitially.Aquarter-cyclelater,thedropisspherical.Afterone-:halfofacycle,thedropisflattenedalonganaxisperpendiculartotheoriginal.Atthree-quartersofacycle,thedropisagainspherical,andafteronefullcyclethedropisbacktoitsoriginalCshape.Thevalidityofthesteady-statedropdeformationsdescribedinSect;on4.1.1afterachangeinaccelerationisrelatedtothefunda-men

88 talfrequencyinequation4.3.Fortimescalesl
talfrequencyinequation4.3.Fortimescaleslessthanone!fundamentalperiod1/f2thesteady-statebehaviorwillnotaccurately)•predictthetransientdropdeformation.Fortimescalesofseveralperiodsthesteady-statebehaviorwillbetheaveragedeformationintime.Aftermanyoscillations,viscousdampingwillcausetheoscillatlonstodecayandthedropdefomatlonwillbeaccurately/determinedbythesteady-statesolution.,:!ItisInstructivetoconsiderthepossibleeffectonthebacksca_ter:._".:cross-sectionofthefundamentalosclllatlon.ThechangeIndropshape!willcauseachangeinthebackscattercross-sectionviatheGans|_theory.Thecross-sectionwilloscillateatthefrequencyoftheitl1982022556-081 "ii:l_,_i,_?.O=Q_ALITYFigure4.2:Fundamental

89 oscilla_ionfrequency:300versusDf2eq(Hz):
oscilla_ionfrequency:300versusDf2eq(Hz):-200100,Oeq(ram)r1982022556-082 droposcillation.AnexampleofaperiodicaxisratioA/Bendtheresultingbackscattercross-sectionobisshownInFigure4.3.:_Theoscillationofc_bisapotentialsourceofdopplernoisein°meteorologlcaldopplerradar.The_boscillationresultsInsidebandsshiftedby_f2'asisshowninFigure4.4.ThisIsaltsourceofconfusioninthatf2isofthesameorderasthedopplerIshiftfromparticlesmovingatmeteorologicalvelocities.Thesei!'f_lsesidebandsignalscouldcauseabroadeninginthevelocityid|stributionfunctionobservedinconditions,suchasturbulence,which!++couldexcit++droposcillations.4.|.3InstabilityandDropBreak-UpIIntheprevioussectionitwasindic,3tedthatsur

90 facetensionistheprimaryrestoringforcemai
facetensionistheprimaryrestoringforcemaintainingdropletequilibrium.Whenother+forces,suchasthehydrostaticpressureresuitin£fromaerodynamicaccelerationbyanotherfluid,becomescomparabletothesurfacetension,thenthepossrbilltyofdropletinstabilityandbreak-up+,mustbeconsidered.Dropbreak,e+isImportant_.omodelsofatmos-jpherlcprecipitationasalimitonthemaximumsizeofdropsand,asi,Langmuirpointedout]t8thatthesmalldropletsresultingfrombreak-_pareamajorsourceofcondensationnucleiinwarmrain,i406servatlonsofdropbreak-uphavebeenmadebyHathewsandHason49i+andother_Oondrops_allingatterminalvelocity.Dropswerei'+stableIntheirro_+,_ly-oblateshapeuntiltheyreachedacertain+criticalbasediameter,whereas

91 mallconcavedepres,_lonInthebasedeepenedu
mallconcavedepres,_lonInthebasedeepeneduntiltheresultingbag-likeshapeevcntuallyburst.This++1982022556-083 Je,'_,_p_,,la';,:_.-'")_�_.":e_pez!Je[odlet[eaed_.-e/v1982022556-084 84L=:iinu|fdFrequency.IQuiescentdropiiiird'f2fdfd¤FrequencyDroposcillatingatthefundar,_ntalfrequencyf2"Figure_._Dopplerspectraforquiescentandoscillatingdrops.1982022556-085 processisknownasthe"bagbreak-upmechanism"andisshownschemati-callyinFigure1t.5•Measurementsofthemaximumstablebasediameteratterminalvelocityvarybetween6and9mmdependingonfactorssuchasturbulence•Theoretically,therearetwobasicapproachestothebreak-upproblem•Thefirstisessentiallydimensional.ItdefinestheWeber+'4numberNastherat

92 iooftheaerodynamicpres-_,Jreonthedroptow
iooftheaerodynamicpres-_,Jreonthedroptowe,:+thesurfacetensionstress+:PaAv2re]N=(4.4):.we0iwhereVreIistherelativewindvelocity,PaisthedensityofairandAisthediameterofthebaseofthedrop.ForvalueslargerthansomecriticalWebernumberthedropsareconsideredtobeunstable.iThecriticalvalueofNhasbeenmeasuredtobeoforder1049,50we+'Thesecondtheoreticalapproach,whichisalsoingoodagreementwithexperimentalresults,issomewhatmoreelegantinthatitconsidersthephysicalmechanismforbreak-up•In1964Komabayasietal.43recognizedthatthebottomsurfaceofalargedropfallingatterminal.:velocitywasinametastablestatewithaheavierfluid(water)beingsupportedinagravitationalfieldbypressureoveralighterfluid(air).Thebottoms

93 urfacewould,therefore,besubjecttogravita
urfacewould,therefore,besubjecttogravitationalorRayleigh-Taylorinstabilitleswhenperturbationsoccurredatthe%bottomsurface.Komabayasietal.assumedthattheperturbationswouldoccuraslinearcapillary-gravitationalwavesandthatthecriticalsizefor+!i.1982022556-086 +':-.+'".-,."+,"Q'bAi.ITYGBi"_,,-.'''.....,_.-/,,II!X/'TB,6RelativeWindFigure4.5Schematicrepresentationofthe"BagBreakupMechanism".1982022556-087 ¢:87.:dropbreak-upwaswhenthebasediameterAwasequaltohalfthecapillary-gravitationalwavelength.Fortypicalvaluesofthesurfacetension,theabovepredictsacriticalbase:llameterof8.55mmforfreely-t'al1ingdrops.TheKomabayasimodelwasfurtherrefinedbyKlettin1970,45whorecognizedthat,ingeneral,dr

94 opsflattenwithaxlalsymmetryand,•therefor
opsflattenwithaxlalsymmetryand,•therefore,instabilityshouldoccurwhenthebasediameterAmatchestheresonancesofcirculargravitationalwavesonthebottomsurface.•51FollowingthenotationofYihincylindricalcoordinates,theequationfortheperturbationofthebottomsurfaceofthedropS(t,p$)isS(t,p,_b)=CJn(kgp)cos(n_))exp(ozt)(4.5)whereCisaconstant,_istheinitialgrowthrateoftheinstabilityandJistheBesselfunctionofnthorder.The,nelgenvaluekisthewavenumberofthesurfacewaveandisdeter-g.1minedbythefol!o_vlngboundaryconditionsatthedropedgeiJJn(kgp)=0P=_A2(II.6)_.dJnArT"(kgp)"0P=T1_'7)':Theallowedsolutionsfortheperturbations,therefore,aretheBesselfun_1onmodesforac;rcularmembrane.Valuesofthezeroes.,_kA:+

95 areshownfordlf4_erentmodesinTable4.1.,,,
areshownfordlf4_erentmodesinTable4.1.,,,ThegrowthrateforeachmodecanbefoundbybalancingforcesQ|,1982022556-088 88Table4.1Thezeroe_ofBesselfunctionsandtheirderivatives(kgA),fromAbramowitzandStegunkA•_9_--2dJn(X)=0Zero#BesselFunction#Jn(X)-0d-_102.4050.000205.5203.8321I3.832!.8412I7.0165.331!I25.1363.054z28.4176.706!136.380_.201234.7618.015o1982022556-089 89OR:C2'.,_.LF,'._7-i$OFPOORQUALITYatthelowersurfaceofthedrop.Assumingthedropisbeingacceleratedataratea,thegrowthratecanbewrittenas51±2o(kA/2)a(pw-pa)(A/z)2=[-(kgA/2)2](4.8)(Pw+Pa)(A/2)3oNegativevaluesof(2implyoscillatorysolutionsforthepertur-bationwhilepositivevaluesimplyinstability.Themostunstable%modeIsthereforetheonew

96 hoseeigenvaluekgA/2maximizesequation4.8.
hoseeigenvaluekgA/2maximizesequation4.8._Onceinstabilityoccurs,theperturbationsgrowquicklyandtheperturbationfunctionS(t,p,¢}isnolongervalid;however,thegeneralcharacteristicsoftheBesselfunctionoriginsshouldbeobservableeveninseverely-perturbeddrops.The"bagbreak-upmechanismI'is,therefore,consideredtobegeneratedfromaperturbationconsistingofthemodewhichcorrespondstothefirstzerooftheaxisymmetricBesselfunctionJ0"ItshouldbenotedthattheJ0n,odeisnotnecessarilythemostunstable,aspredictedbyequation(4.1_).Thisisduetointernalflowinthedrop,whichhasbeenshowntooccurwlthc,taxialsymmetry.Thisflowtendstosuppressmodeswhicharenon-symmetric.TheJois,indeed,themostunstableaxisymmetricmodepredi

97 ctedbyequation4.8forthoseconditionswhere
ctedbyequation4.8forthoseconditionswhere'roagbreak-up"wasobserved.inspectionofequation_.8indicatesthatforlargerval_sof_accelerationordropletsizes,higher-ordersymmetricmodesshould:becomeunstableandthat,ifinternaldropletcirculationcanbesuppressed,thennon-symmetricmodesshouldalsobecomeunstable.1982022556-090 90Someofthesemodeshave,indeed,beenobservedandarepresentedinSection4.3._,4.2ExperimentalTechnieuesforWindTunnelObservationsofDropletbShapeandVelocity!Inordertovalidatetheextension,toarbitraryacceleration,ofthetheoriesofdropdeformationandinstability,exper:mentswereLicarriedouttomeasurethesephenomenaforvariousaccelerations.Dropswereobservedphotographicallyinawindtunnel.T

98 hesmalldropsizesandshorttimescalesrequir
hesmalldropsizesandshorttimescalesrequiredtheapplicationoftechniquesofhigh-speedstroboscopicphotography.Theexperimentalsetupandphoto-graphictechniqueswillbedescribedhere.ExperimentswerecarriedoutintheMITlow-turbulencewindtunnellocatedinBuilding17A.Thetunnelhascross-sectionaldimensionsofIft.by1ft.inthetestsectionandiscapableofvelocitiesfrom0to80mph.Anoptically-clearplexiglasstestsectionwasconstruedtoallowphotographicmeasurements.Tunnelvelocitywasmeasuredbyapltot-staticprobejustupstreamofthelocationwheredropletswereintroducedintotheflow.'iThephotographicsetupisshownschematicallyinFigure4.6.Dropswerephotographedusingashadowgraphtechniquetominimizeexposuretime.Thestrobehad

99 amaximumintensityon-cencerof18x106luxato
amaximumintensityon-cencerof18x106luxatonemeterandhadafullwidthathalfmaximumofeither.8or3IJsec,dependingontheintensitysetting.Thestrobeoutput#/wasdiffusedbyabuffedmylardiffuserscreenwhichbacklitthedrops.r,!i,Thephotographsweretakenbya35ram.camerawitha"macro"lensIi-I1982022556-091 i,91P",'_'-ISOFPOORQUALITY,#CameraDiffuserScreen•dO000r_:0ooOStrobe0_oVooMacru-Lensgtnd//_TunnelTunnelWallVelocity;Figure4.6Schematicdiagt-amoftheFhotographicSet-Up.1982022556-092 92Icapableof!to1magnification.Toprovidemaximumresolution,an!|extremely-fine-grainaerial-photographyfilmwasused(tradename:IKodakTechnicalPan)withahigh-contrastdeveloper(D-19).The!resolutioncombinedwiththissetupwasmeas

100 uredtobetterthan5mmforhigh-contrastobjec
uredtobetterthan5mmforhigh-contrastobjects.tThestrobewastriggeredbyasignalfromafiringcircuit.Forimeasurementsofdropletvelocity,twostrobescouldbefiredsequentiallyitoprovidetimeofflightinformation.ThedelaybetweenstrobescouldIbesetat12,2t_,(;4,93,or124laser,aswasappropriatetothe;velocitiestobemeasured,iThecamerawasgenerallyoperatedwiththeshutterheldopen(bulb?setting)toavoidfocal-planeshuttereffects.Thef-stopswere)determinedexperimentallyandarangeoff-stopswereshotoneach:!photographicrun.Atthebeginningofeachrolloffilm,thealignment!grldonthediffuserscreen,usedtohelpdeterminecameraposition,was:,photographed.Amillimeterscalewasalsophotographedinthefocalplaneprlortoeachruntopro

101 videlengthcalibration.Thephoto-graphicda
videlengthcalibration.Thephoto-graphicdatawasanal,,zedbymicroscopicmeasurementdirectlyfromthefilm.Intheexperimentsdiscussedinthefollowingsection,astreamofwaterdropletswasinJecte_',.,rticallyIt,tothehorizontalflow.Thesetupisshown_,;_|9'J_./_,.w',ichschematicallyviewstheinjectionapparatusalon(jthephotographicaxis.Thedropletstreamwasgeneratedbythenozzle,showninFigure4.7b,whichconsistedofa1.27mmbsveledorificeinaflatplate.Thebevelwaslocatedonthewatersideoftheplatetosuppressnozzleinstabilitieswhich1982022556-093 .+93OF__.....+OFPO_Rf_UALIIY++@=:=+Nozzle"]_Valve-TaPressure:Drop]etGauge+StreamyInjectionTubeTunnelf_PhotographicVelocity_JRegionFigure4.7aExperimentalSet-Uptophoto

102 graphdrooletdeformation.JUL00+0Figure4.7
graphdrooletdeformation.JUL00+0Figure4.7bSchematicdetailofnozzleusedindeformationexperiment.C1982022556-094 94causedthestreamtowanderoutofthefocalplane.Thestreamwa_shieldedfromthetunnelflowuntilthemidpointofthetunnelbya1.9cmplexiglassinjectiontube.Thiswasdonetoavoidfloweffectsnearthewall.Thenozzle_srunwithtapwaterat.either15or30PSIandlocated.!J55.5cmabovethebottomoftheinjectiontube.Thevelocityofthe:dropsatthebottomofthetubewasmeasuredtobe10.1_m/seeat15PSIand15.6m/secat30PSI.Themeanequivalentdropletdiameterswere1.41and.98ram,respectively.Thedropswerenotparticularlyquiescentatinjectionduetothelackofsufficientfalldistancebetweenthenozzleandtheinjectionpointforoscillations

103 ,generatedJatthenozzle,tobedamped.Thevel
,generatedJatthenozzle,tobedamped.Thevelocityshearbelowtheinjectiontubewasmeasuredbyamovablep_tot-staticprobe.Theresultsforallvelocitieswereverysim|lartothoseshowninFigure4.8forafree-streamvelocityof60mph.Thevelocityincreaseswithdistancebelowtheinjectiontubeuntil,atIcm,theflowvelocityisatthefree-streamvalue.Dropletswereobservedbetween1and3cmbelowtheinjectiontubei2InthephotographregionshowninFigure4.7a.InthisreTlon,thedre_shavenothadtimetoacquireappreciablehorizontalvelocity.Dropletaccelerationisinferred,therefore,byassumingthatthe.,_horizontalvelocityiszeroandthatthevelocityoftherelativewindthatthedropletfeels,Visthesumoftheinjectionvelocityrel'Vln/andthefreestreamtunn

104 elvelocityUJ_._1982022556-095 •95"OFpOOR
elvelocityUJ_._1982022556-095 •95"OFpOORQUALITYFigure4•8Flowvelocityversusdistancei:belowthe;,jectiontube•(FreeStreamVelocity=6Omph""60'E-iU-1120,"I'?'•iilllll.5i.o(.5i.oDistanceBelowtheInJectionTube(cm)t,°1982022556-096 96"22U2VreI=Vinj+(4._))TheaccelerationacanbecalculatedfromthehydrodlnamicforceequatlmtI_a=_-(CdPVreI(4.10)whereHwisthemassoftheclrnp,AisthebasediameterandCdisthedragcoefficient.Thedragcoefficientsareextrapolatedfromthe,measuredvalues_orfallingdropsandarediscussedinmoredetailinChapter5.Thevalu_ofR|_',obtainedfromthewphotograph|tally-measureddimensionsofeachdropbyassumingthat+thedropsareoblateellipsoids.4.3ExperimentalResultsOropsnbservedinthephotographi

105 cregio,',,describedInSection'.1t.2,1to3¢
cregio,',,describedInSection'.1t.2,1to3¢mbelo_theInjectiontube,werefoundtoexhibitt++o:typesofbehavior.Forsma,Idropsorlowtunnelvelocities,thedropsflattenedintooblateellipsoidswiththeaxisofrevolutlenalignedwiththedirectionofaccelerationresultingfromtherelativeflowvelocityVreI.For!_rgerdropsorhighervelocities,thedropsilsodeformedInitiallyIntooblateellipsoidsbuttheellipsoidsbecameunstableandthedropsbrokeup.DropsofthefirstkindaredescribedInSection_.3.1andtheunstabledropsaredescribedInSection11.3.2.1982022556-097 974.3.1DropDeformationTheaxisratiosA/Bofdropsdeformedintooblateellipsoidsisplottedagainsttheequivalentaccelerateddiameter(a/g)l/2DeqinFigure4.9.Thedeformationsaremu

106 chgreaterthanthosewhichwerepredictedbyth
chgreaterthanthosewhichwerepredictedbythesteady-statetheoryofSection4.1.Thereasonforthediscrepancycanbeseenbyconsideringthatthetimethedropsareinthephotographicregionisapproximately2msec.Bycomparison,:ithefundan_ntalfrequencyofthemeansizedrop(Deq=1.41ram)is205Hzwhichcorrespondstoaperiodof4.9msec.Thedropsare,therefore,inthephotographicregionforlessthanone-halfofacycle.Theobservedbehaviorofthedropsis,therefore,clearlytransientinnatureratherthansteady-state.Giventheabove,somei_terestingconclusionscanbedrawnfromthedatainFigure4.9.Whilethere_squiteabitofscatterinthedataduetothetransientbehaviorofthedrops,thereseemstobeaclearly-definedlimittothetranslentdeformations.Thisisind

107 icatedinFigure4.9.Thislimitisthemaximumv
icatedinFigure4.9.ThislimitisthemaximumvalueoftheoscillatingaxisratioAJBofadropwhichhassuddenlybeenacceleratedataratea.ThemaximumaxisratioisjustwhbtIs¢requiredtocalculatetheabsorptionandscatteringcross-sectionsfordropswhichexperiencesuddenaccelerationduetovelocityshear,suchasinturbulen:eornearanairfoil.4.3.2InstabiIityandDropSreak-UpAttunnelvelocitiesabove40mph,dropletswereobservedto"_becomeunstableinthephotographicregion,iSuetothehighvelocity,lL1982022556-098 98;OFPOOROUALI_1982022556-099 +-,,.,,-+L_+,-....+99•shearandtheresultingaccelerationfeltbythedroplets,higher-order__31Besselfunctionperturbations,predictedbyKlettwereobservedalongwiththe'*bagbreak-up"modeobserved

108 byothers.Thetheoretical+modesarediscusse
byothers.Thetheoretical+modesarediscussedinSection4.1.3.Atleastsixdifferentperturbationmodeswerepositively:Identified.TheseBesselfunctionmodesaredenotedbylettersInTable4.2alongwiththeboundaryconditionsandtheeigenvaiueskgA/2attheboundary.PhotographsofeachofthesixmodesareshowninFigure4.10wheretherelativewindisverticalinthephotographs.,TheaxisymmetricmodesAandCcorrespondtothefirstandsecondzeroesoftheJ0Besse]functionatthedropletedge.TheBmodetcorrespondstothefirstnontrivia]zeroofthederivativeofJ0on!_:theedge.InFigure4.llaplotoftheJo(kP)surfaceisprovidedI?toaidInvisualizatlon."NodeOcorrespondstothefirstzeroofthenon-symmetricfunctioni:Jl(kp)cos_.AsurfaceplotofthisfunctionIspr

109 ovidedinFigure4.12.Theperiodicdependence
ovidedinFigure4.12.Theperiodicdependencewithazimuthalangle,cos_,isclearlyvisibleinthephotograph.HodesEandFcorrespondtotheJ2(kp)cos2__ndJ3(kp)cos3_functions.Theyarealsomost+clearlydiscernedbytheirperiodicdependenceon_.HodeEis,.:twiceperiodicinoneazimuthalrevolution,whileI_odeFisthreetimes:periodicinonerevolution.TheInstabilitythresholdforeachmodecanbedeterminedfromthegrowthrateequation4.8.Itis?1982022556-100 lootOFPOORQUALITYI.i"-..................IIt_MODEAMODEDJiiij_,.....MODEBMODEEcMODECMODEFFlgure4.10Photographsofdropletsbecomingunstable,,;tnthesixobservedBesselfunctionmodes.4i.1982022556-101 I01oFPOORQUAMTy1982022556-102 /Ii!1021982022556-103 r,_.,¢,,i103ORIGi._'p,L

110 P_G_ISOFPOORQUALITYaa(Pw40-pa)k2�
P_G_ISOFPOORQUALITYaa(Pw40-pa)k2�(kgA/2)2(4.Ii)wherekgA/2istheeigenvaluefortheparticularmode,listedinTable4.2.Theaccelerationcanbefoundfromtheforceequation4.10.Thisyield,arelationshipfortheminimumunstablebasediameterAintermsoftherelativewindvelocityVreI.A)"2[,n.CdPa(2pa.pw)]1/4(kgA/211!2[Vrel]-l/Z(4.12)ObservedvaluesofAandVreIareplottedforthethreeiasymmetricmodesalongwiththeinstabilitythreshold(equation4.12)inFigures4.]3,4.]4,and4.15.ThethresholdwascalculatedassumingCd-1andMwwasequaltothemeanvaluecorrespondingtoa0.98mmdiameterdrop.Thereisgoodagreementwiththetheoreticalthresholdasallobserveddropsexceedthecriticalbasediameterfortheirparticularmode.Itshouldbenotedt

111 hatthereasonthat:thethreeaxisymmetricmod
hatthereasonthat:thethreeaxisymmetricmodeswerechosenforanalysiswassimply;itheprevalenceofdataforthesemodes,duetotheireaseofidentification.Thenon-symmetricmodeswhichwereobservedalsofulfilledtheinstabilitycondltion(equation4.11)butlessdatawas]availableduetomodeidentificationambiguitiesatcertainvlewlng.angles.'_lInconclusion,thesteady-statepredictionsofdropdeformation''"wereseentounderestimatetheobservedvaluesofaxisratioA/B.AmaximumtransientvalueofA/BwasdetemlnedandisshownIn:11"!1982022556-104 4."o_104Table4.2Surfacefunctions,boundaryconditions,andeigenvalues_ofob3ervedmodesJHodeSurfaceFunctionBoundaryConditionElgenvalue(kA/2)miiiJo(kp)JoA-02.405:,dJ0.__)_:BJ0(kP)d-'_"(-0

112 3.832CJo(kP)Jo(-_)-O*5.520DJl(kp)c°sJl(-
3.832CJo(kP)Jo(-_)-O*5.520DJl(kp)c°sJl(-_-)"03.832',zEJ2(kp)cos2J2(--_)-05.136"__tkA_,FJ3(kP)cos3do"T'"011.201._iii|i,i•i,iii*ThisboundaryconditioncorrespondstothesecondzerooftheJoBesselfunctionlI,1982022556-105 '3.0•Ve_,_2.5•L.q-10••__•v,2.0"Inst.Threshold_-a:-••,__._._.....Figure4.13Instability"t'hresholdandodservedvaluesofbasediameterversusVreI.1.0HODEA|0.5I|iUi|ii_oz_3b_sVre!(_/sec)•\/1982022556-106 1060:3,5•03,Q••!°•-j•••_2,5•i1_••I:mInstThreshold.,_,,_----..1,5Fi_gure4.14InstabilitythresholdandobservedvaluesofbasediameterversusVreI.1.0MODEB0,5IIm••il.20253035VreI(H/see)•I11982022556-107 Io7OP,iG_L"_LP_G_IS",1OFPOORQUALITYtII'h.O,•1•3.5•'!"!i•!3.0••!'_••8z.5•==Ins

113 t.Threshold"_'_"----.....----..e)2.01.5F
t.Threshold"_'_"----.....----..e)2.01.5Figure4.15InstabilitythresholdandobservedvaluesofbasediameterversusV1.0re|"HODECe.nm==_a,m,mm_O.S20253035VreI(Wsec)1982022556-108 IO8Figure4.9.TheBesselfunctionperturbationsgivingrisatodropInstabilitypredictedbyKlett1t5havebeenobservedexperimentally.Theperturbedshapesandtheconditionsforinstabilitywerefoundto,_greewet|withthepredictedva|ues_I?1982022556-109 109C!4_PTER5DROPLETTRAJECTORIESTheassessmentofthetrajectoriesofdropletsdrivenbyanotherfluidIsanimportantpartofmanyfieldssuchasaircraftIcing,cloudphysicsandcombustionphysics.Acomputercodewhichsimulatestwo-dimensionaldroplettrajectories,eitherinafreely-flowingfluidornearanobjectsu

114 chasanairfoil,hasbeenwrittenandIsdescrib
chasanairfoil,hasbeenwrittenandIsdescribedinthlschapter.InSection5.1thecomputercodeisdescribed.In_Section5.2windtunnelexperimentsdesignedtovalidatethesimulationarediscussod.InSection5.3someresultsofcaIculatlorsfortrajec-toriesnearanairfoilarepresented.ThesimulationIsextendedfurtherInChapter6,wherethemicrowaveheatingofdropletsnearanairfoilisconsidered.,5.1Computer$lmu|ationofDropletTrajectoriesThefirstattemptstocalculatedroplettrajectoriesInf_owfie|dsperturbedbyobjectsweremadeIn1940.Glauert53studiedtrajec-ctorlesnearcircularcylinderswhichbecamethebasisofInstrumentation54tomeasuretheliquidwatercontentofclouds.KantrowltzstudiedtraJectorle,_andtheirrelationshiptoaircraftIc

115 ing,BothGlauert\andKantrowltzintegratedt
ing,BothGlauert\andKantrowltzintegratedthedropletdlfferentlalequationofmotionanalytlcally.Theapproximationsrequiredtodothlslimitedtheuse-fulnessofthesolutions.In19115,LangmulrandB|odgett55developedamoregeneralapproachforthetrajectoriesnearslmp;egeomtricalshapeswhichrequiredtheuseofadifferentialanalyzer."1982022556-110 110Thetechniqueofiteratlvelysolvingthedropletequationsofmotion,byfinitedifferencetechniques,applicabletoarbitrarytwo-5Gdimensionalflowfields,wasproposedbyBergrunin1947.Thecomputationsweredonebyhand,whichmadecalculatingevenone\:trajectoryamajoreffort.In1953ao-oupledbyBrunandRlnaldodevelopedananalogcomputerwhichmechanicallyintegrateddroplet57-59trajectories

116 .Theycomputedtrajectoriesnearcylindersan
.Theycomputedtrajectoriesnearcylindersand3NACA65,-200seriesairfoils.Theseanalogcomputations,whileimpressive,werelimitedduetothedifficultyinmodifyingtheprogrammedflowfields._Thecalculationofdroplettrajectories,bytheit_rativesolutionofthedropletequationofmotion,Iswell-suitedtothecapabilitiesofmoderndigitalcomputers.Verylittleworkhas,however,beendoneinthisarea.Thosecodeswhichdousedigitalcomputershavegenerallybeendevelopedbyaircraftcompaniesprivatelyandareproprietary•:AcomputercodetocalculatedroplettrajectoriesInanarbitrarytwo-dimensionalflowfieldhasbeenwrittenandIsdescribedinthissection•Section5.1.1discussesthedropletequationofmotion.InSection5.1.2theIterationalgorithmisp

117 resentedwithaflowchartofthecode.Finally,
resentedwithaflowchartofthecode.Finally,inSection5.1.3adiscussionoftheappropriatedragcoefficients,tobeusedforwaterdroplets,ispresented.5.1.1DropletE,.quationsofMotion,Theequationofmotionforadropletbeingdrivenbyanotherfluid_canbewrittenfromthehydrodynamicdragequationasCd._._.1"TPaA_lu-_1(_"v')(5.1)r1982022556-111 ORIG,,,--PAGEiS!!IOFpoORQUALITY,•,+..L--lbWhereaandvaretheaccelerationandvelocityvectorsofthedrop,'tuisthevelocityvectorofthefluid,_isthe.massofthedrop,-Paisthedensityofair,A_Listheareaofthedropperpendicular:\totherelativewindandCdisthedragcoefficientofthedrop.Itisnotedthatv_l-I_-_1(5.2)\isthemagnitudeoftherelativewindfeltbythedrol_aswasdiscussed,+|tlChapter4.T

118 hedropmassisD.+...-(5.3)i:+whereOeqIsthe
hedropmassisD.+...-(5.3)i:+whereOeqIsthediameterofanequivolumetricsphereandPwisthe",i_densityofwater.Inordertosimplifytheuseofequation5.1,itisassumedthat:1theareaAa.ofthedropletisthevalueforanequivolumetrlcsphere+andthatanydeviationfromsphericitywlllbeincludedinthedragcoefficientCd.Therefore:1BA_-lt(-_)2(5.4).+2,!Combiningequations5.2,5.3,and5.4withequation5,1yields:iPaVrel(_._(5.5)"!i;"3Cd_Deq,I+i10&1982022556-112 I12OR!GI,_'_4.....OFpOG,_Q_;ALITY_._TheReynoldsnumberRefordropletsisdefinedas::.Pa(5.6)Re--_OqVr_!wherepisth:dynamicviscosityofairplottedagainsttemperature60inFigure5.1fromthevaluesintheSmlthsonlanMeteorologicalTables.Theequationofmrtlon(equation5.5)canbewri

119 tteninthesimplified;formas]i_CdW_-(-,-_)
tteninthesimplified;formas]i_CdW_-(-,-_)K]--('_-v_(5.7)wherePI;Ka.__'wOeq2(5.8),,,whichisconstantoveraparticulartrajectory.Equation5.7Istheequationofmotionusedinthetrajectorycode.IthasaformsimilartotheequationofmotionfirstproposedbyCdRe55LangmuirandBIodgett.Thecoefficlent(--_--)ispresumedtobeafunctionoftheReynoldsnumberandisdiscussedinsection5.1.3;KIsconstant(assumingOremainsconstant);and(_-_istheaeqrelativewindfeltbythedroplets.:11982022556-113 0,__'''_:'_r-,',-,_--oFFCC,z,CQU_L|i'y\/:_..I.9'Figure5.1Dynamictemperature,viscosityversus./----/1.8"T:X,1.7',_21.6''.....,jill,L:-20-I00IO203040Temperature(Oc)1982022556-114 •-+.,..•+....|1145.1.2IterationAlgorithmtiAflowdiag

120 ramfortheiterativecomputationoftwo-dimen
ramfortheiterativecomputationoftwo-dimension.aldroplettrajectoriesisshownInFigure5.2.ThreegroupsofInputparametersarerequired.Theyaretheflowfield,theInitialdropletconditionsandtheatmosphericconditions.Thecodecalculatestrajectoriesforanarbitraryknowntwo-dlmensionalflowfield.Thevelocityfield_(x_),wherexisposition,mustbeInputeitherthrough+;!flowequationsorbymeansoftabulatedvalues.Iftrajectoriesnear_bodiesarebeingcalculated,thenthepositionofthebodysurfacesmust4tbeincluded.iTheInitialconditionsforthedropletsaretheequivalent,tlameterOtheInitialpositionX'_oandtheInitialvelocityvO.Theieq'atmospherictemperatureandpressurearerequiredtocomputealr•+I,Idensity,viscosity,andKiiI-a,On

121 cetheinitialconditionshavebeendetermined
cetheinitialconditionshavebeendetermined,theequationofIn_)tlon(equation5.7)lssolvedattheInitialposition.Thevelocity.4b.aivandI_sitlonxarethenIncrementedonetimestepAtbythefirstorderEulerIntegrationequations.Ib.Jbxi=viAt+xI(5.9):vi+I=aiAt+vi(5.10)tqtThevalueofAtwastakentobeI0mlc_secondsInmostofthesimulations,althoughitcouldbevariedISnecessary.Oncethe_sitionandveloc!tyhavebeenIncremented,thecodechecksforhydrodynamicstabilityofthedrop(asdeterminedbythe!1982022556-115 115Off:G:;;':;.:-;SOFPOORQUALIFYInitlalpositlonAmbienttemp.,InitialvelocityAnbientpressureDr°plelsize!Calculateacceleration_Bodyshape,-CdRe,,--__BodysizeIa"2-EE-_a'v"u'Freestrea,_veI,J,_IstOrderEulerInt.Scheme

122 Steptime_-increment_i"vi&t
;iAt_1+1"_1&t
Steptime_-increment_i"vi&t
;iAt_1+1"_1&t+"_iii.l+ll"yeFldstrajectoriesIISthedr°phydr°dynamlca|ly[.stable?++Figure5.2Flowdiagramfortrajectorycode.L1982022556-116 116stabilityconditionsinChapter4)andforcollisionwiththebody.If,:eitheroftheaboveconditionsoccur,thenthecodekicksoutofthetIterationloopandoutputstrajectoriestothatpoint.Ifthedrop_emainsstableandnocollislonsoccur,thenthecodeo/clesbackand:calculatesaccelerationatthenewdropletposition.Thecodethenprogressesthroughtheloop.Valuesofdropletvelocityandaccelerationcanalsobeoutputalongwiththetrajectories.AdditionalforcessuchasgravitycanbeIncludedintheacceleratlonequation.5.1.3DragCoefficientsInpreviouscalculationsofdroplet

123 trajectories,Investigatorshave5354,useda
trajectories,Investigatorshave5354,usedavarietyofdragcoefficients.GlauertandKantrowitzassumedStokes(viscous)flowaroundsphericaldroplets.LangmulrandBiodgett55•observedthatataircraftvelocitiesandmeteorologicaldropletdiameters,Stokes'lawdoesnothold.Theyproposedusingdragcoefficientsmeasuredforsolidspheres.iInordertocheckthevalidityoftheLangmuirandBlodgettdrag'CdRe!coefficientsCd,orequivalently(2--_--),thecodewasusedtocompute'!theterminalfallvelocityofdropsintheearthtsgravltatlonalfield.TheresultsareshownInFigure5.3,alongwiththewell-acceptedexperimentalresultsofGunnandKlnzer.61TheLangmulrandBlodgett:coefficientspredictahigherterminalvelocitythanisobservedexperimentally.Ther

124 easonforthediscrepancyisthattheLangmuir!
easonforthediscrepancyisthattheLangmuir!andSlodgettcoefficientsneglectsucheffectsasdropdeformationanditheIncreaseddragduetoturbulenteddiesdownstreamofthedrop.TheseeffectsIncreaseCdandreducetheteminalvelocity.l,,InordertoImproveontheLangmuirandBIodgettcoefficients,itI1982022556-117 117ORIG.T,_:q,LPP,_EIS•OFP03._C_U/;L!TY,PSSsSFigure5..)SfsS1',Temlnalvelocityofdrops(R/see)S_versusdiameter(mm).SsssSJ1sSJS///•ooo.ooooo//080""I/ojo°S/o:;"/o8SSo/o•0C_GunnlindKlnzer(exot.)S/_,......CalculatedbythecodeS0usingLIngmuirandBlocl_tt/0Idrogcmffl¢lents.2/d1982022556-118 ri'•118iwasdecidedtoInverttheGunnandKinzerexperimentalresultstodetermineasetofempiricallymodlfleddragcoefficients.T

125 hemodlfledCdRevaluesof(-_)areplottedagai
hemodlfledCdRevaluesof(-_)areplottedagainstReynoldsnumberInFigure5.4alongwiththeLangmuirandBIodgettvalues.Theex_rlmentalvalues-cutoffatRe-3500,duetodropbreak-up.ItisassumedthattheCdRevaluesof(--_)canbeextrapolatedtohigherReynoldsnumbers.Therefore,fromFigure5.4iCdRe=1.(;99x10-5(Re)l'92Re&#x Tj ;ć ;� TD;&#x /SS; 10;&#x Tf ;&#x 000;3500(5.11)Usingthemodifieddragcoefficients,theterminalvelocitywas'_calculatedbythecodeandIsplottedInFigure5.5.Thevalues,asexpected,agreewellwiththeGunnandKlnzerdatafortheatmosphericconditionsunderwhichtheexperimentswereperformed.Thevaluesoft1080mbpressureand20°Caretypicalsurfaceconditions.Figure5.5alsoIncludesterminalvelocitiescalcu

126 latedfortypicalIcingcondltionsof750mband
latedfortypicalIcingcondltionsof750mbandO°C,wherethelowerairdensitycauseslessdragandaresultlnglyhigherterminalvelocity.TheabovemodifieddragcoefficientshavebeenIncorporatedintothetrajectorycode.ForReynoldsnumbersgreaterthan3500,equation5.11isused.ForReynoldsnumberslessthan3500,thecodeInterpolatesbetweentabh_dCdRevmlue_of(--_--).Thevalueswhichhavebeenused.ereIistadInTable5.1forreference.1982022556-119 119OFPO0_QUALIT_60(t,I')Figure5.4DragcoefficientversusReynoldsnumber.II40(/II,,lII_"tnferr'edfrom/'I200GunnandKinzerIodataI/ii/'Sltjs/tlIIIs//A)60"/fs/40./_LangmuirandBlodg_tt/values',:;-/20-••A_SJSjSI0_'_"400600100020004000600010000Re1982022556-120 +,.+_It120IIt+"I1IP,t+.,

127 +,,'C+.:+_LIIY'I:,+j_......_,*•]+,+t'1J•
+,,'C+.:+_LIIY'I:,+j_......_,*•]+,+t'1J•FI9ure5.5tt+:TERMINALFALLVELOCITYOFDROPS(METERS/SEC)VERSUSDIAMETER(ram.)i,,,_10",/s_.e...o..O"i8,Ofi,/".o"i/°o.r"i/.o/"',,+I+e/,:**•p-//'._.•••oGunn&Klnzer'(expt.),,/_'4o_l_Bnb20°C-.1485cm2/s4_e°*.-------Code(modifiedCd)SurfaceCond|tlonst1080rob20°C-.1485:m:elsi/_-------Code(modifiedCd)lctngConditions;_'_760ml)ZO°C-.178_m21s2/¥lQt1t%(-_,_+1982022556-121 "-i121OFPUO__C'ALII"YTable5.1Valuesof(CdRe/24)asafunction,ofReynoldsnumber.,a•ReCdRe/Z4ReCdRe/Z&-i0.001.0068.73.6E4ii0.051.00998.?4.4090,i1.013134.05.170i0.21.037175.05.943I0.41.073__.0.06.6830.61.103269.07.5200.81.142372.0_.4801.0].1764_3.011.47_'1.71.201603,0!3.691.41.225731.016

128 .08!1.61,240066.01B._6"1.81.2671013.021.
.08!1.61,240066.01B._6"1.81.2671013.021.27_2.01.2851.164.024.01I2.51.3321313.027.031-3.01.3741461.030,32i3.51.4_21613.033.014014 71 64.o37.56i5.01.5131915.04!.49iI6.01.572206&.04_,.54_,jO.O1678_11.05012]I10.01.7822357.054.83,12.01.9012500.059.89:,14.02.0092636.0&5.2416.02.109..._?_..071.0318.02.1982905.076.8620,02.2913033.083,4125.02.4093164.0g_'.TB30.02.6?33293.0o_.0535.02.0513423.0103.6940.03.0133549.0111.05"50.03.3271982022556-122 t1225.2WindTunnelValidationofComputerTrajectories_Thecomputer-genera_ddroplettrajectorieshavebeencheckedbycomparisonwithwindtunnelmeasurementsofdropletvelocity.In•_•Section5.2.1,velocitymeasurementsofdropletsinjectedIntoafreely-flowingwind

129 tunnelarecomparedwithpredictedvalues.InS
tunnelarecomparedwithpredictedvalues.InSection:5.2.2dropletsareobservedjustupstreamofacylinderwheretheflowfieldlsspatiallyvaryingbutwellknown.Themeasuredvelocitiesarecomparedwiththepredictedvaluesand9ooJagreementisobtai_ed.5.2.1DropletsInjectedintoaUniformFlow•Inordertoverifythecomputer-predictedtrajectories,dropletswereobservedintheset-upshowninFigure5.6.DropletswereInJectedIntoafreely-flowingtunnelperpendiculartothe.flow.Dropletvelocltycomponentsalongthetunnelaxisweremeasured125cmdownstreamoftheinjectionsite.ThemeasurementsweremadebythedoublestrobephotographictechniquediscussedInSection4.2andthewlndtunnelwastheHITIft.x!ft.facilitydescribedInthatsection.Theobserveddro

130 pletvelocitiesforthreevaluesofthetunnelf
pletvelocitiesforthreevaluesofthetunnelfree-streamvelocity(45mph,60¢pl.,75mph)areshownInFigure5.7alongwlththecomputer-calculatedvalues.ThereIsgoodagreement_overtheobserved0.15tot.Ommrangeofequivalentdropletdiameters.Attheobservationsitethesmalldroplets,duetotheirlowInertia,areclosetothefree-streamvelocity,whilethelargerdropletsareslowerandstillaccelerating.Thediscrepancybetweenobservedandpredictedvaluesofvelocityareattributedtodifficultyinmeasuring"1982022556-123 123r__I",2s_�.PI,o,,.1.__"t.-'_,,,_JFree...---_"//pStreamVeI.!Photographic-RegiorlIiiI.|",LowertunnelwallFigure5.6Experimentalsetupfordropletvelocitymeasurement_inauniformflow.&!1982022556-124 ......+..

131 .ID,124u,,,_......QI.i_LIT¥OFpO0_III.--I
.ID,124u,,,_......QI.i_LIT¥OFpO0_III.--IIIIIIII!o#II.-:III._.IIIIII|IaI..a06ll__,_eIIIII!!I:i;gloeI,,o_,,,_..Q.8-iII&,,11I0.0m,.?7*""_i/-;_"'/�_i_�,"(]!)_¢0I,f_:'-!_,"o,6,'_,_/,_!,_/11.!••14btSt#PIosoI,_,IeII,IiIIIiIiiI....eI•,_ilt1_-(qclU)fitoOleA1982022556-125 125'iD,slightvariationsinthetunnelvelocity,andperturbedInitialeq•conditionsduetodropletbreak-uporotherInJectioneffects.i•i5.2.2Droplet.TraJectoriesNearaCylindert!InordertoconfirmthatthecomptuercodeaccuratelypredictedItrajectories,Inregionsofspatially-varyingflow,observationsofdropletvelocityjustupstreamofacyllnderweremadeinthewindtunnel.Acylinderwaschosenbecausethetwo-dimenslonalflowaheadofacyllnd

132 erIswellknowntohaveasimpleanalyticalform
erIswellknowntohaveasimpleanalyticalform.AnexampleofthetrajectoriesgeneratedbythecodeIsshownInFigure5.8.Inthiscasethecylinclerhada10canradiusandthe?!free-streamvelocitywas60m/secflowingtotheleft.Fourdropletswithdiametersof5,!0,20and40micronswereIncidentonthecylindertAfromaposition1.25cmabovethestagnationstreamline.TheeffectsofInertiacanbeclearlyseenasthesmallerdropletsareturnedbytheflow,whilethett0-mlcrondropcontinuesonandImpactsthecylinder.TheeffectsofInertiaontheImpingementtrajectorieswlllIbediscussedfurtherinSection5.3.Theexperimentalset-upusedtomeasurevelocitiesneartheJi_cylinderwasverysimilartothatdescribedinSection5.2.1endIsshowninFigure5.9.Thephotographicregionw

133 ascenteredabovethestagnationstreamlineah
ascenteredabovethestagnationstreamlineaheadofthecylinder.TraJectoriesaheadof:_twocylindersofdiameters11.25andZ,4anwereobserved,ThecylinderswerepaintedblacktominimlzestrayreflectionsInthephotographs.Thefree-streamvelocitywasvariedfrom1t5to75mph.fDropletswereinjectedperpendiculartotheflow125anupstreamofthef,i"1982022556-126 12C!!!I"'OFPO02¢_'-.'--:_•(S£1NflN3ill)SIXV_VOILa3h1982022556-127 127OFPOORQJALITYogI"125cm-.////Free_______._StreamvelCyIinderv"--_'Phot;._raphICRegionlILowertunnelwallFigure5.9Experimentalset-upfordropletve'acitymeasurementsaheadof:acylinder.1982022556-128 •q..128cylinderaxisandthevelocitiesweremeasuredbythedoublestrobe:technique.Severalhundreddropl

134 etvelocitieswereobservedandcomparedtothe
etvelocitieswereobservedandcomparedtothecomputedvalues.ThemeasuredandcomputeddropvelocltleswereFoundtoagreewlth3standarddeviationoflessthan_;ofthepredictedvelocityforaccurately-photographeddroplets.tAs•naside,itwasobservedthatwhenI•,gedropletsimpactedth•cylinder,theysplashedInratherconstantpatterns.An•xampleisshowninFigure5.10•.Theresultoftheimpactofalargedropiwas•collectionofsmallerresidualdropletswithvel_cltlesoforderIone-tenthofthefree-streamveloc|tydirectedradially_•yfromtheiIpointoffirstcontactwiththecylinder.Splashpatternsofthisssort•rethoughttobethecauseofthedouble-hornedIcebuild-up_sk•tchedinFigure5.10bobservedonalrfoilsundercertainicingcondltions.:Inconclusion

135 ,theexperimentalevidencesupportsSheresul
,theexperimentalevidencesupportsSheresultsof:thecomputersimulations.Thecode,can,therefore,beappliedtoothersituationsofInterestwithsomeconfidenceIntheresults.5.3SimulationResultsNeartheLeadingEdgeofanAirfol,.IInthissection,resultsofdroplettrajectorycomputationsnear•nairfoil•represented.Th•airfoilI••dlngedgeisslmul•t•dby•two-dimensionalhalfbody,anexampleofwhichisshownInFigure5.11.Thehalfbodywaschosenbecauseitcloselyresemblesthefrontofanairfoilandthevelocityfieldequationshavea#•rticularlysimpleform.Iftheflowisassumedtobeinviscid,thenthevelocityIIJ1982022556-129 Figure5.10aPhotographofalargedropImpactingthecyl-Indersurface.'_/,)ll1982022556-130 ic;.!,._]-.;,_LF.r,C,7..!'3:

136 OFPOORQUALITYFigure5.105Sketchof•"double
OFPOORQUALITYFigure5.105Sketchof•"doublehorn"icet,_tion.i1982022556-131 !_lI131OFPOORQUALITYi:iFreeStream.XXi•.(XSI_hIIt|Figure5.11Viewofthe2Dhalfbody.1982022556-132 :132:;potentialfunctioneissimplythesumofuniformflowinthex-"62'directionandasourceattheorigin_:4x2"4_(x,y)=U(x+xsIn+y2)(5.12)where-xIsthexcoordinateoftheleadingedge'S,zx-h/_r(5.13)S_.L;_wherehisthehalf-thicknessofthebody.Thevelocityvisrelatedto_byxy)-(x,y),Thehalfbodyequationscanbeusedtosimulatedropletimpingementontoanairfoil•TheonlyweaknessIstheassumptionofan0°angleofattack_implicitinthehalfbodyflowmodel.EffectssuchasdropletInertiaanddropletcollectionefficiency,whlchareweakfunctionsof_,canbeadequatelyinves

137 tigated.InordertoItudy"effectswhichdepen
tigated.InordertoItudy"effectswhichdependstronglyon_,theparticularflowfieldofinterestmustbeinsertedinthecode.InSection5.3.],sometwo-dimensionalImpingementtrajectoriesLarepresented.Someresultsondropletcollectionefficiency,andtheireffectontheImpingingmassdistributionfunction,arediscussed.InSection5.3.2,theeffectoftheairfoilonthekinematicsofdropletsisstudied.Finally,InSection5.3.3somaadditional,_simulationsarepresentedwhichdemonstratetheflexibilityofthemodel.•'k_•1982022556-133 ,,,C1335.:3:1Two-DimensionalImpingementTraJectoriesInFigure5.12,thetwo-dlmenslonaltrajectoriesareplottedfordropletsofvariousequivalentdiameter.Theeffectofdropletinertiaisapparent,asitwasInthecaseof

138 thecylindersInSection5.2•!Sma|ldropletsa
thecylindersInSection5.2•!Sma|ldropletsaresweptbythealrfoi],whiletheinertiaofthelargedropscausesthemtoresistchangeindirectionandtheyImpacttheairfoil•IntheexampleinFigure5.12,thefree-streamvelocitywas60m/sactotheleft,andthebodythicknesswas20ca.Theatmosphericconditionsweretypicalicingvaluesof750mbpressure(equivalentaltltudeapproximately10,000feet),-lO°Ctemperature,andgravltationalaccelerationwasneglected.Inthefollowing,theseconditionswll!tbeassumedunlessotherwisenoted.'InFigure5.13,thetangentialtrajectorytothehalfbodyisdrawnfora20-microndroplet.TheInitialverticalpositionofthetangentialtrajectorydefinestheheightoftheImpingementwindow;hi(l)eq).TheImpingementwindowisthezone

139 inwhich,foragivensize,:alldropletsImpact
inwhich,foragivensize,:alldropletsImpactthebody.Forthesymmetricalcaseofthehalfbody,.theImpingementzoneiscenteredaroundthestagnationstreamline.Thereforetheheightoftheimpingementwlndowhi(Deq)isJusttwicetheseparationofthetangentialtrajectoryandthestagnationstreamlinewellaheadofthebody.Fornon-symmtrlcalcases,theh$lghtoftheimpingementwindowIstheInitialseparationbetweentheupperendlowertangentialtrajectories.InFigure5.14,halftheheightoftheImpingementwindowisplottedagainstOeqfora20-cmthickhalfbody.Thefree-streamvelocitiesstudiedwereI10,60and80m/sac.Itisinterestingtonotethat1982022556-134 L134.+1I(SIINNN3BI)SIXVIV31_3h.1982022556-135 135J"_1omcl._ALpaG_is"OFpoORQUALITY_"_'•(SII

140 NnN3BI)SIXVqV3II_13^1982022556-136 136OR
NnN3BI)SIXVqV3II_13^1982022556-136 136ORIGINALPAGEISOFPOORQUALITYFIGURE5.14IMPINGEMENTWINDOWVS.EQUIVALENTDIAMETERft.88"g,II)ItI!I.25.5B.75.1gg.125.15B.J1;EOUIVALENTDIAMETER(MICRONS)1982022556-137 OR_C.,"--;"-'vr.,.,-1.37OFPoorQUALITYdropletswithDeqlessthanI0micronsarealwayssweptbythehalfbodyatthesevelocities.AboveI0micro..,theheightoftheImpinge-mentwindowIncreasesroughlylinearlywlthdiameterupto!00micronswherethehi(Dq)beginstotapertotheasymptoticvalueof10cm.ThislinearbehaviorwasalsonbservedbyBrunetal.intheirsimulations.TheslightIrregularitiesInthehIcurvesarenotphysical.TheyarearemnantofthediscretenatureofthesamplingalgorithmandthereforeshouldbeIgnored.Aseriesoftrajector

141 ies,withvaryingInitialverticalposi_ions,
ies,withvaryingInitialverticalposi_ions,areshownInFigures5.15and5.16for20and40microndroplets.,tThelimitoftheimpingementwindowIsalsoshown.The20-microni!dropletsareclearlymoreinfluencedbytheflowthantheir40-micron,_o0unterparts.dnc(0eq)ThedifferentialcollectionefficiencyoftheairfoildDeqIstheratioofnumberofdropletsofagivensizeactuallyimpactingthebodytothenumberofdropsofthatsizeInitiallyInthevolumesweptoutbythebody.ItcanbewrittenIntermsoftheheightoftheImpingementwindowasLdnc(Deq)h.i(Oeq)_:.-,,(5.15)dOHeqiwhereHisthethicknessofthebody.Thedifferentialcollection,efficiencyIsrelatedtothetotalcollectionefflcler,_/rtcbyIntegratingoverthedropletsizedistributionf(_q)_'-1L'I"chicl;n

142 c"J qeqf(oeq)doeq(5.16)'0t1982022556-138
c"J qeqf(oeq)doeq(5.16)'0t1982022556-138 (SIINnH301)SIXV_V3IIB3A1982022556-139 (SIINNNOBI)SIXVlVOI£_3A1982022556-140 LOR!GINALPAGEIS;140OFPOORQUALITY:Sometypicalmid-clouddistributionfunctionsareshowninFigure5.17.Forcumulusclouds,thefunctionsarefairlysharply:peaked.Themeandropletdiameterisoforder20micronsandtendstoIncreasewiththesizeandageofthecloud.Stratuscloudstendtohavebroaderdistributionfunctionswithsimilarvaluesofthemeandropletdiameter.AnapproximateformoftheclouddistributionfunctionistheKhrglan-Hazandlstribution6Deq2f(Oeq)-const,exp{3Deq/Deq}(5.17)!ii!where_isthemeaneffectivedropletdiameter.JeqdHCThedistributionofimpingingmassd_-canbecalculatedfromeq1thesizedistrib

143 utionfunction,thedifferentialcollectione
utionfunction,thedifferentialcollectionefficiencyandthevolumeofthedrop.dHcII".(0eq)f(Oeq);d-_eq-Vnc(5.18)Combiningequations5.17and5.18andnotingthattlcisroughlyproportlonaltoOeqforcloud-sizedroplets,andthatthevolumeVIsproportionalto03yields:eqdHcOeq6-const,exp(30eqlOe-'_q(5.19)_,eqAnexampleoftheKhrglan-HazandlstributlonfunctionandtheimpingingmassdistributionfunctionisshownInFigure5.18forequalto20microns.Itisinterestingtonotethatthepeakvalueofthe!I't1982022556-141 I141G,,..,+_-[3OF......._+._+..,,_++y1982022556-142 1112OR|GI_,LrL..r,.-...-',.:,:.:OFPOORQUALh"/1982022556-143 143ORIGIN_,LP_..-"i_OFPOCRQUALITYmessdistributlonfunctionoccursat40microns,whichistwicethemeandiam

144 eter_'-inthecloud.Indeed,theextremevalue
eter_'-inthecloud.Indeed,theextremevalue-eqconditionforequation5.19yields-z5--(5.19)'_OeqIdR¢eqmaxdTeqForraindrops,thedifferentialcollectionefficiencyisapproxi-matelyunity.Theacceptedslzedistributionfunctionforrain664IstheMarshall-Palmerdistribution3furtherrefinedbyAtlas.f(0eq)-const,exp{-3.67Oeq/Oe-'_}q(5.20)ThisresultsInanimpingingmassdistributionofdRc3exp{-3.67--}(5.21)dT"const.DeqOeq/Deqeqwiththemeanvalueof0forrainbeingoforderseveralmilli-eqmeters.?/5.3.2KinematicsofDropletsontheStagnatlonStream,neThegeneraleffectofthehalfbodyflowfieldonthekinematicsofimpingingdropletscanbeobservedinthecaseofdropletswhichi.flowonthestagnationstreamline.Dropletsonthestagnation+strea

145 mlineareparticularlyconven!ent,Inthatthe
mlineareparticularlyconven!ent,Inthatthekinematicsare.assentiallyone-dimensional.InFigure5.19,theaccelerationofdropletsisplottedagainst1982022556-144 144ORIGI?!ALPACEISOFPOORQUALITYFIGURE5.1gACCELERATIONVSDISTANCEONSTAGNATIONSTREAMLINE.\3.j-2"':V_I-5MICRONS:IMICRONS"iUMICRONS'48MICRONS:1B2B3848'DISTANCEAHEADOFHALFBODY(CM),I1982022556-145 4145distanceaheadofthehalfbody,forequivalentdiametersof5,10,20.and40microns,fortheflowconditionslistedinSection5.3.1.Theac_leratlon,whichisactuallyadecelerationinthatitactstoslowthedroplets,istheresultoftheslowingoftheflowduetothepressuregradientaheadofthebody.Dropletsflowingtowardstheairfoilbegintofeelsomeacceleration30to40cmaheadofth

146 ebody.Theaccelerationincreasesuntileithe
ebody.Theaccelerationincreasesuntileitherthelargerdrops(20and40microns)impactthebody,orthesmallerdrops(5and10microns)slowtoapproximatelytheflowvelocltyafewcmaheadofthebody.Themaximumvalueoftheaccelerationisontheorderofseveralthousandrgunits.Thehighvaluesofaccelerationthatthedropletsaresubjectto,resultintwomajoreffects.Thefirsti_hydrodynamic,inthattheeffectiveaccelerateddiameter(a)l/2_Deq,disc,'ssedinChapter4,isIncreasedconsiderably.Thevalueof(_)1/2isplottedagainstcdistanceinFigure5.20,forthedropletsdescribedabove.Thepeakvaluesof40to55areobserved,Implyingthatfactorssuchasdropdeformationandpossibiyinstabi!itywi11becumeImportant.itshouldbenotedthatthedropletsintheexamplei

147 nFigure5.20remainbelowtheinstabilitythre
nFigure5.20remainbelowtheinstabilitythresholdfor(g)l/2DeqofapproximatelyIon.•ThesecondeffectofthestrongaccelerationIstoincreasethe:cMelltimeofdropletsIntheregionJustaheadofthebodyThis'canbeseenInFigure5.21wherethedropletvelocityIsplotted.Thedropletsareseentoslowappreciablyseveralcentimetersaheadofthe.:airfoil•Thedwelltimeand,assumingauniformexternaldroplet••%1982022556-146 !OFpoORQ:/',L|_'YFIGURE5.20DIAMETERINCREASEDUETOACCELONSTAGNATIONSTEAMLINE•80.•p5&40,30.5MICRONS10MICRONS!20,20_'ICRONS4_MICRONS:10.B•••|1020_4DISTANCEAHEADOFHALFBODY(ON)tiI1982022556-147 JB_,..-......................_..............|147OR{GIN_LP_C_.ISOFPOORQUALITYFIGURE5.21VELOCITYVSDISTANCEONSTAGNAT

148 IONSTREAMLINE'\88.84m.4mMICRONS_!30.MICR
IONSTREAMLINE'\88.84m.4mMICRONS_!30.MICRONSvMICRONSi5MICRONS2g.lB.ml1B2{{_lE48DISTANCEAHEADOFHALFBODY(CM)1982022556-148 OFPO0!:Q.};ILtYY!_,field,thedropletconcentration,wlllthereforeIncreaseInthlsiregion.I•}Itshouldbenotedthattheobservationsmadefordropletsonthe=stagnationstreamlinearebynomeanslimitedtothatspecialcase.tIDropletsinthegeneralregionaheadoftheairfoilshouldexhibit'isimilarbehavior,althoughthedirectionofaccelerationwillbe;ivariable.Infact,thezoneofn,3ximumaccelerationIsactuallylocatedslightlyoffthestagnationstreamlineJustahe.ofthebody.5.3.3AdditionalSimulationsIInthlssection,someadditionalsimulationsarepresentedtoIllustratetheflexibilityofthecode.Thefirstcase

149 ofInterestwasthatofalargedropImpactingth
ofInterestwasthatofalargedropImpactingthebody.AswasnotedinSection5.2,thelargedropsplashesIntoacollectionofsmallerdropletswithvelocitiesoforderone-tenthofthefree-streamvelocitydirectedradiallyawayfromthepo&ntoffirstImpact.ThesplashwassimulatedbyassumingthatthepointofImpactwasthestagnationpoint.SimulationdropletswereInitiatedatthispointwithInitialvelocitiesof6m/sec(one-tenthofthe60m/sacfree-streamvalue).TheanglebetweentheInitialvelocityandthe',stagnationstreamlinewasvariedfrom0°to90°.ExamplesoftheresultingtrajectoriesareshownInFigure5.22forthecasesof30°,'60°,and90°.Forthe60°and90°trajectories,thedropletsereclearlysweptawaybytheflow.Forthe30°traJectories,thedropletsreturn

150 backandJustgrazesthebodyapproximatelyIom
backandJustgrazesthebodyapproximatelyIomalongthebody.Theoriginofthedouble-hornedIcingshapeshowninFigure5.|Obl1982022556-149 i149,OFPO0_,QUALITYCI_'!FIGURE5.22.I,SPLASHTR^3ECTORIES-2BMICRONDROPLETS.|1iHORIZONTAl.AXIS(!{:NUNITS)1982022556-150 r"'!•,!15OcanbeInferredfromthesimulationresults.Ifthedropletsdon't:freezeImmediatelyonimpact,mostoftheresidualsplashdroplets_wlllbesweptawayuntilsomeIcebeginstoformoffthecenterline.Thesplashdroplets,whicharemovlngessentiallytangentlallytothebody,willImpactatthlspointandthehornwillbegintogrow.Thesecondhornissimplyamanifestationoftheflowsymmetryaboutthestagnationstreamline.•AnothercaseofInterestIstheeffectondroplettrajectoriesofblowin

151 gairouttheleadingedgeoftheairfoll.Figure
gairouttheleadingedgeoftheairfoll.Figure5.23showsaseriesof40-microntrajectorieswithandwithoutblowing.Theblowingvelocityattheleadingedgeis36m/secinthiscase,andthefree-streamvelocityIs60nVsec.FromFigure5.23ItisclearthatItisindeedpossibletopreventdropletsofagivenslze._fromImpingingontoanairfoilbyblowing.Thefeasibilityoficepreventionbysuchatechniquerequiresanassessmantoftheaerodynamicpenalty,ifany,resultingfromsuchblowingandanassessmentofthepowerrequired.1982022556-151 I!'-v_.,..._,_,.'.T2._IS:1OFPC'O!_QL;ALITYILJFIGURE5,23MI_IZONT^LAX_$(10_I._IT_}40NZCRONTR^Jg_CRIES_ITHLgADINGE_BLO_INTSlo_Ingvelocityr36m/sHOR|ZG_TN._,IS(laCMi_:TS)1982022556-152 152CHAPTER6COMPUTERSIMULATI

152 ONSOFDROPLETHEATINGTheproblemofheatingwa
ONSOFDROPLETHEATINGTheproblemofheatingwaterdropletsbymicrowaveradiationinthe!vicinityofanairfoilisstudiedinthischapterbymeansofacomputer!lsimulation.Thesimulationis,essentially,anamalgamationofthei!r£sultsoftheprecedingchapters.InSection6.1thestructureof_:heiiT;£co_isdescribed.Section6.2discussesthemodelusedtomaintaindropletenergybalanceinthesimulation.Finally,inSection6.3,•someresultsofthedropletheatingsimulationarepresented,it6.1DescriptionoftheSimulation_nthissectionthedropletheatingcodeisdescribed.Thecodewaswrittenprimarilyasameansforobtaininganalysisanddesigncriteria.foraircraftanti-icingschemeswhichemploymicrowavepreheatingofwaterdropletspriortoimpact.Asaconseque

153 nce,theresultsofthec_depresentedinthscha
nce,theresultsofthec_depresentedinthschaptercenterarounddropletswhichareheatedbyJmicrowavefieldsastheyflowinthevicinityofanairfoil.Thecodeis,actually,moregeneralthanthisspecificapplicationandcouldjustaseasilybeappliedtoproblemssuchastheextinctionofsolarradiationbyclouddropletsortheattenuationofradarorcommunicationsignalsintheatmosphere,AflowdiagramofthecodeisshowninFigure6.1.Theprimarystructurecftheheatingcodeisthedroplettrajectorycodediscussedin._Section5,1.Twosectionsareaddedtothetrajectorycodetoa_countfordropletheating.ThefirstaccountsfortheheatingeffectofthemicrowavefieldbycaiculatlngthePoyntingfluxatthedropletIocatlo_.,1982022556-153 !5,_riD,-,,,,,i',",Gc"ISOFPOOR

154 QUALITYInitialposltionAmbienttemp.Initia
QUALITYInitialposltionAmbienttemp.InitialvelocityAnCientpressureDropletsizeII_.-CdRerlelcI_---BodysizeI_=_(_""_)Freestreamvel.lst'0rderEulerInt.SrhemeSteptlmeIncrement+-_iAt-Xi+l"+xi&t_i+l=_iAt+_iyeildstrajectoriesIISthedr°phydr°dyNaml(_al|Y_stable_"-@._.i_i+i'CalculateabsorbtionElectromagneticcross-sectioncsandfieldPoyntingfluxa_x.modelI-I'ncremen_"--/dropletparametersHeattransferdTdlossratesTd,l+l"(t'_-")iAt+TiHi+I•(_-tH)lAt+HIf'_,l+l"(d-'_')lAt4-fw,ii1__._Doesdroplethltthebodyiattimetl+I1Figure6.1Flowdiagramforheatingcode.1982022556-154 )154£'givensomemodeloftheelectromagneticfieldandcalculatingthedropletabsorptioncross-section.Thecross-sectioniscalculatedbyFirstass

155 umingRayleighabsorption(equation2.17).Th
umingRayleighabsorption(equation2.17).Thecodethenchecksfordropdeformationandothernon-Rayleigheffectsandcorrectsthecross-sectionsbythemultiplicativeFactorsdiscussedinSectr,,n).!.1.TheFollowingnon-:Rayleigheffectsareincludedincalculatingtheabsorptioncross-sectioni|-0a:largedroplets(Mietheory)non-sphericaldroplets(Ganstheory)phasechangeeffectsthermaleffectsonthedielectricconstantIntheabove,themagnitudeofthedropaxisratioA/Bistakentobe50_;ofthemaximumtransientdeformationlimltdiscussedinSection3.3.Giventheabove,itlspossibletocalculatetherateofabsorption'ofmicrowaveenergybythedropas,tdQrfd--i""os(6.11a_6whereSisthemagnitudeofthePoyntlngvectoratthedroplocation.Thesecondadditio

156 nalsectionIntheheatingcodecalculatestheI
nalsectionIntheheatingcodecalculatestheIncrementaltemperaturechangeoveronetimestep&tbyconsideringthetotalheatbudgetofthedrop,Includingsucheffectsasevaporation,miring,diffusiveheattransferandventilation,alongwiththemicrowaveheatlngofequation6.1.Theheatbudgetofadropis:discussedinmoredetailinSection6.2.Considerationofthedropletheatbudgetyieldstherateofchangeofsuchdropletparametersasthedroplettemperature,Td,thedroplet1982022556-155 a¢r,+,'++/%155OFPOORQUALITYmass,M,andthedropletmassfraction,f.TheseparametersareintegratedateachtimestepAtbytheEulerintegrationschemei:discussedinSectionS.I!dTdTd,i+l"(-_)iAt+td,i(6.2)°dM:"Mi+l"(_)iAt+Mi(6.3)fw,i+i"(-_)iAt+fw,i(6.4)Thedropletpar

157 ametersTd,Mandfarethereforefree-runningp
ametersTd,Mandfarethereforefree-runningparametersintheheatingcode.TheinitialvalueofTdisgenerallyassumedtobetheambienttemperatureT.Theinitialmassfractionanddropletmassareinput:a:parametersinthecode.ThedropletmassisrepresentedbythediameterofthewatersphereofequivalentmassDeq.0_-'(..s),:Asinthetrajectorycode,thedropletparametersandpositionare,IncrementedthroughtheloopuntilthedropleteitherImpactsthebody,)missesthebodyorbecomesunstable.2Thecodeparameterswhichwillbeassumedinthefollowing,unlessotherwisenoted,are:'Body-half-body20cmthickFreeStreamVelocity-60m/secAtmosphere=Icingconditions(750mb,-20°C),1982022556-156 156OR!C:t."":"OFF-O0_IQ:JALITYTheelectromagneticFieldmodelandt

158 hecoordinatesystemusedInthecodeareshowni
hecoordinatesystemusedInthecodeareshowninFigure6.2.ThefieldconsistedofaboundsurfacewavewhichpropagatedalongtheleadingedgeoftheairfoilperpendiculartotheFlow.Thefieldstrengthwasexponentiallydecayingwithdistanceaheadoftheairfoilandhadacos20angulardependencecenteredaboutthestagnationstreamline.ThefielddirectionisassumedtobeperpendiculartotheradialdirectioninFigure6.2.Thereforewe2I_1-E0exp{-(r-,s)/¢}cosz(_0)(6.6)where8-arctanx(6.7)Y_x2r,=+y2(6.8)00IsareferenceangledefiningtheangularwidthoftheField,E0isthemaximumelectricFieldstrengthatthesurface,ZIstheexponentialdecaylengthoftheFieldandxsisthexcoordinateoftheleadingedgeofthehalfbody.Onthestagnationstreamline,theelectricfield

159 simplifiesto.?I_'1-E0exp{-(x-Xs)/(:}(6.9
simplifiesto.?I_'1-E0exp{-(x-Xs)/(:}(6.9)"whlchistheexponentialdecayassociatedwithaplanardielectricsurfacewaveguide.15Theexponentlationlength_ofsuchwaveguldes1'canbecontrolledbythethicknessofthedielectriccoating.TheeffectofvaryingtheaboveelectromagneticfieldparametersandthewavelengthXisstudiedinSection6.3.:'l1982022556-157 157.,_,,,;"-'":_3OFPOORQUALt_Yy;¢Y(y)Ex,:_:'Figure6.2Coordinatesystemusedfortheelectromagnetic:fieldmodel.,i11982022556-158 7rORIGINALP,_,C_IS'158OFPOORQUALITY•6.2DropletEnerg_BalanceInordertocalculatetherateofchangeofsuchdropletparametersastemperature,Tjdropletmass,_:.andmassfraction,_itlsnecessarytoobservetheheatbudgetofthedrop•Theheatbudgetis\dQtd

160 dQrdQrfd%!t+(6IO)d--_"-'_'-++dt-at'-wher
dQrdQrfd%!t+(6IO)d--_"-'_'-++dt-at'-wheredO.td-_-"mtotalheatchangeindrop(6.11)::dQrfd-"_-=heatchangeduetoradiation(6.12)-heatchangeduetoevaporation(6.13):dQmelt_'_t-heatchangeduetomelting(6.14)dO.tdmheatchangeduetothermaldiffusion(6.15)Thetotalrateofchangeofheatisrelatedtothechangeindroplettemperaturebythespeciflcheatatconstantpressure(Cp-Ical/gm°C-4.186xI07erg/gm°C)andthemassNbydQtdTmCpM_(6.16)_TherateofchangeofheatduetoradiationalheatingwaswrittenIn..$ectlon6.1asdO,rfd-'T-"°aS(6.17)1982022556-159 fORIGINALPAG_IS159OFPOORQUALITY'_:whereoaistheabsorptioncross-sectlonofthedropandSisthe•,:.Poyntingflux.':Thechangeofheatduetodropletevaporat;onistdt-Lwv_t'dM"Lwvfv(d_)(6.I8

161 ).,0:.whereLwv-thespecificheatofevaporat
).,0:.whereLwv-thespecificheatofevaporationofwater(Lv-600cal/gm-2.5xI0IOerg/gm)dM_--therateofchangeofmassduetovapordiffusion:dMdM":":(_t)O-thevalueof_forastationarydropY"mtheventilationcoefficientwhichaccountsFortheenhanced•V"vapordiffusionduetoairflowaroundthedropletdM"rThevalueof(_tt)Omaybefoundbysolvingthesteady-state,dlffusionequation_Pv:=Dv Zpv(6.18)wherePvIsthewatervapordensityandDvisthevapordiffusion:rcoefficient.Calculatingtherateoffluxofwatervaporthroughsomedroplets6closedsurfacearoundthedropyields,forsphericalJD_1"4_Dv(-_"q')("(6.20)"i(_tt)0Pv,aPv,d),::iwhereOeql2IstheradiusofthedropandPv,a'Pv,dere_heawnblentanddropletsurfacevapordensltles,Iftheeffectofcurvat

162 ureonPv,dcanbeIgnoredandtheamb!3ntatmosp
ureonPv,dcanbeIgnoredandtheamb!3ntatmosphe_issaturatefwlthJI"_'_-_'_"""1982022556-160 ....,-,,L--..............+..........;m160ORIGIN_:LF::_:_i3OFP,.,,,_2O';ALITY.watervapor,thenPand0aresimplythesaturationvaporviavjd,_pressuresoverwaterandd¢,_-ndonlyontemperatureappropriatetothepressuretobefound,ValuesofsaturationvaporpressureversustemperatureareplottedinFigure6.3fromtheSmlthsonianHeteorological6OTables.Valuesoftheventilationcoefficientfv'whichistheincreasein,|vapordiffusionduetoairflowbythedrop,havebeendeterminedempiricallybyBeardandPruppacher65tobe_:f-=0.78+0.308N113Rel/2(6.21)VSC,VwhereReistheReynoldsnumberandNsc,vistheSchmidtnanber-sc,vPaDv(6.22)_pisthedynamicvisco

163 sityofairandPaisthedensity.Thevalue66r'0
sityofairandPaisthedensity.Thevalue66r'0fthevapordiffusionistakentobe2___a__3OK)l.gh1013.-ab)(6.23)Dv"0.211((Pa,twhereTaandPaarethe_r_ientvaluesoftemperatureandpressure,Theheatlossfromthedropcanbefoundbynotingthatthethermali.:andvapordiffusionequationshaveidenticalform.Therefore,by_"?analogytothevapordiffusioncase,_dQtdD-ThlllTka(-_)(Ta-Td)(6.24)i,;!wherekaisthethermaldiffusioncoefficienttakentobe_:.:2.4x103erg/cmsec°Cinairand"{rhistheheatventilationcoefficient,whichlstakentobeequivalentto_vfol|owingthe+6_exampleofPruppacher.1982022556-161 J161!.lIur_.....\-\%\4_N.T\ii.-\t',,q_3_4.11uL"1I\°\i0i•_-!1.I,-�a_1982022556-162 i162........._........:5OFPOORQUALm1(,.-vT

164 hechangeofheatduetomeltingoficewithinthe
hechangeofheatduetomeltingoficewithinthedropcanbe,:relatedtothechangeinmassfractionfwbythelatentheatoffusionLlw(Liw=80cal/gm=3.36xl0_erg/cm)dfd%eltwdt=LiwM_(6.25):Intheheatingcode,nomeltln9isassumedtotakeplaceunt'lthedropletisatoraboveO°C,atwhichpointthetemperatureremainsconstantandallexcessgoesintomeltingiceuntilfbecomesunity.wByconsideringtheabove,itispossibletowritetherateofchangeequationsforthevariousparametersofinterestTd¢O";dTddt"--CIM[°aS+4_r(Oeq/2)Tv(LwOv(Pv,a-Pv,d)+ka(Ta"Td))]'P(6.26)Td=0dfw.i[oS+41T(Deq/2)_(LwvDv(Pv,aPv,d)+ka(TaTd))]:=Liwa""(6.27)dM"41T(Deq/2)TvLwvDv(Pv,a"Pv,a)(6.28)-Inordertocheckthedropletparametersectionofthecode,the_lingandevaporationofaI

165 _dropletwassimulated.Thedropletwasgivena
_dropletwassimulated.Thedropletwasgivenaninitialtemperatureof20°Cinanambientsaturate.J'-atmosphereofO°C.Thetwocasesofastationary,unventilateddrop%andafreely-falling,ventilateddropweresimulated.TheresultsareshowninFigures6.4and6.5.Theventilateddropisseentoequilibrateinseveralseconds,whiletheunventilateddroprequirest1982022556-163 5.FREELYFALLING-VENTILATEDt1982022556-164 ii164OFpr,r"_qL!;_Ll'i'y-.,FIGURE6.5'..4MASSCHANGEFORCOOLINGDROP(INITIALLY20C)"1000.AMBIENTTEMP0Ci998.,•i"_=996,VZ994.,--,FREELYFALLING-VENTILATEDQ_1992.!t;I9gB.tIttIIIIiB.1.2.3.4.5.6.7.8.TIME(SEC)1982022556-165 165severaltensofseconds.ThisisgenerallyconsistentwiththeexperimentalresultsofKinzerandGunn,6

166 7whofoundthatai.35mm+dropcooling7.5°Cins
7whofoundthatai.35mm+dropcooling7.5°Cinsubsaturatedairtook4.4secondstoequilibrate.+'ThereIs,therefore,someconfidencethatthecodereasonablyapproximatesthechangesindropletparameters,i6.3SimulationResultsI+InFigure6.6,droplettemperatureIsp;ui.L=dversusdistance!+taheadofa20-cm-thickhalf-body,fordropletsimpingingalongtheiostagnationstreamline.Inthissimulation,tP_ewavelength).wasI+Icm,themaximumfieldstrengthE0was7.5kV/cm,theexponentlation!lengthofthefieldlwaslOcm,andthefree-streamveloci,'.ywasI60m/sac.Dropletsof20,40,i00and1000microndiametersareseentoheatfromtheambienttemperatureof-20°CtovaluesgreaterthanO°Cuponimpact.Itshouldbenotedthatal1thethermodynamiclossessuchasevaporat

167 loranddiffusionhavebeenincludedandthatth
loranddiffusionhavebeenincludedandthatthemaximumelectricfieldIsone-halfoftheairbreakdownvalueof6815kVtcm.Itisclear,therefore,thatfortheabovec_ndltionsdropletscanbeheatedtoabovefreezingbeforeImpactingtheairfoil.+FurtherinspectionofFigure6.6showsthatthesmallestdrop'i'(20microns)Iswannedtheleast.Thisisexpectedevcnthoughthe_:smallerdropletsdwellsomewhatlongerInthehighfieldretlonduetoI,4theirlowInertia.Thisisduetothesmalldropsbeingpoorabsorbersj,,tbycomparisontothelargerdrops.Thetrade-offbetweendwelltime!andabsorptioncross-sectioniswhatcausesthebunchingofthe20,401'!and100microntemperaturesinFig_re6.6,whil,-,fortheIO00micron!+dro_thestrongnon-Rayleighabsorptioneffectc1,wnlna

168 te.+ti1982022556-166 "_166,'_OFPOCRQUaLi
te.+ti1982022556-166 "_166,'_OFPOCRQUaLiTYb:,FIGURE5.Bt!i{iDROPLETHEATINGBYMICROWAVESAHEADOFHALFBODY20.fo10.vi0.,_00MICRONS00MICRONS40MICRONSm'xMICRONSPw-10.QOISTANCEAHEADOFBODY(10CMUNITS)r:11982022556-167 167i|TheeffectofvaryingtheelectricfieldparametersIsshownfor20"Imicrondropletsimpingingonthestagnationstreamline.The20microndrop-!;Jletshavebeenchosenasaworst-casedesignpoint,inthatdiameterslessthan20m[cro,swereshowninSection5.3tohavealowcollectionefficiencyandtheirheatingis,therefore,lesscritical.InFigure6.7thedroplettemperatureatimpactisplottedversusthemaximumelectricfieldstrengthforawavelengthofIcmandseveralvaluesoftheexponentiationlength,Forfieldstrengthsabove2.5k

169 V+thefinaldroplettemperatureincreaseslin
V+thefinaldroplettemperatureincreaseslinearlywithelectricfieldatarateofapproximately4.2°C/kV.Theexponentiationlengthisseentohaveaweakeffectonthefinaldroplettemperature,asthemosteffectiveheatingoccursclosetotheairfoil.Sincetheabilitytoheatdropsdoesnotdependonlongexponentiationlengths,thenit+Isclearlyadvantageoustokeeptheelectricfieldcloselyboundtotheairfoilinordertominimizetheheatingofdropletswhichmisstheairfoil._InFtgure6.8,thedroplettemperatureatImpactisplottedagainst'-wavelengthforseveralvaluesoftheelectricfieldandalO-cmi'exponentlationlength.Thefinaltemperatureincreasesquicklywithidecreasingwavelengthfor_,lessthan3cm.Finaltemperaturesat._freezingareobtainableforAles

170 sthan2can.Thefieldstrengthrequiredtoheat
sthan2can.ThefieldstrengthrequiredtoheatdropletsIsseenInFigur_6.7++_+'mtohaveamaximumvalueontheorderof7.5kV/cmneartheairfoil,+surfacewith_,-Icm.Thepowerfluxforaplanewaveof7.5kV/cmcanbecalculatedfromthePoyntlngvector',c(6.29)it.......L+1982022556-168 168ORIG!'n;_L_:,',___L_OFPOOR(_L;;_UTYFIGURE5.7'|DROPLETTEMPATIMPACTVS.MAXELECTRICFIELDi20.!20MICRONDROP!1CMWAVELENGTH1982022556-169 _69OP,;G::_._LPAC_ISOFPOOI__UALITYFIGURE6.8DROPLETTEMPERATUREATIMPACTVS.WAVELENGTH2g.^MAXELECFIELDuvQ10.13KV/CM=u10KV/CM=,7KV/CMPKV/CM0.1KV/CMwxWm-10.Q°\+-28,B.2,4.6,8.1_,WAVELENGTH(CM)•+1982022556-170 I170_,tobe74kW/cm.Thishighpeakpowerdensitylmplieslargepowerflo_In..themicrowavecircuit.While

171 thishighpowerisnotlostfromthesystem,]ias
thishighpowerisnotlostfromthesystem,]iasitrecirculates,itcreatespowerhandlingproblems.Itmaythereforebeadvantageoustokeepthefieldstightlyboundandtooperateatshorterwavelengthswherethemaximumelectricfieldsarereduced,asisimpliedbyFigure6.8.Itshouldalsobenotedthattheelectromagneticfieldsarenotnecessarilyplanewavesanditmaybepossibletocreateafield"rstructurewhichmaximizestheelectricfieldwhileminimizingthemagneticfield.Thiswouldreducethecircuitpowerfluxfromequation6.29,whilestillheatingthedropletsduetothenon-magnetlcnatureofwaterdropletabsorption.InFigures6.9andG.10,theeffectoftheambienttemperatureandtheflowvelocityareplottedfor20micronand100microndroplets.Themaximumelectricfi

172 eldis7.5kV/cmwithI-Icanona10cmexponencia
eldis7.5kV/cmwithI-Icanona10cmexponenciationlength.InFigure6.9withafree-streamvelocityof60m/sac,thefinaldroplettemperaturedecreaseswithambienttemperature,aswouldbeexpected.Itisnotedthatthe20micron:dropimpactsatabovefreezing,evenforambienttemperaturesof-30°Candthatthefinaltemperatureofthe100microndropaveragesapproximately/4°warmerthanthe20microndrop.Theeffectofthefree-streamvelocity,atafixedambienttempera-tureof-20°C,isshowninFigure6.10.Asexpected,thehighvelocitiesprovidelesstimetoheatthedropsandtheirfinaltemperaturesarecorrespondinglylower.Theli°Cto5°Cdifferenceitbetweenthe20andi00microndropsisagainobserved.i1982022556-171 FIGURE5,gDROPLETTEMPATIMPACTVS.AMBIENTTEMP_B0.

173 :Q-3B.-2B.-IB,_.AMBIENTTEMPERATURE(DEGC)
:Q-3B.-2B.-IB,_.AMBIENTTEMPERATURE(DEGC)(1982022556-172 172OFPOOffQUALITYFIGURE6.10DROPLETTEMPATIMPACTVS.FREESTREAMVEL.20.8%100MICRONS10._20.......!I:40.50.60.70.80.i?FREESTREAMVELOCITY(M/S).i1982022556-173 173InFigure6.11themassfractionisplottedversusdistanceahead,ofthebody.ThevalueoftheelectricfieldparametersIsthesameasabove,andthefree-streamvelocityis60m/sec.TheInitialvalueoffwIs0.1.Thelarge(1000micron)dropistotallyme.ltedbythetimeItiswithin6cmofthebody,whilethesmallerdropsarestillpartiallyfrozenuponimpact.Thisisnotconsideredaproblem,inthatmostmix_d-phaseparticlesintheatmosphereareprecipitation;particleswlthdimensionsontheorderofmillimeters._,_Anexampleofdropletswhi

174 chimpingefrompositionsotherthanthestagna
chimpingefrompositionsotherthanthestagnationstreamlineisshowninFigure6.12.Droplettemperature:_atthenoseoftheairfoil(x-Xs)Isplottedagainstinitialseparationfromthestagnationstreamline.ThedropletsizeIs20mi,-:-onsandthereferenceangleB0is45°.Thelimitoftheimpingementwindow_Isalsoshown.Theplotissomewhatmisleading,inthattherewillbesomeadditionalheatingfordropswhichimpactbehindthenoseoftheairfoilandthefinaltemperaturealsodependsondiffusivelosses.Itisclear,however,thatthedropletswhichareheatedthemostarethosepwithintheImpingementlimit,andthatbyJudiciouschoiceoftheelectricifieldtheselectiveheatingofonlytheimpactingdropletscouldbemaximized.Inconclusion,acomputersimulationhasbeenmad

175 eofdropletsbeingheatedbymicrowaveradiati
eofdropletsbeingheatedbymicrowaveradiationastheyImpingeontoanairfoil.Thesimulationutilizestheresultsoftheprecedingchapters.The_resultsindicatethatItisIndeedfeasibletoheatdropletstoabove;._freezingpriortoimpactundermostIcingconditions.Theeffectofvariouselectromagneticfieldandflowparametersondropletheating1982022556-174 174G....:..FIGUREB.11"#t*"MASSFRACTIONOFWATERINDROPVS.POSITION:1982022556-175 _175_"_.....FIGURE5.12¢i:FINALDROPLETTEMPVS.INITIALPOSITION20o1982022556-176 jI'176=hasbeencalculated,andtheresultsarepresentedasanaidtothe4!designandanalysisofanti-icingsysten_whichmightemploytheitechniquesofmicrowaveheating.Theresultsindicatethatwavelengthslessthan2cmandmaximu

176 mfieldstrengthsgreaterthan7kV/cmarerequi
mfieldstrengthsgreaterthan7kV/cmarerequiredtoheatdropletstoabovefreezingpriortoimpact.Whiletheirequiredfieldstrengthishigh,itislessthanthebreakdownvalueof|',air,andthesimulationindicatesthatthefieldscanbekeptquitecloselyboundtotheairfoilwithlittledegradationinperformance.i,!i1982022556-177 r177CHAPTER7CONCLUSI0NSTheprimarygoalofthisthesiswastounderstandthephysicsInvolvedinadvancedmicrowaveanti-icingsystemsandtodeterminetheirfeasibilityintermsoftheabilitytoheatsupercooledwaterdropletstoabovefreezingpriortoimpact.Correspondingly,thisquestionwasansweredbymeansofacomputersimulationinChapter6•_TheconclusionisthatItisindeedpossibletopre-heatwaterdropletstoabovefreezingundera

177 nticipatedicingconditions,aslong.'asthem
nticipatedicingconditions,aslong.'asthemaximumelectricfieldexceeds7kw/cmandthewavelengthis:_lessthan2cm.Detaileddesigncurvesfortheelectricfield;parametersareprovidedinChapter6.I;i1Theworkwassuccessfulindemonstratingitsoriginalgoal,]tI.e.thatsupercooledwaterdropletscanbeheatedtoabovefreezingaheadoftheairfoil.Inordertocreatethephys!cally-reallsticsimulationfromwhichtheaboveconclusionsweredrawn,somei"preliminaryworkwasnecessary.Theoriginaltechniques,methodology_,_:andresultswhichweredevelopedaremostlikelyassignificantas_,titheoriginalgoal•Theseresultswillbereviewedbrieflyby,.chapter.1InChapter3,atechniquewasdevelopedtoaccuratelymeasuretheabsorption¢ross-sectlonofasingledr

178 opletbyitseffectontheQofalo_lossresonant
opletbyitseffectontheQofalo_lossresonantcavityUsingthistechnique,the"';theoryofabsorptionbywater-coatedicespheresofAdenandKerkerl,wasexperimentallyconfirmedforthefirsttime.Inaddition,anb1982022556-178 ..1781extensionoftheAdenandKerkertheorytomeltingoflrregularlt-I1.|t;shapedparticleswasmadeandconfirmedexperimentally.Finally,iinChapter3,anextensionoftheresonantcavitytechniqueallowedthedielectricpropertiesofsupercooledwatertobemeasured,at_,-2.82cm,downto-17°C.ThisnewdataisshowninFigure3.12.;InChapter4,thehydrodynamicsofacceleratedwaterdropletswereobservedinthewindtunnelbymeansofhigh-speedstrobephotography.Thetransientdeformationlimitofthemajor-to-minor"axisratiowasexperi

179 mentallycleterminedfordropletssuddenlyac
mentallycleterminedfordropletssuddenlyacceleratedbyanotherFluid.TheresultisshowninFigure4.9.Inaddition,thetheoreticalpredictionofKlett45astotheformof:Raylelgh-TaylorinstabilityinwaterdropletswasunambiguouslyconflmedastheFirstsixBesselFunctioninstabilitymodeswereidentifiedphotographically.ThisexperimentalevidenceverifiesthetheoryofthemaximumstabledropsizeIntheatmosphere.Theunstablebreak-upoflargewaterdropletscauses*.hetruncationatapproximately5mmobservedintheatmosphericexponentialdropletsizedistributionofHarshallandPalmer.In_hapter5,acomputercodewaswrittentocalculatedroplettrajectories,Inanygivenflowfield,andwasexperimentallyverified.ThecodeIsusefulinanalyzingthesuscept

180 ibilityofiairfoilstoIcingandtoanalyzingn
ibilityofiairfoilstoIcingandtoanalyzingnewicepreventionconcepts.Withtheassistanceofthecodeitwasdeterminedthat,forclouddroplets,',thelargerdropscontributeappreciablymoretoIcingduetotheirIncreasedcollectionefficiencyandtheirIncreasedmass.In1982022556-179 I__i•'.__,J179:dddition,themechanismofthe"double-horned"icingshapesresultingfromlargesplashingdrop!etswasobservedandmodeled.Finally,inChapter6theaboveresultswereamalgamatedint.acomputersimulation,whereitwasseenthatitisindeedpossfbletoheatdropletstoabovefreezing,evenwhenevaporativeandconvectivelossesareincluded.FromthedesigncurvesofChapter6,someconclusionscanbedrawnastothepracticalrequirementsofamicrowaveantl-lcingsystem.

181 InFigurp-6.8wherethestronginfluenceofthe
InFigurp-6.8wherethestronginfluenceofthewavelengthondropletheatirgisdisplayed,itisclearthattheminimalwavelengthisoptimalintermsofheatingandthatawavelengthoflessthan2cmisrequired.Balancingthedesira-bllltyofshortwavelengthsisthefactthatmicrowaveequipmentcoststendtoIncreasemonotonicallywithdecreasingwavelength.Thesefactorsindicatethatapractical,_nti-iclngsystemwouldoperatewithawavelengthofapproximatelyIcm.InChapter6,forX-1cmitwasfoundthatarelativelyhlghfieldstrengthof_roximately7.5kVwasrequiredtoanti-iceinallanticipatedicingconditions.Thisfleldstrengthisone-halftheatmosphericbreakdownlimit.EventhoughthefieldsLrengthwasrequiredtobelarge,intheresultsofChapter6Itwasfoundt:_t

182 itneednotbeextensive,asatightly-boun"(.i
itneednotbeextensive,asatightly-boun"(.i_-10an)surface'_:wavewasfoundtoperformaswella_aloosely-boundwave(_=40cm).;,Thisresu"Indicatesthatitshouldbepossibletobequiteefflclert!Inonlyheatingtheimpactingwaterdroplet_.Withthecriteriafor-:dropletheatlngdeterminedb_.,theaboveresults,thepractlcalityofmicrowaveanti-icingsystemsdependsolelyonengineering1982022556-180 IiimI,efficiencyc_nslderationsandtheabilityofthemicrowave_ystem-tobeintegratedwithothersystems,inahybriddesign,asdiscussedInChapter!,tosolvetherunback-refreezep.,'nblem.}?2i1982022556-181 '_';i81.!.LISTOFSYMBOLSii'iAdropmajordiameter1iAdropareaperpendiculartotherelativewindA/BellipsoidaxisratioiIasphereradius_iadrop

183 accelerationvector'!magneticandelectrice
accelerationvector'!magneticandelectricexpansioncoefficientsofthe_an'bnscatteredwaveBdropminordiameterBmagneticfieldvector_IincidentmagneticfieldvectorCddragcoefficientCspecificheatofwateratconstantpressurePcspeedoflight!OcylindricalcavitydiameterDequivolumetricspherediametere:lm.DmeanvalueofDeqeqr,vwatervapordiffusioncoefficientelectricFieldvector1982022556-182 W.••b-w,.182_iIncidentelectricfieldvector"E0maximumelectricfieldstrengthEzelectricfieldstrengthinTMoI0modeeeccentricityofanellipsoid_0polarizationvectorFstructurefactorffrequency:focavityresonantfrequencyfdDopplerfrequencyfdroposcillationfrequencyofthenthmodenfmassfractionofwaterWC"fhaverageheatventilationcoeff

184 icientaveragevaporventilationcoefficient
icientaveragevaporventilationcoefficientVg,g'Gansfactors._gaccelerationduetogravityHairfoilthicknessH_magneticfieldstrengthinTH010modehbodyhalfthickness1982022556-183 =,183hIImpingementwindowheightthJnBesseIfunctionnKdielectricparameterKconstantinthedropletequationofmotionawavevectorKthermaldiffusioncoefficienta;kgravitationalwavenumberg_0i,cldentwavevector.t=kscatteredwavevectorSLcavitylengthLiwlatentheatoffusionJLlatentheatofevaporationWVexponentiallengthoftheelectricf:eldMdropmassM.massofIceII_massofwaterdMcImpingingmassdistributionmcomplexrefractiveindex•i11982022556-184 184INnumberofscatterersNWebbernumberweNscSchmldtnumber•nmodenumbernrealpartoftherefractive;ndex

185 t_unitvectorinthedirectionofpropagationP
t_unitvectorinthedirectionofpropagationPpowerPaambientpowerP,P'geometricalfactorsforellipsoid_:Pdpowerdissipatedbyadrop,.tPpowerdissipatedbythecavitywallsW"Pmin'Pmax_inimumandmaximumpowerreceived,Qqualityfacto•:.QheatrateofchangeofheatqchangeinwavevectorRralnfa_lrateReReynoldsnumber•radialcoordinate'a1982022556-185 .185-rinterpartlclespacingSsurfacefunctionPoyntingvectorSlincidentPoyntlngvectorTtemperature4Tambienttemperaturea:T(t)cavitytemperatureTddroplettemperaturettimetotimemeltingbeginstftin_meltingends_.Uenergystoredinthecavity-Ufreestreamvelocity_-ufluidvelocityVdropletvolumeVre!relativewindvelocity_:.Lvdropletvelocity'.x,y,zCartesiancoordinates1982022556-186 .p

186 186-=thxjpositionofthejscattererxhalfbod
186-=thxjpositionofthejscattererxhalfbodyscalelength'_2Tra/),•_instabilitygrowthrate?O[angleofattack,.,£fulIwidthathalfmaximum¢complexdielectricconstant_realpartofc"negativeoftheimaginarypartoferlcairfoilcollectionefficiency(_angularcoordinate,_80heatingcodereferenceangle1:knegativeoftheimaglr.-_ypartofmXwavelength_'),Harshal1-Palmercoefflclent1Jdynamicviscosity._-ellipsoidaxisofrevolution_,rl,Xellipsoidcoordinates?1982022556-187 !iI187!pdensltyI;!I'PidensityoflcePwdensityofwaterPvdensityofwatervapor::p,¢,zcylindricalcoordinates,.asurfecetensioni;0CrOSS-Section!nbbackscattercross-sectionaabsorptioncross-section,'_sscatteringcross-sectionattotalcross-sectionidodlffertia

187 lcross-section2¢velocitypotentialfunctio
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