- Email: [email protected]
Measurements of wall shear stress in particle beds when vibrations are imposed vertically along the direction of shear
Measurementsof wall shear stress in particle beds when vibrations are imposedverticallyalong the direction of shear AKIYAMA andTAKESHI SHIMOMURA TETSUO Harnamatsu Chemical Shizuoka 432, Japan University, of Engineering, Department 20August 1992 forAPTFebruary 24 1991; Received accepted vibrations beds when stress invibrating were made ofthewall shear Abstract-Measurements particle ofparticles, and ofshear. This work deals with five kinds thedirection were imposed vertically along in the 20-80 Hz vibration wall shear stress on vessel examines thedependency ofthe size, frequencyrange theeffect 2-8.Inorder toexamine togravitational acceleration intherange ofvibratory andtheratio wasmade between a comparison stress onvibrating ofthewall shear beds, experimentally particle ones. When thewall calculated airpressures atthebottom ofthebedandtheoretically measured and inmodel theagreement between model calculations shear stress was predictions incorporated deteriorated theagreement inbeds oflarge measurements beads; however, significantly glass improved size. with decreasing particle progressively NOMENCLATURE diameter mean particle [,um] ds innerdiameter ofvessel [m] oftube,respectively [m] D;,Do innerandouterdiameters ofvibration [Hz] frequency f duetogravity acceleration [m/s2] 9 L bedheight [m] particle bed[Pa] beneath vibrated PL airpressure [Pa] pressure Po atmospheric bedandbottom ofvessel ofairgapbetween s thickness [m] vibrating particle t time[s] time[s] T cycle vessel x ofvibrating [m] displacement ofvessel [m] amplitude displacement xo half-wave beddepthwithin tube[m] Zj vessel ofimmersed tubewithin [m] zo depth Greek letters tothatduetogravity A ratioofvibratory acceleration [-] ofparticle bed[kg/m3] pb bulkdensity wallshearstress T (=tainFigs9-12)[Pa] shear stress obtained fromequation (1)uponsetting ti = fo= Ta[Pa] Ta mean ofTaover40 <_ average z; 60mm[Pa] t.
130 withd(s + x)ldt2 wallshearstress overziandisassociated averaged [Pa] T dtlT Ti dt/T [Pa] [Pa] z; ti J0 r z; withd2s/dt2 shearstress overzoandisassociated [Pa] averaged To wall TT dt/T[Pa] to J 0To w [s-'] angular frequency Q [-] =4Ta/PbDig 1.INTRODUCTION withparticulate ofindustrial Vibrated bedsareusedina number processes dealing of vibrations to fluidized beds(vibro-fluidized materials. bed)has Application heat-transfer rates[1-3]. Thevibro-fluidized bed beenreported toenhance recently havealso hasbeenusedforheatexchanger andindrying [6,7].Vibrations [4, 5] offlow of beenapplied toapneumatic conveyor [8, 9], compaction [10,11],control cohesive suchasclassification, grinding, [12],andmanyotherprocesses powders of these andmixing discharge, separation [13,14].Theperformance storage, duetotheflowpromotion ofbulksolids industrial isimproved mainly processes Sincetheflowcharacteristics of bulksolidsare whensubjected to vibrations. interest in thereis a practical influenced friction, byparticle-wall considerably thiseffect. understanding thewallfriction orshearstress in authors haveinvestigated Numerous [14-18] forcasesin which vibrations wereapplied eitheralongthe particulate systems tothedirection of toit,butalways atrightangles direction ofthewallortraverse method tomeasure thewall shear.Ina previous a novel paper[19],wedescribed when vibrations were the stress inbedsofglass beads shear imposed vertically along isanextension oftheabove fordifferent direction ofshear. Thepresent study study innumber andranges thanthosestudied kinds ofparticles withparameters greater thisstudy examines theeffect ofthewallshear intheprevious Furthermore, paper. bedsbycomparing thecalculated andmeasured air stressonvibrating particle ofthebed. atthebottom pressures APPARATUS AND METHODS 2.EXPERIMENTAL isschematically shown inFig.1.Itconsists ofanelectroTheexperimental set-up The tubeandacrylic vessel alongwithinstruments. vibrator, acrylic magnetic inFig.1areexplained intheNomenclature. Weusedthreesizeranges of symbols oftenryu sands Thephysical andtwosizeranges (irregular particles). glassbeads oftheseparticles aregiven inTable1. properties 20and80Hz,andtheratioofthe was between Thevibrational frequency fvaried Awasvaried 3and8bychanging togravitational acceleration between the vibratory wasvibrated withparticles toasetheight, thevessel Afterloading bythe amplitude. weresetbyanoscillator vibrator. Thefrequency andamplitude electromagnetic inthe Theacrylic tubewasthenimmersed andamplifier, vertically respectively. asillustrated inFig.1,thesurface bedandheldstationary. Then, vibrating particle anequilibrium leveloftheparticle bedwithin thetuberoseorfellandreached
Ti
131 Table 1. ofparticles Physical properties
normal totheparallel divided ° Minimum distance between two planes. parallel planes bythedimension
1.Schematic ofexperimental apparatus. diagram Figure
132 within 1min:it stabilized at a leveldifferent fromthatofthe position, usually bedoutside thetube.Theratioz;/zowasa function of meanparticle particle diameter stress canbedetermined as,A, f,D,andzo.Thewallshear bymeasuring ziandzoinFig.1. 3.FORCE BALANCE Thetime-averaged forcebalance overonecycle ofthevibrating bedwas particle shown tosatisfy (1)[19]: equation Whereti is the wallshearstressaveraged overzi andis associated with andtoisthataveraged andisassociated withd2s/dt2. Note d(s + over zo thattheforceassociated withvessel reduces tozerowhen displacement averaged overonecycle.It should alsobenotedthatequation onthe (1)wasderived thatthepressure isuniformly atanycross-section andthat assumption distributed, themagnitudes ofZiand zo aresufficiently thanthatofs. greater Theupper termontheright-hand sideofequation signofthesecond (1)istobe usedwhen when signs zi< zo and , thelower zi> zo The . riseandfallofz;relative toz.appears tobedetermined ofparticle movement around thetube bythepattern Thedirection ofshear stress isdependent onthepattern ofparticle movement [19]. onthesinusoidal vibrational motion. Thissuperimposed superimposed particle inmagnitude movement isinduced between theinterparticle and bythedifference frictions. particle-wall There aretwounknowns inequation (1),i.e.tiandto.Inprinciple, tiandt. can becalculated twosetsof zoandz;,obtained withvessels ofdifferent bypairing 2andanother fromTable Thisprocedure, size,onesetfromTable 3,forexample. intheresulting risetoconsiderable over however, gives variability ti andTovalues thebeddepthinvestigated. inthevalues Thisvariation andToisexpected, of 1'1 because thedisturbance ofsolidmovement tubewillchange with bytheimmersed sized vessels. Infact,themagnitude oft. mayvarywithDeandbed differently ofthepresence orabsence ofthetube.Thusonlyanaverage height irrespective shearstress canbeobtained fromequation za,which (1)bysetting ti = t. = Ta, Table 2. =190 ofwall shear stress anddepths ofparticle beds L=100 Magnitude mm, mm) D¡=80mm, (Dc
133 Table 3. ofparticle beds L=100 ofwall shear stress anddepths mm, mm) Magnitude (D,=140 D;=80mm,
ofthe willbediscussed hereafter. Weintroduce below a parameter, indicative ofthewallshearstress andthegravitational relative acceleration, importance HereQistheratioofthesecond termtothefirsttermontheleft-hand sideof insubsequent willbegiven asa frame ofreference figures. equation (1).Itsvalue Thestatic shearstress tothedynamic one(La inthisstudy) isdifficult to equivalent because thestaticnormal stress tothedynamic onesacting on obtain, equivalent threedifferent surfaces andoutside ofthetube,andthevessel arevery wall) (inside difficult tospecify. 4.RELATION BETWEEN D ,DiAND ds within bedgenerally riseinthecentral Theparticles thevibrating particle region, thesidewalls. Thiscirculating motion ofparticles andtravel down along appears differences between thebedsinside andoutside thetube. toinduce thesurface level Itseems tochoose sucha sizecombination ofthevessel andthe desirable, therefore, beleastlikely todisturb thenatural ofparticles. tubethatwould circulating pattern bedwithin thering(between thevessel andthetube)issurrounded Theparticle by walls whose surface areaexceeds twice thesurface areaofthewallincontact with theparticles inthetube,thusthecross-sectional areaoftheringshould besomewhat indicated thatto select a largerthanthatofthetube.Preliminary experiments combination ofDcandD;isthekey,andismuch moreimportant thanto proper select a proper inmeasuring method. D;forgiven as,tosucceed Tabythepresent 5.RESULTS S. l.Effect of D ,D¡andL Toillustrate theeffects ofDcandL onra,representative dataforglassbeads of inTables 2-5.Excluding thevessel arepresented size,theexperimental d, = 332,um conditions which ledtothedataofTables 2and3areidentical. Likewise, excluding theexperimental conditions forTables 3-5areidentical. There isnot thebedheight, much difference intheTavalues between Tables 2and3 forf s 50Hz.However,
134 Table 4. L=120 stress anddepths ofparticle beds ofwall shear mm, mm) Magnitude (D,=140 D¡=80mm,
5. Table ofparticle L=140 ofwall shear stress anddepths beds mm, mm) Magnitude (D,=140 D,=80mm,
2become in differences withthose verysmall compared betweenand zoziinTable 3at f= 80Hz(inreality 60Hz,notshown andeven Table reverthough), Thissharpdecrease indifferences salsintherelative ofzoandzioccur. magnitude between to occurwhen thepattern ofparticle movements inthe zoandziseems oftheimmersed tubebecomes Athighfrequencies neighborhood highly disorderly. theregular vertical movements ofparticles inthebedofDs= 190 mm tendtobe Thelatterbedisthus disturbed moreeasily thanthatinthebedofD,= 140 mm. tomeasure overa wide ofparameter values. range preferable -ca There islittledifference values between Tables 3and4,except at f= 50Hz, in za when thebedofL = 100 mm lessstable, i.e.differences between appears zoand z; values become smallcompared withthoseinthebedofL = 120 mm. Although = lessstable thanbedsof it isnotshown hereforbrevity, bedsofL 80mmwere inTable5 (L= 140 mm) areslightly L = 100 mm. AtA= 5,ravalues greater AtA= 2, however, inbedsof thanthoseinTable4 (L= 120 mm). Tavalues L = 140 mm smaller thanthosein bedsofL = 120 mm weresignificantly (not Thisdifference inthebehavior between bedsofL = 120and140mmmay shown).
135 result intheirviscoelastic fromthedifference thedegree ofwhich module, usually withincreasing bedheight. increases theTaindeepbedsisgenerally Consequently, fromthatin shallow different beds.Similar features wereobserved in bedsof Tubesandvessels of othersizessuchasD;= 40and60mm, otherparticles. andD,= 160 mm werealsotested,butthecombination of Di= 80 mm and to bethemostsatisfactory formeasuring appeared D,= 140 mm raoverwide ofparameters. Theabove bedsofdifferent dimensions ranges comparisons among = 140mmand andheights ledustomeasure inbedsof D;= 80mm, Dc Taprimarily L = 120 mm, results ofwhich arereported below. experimental 5.2.Relation between Taandf It is apparent fromTables 2-5thatendeffects become whenthe significant immersed orwhen itsendistooclose tothevessel depthofthetubeistoosmall bottom. Thusanarithmetic wastakenover40 szi 60 mm and average Of -[a denoted asza.Itisplotted inFigs2-6,wherein theQvalues are against frequency asa frame ofreference. Thetavalues smaller than20Paarenotshown inthe given because thepresent method isnotsuitable tomeasure sucha small figures magnitudeoftaaccurately. Thereis a general inFigs2-6thatfora given with tendency A,ia decreases someexceptions A= 4and5 when increasing frequency, exist,notably although in Fig.3.Thisirregular bedbehavior fromresonance mayhavepartlyresulted effects. It is apparent fromFigs2-4that zadecreases withdecreasing particle whenbothA andf aresmall, thefainbedsof as= 99,um is size,andexcept small withthose inbedsoflarger Thezainbedsof considerably compared particles. sands isgenerally seentodecrease moresharply withincreasing tenryu f thanthat inbedsofsimilar-sized Itisapparent fromFigs2-6that,at a spherical particles. smallfrequency, becomes f = 20Hz,theeffectof intheforcebalance nearly totheeffect ofgravitational acceleration foralltheparticles for equivalent except thesmall beads. glass
2.Relation between Figure f beads, faand(glass ds=99 um).
136
between 3.Relation beads, 11m). f (glass d,=227 taand Figure
4.Relation between beads, f (glass ds=332,um). Figure faand
52.Relation between f sands,=227 ds11m). taand(tenryu Figure
137
6.Relation between Figure f (tenryu sands, iaand ds=332um). 5.3.Relation between taandA results therelation between in Representative showing faandAarepresented thatthe Fig.7for f= 30HzandinFig.8for f= 50Hz.BothFigs7and8indicate sands issmaller forA< 3,buttendtobelarger forA> 5than zainbedsoftenryu thatinbedsofsimilar-sized beads. ItcanalsobeseenthatforA> 4,theta glass inbedsoflarge maintains alarge whereas thatinbeds particles value, (ds= 332 pm) ofsmall decreases Inthecaseofintermediate-sized particles (ds= 99,um) sharply. thefa -Arelation forglass beads isconsiderably different particles (ds= 227,um), fromthatfortenryu ontheapplied sands, depending frequency. When=f50Hz, thebehaviors oftainbedsofglassbeadsandsimilar-sized sandsarenot tenryu much different. Incontrast, inbedsofglass beads decreases when=f30Hzthe za
7.Relation between Figure t. andA(f=30 Hz).
138
8.Relation between Figure faandA(f=50Hz). withincreasing thetainbedsoftenryu sandsincreases with A,whereas sharply AuntilA= 4,thendecreases forA >_ 7.Thisdifference increasing onlyslightly intheperformance between sandsandglassbeadsseems to havearisen tenryu intheirinterparticle forces. Theinterparticle forces fromthedifference primarily roleindetermining themagnitude offg,because playanimportant theystrongly affect thepattern ofparticle movements onwhich thetaisdependent. to Compared andsimilar-sized are tenryusands,glassbeadsof d, = 227,um largeparticles theparticle movements tobecome under mobile, causing disorderly highvibratory andhence f values seem tohavebeenreduced tosmall values. intensities, WALL AIR 6.THE EFFECT OFTHE SHEAR STRESS ON PRESSURE ATTHE BOTTOM OFVIBRATING PARTICLE BEDS Inanearlier a comparison between calculated and paper [20]wemade theoretically determined atthebottom ofvibrating beds.The experimentally pressures particle theoretical model assumed thattheparticle bedisinelastic andmoves asa single withnegligible wallfriction. Themodel pistonof constant voidage predictions well withmeasurements thevibratory waslessthen7g,the when acceleration agreed 40and70Hz,andthebedheight waslessthan0.10mforbeds between frequency of largeandintermediate and glassbeads(ds= 322and22711m, respectively), 0.07mforbedsofsmall beads AsseenfromFigs2-6, Ta values glass (ds= 99,um). tendtobecome 40Hz.Thustoexamine theeffect oft. increasingly largefor f <_ onvibrating atthevessel particle beds,theairpressure base,PL,wascalculated measured (intheregion f40 sHz),incorporating tavalues. Theequation inflight, ofmotion oftheentire is: bed,when
139
9.Effect ofwall shear stress onairpressure beneath bed A=3, Figure vibrating particle 11m, (d,=332 L=0.06 m). f = 30 Hz, where L isthebedheight andPoisatmospheric Positive were pressure. tavalues usedwhends/dt>0 andnegative wereusedwhends/dt< 0. Other favalues aswellasthedetailed solution method canbefound elsewhere governing equations [20]. Thecalculated andmeasured arecompared inFigs9-12,which indicate PLvalues thatwhen t isincorporated inthemodel theagreement between model predictions andmeasurements It isalsoevident fromFigs9-12that significantly improves.
10.Effect ofwall shear stress onairpressure beneath bed A=5, Figure vibrating particle (ds=332 pm, L=0.10 m). f 20Hz,
140
11.Effect ofwall shear stress onairpressure beneath bed A=4, Figure vibrating particle (ds=227 um, m). f =30Hz,L=0.08 withthebedsofas= 332,um, theagreement inthebedsofas= 227,um compared islesssatisfactory. Itisnotshown hereforbrevity, butinbedsofds= 99,um and theagreement irregular particles (intheregion f :540Hz)wasratherpooreven when inthemodel. fawasincorporated Thissuggests thattherearefactors otherthenta thatcausethedeviation of frommeasurements. Theeffects ofsuchfactors become predictions increasingly withdecreasing size.Ithasbeenshown important particle [21]that,inthecaseof
12.Effect stress onairpressure beneath bed A=5, ofwall shear Figure vibrating particle (d,=227 um, f = 20 Hz, L = 0.10 m).
141 dueto a viscoelastic model irregular particle beds,predictions agreewellwith measurements when values havebeenchosen foradjustable appropriate parameters inthemodel. 7.CONCLUSIONS Thedependence ofthewallshear stress onparticle andintensity species, frequency ofvibrations wasstudied forthefirsttimeforthecasewhenvibrations were of shear.Within theexperimental imposed vertically alongthedirection range studied thefollowing observations werenoted. theparticle thewallshearstress, becomes. (1)Thelarger size,thelarger decreases withincreasing A,tagenerally when resonance (2)Given frequency except occurs. is smaller forA< 3, but (3)Thetainbedsofirregular particles (tenryu sands) tendstobelarger forA> 5thanthoseinbedsofsimilar-sized glassbeads. in bedsof largeparticles maintains forlargeA values, (4)The za largevalues whereas thetainbedsofsmall decreases withincreasing A. particles sharply oftaintheforcebalance becomes (5)Ata small frequency, f = 20Hz,theeffect totheeffect ofgravitational acceleration foralltheparticles nearly equivalent forthesmall except glassbeads. themeasured wallshear stress wasincorporated inmodel calculations the '(6)When between calculated andmeasured atthebottom ofthebed agreement pressures in bedsof largeglassbeads(ds= 332,um), butthe improved significantly deteriorated withdecreasing size. agreement progressively particle REFERENCES 1.K. Malhotra andA.S.Mujumdar, Immersed surface heat transfer inavibrated fluidized bed. Ind. Chem. Res. 1987. 26,1983-1992, Eng. 2.K.Takahashi and K.Endoh, Effect ofvibration onforced convection mass transfer. J.Chem. Eng. 1989. Jpn22,120-124, 3.K.Takahashi andK.Endoh, Anew correlation method fortheeffect ofvibration onforced convection heat transfer. J.Chem. 1990. Eng. Jpn3,45-50, 4.C.W.Cheah, D.E.Hirt, Y.A.Liu and A.M.Squires, Avibrofluidized-bed heat forheat exchanger from ahotgasI.Feasibility ofapilot-scale Powder Technol. recovery study system. 55, 257-267, 1988. 5.C.W.Cheah, D.H.Hirt, Y.A.Liu and A.M.Squires, Avibrofluidized bed heat forheat exchanger from ahotgasII.Heat-transfer evaluation ofapilot-scale Powder Technol. recovery system. 55, 1988. 269-276, 6.V.Chlenov andN.Mikhailov, beds. Izdatel'stvo Fluidized 1972. Vibrating Nauka, Moscow, 7.M.Bukareva, V.Chlenov andN.Mikhailov, offerrite invibroboiled powder layer. Drying 1967. Poroshkovaya 8,85-88, Metallurgiya 8.T.Kano, Reduction ofpower inpneumatic ofgranular materials. Bulk Solid consumption conveying 1985. 5,663-669, Handling 9.O.Molerus andW.Siebenhaar, Chem. Vibration induced Technol. pneumatic conveying.Eng. 14, 1991. 141-145, 10.J.E.Ayer andF.E.Soppet, J.Am. Ceram. Soc. 1969. Vibratory compaction, 52, 414-416, 11.G.L.Messing andG.Y.Onoda, inbinary relations Inhomogeneity-packing density powders. J.Am. Ceram. Soc. 1978. 61, 363-366, 12.G.W.J.Wes, S.Stemerding andD.J.vanZuilichem, Control offlow ofcohesive powders by means ofsimultaneaous aeration andvibration. Powder Technol. 1990. 61,39-49,
142 13.I.Gutman, Industrial Uses Vibration. London: Business 1968. of Mechanical Book, 14.A.W.Roberts, Handbook Science andTechnology. New York: Van Nostrand Reinhold, of Powder 6,1984. Chap. 15.P.C.Arnold andA.S.Kaaden, wall friction vibration. Powder bymechanical Reducing hopper 1977. Technol. 16, 63-66, 16.A.W.Roberts and O.J.Scott, Aninvestigation into theeffects ofsinusoidal and random vibrations onthestrength andflow ofbulk solids. Powder Technol. 1978. 21, 45-53, properties Afriction 17.A.J.Matchett, model fortheeffects ofsinusoidal vibrations shear stress in unpon Powder Technol. 1986. systems. 47,1-8, particulate 18.A.J.Matchett, Atheoretical inavibrating, model ofshear non-cohesive solid. Powder particulate 1991. Technol. 65,81-88, 19.T.Akiyama and T.Shimomura, ofwall shear stress invibrating beds. Powder Investigation particle Technol. 1991. 66,243-247, 20.T.Akiyama andH.Kurimoto, model of vibrated beds. Chem. Sci. Compressible gas particle Eng. 1988. 43,2645-2653, 21.T.Akiyama andH.Yamaboshi, Behavior ofvibrating beds ofirregular Powder Technol. particles. 69, 163-169, 1992.
130 withd(s + x)ldt2 wallshearstress overziandisassociated averaged [Pa] T dtlT Ti dt/T [Pa] [Pa] z; ti J0 r z; withd2s/dt2 shearstress overzoandisassociated [Pa] averaged To wall TT dt/T[Pa] to J 0To w [s-'] angular frequency Q [-] =4Ta/PbDig 1.INTRODUCTION withparticulate ofindustrial Vibrated bedsareusedina number processes dealing of vibrations to fluidized beds(vibro-fluidized materials. bed)has Application heat-transfer rates[1-3]. Thevibro-fluidized bed beenreported toenhance recently havealso hasbeenusedforheatexchanger andindrying [6,7].Vibrations [4, 5] offlow of beenapplied toapneumatic conveyor [8, 9], compaction [10,11],control cohesive suchasclassification, grinding, [12],andmanyotherprocesses powders of these andmixing discharge, separation [13,14].Theperformance storage, duetotheflowpromotion ofbulksolids industrial isimproved mainly processes Sincetheflowcharacteristics of bulksolidsare whensubjected to vibrations. interest in thereis a practical influenced friction, byparticle-wall considerably thiseffect. understanding thewallfriction orshearstress in authors haveinvestigated Numerous [14-18] forcasesin which vibrations wereapplied eitheralongthe particulate systems tothedirection of toit,butalways atrightangles direction ofthewallortraverse method tomeasure thewall shear.Ina previous a novel paper[19],wedescribed when vibrations were the stress inbedsofglass beads shear imposed vertically along isanextension oftheabove fordifferent direction ofshear. Thepresent study study innumber andranges thanthosestudied kinds ofparticles withparameters greater thisstudy examines theeffect ofthewallshear intheprevious Furthermore, paper. bedsbycomparing thecalculated andmeasured air stressonvibrating particle ofthebed. atthebottom pressures APPARATUS AND METHODS 2.EXPERIMENTAL isschematically shown inFig.1.Itconsists ofanelectroTheexperimental set-up The tubeandacrylic vessel alongwithinstruments. vibrator, acrylic magnetic inFig.1areexplained intheNomenclature. Weusedthreesizeranges of symbols oftenryu sands Thephysical andtwosizeranges (irregular particles). glassbeads oftheseparticles aregiven inTable1. properties 20and80Hz,andtheratioofthe was between Thevibrational frequency fvaried Awasvaried 3and8bychanging togravitational acceleration between the vibratory wasvibrated withparticles toasetheight, thevessel Afterloading bythe amplitude. weresetbyanoscillator vibrator. Thefrequency andamplitude electromagnetic inthe Theacrylic tubewasthenimmersed andamplifier, vertically respectively. asillustrated inFig.1,thesurface bedandheldstationary. Then, vibrating particle anequilibrium leveloftheparticle bedwithin thetuberoseorfellandreached
Ti
131 Table 1. ofparticles Physical properties
normal totheparallel divided ° Minimum distance between two planes. parallel planes bythedimension
1.Schematic ofexperimental apparatus. diagram Figure
132 within 1min:it stabilized at a leveldifferent fromthatofthe position, usually bedoutside thetube.Theratioz;/zowasa function of meanparticle particle diameter stress canbedetermined as,A, f,D,andzo.Thewallshear bymeasuring ziandzoinFig.1. 3.FORCE BALANCE Thetime-averaged forcebalance overonecycle ofthevibrating bedwas particle shown tosatisfy (1)[19]: equation Whereti is the wallshearstressaveraged overzi andis associated with andtoisthataveraged andisassociated withd2s/dt2. Note d(s + over zo thattheforceassociated withvessel reduces tozerowhen displacement averaged overonecycle.It should alsobenotedthatequation onthe (1)wasderived thatthepressure isuniformly atanycross-section andthat assumption distributed, themagnitudes ofZiand zo aresufficiently thanthatofs. greater Theupper termontheright-hand sideofequation signofthesecond (1)istobe usedwhen when signs zi< zo and , thelower zi> zo The . riseandfallofz;relative toz.appears tobedetermined ofparticle movement around thetube bythepattern Thedirection ofshear stress isdependent onthepattern ofparticle movement [19]. onthesinusoidal vibrational motion. Thissuperimposed superimposed particle inmagnitude movement isinduced between theinterparticle and bythedifference frictions. particle-wall There aretwounknowns inequation (1),i.e.tiandto.Inprinciple, tiandt. can becalculated twosetsof zoandz;,obtained withvessels ofdifferent bypairing 2andanother fromTable Thisprocedure, size,onesetfromTable 3,forexample. intheresulting risetoconsiderable over however, gives variability ti andTovalues thebeddepthinvestigated. inthevalues Thisvariation andToisexpected, of 1'1 because thedisturbance ofsolidmovement tubewillchange with bytheimmersed sized vessels. Infact,themagnitude oft. mayvarywithDeandbed differently ofthepresence orabsence ofthetube.Thusonlyanaverage height irrespective shearstress canbeobtained fromequation za,which (1)bysetting ti = t. = Ta, Table 2. =190 ofwall shear stress anddepths ofparticle beds L=100 Magnitude mm, mm) D¡=80mm, (Dc
133 Table 3. ofparticle beds L=100 ofwall shear stress anddepths mm, mm) Magnitude (D,=140 D;=80mm,
ofthe willbediscussed hereafter. Weintroduce below a parameter, indicative ofthewallshearstress andthegravitational relative acceleration, importance HereQistheratioofthesecond termtothefirsttermontheleft-hand sideof insubsequent willbegiven asa frame ofreference figures. equation (1).Itsvalue Thestatic shearstress tothedynamic one(La inthisstudy) isdifficult to equivalent because thestaticnormal stress tothedynamic onesacting on obtain, equivalent threedifferent surfaces andoutside ofthetube,andthevessel arevery wall) (inside difficult tospecify. 4.RELATION BETWEEN D ,DiAND ds within bedgenerally riseinthecentral Theparticles thevibrating particle region, thesidewalls. Thiscirculating motion ofparticles andtravel down along appears differences between thebedsinside andoutside thetube. toinduce thesurface level Itseems tochoose sucha sizecombination ofthevessel andthe desirable, therefore, beleastlikely todisturb thenatural ofparticles. tubethatwould circulating pattern bedwithin thering(between thevessel andthetube)issurrounded Theparticle by walls whose surface areaexceeds twice thesurface areaofthewallincontact with theparticles inthetube,thusthecross-sectional areaoftheringshould besomewhat indicated thatto select a largerthanthatofthetube.Preliminary experiments combination ofDcandD;isthekey,andismuch moreimportant thanto proper select a proper inmeasuring method. D;forgiven as,tosucceed Tabythepresent 5.RESULTS S. l.Effect of D ,D¡andL Toillustrate theeffects ofDcandL onra,representative dataforglassbeads of inTables 2-5.Excluding thevessel arepresented size,theexperimental d, = 332,um conditions which ledtothedataofTables 2and3areidentical. Likewise, excluding theexperimental conditions forTables 3-5areidentical. There isnot thebedheight, much difference intheTavalues between Tables 2and3 forf s 50Hz.However,
134 Table 4. L=120 stress anddepths ofparticle beds ofwall shear mm, mm) Magnitude (D,=140 D¡=80mm,
5. Table ofparticle L=140 ofwall shear stress anddepths beds mm, mm) Magnitude (D,=140 D,=80mm,
2become in differences withthose verysmall compared betweenand zoziinTable 3at f= 80Hz(inreality 60Hz,notshown andeven Table reverthough), Thissharpdecrease indifferences salsintherelative ofzoandzioccur. magnitude between to occurwhen thepattern ofparticle movements inthe zoandziseems oftheimmersed tubebecomes Athighfrequencies neighborhood highly disorderly. theregular vertical movements ofparticles inthebedofDs= 190 mm tendtobe Thelatterbedisthus disturbed moreeasily thanthatinthebedofD,= 140 mm. tomeasure overa wide ofparameter values. range preferable -ca There islittledifference values between Tables 3and4,except at f= 50Hz, in za when thebedofL = 100 mm lessstable, i.e.differences between appears zoand z; values become smallcompared withthoseinthebedofL = 120 mm. Although = lessstable thanbedsof it isnotshown hereforbrevity, bedsofL 80mmwere inTable5 (L= 140 mm) areslightly L = 100 mm. AtA= 5,ravalues greater AtA= 2, however, inbedsof thanthoseinTable4 (L= 120 mm). Tavalues L = 140 mm smaller thanthosein bedsofL = 120 mm weresignificantly (not Thisdifference inthebehavior between bedsofL = 120and140mmmay shown).
135 result intheirviscoelastic fromthedifference thedegree ofwhich module, usually withincreasing bedheight. increases theTaindeepbedsisgenerally Consequently, fromthatin shallow different beds.Similar features wereobserved in bedsof Tubesandvessels of othersizessuchasD;= 40and60mm, otherparticles. andD,= 160 mm werealsotested,butthecombination of Di= 80 mm and to bethemostsatisfactory formeasuring appeared D,= 140 mm raoverwide ofparameters. Theabove bedsofdifferent dimensions ranges comparisons among = 140mmand andheights ledustomeasure inbedsof D;= 80mm, Dc Taprimarily L = 120 mm, results ofwhich arereported below. experimental 5.2.Relation between Taandf It is apparent fromTables 2-5thatendeffects become whenthe significant immersed orwhen itsendistooclose tothevessel depthofthetubeistoosmall bottom. Thusanarithmetic wastakenover40 szi 60 mm and average Of -[a denoted asza.Itisplotted inFigs2-6,wherein theQvalues are against frequency asa frame ofreference. Thetavalues smaller than20Paarenotshown inthe given because thepresent method isnotsuitable tomeasure sucha small figures magnitudeoftaaccurately. Thereis a general inFigs2-6thatfora given with tendency A,ia decreases someexceptions A= 4and5 when increasing frequency, exist,notably although in Fig.3.Thisirregular bedbehavior fromresonance mayhavepartlyresulted effects. It is apparent fromFigs2-4that zadecreases withdecreasing particle whenbothA andf aresmall, thefainbedsof as= 99,um is size,andexcept small withthose inbedsoflarger Thezainbedsof considerably compared particles. sands isgenerally seentodecrease moresharply withincreasing tenryu f thanthat inbedsofsimilar-sized Itisapparent fromFigs2-6that,at a spherical particles. smallfrequency, becomes f = 20Hz,theeffectof intheforcebalance nearly totheeffect ofgravitational acceleration foralltheparticles for equivalent except thesmall beads. glass
2.Relation between Figure f beads, faand(glass ds=99 um).
136
between 3.Relation beads, 11m). f (glass d,=227 taand Figure
4.Relation between beads, f (glass ds=332,um). Figure faand
52.Relation between f sands,=227 ds11m). taand(tenryu Figure
137
6.Relation between Figure f (tenryu sands, iaand ds=332um). 5.3.Relation between taandA results therelation between in Representative showing faandAarepresented thatthe Fig.7for f= 30HzandinFig.8for f= 50Hz.BothFigs7and8indicate sands issmaller forA< 3,buttendtobelarger forA> 5than zainbedsoftenryu thatinbedsofsimilar-sized beads. ItcanalsobeseenthatforA> 4,theta glass inbedsoflarge maintains alarge whereas thatinbeds particles value, (ds= 332 pm) ofsmall decreases Inthecaseofintermediate-sized particles (ds= 99,um) sharply. thefa -Arelation forglass beads isconsiderably different particles (ds= 227,um), fromthatfortenryu ontheapplied sands, depending frequency. When=f50Hz, thebehaviors oftainbedsofglassbeadsandsimilar-sized sandsarenot tenryu much different. Incontrast, inbedsofglass beads decreases when=f30Hzthe za
7.Relation between Figure t. andA(f=30 Hz).
138
8.Relation between Figure faandA(f=50Hz). withincreasing thetainbedsoftenryu sandsincreases with A,whereas sharply AuntilA= 4,thendecreases forA >_ 7.Thisdifference increasing onlyslightly intheperformance between sandsandglassbeadsseems to havearisen tenryu intheirinterparticle forces. Theinterparticle forces fromthedifference primarily roleindetermining themagnitude offg,because playanimportant theystrongly affect thepattern ofparticle movements onwhich thetaisdependent. to Compared andsimilar-sized are tenryusands,glassbeadsof d, = 227,um largeparticles theparticle movements tobecome under mobile, causing disorderly highvibratory andhence f values seem tohavebeenreduced tosmall values. intensities, WALL AIR 6.THE EFFECT OFTHE SHEAR STRESS ON PRESSURE ATTHE BOTTOM OFVIBRATING PARTICLE BEDS Inanearlier a comparison between calculated and paper [20]wemade theoretically determined atthebottom ofvibrating beds.The experimentally pressures particle theoretical model assumed thattheparticle bedisinelastic andmoves asa single withnegligible wallfriction. Themodel pistonof constant voidage predictions well withmeasurements thevibratory waslessthen7g,the when acceleration agreed 40and70Hz,andthebedheight waslessthan0.10mforbeds between frequency of largeandintermediate and glassbeads(ds= 322and22711m, respectively), 0.07mforbedsofsmall beads AsseenfromFigs2-6, Ta values glass (ds= 99,um). tendtobecome 40Hz.Thustoexamine theeffect oft. increasingly largefor f <_ onvibrating atthevessel particle beds,theairpressure base,PL,wascalculated measured (intheregion f40 sHz),incorporating tavalues. Theequation inflight, ofmotion oftheentire is: bed,when
139
9.Effect ofwall shear stress onairpressure beneath bed A=3, Figure vibrating particle 11m, (d,=332 L=0.06 m). f = 30 Hz, where L isthebedheight andPoisatmospheric Positive were pressure. tavalues usedwhends/dt>0 andnegative wereusedwhends/dt< 0. Other favalues aswellasthedetailed solution method canbefound elsewhere governing equations [20]. Thecalculated andmeasured arecompared inFigs9-12,which indicate PLvalues thatwhen t isincorporated inthemodel theagreement between model predictions andmeasurements It isalsoevident fromFigs9-12that significantly improves.
10.Effect ofwall shear stress onairpressure beneath bed A=5, Figure vibrating particle (ds=332 pm, L=0.10 m). f 20Hz,
140
11.Effect ofwall shear stress onairpressure beneath bed A=4, Figure vibrating particle (ds=227 um, m). f =30Hz,L=0.08 withthebedsofas= 332,um, theagreement inthebedsofas= 227,um compared islesssatisfactory. Itisnotshown hereforbrevity, butinbedsofds= 99,um and theagreement irregular particles (intheregion f :540Hz)wasratherpooreven when inthemodel. fawasincorporated Thissuggests thattherearefactors otherthenta thatcausethedeviation of frommeasurements. Theeffects ofsuchfactors become predictions increasingly withdecreasing size.Ithasbeenshown important particle [21]that,inthecaseof
12.Effect stress onairpressure beneath bed A=5, ofwall shear Figure vibrating particle (d,=227 um, f = 20 Hz, L = 0.10 m).
141 dueto a viscoelastic model irregular particle beds,predictions agreewellwith measurements when values havebeenchosen foradjustable appropriate parameters inthemodel. 7.CONCLUSIONS Thedependence ofthewallshear stress onparticle andintensity species, frequency ofvibrations wasstudied forthefirsttimeforthecasewhenvibrations were of shear.Within theexperimental imposed vertically alongthedirection range studied thefollowing observations werenoted. theparticle thewallshearstress, becomes. (1)Thelarger size,thelarger decreases withincreasing A,tagenerally when resonance (2)Given frequency except occurs. is smaller forA< 3, but (3)Thetainbedsofirregular particles (tenryu sands) tendstobelarger forA> 5thanthoseinbedsofsimilar-sized glassbeads. in bedsof largeparticles maintains forlargeA values, (4)The za largevalues whereas thetainbedsofsmall decreases withincreasing A. particles sharply oftaintheforcebalance becomes (5)Ata small frequency, f = 20Hz,theeffect totheeffect ofgravitational acceleration foralltheparticles nearly equivalent forthesmall except glassbeads. themeasured wallshear stress wasincorporated inmodel calculations the '(6)When between calculated andmeasured atthebottom ofthebed agreement pressures in bedsof largeglassbeads(ds= 332,um), butthe improved significantly deteriorated withdecreasing size. agreement progressively particle REFERENCES 1.K. Malhotra andA.S.Mujumdar, Immersed surface heat transfer inavibrated fluidized bed. Ind. Chem. Res. 1987. 26,1983-1992, Eng. 2.K.Takahashi and K.Endoh, Effect ofvibration onforced convection mass transfer. J.Chem. Eng. 1989. Jpn22,120-124, 3.K.Takahashi andK.Endoh, Anew correlation method fortheeffect ofvibration onforced convection heat transfer. J.Chem. 1990. Eng. Jpn3,45-50, 4.C.W.Cheah, D.E.Hirt, Y.A.Liu and A.M.Squires, Avibrofluidized-bed heat forheat exchanger from ahotgasI.Feasibility ofapilot-scale Powder Technol. recovery study system. 55, 257-267, 1988. 5.C.W.Cheah, D.H.Hirt, Y.A.Liu and A.M.Squires, Avibrofluidized bed heat forheat exchanger from ahotgasII.Heat-transfer evaluation ofapilot-scale Powder Technol. recovery system. 55, 1988. 269-276, 6.V.Chlenov andN.Mikhailov, beds. Izdatel'stvo Fluidized 1972. Vibrating Nauka, Moscow, 7.M.Bukareva, V.Chlenov andN.Mikhailov, offerrite invibroboiled powder layer. Drying 1967. Poroshkovaya 8,85-88, Metallurgiya 8.T.Kano, Reduction ofpower inpneumatic ofgranular materials. Bulk Solid consumption conveying 1985. 5,663-669, Handling 9.O.Molerus andW.Siebenhaar, Chem. Vibration induced Technol. pneumatic conveying.Eng. 14, 1991. 141-145, 10.J.E.Ayer andF.E.Soppet, J.Am. Ceram. Soc. 1969. Vibratory compaction, 52, 414-416, 11.G.L.Messing andG.Y.Onoda, inbinary relations Inhomogeneity-packing density powders. J.Am. Ceram. Soc. 1978. 61, 363-366, 12.G.W.J.Wes, S.Stemerding andD.J.vanZuilichem, Control offlow ofcohesive powders by means ofsimultaneaous aeration andvibration. Powder Technol. 1990. 61,39-49,
142 13.I.Gutman, Industrial Uses Vibration. London: Business 1968. of Mechanical Book, 14.A.W.Roberts, Handbook Science andTechnology. New York: Van Nostrand Reinhold, of Powder 6,1984. Chap. 15.P.C.Arnold andA.S.Kaaden, wall friction vibration. Powder bymechanical Reducing hopper 1977. Technol. 16, 63-66, 16.A.W.Roberts and O.J.Scott, Aninvestigation into theeffects ofsinusoidal and random vibrations onthestrength andflow ofbulk solids. Powder Technol. 1978. 21, 45-53, properties Afriction 17.A.J.Matchett, model fortheeffects ofsinusoidal vibrations shear stress in unpon Powder Technol. 1986. systems. 47,1-8, particulate 18.A.J.Matchett, Atheoretical inavibrating, model ofshear non-cohesive solid. Powder particulate 1991. Technol. 65,81-88, 19.T.Akiyama and T.Shimomura, ofwall shear stress invibrating beds. Powder Investigation particle Technol. 1991. 66,243-247, 20.T.Akiyama andH.Kurimoto, model of vibrated beds. Chem. Sci. Compressible gas particle Eng. 1988. 43,2645-2653, 21.T.Akiyama andH.Yamaboshi, Behavior ofvibrating beds ofirregular Powder Technol. particles. 69, 163-169, 1992.