Toshiro Miyahara

ミヤハラ トシロウ

宮原 敏郎

宮原, 敏郎

ミヤハラ, トシロウ

Miyahara, Toshiro

AROUS Academic Repository of Okayama University of Science / 岡山理科大学学術リポジトリ

ＴＨＥＢＵＬＬＥＴＩＮＯＦＯＫＡＹＡＭＡＵＮＩＶＥＲＳＩＴＹＯＦＳＣＩＥＮＣＥＮＯ４１Ａｐｐ､169-174(2005）CharacteristicsorFluidFloworFroｔｈｏｎａＰｅｒｈｍｔｅｄＰｌａｔｅTｏｓｈｉｒｏＭＩＹＡＨＡＲＡDepαﾉﾌﾞ"】eﾉ11fq/:,4〃/〃die"7Ｍ〕ノトＦ(ＩＣ"/０'０/Eﾉ79ﾉﾉ'Ｃｅﾉｿﾞ"ｇｏﾉｾIyα"7αＵ〉1〃ｅＭ)ﾉq/Scjel7ce，１－１，Rjdbjcho,Ｏ/(､yαﾉ"α700-0005,k"α〃(ReceivedJuly26,2005;acceptedNovember7,2005）雄y肋、F:Perfbratedplate,Froth,Oasliquidholdup,LiquidcirculationThispaperrepresentsatheoreticalapproachfUrestimatingaxialgas(liquid)ｈｏｌｄｕｐａｎｄｍｅａｎｇａｓ(liquid)ｈｏｌｄｕｐｉｎｔｈｅｆＭｈ(gas-liquidmixturemmedbyblowinggasintoshallowliquid）onaperfDratedplatebymeansofthecalcuIusofvariation・ThepaperalsodescribessomefUndamentalcharacteristicssuchasgas(liquid)holdupand/orliquidcirculationinthefroth，whichareofimportancetothepermrmanceofgas-liquidcontactors･OfparticularemphasesfDrcharacteristicsofthefrotha1℃axialandradialprofilesofgas(liquid)holdup,ｍｅａｎgas(liquid)holdupandliquidcirculationobtainedexperimentally．IntroductionTheoperationofgreatestimportancefbrgas-liquidcontactorssuchasbubblecolumnsan｡/orplatecolumns，inWhichperfbratedplatesarecommonlyused，isthedispersionofgasintoshallowliquidWhengasisdispeIsedintoliquidonaperfbratedplate，gas-liquidmixture，namely，ｆiDthappears・Tbclarifythebehaviorand/orcharacteristicsofthefrothisverydifficultbecausethefluiddynamicsofthesystemisextremelycomplexandpoorlyunderstoodFurthermore，intheapplicationtochemical，petrochemicalorbiochemicalreactions,masstransferinvolvinggaseousreactantsoccursthroughthegas-liquidinterface、Hence，ｇａｓ（liquid）holdupandliquidcimulationinthefiothonaperfbratedplateareofgreatinterestbecauseoftheirdirecteffectsoncolumnefficiency・Therefbre,toMlyunderstandorpredicttheflowbehaviororreactorperfbrmance，effbctsofgas(liquid)holdupandliquidcirculationinthefiDthneedtobeaccountedlnthispaper,atheoreticalapproachfbrestimatingaxialgas（liquid）ｈｏｌｄｕｐａｎｄｍｅａｎｇａｓ（liquid）holdupinthefiPothonaperfbratedplateispresentedbasedoncalculusofvariationInaddition，someimportantcharacteristicspertainingtogas（liquid）holdupandliquidcirculationobtainedexperimentaIlyarebrieflyshownOfparticularemphasesareaxialandradialprofilesofgas（liquid)holdup,ｍｅａｎｇａｓ(liquid)holdupandliquidvelocityinthefiothbubbleandKimanellipticalbubblefbrgasholdupevaluation，theydidnotconsidertheeffbctsofnearwakeofbubble・Inthefbllowing，fimdamentalframeworkfbramodelbasedontheassumptionofasphericalbubbleincludingthenearwakeispresentedtopredicttheaxialgasholdupprofilesandthemeangasholdupHerethefbllowingassljmptionsaremadetosimplifythemodelfbrmulationl)Thebubbleisasphericalcapbubble、2)Theenergyofmixingandtheenergyofdiffilsionbetweenliquidandgasareneglected3)Thefrictionbetweengasandliquidisneglected4)ThedistributionofthegasholdupisunifbnnintheradialｄｉｒｅｃｔｉｏｎＴｈｉｓｉｓｎｏｔａｌｗａｙｓｔｒｕｅｉｎpractice．。ⅡﾉﾘﾘＩｌＭｌＩ/ＩＩＩＩしＸ(ａ）SlrucIureofbubbIe (ｂ）ＥｌｅｍｅｎＩａｒｙｌａｙｅｒｉｎｆｒｏＩｈFiglStructureofbubbleandelementarylayerinfroth１.AxialandMeanCas(Liquid)HoldupL1TheoreticalTreatmentAzbel(1962)andＫｉｍ(1965)havemademodelingattemptsfbraxialandmeangasholdupinthefrothonaperfbratedplate・WhileAzbelassumedasphericalＡｓｓｈｏｗｎｉｎＦｉｇ､１(a),thesphericalcapbubbleisaccompaｎｉｅｄｂｙｌｉｑｕｉｄｖｏｌｕｍｅｙｉｒａｓｔｈｅｗａｋｅ(Collins，1965a，1965b，1966,1967)．ThewholevolumecontainingboththebubblevolumePbandthe170 ToshiroMIYAHARAwakevolumezrisapproximatedasaspherewithradiusqDaviesandTnylor（1950)showedthatO=６０degreeAsshowninFig､１(b),letusconsiderうｉｎｔｈｅｆｒｏｔｈｏｎａｐｌａｔｅ，anelementarylayerwiththeheighMzTherefbre,thegasholdupinthislayeris必-；|'鮫等'１U二号 (7)５ ３Ｓｃ＝云兀α'１ (8)壜｡-㎡(÷-…－㎡,|Ⅷ 血Ｇ巫但一地上⑰２ｍ７’４咽(1) (9)ThetotalenergyofthiselementaryIayerE,iｓWhere〃ｉｓｔｈｅｎｕｍｂｅｒｏｆｂｕｂｂｌｅｓｉｎｔｈｅｕｎｉｔｖolumeThepotentialenergyofthislayeris昨"トル箒("菩凹際叱|α（10）ThesystemshouldbestablewhenE）takesaminimumvalueunderthesubsidiaryconditionthattheliquidholdupisconstant｡E,-(l-gc)p1gz血 (２）lfthebubbleaccompaniesliquidKtｉｍｅｓａｓｍｕｃｈａｓｔｈｅｖｏｌｕｍｅｏｆｔｈｅｓｐｈｅrewithradiusa，ａｎｄｔｈｅｖｏｌｕｍｅＫ,onebubbleisaccompaniedbyliquidｏｆ/V）timesasmuchasthebubblevolumeH-jili1(1-層ルーc･nSfandtheboundaryconditionE｡(ん,)-1(11）_＋cosO-Lcos30＋全Ｋ２(3)Ⅳﾉー --.゜Ⅲ十・㎡，２３(12）OntheassumptionthatthekineticenergyofthiselementarylayerdEA,isthesumofboththekineticenergyｏｆｔｈｅｇａｓａｎｄｔｈｅｅｎｅｒｇｙｏｆｔｈｅｌｉｑｕｉdaccompaniedbythebubble，ａｎｄｔｈａｔｔｈｅｇａｓａｎｄｌｉｑｕｉｄｍｏｖｅａｔｔｈesamevelocityうthefbllowingequationisobtainedTherefbreE,(ＥＣ)=LI'F(`c,zルハ。)'，H(ｓｏ)-Ｊ'１℃(飴)土(13）(14）〃;．”1号~｡M÷硬`|耽豈ｐＩＵｉ㎡(÷｡M÷．‘，．)Ｍ‐(p'十Mp,)ECU;`Ｍ（４）AfierintroducingtheLagrangemultiplier人thefbllowingEuler，sequationisobtainedbyusingthecalculusofvariation．旦旦十入旦旦-＝０８８０ ８８０ (15）Thesurftlceareaofthesphericalcapbubbleisexpressedasfbllows SubstitutionofthefimctionFandGｉｎｔｏＥｑ.（15）■glvess-㎡(㎡M-2cM）（５）ThentheenergyofthesurfacetensioMEvisgivenby咽-㎡(sin2伽-2.M)伽（`）、.＿＋￣、(16）ＣＧ＝〔丁一/L－ＯＪＰFurther,substitutionofEq.（12)intoEq.（16)givesA-芸｡-p岬:(化辮等,,)卿ハIftheflowaroundthespherecontainingthebubbleandthewakecanbeassumedtobethepotentialflowうK=1/２(Darwin,1953;Milne-Thomson,1968)．TherefbreCharacteristicsofFluidFlowofFrothonaPerforatedPlate 1７１FromEqs.(11)and(16),substitutingthevalueof入andregardingpﾉ>>肘wecangettheffothheight〃asfbllowsfiomasieveplateThisisprobablyduetostagnationofthedisruptbubblescausedbythegasjetnearthesieveplate・ThemiddleregionhasatendencytoshowaconstantvalueofeﾉathighliquiddepthThisismostlikｅｌｙｄｕｅｔｏｔｈｅｆａｃｔｔｈａｔｔｈｅｂｕｂｂｌｅｒｉsesatthetenninalvelocitybalancedwiththebuoyancy，thedragfbrceandtheinertialfbrceinthisregionTheusualbubblecolumnwithadeeppoolofliquidmaycorrespondtothecontactorwithlaIgemiddleregionFigure3，presentedbymkahashieMﾉ.（1974)，showscomparisonofthecalculatedvaluesfiomEq．(19)withtheexperimcntalonesofeGatUkF682cm/sandvariousliquiddepths・AtlowliqUiddepth，themeasurementsag１℃ewellwiththepredictedonescomparatively・Atlargeliquiddepth，howeveBthevaluesofmiddleregionarenearlyconstant，ａｎｄthepredictedvalueisalittlelalgcAsmentionedabove，thismaybeattributedtotheterminalvelocityofbubbleinthisregionlnFi９．３，thedashedlinesshowtheresultsobｔａｍｅｄｂｙＡｚｂｅｌｗｈｏｄｉｄｎｏｔｔａｋｅｉｎｔｏａccounttheeffectofthenearwakeofbubble．ﾄＨｌ肝周 (18）whereFHﾉlgc2/(9日)．hold叩iｓTherefbre，ｔｈｅａｘｉａｌｇａｓ(19）どＧ＝８６＿ムFromEq.(18),themeangasholdupisobtained邇将 (20）８０＝Therefbre，themeangas-liquidhold叩ratioｉｓ(TnkahasmeMﾉ.,1973） ｎＩＵ母厩Ｌ｜妬 (21）ＺＥｃｍ．Effectofclearliquidheightonliqllidhold叩L2AxialPmfilesofGas(Liquid)Holdup Fig.３ＺＥＣｍ．●、５１０１５２０２５ｚＥ仁、。０５１０１５２０２５３０３５幻 Figure4，obtainedbyTnkahashieM/、（1974)，showsthecomparisonbetweenthemeasurementsandthecalculatedvaluｅｓｏｆＥＧｗｉｔｈｃｈａｎｇｅｉｎＵｌｇ･ａｔH=１０ｃｍ・AtsmallUkMheexperimentalvaluescoincidewellwiththepMictedones;atlalgeUkobothvaluesdeviateinthemiddleregionThisisprobablyduetotheconsiderableliquidentrainment1０ｒＩＯｈｌＰｓ０0５１０１５２０２５０５１０Ｉ５ＺＯ２５３０３５ＺＥＣｍ．ＺＥｇｍ．ＩａｌＩｂｌＩＥＩｌｄｌＦｉｇ､２AxialprofnesofmeanliqUidholdup、Figure2，obtainedbynkahashieM/､（1974)，showsaxialprofilesofthemeanliqUidholduponapelfbratedeﾉ(＝１－８G）measuredbymeansqfγraytechnique・FIDmthesefigures，ｅノdecreasessuddenlynearthesurfhceofaperfbratedplate(bubblefbrmationregion)；thenthedegreeofthediminutionbecomesmild(middleregion);andabovethisregion,８K=1-8G)decreases(upperlayerregion）withincreasinggasvelocityBAthighgasvelocityウｔｈｅｍｉｎｉｍｕｍｖａｌｕｅｏｆｅノａｐｐｅａｒｓａｔａｂｏｕｔ２－３ｃｍ５０，１ｕ碑’一、0 ５１０１５２０ｚ５ＺＥｃｍ．EffbctofgasvelocityonliquidholdupFig.４UBC;35.bIcihね■Ｉ“９■の□》■〃■・鮫Ｕ。｝■。■■寵 Kｗ 随尼、。⑥ 。● ５｡ ７｡ ＩＤ① 1５● 2０ToshiroMIYAHARA172fromｔｈｅｔｏｐｏｆｔｈｅｆｉｏｔｈａｎｄｔｈｅｖｉｏｌｅｎｔｄisturbanceintheupperlayerregion．L3MeanGas-LiquiｄＨｏｌｄｕｐＲａｔｉｏＡｓｓｈｏｗｎｉｎＦｉ９．５includingtheresultsofBraulickαα/、（1965）andYOshitome（1963)，byTMahashieMﾉ.（1974),thevaluesofthemeangas-liquidholdupratio，ｉｎｔｈｅｒａｎｇｅｏｆＨ=3-70ｃｍ，areplottedagainstFroudenumberFrdependingonEq.(21)．Ａｔｌａｌｇｅ凡boththetheoreticalandexperimentalvaluesareingoodagreement・Inthisfigure,thedashedlineistheresultobtainedbyAzbelandthedottedlinetheresultbyKim．ｔｈｏｓｅｏｆｇａｓｈｏｌｄｕｐｉｎｔｈａｔｔｈｅｌｏｃａｌliquidvelocityshowsamaximumattｈｅｓａｍｅｐｏｓｉｔｉｏｎａｓｔｈａｔａｔｗｈｉｃｈｍａｘimumgasholdupoccurs．0.706 ;.;蝿 】ｕ」I［ｌ［&開８．⑩今ｅｅ０．５，Ｕ０．４Ｊ'０３う巴 ユ屈コト； 一面Ｕ、,『…；蝋‘ Iｑ△ ①Ｐ､､１０ｍ８▽．｡１－０５ｍﾉＳｃ０．２０．１DＬＪ００．２０．４０．６０．８１００．２０．４０６０.ＢｒノＲ［－コｒノＲＥ－コFig.６PmfHesofgasholdupandliquidvelocityｎＩＵくぼ‐二一忠ＩAxialclassifIcationofthefiothregimeaspresentedbyMiyaharaejaﾉ.（1989）isillustratedinFig、７．ＨｅｒｅｔｈｅｇａｓｈｏｌｄｕｐａｔｔｈｅｃｅｎｔｅｒｉｓｐlottedasafUnbtionoftheratioofaxialpositiontoliquiddepthAsseenfromthisfigure，thefiDthregimecanbedividedintothreeregionsalongtheaxialdirectionasdescribedabove：bubbIefbrmationregionfbrL/Ｈ<0.3,middlereｇｉｏｎｆｂｒＯ､3<L/Ｈ＜L6andupperlayerregionfbrL/Ｈ＞1.6．TheprofnesofgasholdupandliquidvelocityinthemiddleregionshowadistinctivebehaviorinthattheyhaveamaximumatacertainradialpositionTherefbre，thefbllowingconcemsonlythemiddleregionFfpU9JKgH)ヒーコFig.５GasIiquidholdupratioversusFroudｅｎｕｍｂｅｒFIomFig5，itisobviousthatthetheoreticalequationofthemeangas-liquidholdupratioEq.(21）agreeswiththeexperimentalvaluesbetterthandothoseobtainedbyAzbelandKimAlsoEq.(21）doesnotalwaysrepresentthemeasurementsTherefbre，byconsiderationofthesemeasurements，empiricalequationswereobtainedasfbllows２１」差=６５編ｌ－８ｏ丹＝８．５×１０－４ （22） L/Ｈ＝1.6Ｌ/Ｈ＝０３/￣iI）脇99.-9-潟(1) 一（『『四一四頁》５２ｊ０００［Ｉ］Ｕの囚=と-23編１－ＢＧＦｒ三８．５×１０－４ （23）(1)Bubbleform9tiqnregIon(2)Middleregion(3)UpperlayerregIon2．RadialProfilesofGasHoldupandLiquidCirculationHills（1974）ａｎｄＫａｔｏｈｅｒａﾉ.（1975）showedthatradialpmfilesofgashold叩andliquidcirculationvelocitydonotvaryalongtheaxialdirectionexceptinthevicinityoftheplateandatthetopofthefiFothFigure6,obtainedbyMiyaharaeMﾉ.（1989),showsradialplofilesofgashold叩measuredbymeansofaconductivityplobeandvoidfractionunitandliquidvelocitymeasuredbymeansofthePavlovtubedescribedpreviouslybyHills（1974)．Asillustrated，gasholdupshowsamaximumatacertainradialpositionabehaviornotcommonlyfbundintheusualbubblecolumnwithadeeppoolofliquidMoreoveEprofilesofliquidvelocitｙａｒｅｒｅｌａｔｅｄｔｏ００５0０２１２ＬノＨ［－コFig7Frothcharacteristicsalongaxialdirection３Figure8，obtainedbyMiyaharaerα/､（1989)，showsradialprofilesofgashold叩andliquidvelocityasaparameterofsuperficialgasvelocityinthemiddleregion Althoughofcoursetheparameterisdiffbrent,Fig.９ｓｈｏｗsradialprofilesofgasholdupandliquidvelocityFromthesefigures，６Ａ２０１－１２Ⅱ４６００印処Ｅ］－則，Ｕ幻ＩLiquidvelocity００８ g;8．▽▼＿マ、①△▼ＯＬＱｏｗｗ ０7▽ ▽Ｈ＝０．１０ｍLbc=1.05ｍﾉＳ１１Ⅱ０断▽－△取､の８CharacteristicsofFluidFlowofFrothonaPerforatedPlate １７３Ⅲ」1KⅡ[nUBUuUH蟻０．６ Liquidvelocity ■一ｍＪ●ソ「ｅ⑪ｐ、－○一〔Ｕ《Ｕ００ 蕊ｊ０ ０４■《巴【己【】、■（四「用Ⅲ ７］母Ｈ四日ロ『口期１６０［叩苣□－．℃⑩ ０ ■Ｄ劃ｎｕＭ弧彦8［Ｉ］Ｃｕ 日日弓匠 死ロ 加配油麺魍繩咄叩麺０００００００００ 加犯犯釦釦泥印泥５００００００００１００００００「Ｕｍｊ白】かＬ缶 《ＨＵＲＪ ｍｈｕＬ＝０．０３０ｍ0.6L］０６０．８１００．２０４[－コｒノＲ０．６０．８[-コ００．２０．４０．６０．８１ｒノＲ［－コ００２oＡｎＲ００．２０．４０．６０．８１００．ｚ0.2ｂ ｕ・ｕｕｒノＲ［－コｒノＲ［－コFig.８EffectofsuperficialgasvelocityonprofilesofgasholdupandliquidvelocityFig.９EffectofliquiddepthtocolumndiameterratioonprofilesofgasholdupandliquidvelocityradialpositionsshowingamaximumingasholdupandupwardliquidvelocitymovetowardsthecolumncenterwithincreasingliquiddepthanddecreasingcolumndiameteEnamelyasH/DTincreases．AthigherHⅡＤ刀ｔｈｅｒｅｉｓａｃｅｎｔｒａｌｍａｘｉｍｕｍｉｎｂｏｔｈｇａsholdupandliquidvelocity；thesephenomenaarecommonlynoticedfbrtheusualbubblecolumnwithadeeppoolofliquidTheH/DrvalueatthetransitionfiFomtheplatecolumntothebubblecolumnregimeisdeducedtobearoundunitybHills（1974）andMiyauchieM/､（1970）studiedradialgasholdupinbubblecolumntheoreticallyusingthetheoryofcilCulation・HoweveEthistheoryreqUirestheuseofsomeunknownvariables・Therefbre,MiyaharaeMﾉ.（1989）triedtocolrelateprofilesofgasholdupinthemiddlefiothregionbythefbllowingequations．〃＝0.3月~'＋１．８，忽-0.05"-ＭD"Ｗ,(29）(30）FigurelO，obtainedbyMiyaharaeML（1989)，showsacomparisonofresultscalculatedusingEqs．(24）ａｎｄ（25）withtheexperimentalmeasurements・ThecalculatedcurvescomparefavorablywiththemeasurementsexceptｆｂｒｔｈｏｓｅｗｈｅｒｅＨＤ７＝1.01．Forthesakeofcomparison，theempiricalresultsofKatohemﾉ.（1975）fbrbubblecolumnsareshowninthesaｍｅｇｒａｐｈ1.6－CaIcuIatedfrom苧-(急「…伽|:舌二二Ｊi二判…伽ｗｈｅｒｅｎ,ＥＣＣ,８０",,〃ａｎｄ〃，aregivenasfbllows(美W-OMDm｡(2`)1.4２１８０［‐］・湿一狂0.60.40.2(27）eG〔-０．４６ ＯＯＺＱ４０．６０．８１ｒノＲ［－コ２２ﾛｰﾆﾆﾕｰ0.3-0.053F７－０８２８℃,,，FiglOComparisonbetweencalculatedandexperimentalmeasurements(28）174 ToshiroMIYAHARAConcludingRemarkslnperfbratedplatecolumnswithgasbubblesasadiscretephase，ｔｈｅｆｉｏｔｈｒｅｇｉｍｅｆｂｒｍｅｄｂｙｂｌｏｗｉｎｇｇasintoshallowliquidonaperfbratedplatecanbedividedintothreeregionsalongtheaxialdirection：bubblefbrmationregion，middleregionandupperlayerregionTheaxialprofileofgasholdUpbasedontheoreticalanalysisincludingtheeffectofthenearwakeofbubbletakingintoaccounttheEuler，sequationbymeansofthecalculusofvariationagreeswiththemeasurementsbetterthanthatobtainedbyAzbeLThemeangasholdupisbetterexprcssedbythetheoreticalequationderivedfiomtheaxialprofileoｆｇａｓｈｏｌｄ叩thantheresultsobtainedbyAZbelandKimHoweveMhemeasurementsbehaveslightlydifferentlyffomthetheoreticalequationTherefbre，theempiricalequationswereobtaiｎｅｄＡｓｆｂｒｔｈｅｒａｄｉａｌｇａｓｈｏｌｄｕｐａｎｄｌiquidcirculation，ｔｈｅｒｅｉｓａｍａｘｉｍｕｍｉｎｔｈｅｇasholdupatacertainradialpositionwithacentralminimumunlikethebehaviorintheusualbubblecolumnThisbehaviorcolTespondstothatofradialprofilesofliquidvelocityBSigmficantinfbrmationongashold叩andliqUidcirculationintheusualbubblecolumnisavailableintheliterature，Ｍｏreworkisneeded，howeveLtogeneralizethefluidflowdynamicsoffiothonaperfbratedplate．＝bubblevolume,ｍ３＝ｗａｋｅvolume,ｍ３＝distancefiomaperfbratedplate,、９％瓜Ｚ＝gasholdup＝meangasholdUP＝gasholdupatthecenter＝maximumgasholdup＝liqUidholdup＝angle,rad＝Lagrangemultiplier＝gasdensity§ｋg/ｍ３＝liqUiddensityｩｋg/ｍ３＝surfacetension,Ｎ/、妃一町岫蜥ｑ０入ｐｐｏｇＩReferencesAzbel,ｕｓ:Ｋｈｉｍ・Pro、.,No.1１（1962);ん妃、、Ｃ７ｉｅ腕.肋9.,3,319(1963）Braulick,ＷＪ.,J・RFairandBLemer:４１．１凪１，７３(1965）Collins,Ｒ､:エF7iｲＭＭｂｃｈ.,22,763(1965a）Collins,Ｒ:die"zDzgSbj.,２０，８５１（1965b）Collins,ＲＭ肋尅MbcA.,25,469(1966）Collins,ＲＭ川尅Ｍｂｃｈ.,28,97(1967）Davies,Ｒ､MandSirGeoffreynlylor:Ｐｍｃ・ＲｑｙＬＳｂｃ.(London),Ａ200,375(1950）Darwin,Ｃ:Ｐｍｃ・Ctz肋.Ｐﾉbjﾉ.,49,342(1953）Hills,』.Ｈ:Dα"Ｍ７ｚ川die､､肋g瓜５２，１（1974）Katoh,Ｙ,Ｓ,NishinakaandS・Morooka:KngzzA皿肋gzJ肋Ｒｏ"62"zsh",1,530(1975）Ｋｉｍ,ＳＫ.:ＨＷａＡａﾉＭ(ＷａＨＷａＡａｋＫｂ'Zg叩,No.6,399(l965Morem.Ｃ舵加Ｅ'29.,6,634(1966）Milne-Thomson,ＬＭ:``TheoreticalHydmdynamics,''240(1968）Miyahara,Ｔ,mTBuzaki,1ESaito,YMatsubaandTnlkahashi:７１ｌｉｅｄｉｅ〃cαﾉ助gmeemZgJb"、α４40,21(1989）Miyauchi,nandCShyu:KtzgZJ肋KDgzz肋,３４，９５８（1970）Takahashi,Ｔ,Ｒ・MatsunoandTMiyaharaMdiel"・助gJZZpα",６，３８(1973）nlkahashi,Ｔ､,TMiyaharaandK・Shimizu:Ｊ:Ｃｈｅ碗．助ｇＪｔＺｐα",７，７５(1974）YOshitome,Ｈ:ｈＪｇｎ胸ＫＤｇｎ肋,27,27(1963）Notationa＝radiusofsphericalcapbubble,ｍＤ７＝columndiametenmciEt＝kineticenelgyofelementarylayerperunitcross-sectionalarea,ｋg/ｓ２必＝potentialeneWofelementaIylayerperumtcross-sectionalarea,ｋg/s2CZE，＝energyofsurfacetensionofelementarylayerperunitcross-sectionalarea,ｋg/ｓ２Ｅ１＝totalenergyperunitcross-sectionalarea，ｋg/s２Ｆ＝fimctiondefinedbyEq.(13),kg/(ms2）丹＝Fmudenumber{=､【;g/(gH)｝９＝gravitationalacceleration,ｍ/s２Ｇ＝fimctiondefinedbyEq.(14),kg/(ms2）Ｈ＝liqUiddepth,ｍﾉi／＝fiothhcight,ｍＬ＝measulingheight,ｍ〃＝numberofbubbleinunitvolume,ｍ－３〃＝exponentinEq.(29）"’＝exponentinEq.(30）ﾊル＝constantdefinedbyEq.(3)尺＝columnradius,ｍア＝radialdistance,ｍｌｂ＝７atmaximumgashold叩,ｍ８＝surfaceareaofasphericalcapbUbble,ｍ２Ｄｂ＝bUbblerisevelocity9m/SUk。＝superficialgasvelocitybm/ｓ"ノ＝upwardliquidvelocity;ｍ/ｓ

Characteristics of Fluid Flow of Froth on a Perforated Plate

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Characteristics of Fluid Flow of Froth on a Perforated Plate

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