流體力學(xué)與傳熱課件Heterogeneous-Flow-and-Separation

3.HeterogeneousFlowandSeparation

3.1FlowPastImmersedObjectsDefinitionofDragCoefficientforFlowPastImmersedObjects3.HeterogeneousFlowandSepa1.IntroductionandtypesofdragTheflowoffluidsoutsideimmersedbodiesappearsinmanychemicalengineeringapplicationsandotherprocessingapplications.Forexamplesettling,dryingandfiltration,andsoon.1.Introductionandtypesof2.Dragcoefficient

Correlationsofthegeometryandflowcharacteristicsforsolidobjectssuspendedinfluidaresimilarinconceptandformtothefrictionfactor-Reynoldsnumbercorrelationgivenforflowinsideconduits.2.DragcoefficientCorrelaInflowthroughpipes,thefrictionfactorwasdefinedastheratioofthedragforceperunitareatotheproductoffluiddensityandvelocityhead.Inflowthroughpipes,the

ForflowpastimmersedobjectsthedragcoefficientisobtainedbysubstitutingCDforthefrictionfactorKfinequation(1.4-32

)3.1-1

thereforeForflowpastimmersedobjecTheReynoldsnumberforaparticleinafluidisdefinedasFromdimensionalanalysis,thedragcoefficientofasmoothsolidinanincompressiblefluiddependsuponaReynoldsnumberandthenecessaryshaperatios.Foragivenshape3.1-2

TheReynoldsnumberforapartDragcoefficientsoftypicalshapesForeachparticularshapeofobjectandorientationoftheobjectwithrespecttothedirectionofflow,adifferentrelationofCDversusReexists.CorrelationsofdragcoefficientversusReynoldsnumberareshowninfigure.Dragcoefficientsoftypicals流體力學(xué)與傳熱課件Heterogeneous-Flow-and-SeparationThesecurveshavebeendeterminedexperimentally.However,inthelaminarregionforlowReynoldsnumbers,lessthanabout1.0,theexperimentaldragforceforasphereisthesameasthetheoreticalStokes'lawequationasfollows:3.1-3

ThesecurveshavebeendeteCombiningEqs.(3.1-1)and(3.1-3)andsolvingforCD,thedragcoefficientpredictedbyStokes'lawis3.1-4

CombiningEqs.(3.1-1)and(3.ThevariationofCDwithReisquitecomplicatedbecauseoftheinteractionofthefactorsthatcontrolskindragandformdrag.Forasphere,astheReynoldsnumberisincreasedbeyondtheStokes'lawrange,separationoccursandawakeisformed.ThevariationofCDwithReisFurtherincreasesinRecauseshiftsintheseparationpoint.AtaboutRe

=3×105thesuddendropinCDistheresultoftheboundarylayerbecomingcompletelyturbulentandthepointofseparationmovingdownstream.FurtherincreasesinRecauseIntheregionofReabout1×103to2×105,thedragcoefficientisapproximatelyconstantforeachshapeandCD=0.44forasphere.IntheregionofReabout1×103.1.2FlowthroughBedsofSolids

1.Introduction

Asystemofconsiderableimportanceinchemicalandotherprocessengineeringfieldsisthepackedbed,whichisusedforafixed-bedcatalyticreactor,adsorptionofasolute,absorption,filterbed,andsoon3.1.2FlowthroughBedsofSolInthetheoreticalapproachused,thepackedcolumnisregardedasabundleofcrookedtubesofvaryingcross-sectionalarea.ThetheorydevelopedinChapter1forsinglestraighttubesisusedtodeveloptheresultsforthebundleofcrookedtubes.Inthetheoreticalapproachus2.LaminarflowinpackedbedsCertaingeometricrelationsforparticlesinpackedbedsareusedinthederivationsforflow.Thevoidfractionεinapackedbedisdefinedas2.LaminarflowinpackedbedsThespecificsurfaceofaparticleavisdefinedasForasphericalparticle,,Thespecificsurfaceofapartwhereaistheratiooftotalsurfaceareainthebedtototalvolumeofbed(voidvolumeplusparticlevolume)Since(1-ε)isthevolumefractionofparticlesinthebedwhereaistheratiooftotalTheaverageinterstitialvelocityinthebedisu'andisrelatedtothesuperficialvelocityubasedonthecrosssectionoftheemptycontainerby3.1-9TheaverageinterstitialvelocTodeterminetheequivalentchanneldiameterDe,thesurfaceareafornparallelchannelsoflengthLissetequaltothesurface-volumeratiotimestheparticlevolumeS0L(1-ε).3.1-6whereS0isthecross-sectionalareaofthebedTodeterminetheequivalentchThevoidvolumeinthebedisthesameasthetotalvolumeofthenchannels3.1-7CombiningEqs.(3.1-6)and(3.1-7)givesanequationforDe3.1-8ThevoidvolumeinthebedisForflowatverylowReynoldsnumbers,thepressuredropshouldvarywiththefirstpowerofthevelocityandinverselywiththesquareofthechannelsize,inaccordancewiththeHagen-Poiseulliequationforlaminarflowinstraighttubes.ForflowatverylowReynoldsTheequationsforu'(equation3.1-9)andDe(equation3.1-8

)areusedintheHagen-PoiseuilleequationTheequationsforu'(equationor3.1-11ThetrueLislargerbecauseofthetortuouspath.Experimentaldatagiveanempiricalconstantof150for72or3.1-11ThetrueLislargerEquation(3.1-12

)iscalledtheBlake-Kozenyequationforlaminarflow,voidfractionslessthan0.5,effectiveparticlediameterDp,andRep<10:3.1-12

Equation(3.1-12)iscalledt3.2MotionofParticlesthroughFluidsManyprocessingsteps,especiallymechanicalseparations,involvethemovementofsolidparticlesorliquiddropsthroughafluid.Thefluidmaybegasorliquid,anditmaybeflowingoratrest.3.2MotionofParticlesthrougtheestimationofdustandfumesfromairorfluegas,theremovalofsolidsfromliquidwastes,andtherecoveryofacidmistsfromthewastegasofanacidplantExamplesare：theestimationofdustandfumThreeforcesactonaparticlemovingthroughafluid:theexternalforce,gravitationalorcentrifugal;Threeforcesactonaparticle(2)thebuoyantforce,whichactsparallelwiththeexternalforcebutintheoppositedirection;and(3)thedragforce,whichappearswheneverthereisrelativemotionbetweentheparticleandthefluid.(2)thebuoyantforce,whichaMechanicsofparticlemotion

Themovementofaparticlethroughafluidrequiresanexternalforceactingontheparticle.Thisforcemaycomefromadensitydifferencebetweentheparticleandthefluid.MechanicsofparticlemotionItalsomaybetheresultelectricormagneticfields.Inthissectiononlygravitationalorcentrifugalforces,whicharisefromdensitydifferences,areconsidered.ItalsomaybetheresulteInthegeneralcase,thedirectionofmovementoftheparticlerelativetothefluidmaynotbeparallelwiththedirectionoftheexternalandbuoyantforcesInthegeneralcase,thedirecOnlytheone-dimensionalcase,wherethelinesofactionofallforcesactingontheparticlearecollinear,isconsideredinthissection.Onlytheone-dimensionalcaEquationforone-dimensionalmotionofparticlethroughfluid

ConsideraparticleofvolumeVp,densityρpmovingthroughafluid.Threeforcesactingonaparticleare:(1)externalforce:Fe=m

ae(2)buoyantforce:(3)dragforce:FeFbFDEquationforone-dimensionalmThentheresultantforceontheparticleisFe-Fb–Fd,(3.2-1

)Theaccelerationoftheparticleisdu/dt,ThentheresultantforceonthsubstitutingtheforcestoEq(3.2-1

)gives(3.2-5

)substitutingtheforcestoEq(Theterminalvelocityisfoundbytakingdu/dt=03.2-5TheterminalvelocityisfoundMotionfromgravitationalforce

Iftheexternalforceisgravity,ae

isg,theaccelerationofgravity,andEq.(3.2-5)becomes3.2-6MotionfromgravitationalforcMotioninacentrifugalfield

Theaccelerationfromacentrifugalforcefromcircularmotionisae

=rω2

(3.2-7)

(3.2-8)MotioninacentrifugalfieldTerminalvelocity

Thedragalwaysincreaseswithvelocity,theaccelerationdu/dtofaparticledecreaseswithtimeandapproachestozero.Theparticlequicklyreachesaconstantvelocity,whichisthemaximumattainableundercircumstances,andwhichiscalledtheterminalvelocityut.TerminalvelocityThedragaTheequationfortheterminalvelocityut

isfound,forgravitationalsettling,bytakingdu/dt=0.ThenfromEq.(3.2-6),3.2-9TheequationfortheterminalInmotionfromacentrifugalforce,thevelocitydependsontheradius,andtheaccelerationisnotconstantiftheparticleisinmotionwithrespecttothefluid.InmotionfromacentrifugalfInmanypracticalusesofcentrifugalforce,however,du/dtissmallincomparisonwiththeothertwotermsinEq.(3.2-8)InmanypracticalusesofcentIfdu/dtisneglected,aterminalvelocityatanygivenradiuscanbedefinedbytheequation3.2-10Ifdu/dtisneglected,atermiDragcoefficientThequantitativeuseofEqs.(3.2-5)to(3.2-10)requiresthatnumericalvaluesbeavailableforthedragcoefficientCD.FigureshowsthedragcoefficientasafunctionofReynoldsnumber.DragcoefficientThequantitatiThedragcurveshowninfigureapplies,however,onlyunderrestrictedconditions.Theparticlemustbeasolidsphere,itmustbefarfromotherparticlesandfromthevesselwallsothattheflowpatternaroundtheparticleisnotdistorted,andtheparticlemustbemovingatitsterminalvelocitywithrespecttothefluid.ThedragcurveshowninfigureWhentheparticleisatthesufficientdistancefromtheboundariesofthecontainerandfromotherparticles,sothatitsfallisnotaffectedbythem,theprocessiscalledfreesettling.Ifthemotionofparticleisimpededbyotherparticles,whichhappedwhentheparticlesareneareachothereventhoughtheymaynotactuallybecolliding,theprocessiscalledhinderedsettling.WhentheparticleisattheIftheparticlesareverysmall,Brownianmovementappears.Thiseffectbecomesappreciableataparticlesizeofabout2-3μmandpredominatesovertheforceofgravitywithaparticlesizeof0.1orless.IftheparticlesareverysTherandommovementoftheparticletendstosuppresstheeffectoftheforceofgravity,sosettlingdoesnotoccur.ApplicationofcentrifugalforcereducestherelativeeffectofBrownianmovement.Therandommovementofthemovementofsphericalparticles

Iftheparticlesarespheresofdiameterdp(3.2-11

)(3.2-12

)AndmovementofsphericalparticleSubstitutionofmandApfromEq(3.2-11

)and(3.2-12

)intoEq(3.2-9)and(3.2-10)gives(3.2-13

)(3.2-13a)andSubstitutionofmandApfTheterminalvelocitiesatthedifferentReynoldsnumberIntheory,stokes’lawisvalidonlywhenReisconsiderablylessthanunity.Eq.(3.2-13

)maybeusedwithsmallerrorforallReynoldsnumberslessthan1.Theterminalvelocitiesatthe

Forgravitysettlingofaspheres,atlowReynoldsnumbers,thedragcoefficientvariesinverselywithRe.(3.2-14

)(3.2-16

)andsubstitutingEq(3.2-14

)intoEq(3.2-13

),givesForgravitysettlingofasp

Equation(3.2-16

)isknownasStokes′law,andappliesforparticleReynoldsnumberslessthan1.0.substitutingEq(3.2-14

)intoEq(3.2-13a),gives(3.2-20

)Equation(3.2-20

)canbeusedtopredictthevelocityofasmallsphereinacentrifugalfield.Equation(3.2-16)isknownaForRe>1000,thedragcoefficientisapproximatelyconstantat0.40to0.45,andlets(3.2-19)so

theequationisCD=0.44ForRe>1000,thedragcoeffEquation(3.2-19)isNewton’lawandappliesonlyforfairlylargeparticlesfallingingasesorlow-viscosityfluids.Equation(3.2-19)isNewton’CriterionforsettlingregimeToidentifytherangeinwhichthemotionoftheparticlelies,thevelocitytermiseliminatedfromReynoldsnumberbysubstitutingutfromEq.(3.2-16

)togive,fortheStokes’lawrange（3.2-21

）CriterionforsettlingregimeRe=K3/18.Re<1.0,toprovideaconvenientcriterionK,let（3.2-22

）Re=K3/18.Re<1.0,toprovide

Then,fromEq(3.2-21

),Re=K3/18.SettingRe=1.0andsolvinggivesK=2.6.Ifthesizeoftheparticleisknown,KcanbecalculatedfromEq(3.2-22).IfKsocalculatedislessthan2.6,Stokes’lawapplies.Then,fromEq(3.2-21),Re=

SubstitutionforutfromEq.(3.2-19)showsthatfortheNewton’slawrangeRe=1.75K1.5.IntherangebetweenStokes’lawandNewton’slaw(2.6<K<68.9),theterminalvelocityiscalculatedfromEq(3.2-13

)usingavalueofCDfoundbytrialfromFig.Settingthisequalto1000andsolvinggivesK=68.9.ThusifKisgreaterthan68.9,Newton’slawapplies.SubstitutionforutfromEqproblem1SettlingofasphericalparticleinaairisfollowedStokes’law,ifthetemperaturechangesfrom25to50℃,theterminalvelocitywill();

ifsettlinginliquid,theterminalvelocitywill()

Theterminalvelocityisthevelocitythattheaccelerationthataparticlemovesthroughthefluidapproachesto()problem1Settlingofasphericaproblem2Asinglesphericalparticlesettlingfreelyinthefluidanditislaminarflow,whentheparticlediameterincreases,theterminalvelocityuwill

;whentheviscosityoffluidincreases,uwill

;ifthefluidisagas,whathappenstou

ifthetemperatureincrease?

problem2AsinglesphericalparAsshownbyequations(3.2-16

)and(3.2-19),theterminalvelocityutvarieswiththesquareofdiameterofparticleinthe()range,whereasinthe()rangeitvarieswith0.5powerofthediameterofparticleForagivenpackedbed,Blake-Kozenyequationindicatesthattheflowis()tothepressuredropand()proportionaltothefluidviscosity.Asshownbyequations(3.2-161、試計算直徑為30μm的球形石英顆粒（其密度為2650kg/m3），在20℃水中和20℃常壓空氣中的自由沉降速度。Tocalculatetheterminalvelocityofasphericalquartzparticle,30μmindiameterand2650kg/m3indensity,settlinginthewaterandtheairatthetemperatureof20℃,respectively.1、試計算直徑為30μm的球形石英顆粒（其密度為2650kgSolution:d=30μm、ρs=2650kg/m3（1）μ=1.01×10-3Pa·sandρ=998kg/m3forwateratt=20℃checkItisfollowedstokes’lowut=8.02×10-4m/sSolution:d=30μm、ρs=2650kg/m（2）μ=1.81×10-5Pa·sρ=1.21kg/m3for

theairatt=20℃Assumingthatthetypeofflowingisfollowedstokes’lawcheckut=7.18×10-2m/s。

（2）μ=1.81×10-5Pa·sρ=1.3.HeterogeneousFlowandSeparation

3.1FlowPastImmersedObjectsDefinitionofDragCoefficientforFlowPastImmersedObjects3.HeterogeneousFlowandSepa1.IntroductionandtypesofdragTheflowoffluidsoutsideimmersedbodiesappearsinmanychemicalengineeringapplicationsandotherprocessingapplications.Forexamplesettling,dryingandfiltration,andsoon.1.Introductionandtypesof2.Dragcoefficient

Correlationsofthegeometryandflowcharacteristicsforsolidobjectssuspendedinfluidaresimilarinconceptandformtothefrictionfactor-Reynoldsnumbercorrelationgivenforflowinsideconduits.2.DragcoefficientCorrelaInflowthroughpipes,thefrictionfactorwasdefinedastheratioofthedragforceperunitareatotheproductoffluiddensityandvelocityhead.Inflowthroughpipes,the

ForflowpastimmersedobjectsthedragcoefficientisobtainedbysubstitutingCDforthefrictionfactorKfinequation(1.4-32

)3.1-1

thereforeForflowpastimmersedobjecTheReynoldsnumberforaparticleinafluidisdefinedasFromdimensionalanalysis,thedragcoefficientofasmoothsolidinanincompressiblefluiddependsuponaReynoldsnumberandthenecessaryshaperatios.Foragivenshape3.1-2

TheReynoldsnumberforapartDragcoefficientsoftypicalshapesForeachparticularshapeofobjectandorientationoftheobjectwithrespecttothedirectionofflow,adifferentrelationofCDversusReexists.CorrelationsofdragcoefficientversusReynoldsnumberareshowninfigure.Dragcoefficientsoftypicals流體力學(xué)與傳熱課件Heterogeneous-Flow-and-SeparationThesecurveshavebeendeterminedexperimentally.However,inthelaminarregionforlowReynoldsnumbers,lessthanabout1.0,theexperimentaldragforceforasphereisthesameasthetheoreticalStokes'lawequationasfollows:3.1-3

ThesecurveshavebeendeteCombiningEqs.(3.1-1)and(3.1-3)andsolvingforCD,thedragcoefficientpredictedbyStokes'lawis3.1-4

CombiningEqs.(3.1-1)and(3.ThevariationofCDwithReisquitecomplicatedbecauseoftheinteractionofthefactorsthatcontrolskindragandformdrag.Forasphere,astheReynoldsnumberisincreasedbeyondtheStokes'lawrange,separationoccursandawakeisformed.ThevariationofCDwithReisFurtherincreasesinRecauseshiftsintheseparationpoint.AtaboutRe

=3×105thesuddendropinCDistheresultoftheboundarylayerbecomingcompletelyturbulentandthepointofseparationmovingdownstream.FurtherincreasesinRecauseIntheregionofReabout1×103to2×105,thedragcoefficientisapproximatelyconstantforeachshapeandCD=0.44forasphere.IntheregionofReabout1×103.1.2FlowthroughBedsofSolids

1.Introduction

Asystemofconsiderableimportanceinchemicalandotherprocessengineeringfieldsisthepackedbed,whichisusedforafixed-bedcatalyticreactor,adsorptionofasolute,absorption,filterbed,andsoon3.1.2FlowthroughBedsofSolInthetheoreticalapproachused,thepackedcolumnisregardedasabundleofcrookedtubesofvaryingcross-sectionalarea.ThetheorydevelopedinChapter1forsinglestraighttubesisusedtodeveloptheresultsforthebundleofcrookedtubes.Inthetheoreticalapproachus2.LaminarflowinpackedbedsCertaingeometricrelationsforparticlesinpackedbedsareusedinthederivationsforflow.Thevoidfractionεinapackedbedisdefinedas2.LaminarflowinpackedbedsThespecificsurfaceofaparticleavisdefinedasForasphericalparticle,,Thespecificsurfaceofapartwhereaistheratiooftotalsurfaceareainthebedtototalvolumeofbed(voidvolumeplusparticlevolume)Since(1-ε)isthevolumefractionofparticlesinthebedwhereaistheratiooftotalTheaverageinterstitialvelocityinthebedisu'andisrelatedtothesuperficialvelocityubasedonthecrosssectionoftheemptycontainerby3.1-9TheaverageinterstitialvelocTodeterminetheequivalentchanneldiameterDe,thesurfaceareafornparallelchannelsoflengthLissetequaltothesurface-volumeratiotimestheparticlevolumeS0L(1-ε).3.1-6whereS0isthecross-sectionalareaofthebedTodeterminetheequivalentchThevoidvolumeinthebedisthesameasthetotalvolumeofthenchannels3.1-7CombiningEqs.(3.1-6)and(3.1-7)givesanequationforDe3.1-8ThevoidvolumeinthebedisForflowatverylowReynoldsnumbers,thepressuredropshouldvarywiththefirstpowerofthevelocityandinverselywiththesquareofthechannelsize,inaccordancewiththeHagen-Poiseulliequationforlaminarflowinstraighttubes.ForflowatverylowReynoldsTheequationsforu'(equation3.1-9)andDe(equation3.1-8

)areusedintheHagen-PoiseuilleequationTheequationsforu'(equationor3.1-11ThetrueLislargerbecauseofthetortuouspath.Experimentaldatagiveanempiricalconstantof150for72or3.1-11ThetrueLislargerEquation(3.1-12

)iscalledtheBlake-Kozenyequationforlaminarflow,voidfractionslessthan0.5,effectiveparticlediameterDp,andRep<10:3.1-12

Equation(3.1-12)iscalledt3.2MotionofParticlesthroughFluidsManyprocessingsteps,especiallymechanicalseparations,involvethemovementofsolidparticlesorliquiddropsthroughafluid.Thefluidmaybegasorliquid,anditmaybeflowingoratrest.3.2MotionofParticlesthrougtheestimationofdustandfumesfromairorfluegas,theremovalofsolidsfromliquidwastes,andtherecoveryofacidmistsfromthewastegasofanacidplantExamplesare：theestimationofdustandfumThreeforcesactonaparticlemovingthroughafluid:theexternalforce,gravitationalorcentrifugal;Threeforcesactonaparticle(2)thebuoyantforce,whichactsparallelwiththeexternalforcebutintheoppositedirection;and(3)thedragforce,whichappearswheneverthereisrelativemotionbetweentheparticleandthefluid.(2)thebuoyantforce,whichaMechanicsofparticlemotion

Themovementofaparticlethroughafluidrequiresanexternalforceactingontheparticle.Thisforcemaycomefromadensitydifferencebetweentheparticleandthefluid.MechanicsofparticlemotionItalsomaybetheresultelectricormagneticfields.Inthissectiononlygravitationalorcentrifugalforces,whicharisefromdensitydifferences,areconsidered.ItalsomaybetheresulteInthegeneralcase,thedirectionofmovementoftheparticlerelativetothefluidmaynotbeparallelwiththedirectionoftheexternalandbuoyantforcesInthegeneralcase,thedirecOnlytheone-dimensionalcase,wherethelinesofactionofallforcesactingontheparticlearecollinear,isconsideredinthissection.Onlytheone-dimensionalcaEquationforone-dimensionalmotionofparticlethroughfluid

ConsideraparticleofvolumeVp,densityρpmovingthroughafluid.Threeforcesactingonaparticleare:(1)externalforce:Fe=m

ae(2)buoyantforce:(3)dragforce:FeFbFDEquationforone-dimensionalmThentheresultantforceontheparticleisFe-Fb–Fd,(3.2-1

)Theaccelerationoftheparticleisdu/dt,ThentheresultantforceonthsubstitutingtheforcestoEq(3.2-1

)gives(3.2-5

)substitutingtheforcestoEq(Theterminalvelocityisfoundbytakingdu/dt=03.2-5TheterminalvelocityisfoundMotionfromgravitationalforce

Iftheexternalforceisgravity,ae

isg,theaccelerationofgravity,andEq.(3.2-5)becomes3.2-6MotionfromgravitationalforcMotioninacentrifugalfield

Theaccelerationfromacentrifugalforcefromcircularmotionisae

=rω2

(3.2-7)

(3.2-8)MotioninacentrifugalfieldTerminalvelocity

Thedragalwaysincreaseswithvelocity,theaccelerationdu/dtofaparticledecreaseswithtimeandapproachestozero.Theparticlequicklyreachesaconstantvelocity,whichisthemaximumattainableundercircumstances,andwhichiscalledtheterminalvelocityut.TerminalvelocityThedragaTheequationfortheterminalvelocityut

isfound,forgravitationalsettling,bytakingdu/dt=0.ThenfromEq.(3.2-6),3.2-9TheequationfortheterminalInmotionfromacentrifugalforce,thevelocitydependsontheradius,andtheaccelerationisnotconstantiftheparticleisinmotionwithrespecttothefluid.InmotionfromacentrifugalfInmanypracticalusesofcentrifugalforce,however,du/dtissmallincomparisonwiththeothertwotermsinEq.(3.2-8)InmanypracticalusesofcentIfdu/dtisneglected,aterminalvelocityatanygivenradiuscanbedefinedbytheequation3.2-10Ifdu/dtisneglected,atermiDragcoefficientThequantitativeuseofEqs.(3.2-5)to(3.2-10)requiresthatnumericalvaluesbeavailableforthedragcoefficientCD.FigureshowsthedragcoefficientasafunctionofReynoldsnumber.DragcoefficientThequantitatiThedragcurveshowninfigureapplies,however,onlyunderrestrictedconditions.Theparticlemustbeasolidsphere,itmustbefarfromotherparticlesandfromthevesselwallsothattheflowpatternaroundtheparticleisnotdistorted,andtheparticlemustbemovingatitsterminalvelocitywithrespecttothefluid.ThedragcurveshowninfigureWhentheparticleisatthesufficientdistancefromtheboundariesofthecontainerandfromotherparticles,sothatitsfallisnotaffectedbythem,theprocessiscalledfreesettling.Ifthemotionofparticleisimpededbyotherparticles,whichhappedwhentheparticlesareneareachothereventhoughtheymaynotactuallybecolliding,theprocessiscalledhinderedsettling.WhentheparticleisattheIftheparticlesareverysmall,Brownianmovementappears.Thiseffectbecomesappreciableataparticlesizeofabout2-3μmandpredominatesovertheforceofgravitywithaparticlesizeof0.1orless.IftheparticlesareverysTherandommovementoftheparticletendstosuppresstheeffectoftheforceofgravity,sosettlingdoesnotoccur.ApplicationofcentrifugalforcereducestherelativeeffectofBrownianmovement.Therandommovementofthemovementofsphericalparticles

Iftheparticlesarespheresofdiameterdp(3.2-11

)(3.2-12

)AndmovementofsphericalparticleSubstitutionofmandApfromEq(3.2-11

)and(3.2-12

)intoEq(3.2-9)and(3.2-10)gives(3.2-13

)(3.2-13a)andSubstitutionofmandApfTheterminalvelocitiesatthedifferentReynoldsnumberIntheory,stokes’lawisvalidonlywhenReisconsiderablylessthanunity.Eq.(3.2-13

)maybeusedwithsmallerrorforallReynoldsnumberslessthan1.Theterminalvelocitiesatthe

Forgravitysettlingofaspheres,atlowReynoldsnumbers,thedragcoefficientvariesinverselywithRe.(3.2-14

)(3.2-16

)andsubstitutingEq(3.2-14

)intoEq(3.2-13

),givesForgravitysettlingofasp

Equation(3.2-16

)isknownasStokes′law,andappliesforparticleReynoldsnumberslessthan1.0.substitutingEq(3.2-14

)intoEq(3.2-13a),gives(3.2-20

)Equation(3.2-20

)canbeusedtopredictthevelocityofasmallsphereinacentrifugalfield.Equation(3.2-16)isknownaForRe>1000,thedragcoefficientisapproximatelyconstantat0.40to0.45,andlets(3.2-19)so

theequationisCD=0.44ForRe>1000,thedragcoeffEquation(3.2-19)isNewton’lawandappliesonlyforfairlylargeparticlesfallingingases