化工原理英文教材流体输送阻力系数Flow-past-immersed-bodies课件

化工原理

PrinciplesofChemicalIndustry化工原理FlowpastimmersedbodiesFlowpastimmersedbodiesThediscussioninthisChapteris

concernedabout:1.thelawsoffluidflow;2.factorsthatcontrolchangesofpressureandvelocityoffluidsflowingpastsolidboundariesandwasespeciallyconcernedwithflowthroughclosedpipesandchannels.Emphasisduringthediscussionwasplacedprimarilyonthefluid.ThediscussioninthisChapterThesituationwherethesolidisimmersedin,andsurroundedbyfluidisthesubjectofthischapter.ThesituationwherethesolidDragandDragcoefficientsDrag

——TheforceinthedirectionofflowexertedbythefluidonthesolidDragcoefficients

——Ananalogousfactorusedforimmersedsolids(likelocalfrictionalcoef.)DragandDragcoefficientsDragDragWhenthewallofthebodyisparallelwiththedirectionofflow,showninFig.3-10a,dragforceisthewallshear.DragWhenthewallofthebodFig.7-1showsthepressureandshearforcesactingonanelementofareadAinclinedatanangleof90°-αtothedirectionofflow.DirectionOfflowαPcosαdA(formdrag)Forecefrompressure=pdATwsinαdA(walldrag)TwdAFig.7-1showsthepressureandThedragfromwallshearisτw•sinα•dA(walldrag),and

thatfrompressureisp•cosα•dA(formdrag).Theabovedragsareinthedirectionofflow.Thetotaldragonthebodyisthesumoftheintegralsofthesequantitieseachevaluatedovertheentiresurfaceofthebodyincontactwiththefluid.ThedragfromwallshearisτwThetotalintegrateddragfromwallsheariscalledwalldrag.Thetotalintegrateddragfrompressureiscalledformdrag.化工原理英文教材流体输送阻力系数Flow-past-immersed-bodies课件Inpotentialflow(idealfluids)thereisnowalldrag.Also,thepressuredraginthedirectionofflowisbalancedbyanequalforceintheoppositedirection,andintegraloftheformdragiszero.Sothereisnonetdraginpotentialflow.Inpotentialflow(idealfluidDragcoefficientsIfFDisthetotaldrag,theaveragedragperunitprojectedareaisFD/A,justasfrictionfactorf

isdefinedastheratioofτwtotheproductofthedensityofthefluidandthevelocityhead,sothedragcoefficientdenotedbyζ(CD),isdefinedastheratioofFD/Atothissameproduct,or(7-1)DragcoefficientsIfFDisthTheequation(7-1)isthesameastheequation(5-71)LetΔp=FD/A,andsubstitutingintheaboveequationTheequation(7-1)isthesameForparticleshavingshapesotherthanspherical,itisnecessarytospecifythesizeandgeometricformofthebodyanditsorientationwithrespecttothedirectionofflowofthefluid.Forparticleshavingshapesotshapefactors:Onemajordimensionischosenasthecharacteristiclength,andotherimportantdimensionsaregivenasratiostothechosenone.shapefactors:

Fromdimensionalanalysis,thedragcoefficientofasmoothsolidinanincompressiblefluiddependsuponaReynoldsnumberandthenecessaryshapefactors.(7-2)Fromdimensionalanalysis,tDragcoefficientsoftypicalshapes

InFig,curvesofCD(ζ)vs.Reareshownforspheres,longcylinders,anddisks.Theaxisofthecylinderandthefaceofthediskareperpendiculartothedirectionofflow.DragcoefficientsoftypicalsForlowReynoldsnumbersthedragforceforasphereconformstoatheoreticalequationcalledStokes´law,whichmaybewritten(7-3)FromEq.(7-3),thedragcoefficientpredictedbyStokes´lawis,usingEq(7-1)(7-4)Intheory,Stokes’lawisvalidonlywhenReisconsiderablylessthan1.ForlowReynoldsnumbersthed

Thelawisespeciallyvaluableforcalculatingtheresistanceofsmallparticles,suchasdustsorfogs,movingthroughgasesorliquidsoflowviscosity,orforthemotionoflargerparticlesthroughhighlyviscousliquids.ThelawisespeciallyvaluabMechanicsofparticlemotionTheforceactingontheparticlemaycomefromadensitydifferencebetweentheparticleandthefluid,oritmaybetheresultofelectricormagneticfields.MechanicsofparticlemotionThreeforcesactonaparticlemovingthroughafluid:

Theexternalforce,gravitationalorcentrifugal;Thebuoyantforce,whichactsparallelwiththeexternalforcebutintheoppositedirection;Thedragforce,whichappearswheneverthereisrelativemotionbetweentheparticleandthefluid.ThreeforcesactonaparticleEquationforone-dimensionalmotionofparticlethroughfluidConsideraparticleofvolumevf,densityρfmovingthroughafluid.Threeforcesactingonaparticleare:(1)externalforce:Fe=Vf

ρf

ae(2)buoyantforce:Fb=Vfρae(3)dragforce:FeFbFDEquationforone-dimensionalmThentheresultantforceontheparticleis

Fe-Fb–Fd,Theaccelerationoftheparticleisdu/dt,

(7-25)substitutingtheforcestoEq(7-27)gives(7-30)ThentheresultantforceonthTerminalvelocityThedragalwaysincreaseswithvelocity,theaccelerationdu/dtofaparticledecreaseswithtimeandapproachestozero.Theparticlequicklyreachesaconstantvelocity,whichisthemaximumattainableundercircumstances,andwhichiscalledtheterminalvelocityut.TerminalvelocityThedragalwa(7-31)Theterminalvelocityisfoundbytakingdu/dt=0,thenfromEq(7-31)(7-32)化工原理英文教材流体输送阻力系数Flow-past-immersed-bodies课件Inthegravitationalsetting,ae=g(7-33)

motioninacentrifugalfield,theaccelerationfromcentrifugalforcefromcirclemotionis

ae=rω2wherer=radiusofpathofparticle

ω=angularvelocity,radians/secInthegravitationalsetting,theterminalvelocityis(7-34)theterminalvelocityisdragcoefficientThequantitativeuseofEq(7-33)and(7-34)requiresthatnumericalvaluesisavailableforthedragcoefficientζ.dragcoefficientThequantitThedragcurveshowninFig.7-6applies,however,onlyunderrestrictedconditions.Theparticlemustbeasolidsphere,itmustbefarfromotherparticlesandfromthevesselwallsothattheflowpatternaroundtheparticleisnotdistorted,andtheparticlemustbemovingatitsterminalvelocitywithrespecttothefluid.ThedragcurveshowninFig.7-Whentheparticleatthesufficientdistancefromtheboundariesofthecontainerandfromotherparticles,sothatitsfallisnotaffectedbythem,theprocessiscalledfreesettling.Ifthemotionofparticleisimpededbyotherparticles,whichhappenwhentheparticlesareneareachothereventhoughtheymaynotactuallybecolliding,theprocessis

calledhinderedsettling.WhentheparticleatthesuffiIftheparticlesareverysmall,Brownianmovementappears.Thiseffectbecomesappreciableataparticlesizeofabout2-3µmandpredominatesovertheforceofgravitywithaparticlesizeof0.1orless.IftheparticlesareverysmTherandommovementoftheparticletendstosuppresstheeffectoftheforceofgravity,sosettlingdoesnotoccur.ApplicationofcentrifugalforcereducestherelativeeffectofBrownianmovement.TherandommovementoftheparMovementofsphericalparticlesIftheparticlesarespheresofdiameterdp

(7-35)And

(7-36)MovementofsphericalparticleSubstitutionofVpandApfromEq(7-35)and(7-36)intoEq(7-33)and(7-34)gives(7-37)And(7-37a)SubstitutionofVpandApfromTheterminalvelocitiesatthedifferentReynoldsnumber

Intheory,stokes’lawisvalidonlywhenReisconsiderablylessthanunity.

Eq.(7-4)maybeusedwithsmallerrorforallReynoldsnumberslessthan1.TheterminalvelocitiesattheForgravitysettlingofaspheres,atlowReynoldsnumbers,thedragcoefficientvariesinverselywithRe.(7-38)andsubstitutingEq(7-38)intoEq(7-37),gives(7-40)ForgravitysettlingofasEquation(7-40)isknownasStokes´law,andappliesforparticleReynoldsnumberslessthan1.0.SubstitutingEq(7-38)intoEq(7-37a),gives(7-41)Equation(7-41)canbeusedtopredictthevelocityofasmallsphereinacentrifugalfield.Equation(7-40)isknownasFor1000<Re<200000thedragcoefficientisapproximatelyconstantattheequationsareζ=0.44For1000<Re<2000001000<Re<200000thedragcoefficientisapproximatelyConstantanditisζ=0.44(7-41)theterminalvelocityisasfollows

(7-43)1000<Re<200000CriterionforsettlingregimeToidentifytherangeinwhichthemotionoftheparticlelies,thevelocitytermiseliminatedfromReynoldsnumberbysubstitutingutfromEq.(7-40)togive,fortheStokes’-lawrange

(7-44)CriterionforsettlingregimeTRe<1.0.ToprovideaconvenientcriterionK,Let(7-45)Re<1.0.ToprovideaconvenienThen,fromEq(7-44),Re=K3/18.SettingRe=1.0andsolvinggivesK=2.6.Ifthesizeoftheparticleisknown,KcanbecalculatedfromEq(7-45).IfKsocalculatedislessthan2.6,Stokes’lawapplies.Then,fromEq(7-44),Re=K3/18.SubstitutionforutfromEq.(7-45)showsthatfortheNewton’slawrangeRe=1.75K1.5.Settingthisequalto1000andsolvinggivesK=68.9.ThusifKisgreaterthan68.9,Newton’slawapplies.SubstitutionforutfromEq.(7IntherangebetweenStockes’lawandNewton’slaw(2.6<K<68.9),theterminalvelocityiscalculatedfromEq(7-39)usingavalueofζfoundbytrialfromFig.7-6化工原理英文教材流体输送阻力系数Flow-past-immersed-bodies课件ExampleAliquiddropwithp=1000kg/m3,dp=0.0001m,settlesfreelyinastaticairwith=0.00012kg/m.s,=1.22kg/m3.Ifthesettleobeysstokeslaw,whatistheterminalvelocityut,inm/s?Ifairtempincreases,whathappenstothesettlingvelocity(forlaminarflowRe1)ExampleAliquiddropwithp=化工原理

PrinciplesofChemicalIndustry化工原理FlowpastimmersedbodiesFlowpastimmersedbodiesThediscussioninthisChapteris

concernedabout:1.thelawsoffluidflow;2.factorsthatcontrolchangesofpressureandvelocityoffluidsflowingpastsolidboundariesandwasespeciallyconcernedwithflowthroughclosedpipesandchannels.Emphasisduringthediscussionwasplacedprimarilyonthefluid.ThediscussioninthisChapterThesituationwherethesolidisimmersedin,andsurroundedbyfluidisthesubjectofthischapter.ThesituationwherethesolidDragandDragcoefficientsDrag

——TheforceinthedirectionofflowexertedbythefluidonthesolidDragcoefficients

——Ananalogousfactorusedforimmersedsolids(likelocalfrictionalcoef.)DragandDragcoefficientsDragDragWhenthewallofthebodyisparallelwiththedirectionofflow,showninFig.3-10a,dragforceisthewallshear.DragWhenthewallofthebodFig.7-1showsthepressureandshearforcesactingonanelementofareadAinclinedatanangleof90°-αtothedirectionofflow.DirectionOfflowαPcosαdA(formdrag)Forecefrompressure=pdATwsinαdA(walldrag)TwdAFig.7-1showsthepressureandThedragfromwallshearisτw•sinα•dA(walldrag),and

thatfrompressureisp•cosα•dA(formdrag).Theabovedragsareinthedirectionofflow.Thetotaldragonthebodyisthesumoftheintegralsofthesequantitieseachevaluatedovertheentiresurfaceofthebodyincontactwiththefluid.ThedragfromwallshearisτwThetotalintegrateddragfromwallsheariscalledwalldrag.Thetotalintegrateddragfrompressureiscalledformdrag.化工原理英文教材流体输送阻力系数Flow-past-immersed-bodies课件Inpotentialflow(idealfluids)thereisnowalldrag.Also,thepressuredraginthedirectionofflowisbalancedbyanequalforceintheoppositedirection,andintegraloftheformdragiszero.Sothereisnonetdraginpotentialflow.Inpotentialflow(idealfluidDragcoefficientsIfFDisthetotaldrag,theaveragedragperunitprojectedareaisFD/A,justasfrictionfactorf

isdefinedastheratioofτwtotheproductofthedensityofthefluidandthevelocityhead,sothedragcoefficientdenotedbyζ(CD),isdefinedastheratioofFD/Atothissameproduct,or(7-1)DragcoefficientsIfFDisthTheequation(7-1)isthesameastheequation(5-71)LetΔp=FD/A,andsubstitutingintheaboveequationTheequation(7-1)isthesameForparticleshavingshapesotherthanspherical,itisnecessarytospecifythesizeandgeometricformofthebodyanditsorientationwithrespecttothedirectionofflowofthefluid.Forparticleshavingshapesotshapefactors:Onemajordimensionischosenasthecharacteristiclength,andotherimportantdimensionsaregivenasratiostothechosenone.shapefactors:

Fromdimensionalanalysis,thedragcoefficientofasmoothsolidinanincompressiblefluiddependsuponaReynoldsnumberandthenecessaryshapefactors.(7-2)Fromdimensionalanalysis,tDragcoefficientsoftypicalshapes

InFig,curvesofCD(ζ)vs.Reareshownforspheres,longcylinders,anddisks.Theaxisofthecylinderandthefaceofthediskareperpendiculartothedirectionofflow.DragcoefficientsoftypicalsForlowReynoldsnumbersthedragforceforasphereconformstoatheoreticalequationcalledStokes´law,whichmaybewritten(7-3)FromEq.(7-3),thedragcoefficientpredictedbyStokes´lawis,usingEq(7-1)(7-4)Intheory,Stokes’lawisvalidonlywhenReisconsiderablylessthan1.ForlowReynoldsnumbersthed

Thelawisespeciallyvaluableforcalculatingtheresistanceofsmallparticles,suchasdustsorfogs,movingthroughgasesorliquidsoflowviscosity,orforthemotionoflargerparticlesthroughhighlyviscousliquids.ThelawisespeciallyvaluabMechanicsofparticlemotionTheforceactingontheparticlemaycomefromadensitydifferencebetweentheparticleandthefluid,oritmaybetheresultofelectricormagneticfields.MechanicsofparticlemotionThreeforcesactonaparticlemovingthroughafluid:

Theexternalforce,gravitationalorcentrifugal;Thebuoyantforce,whichactsparallelwiththeexternalforcebutintheoppositedirection;Thedragforce,whichappearswheneverthereisrelativemotionbetweentheparticleandthefluid.ThreeforcesactonaparticleEquationforone-dimensionalmotionofparticlethroughfluidConsideraparticleofvolumevf,densityρfmovingthroughafluid.Threeforcesactingonaparticleare:(1)externalforce:Fe=Vf

ρf

ae(2)buoyantforce:Fb=Vfρae(3)dragforce:FeFbFDEquationforone-dimensionalmThentheresultantforceontheparticleis

Fe-Fb–Fd,Theaccelerationoftheparticleisdu/dt,

(7-25)substitutingtheforcestoEq(7-27)gives(7-30)ThentheresultantforceonthTerminalvelocityThedragalwaysincreaseswithvelocity,theaccelerationdu/dtofaparticledecreaseswithtimeandapproachestozero.Theparticlequicklyreachesaconstantvelocity,whichisthemaximumattainableundercircumstances,andwhichiscalledtheterminalvelocityut.TerminalvelocityThedragalwa(7-31)Theterminalvelocityisfoundbytakingdu/dt=0,thenfromEq(7-31)(7-32)化工原理英文教材流体输送阻力系数Flow-past-immersed-bodies课件Inthegravitationalsetting,ae=g(7-33)

motioninacentrifugalfield,theaccelerationfromcentrifugalforcefromcirclemotionis

ae=rω2wherer=radiusofpathofparticle

ω=angularvelocity,radians/secInthegravitationalsetting,theterminalvelocityis(7-34)theterminalvelocityisdragcoefficientThequantitativeuseofEq(7-33)and(7-34)requiresthatnumericalvaluesisavailableforthedragcoefficientζ.dragcoefficientThequantitThedragcurveshowninFig.7-6applies,however,onlyunderrestrictedconditions.Theparticlemustbeasolidsphere,itmustbefarfromotherparticlesandfromthevesselwallsothattheflowpatternaroundtheparticleisnotdistorted,andtheparticlemustbemovingatitsterminalvelocitywithrespecttothefluid.ThedragcurveshowninFig.7-Whentheparticleatthesufficientdistancefromtheboundariesofthecontainerandfromotherparticles,sothatitsfallisnotaffectedbythem,theprocessiscalledfreesettling.Ifthemotionofparticleisimpededbyotherparticles,whichhappenwhentheparticlesareneareachothereventhoughtheymaynotactuallybecolliding,theprocessis

calledhinderedsettling.WhentheparticleatthesuffiIftheparticlesareverysmall,Brownianmovementappears.Thiseffectbecomesappreciableataparticlesizeofabout2-3µmandpredominatesovertheforceofgravitywithaparticlesizeof0.1orless.IftheparticlesareverysmTherandommovementoftheparticletendstosuppresstheeffectoftheforceofgravity,sosettlingdoesnotoccur.ApplicationofcentrifugalforcereducestherelativeeffectofBrownianmovement.TherandommovementoftheparMovementofsphericalparticlesIftheparticlesarespheresofdiameterdp

(7-35)And

(7-36)MovementofsphericalparticleSubstitutionofVpandApfromEq(7-35)and(7-36)intoEq(7-33)and(7-34)gives(7-37)And(7-37a)SubstitutionofVpandApfromTheterminalvelocitiesatthedifferentReynoldsnumber

Intheory,stokes’lawisvalidonlywhenReisconsiderablylessthanunity.

Eq.(7-4)maybeusedwithsmallerrorforallReynoldsnumberslessthan1.TheterminalvelocitiesattheForgravitysettlingofaspheres,atlowReynoldsnumbers,thedragcoefficientvariesinverselywithRe.(7-38)andsubstitutingEq(7-38)into