Effect of vibration on the stability of a gas-fluidized bed of fine powder

E ec o ib a ion on he s abili y o a gas- luidized bed o ine powde J. M. Val e de, A. Cas ellanos, and M. A. S. Quin anilla Depa amen o de Elec onica y Elec omagne ismo, Uni e sidad de Se illa, A enida Reina Me cedes s/n, 41012 Se illa, Spain 共Recei ed 18 Janua y 2001; e ised manusc ip ecei ed 2 Ap il 2001; published 18 July 2001兲 We ha e in es iga ed he e ec o ib a ions on he s abili y o gas- luidized beds o ine powde s 共pa icle size ⬃10 ␮ m). The powde is uni o mly luidized by an adjus able gas low ha enables us o con ol he a e age solid olume ac ion ␾ 0. The luidized bed is hen subjec ed o a e ical oscilla o y mo ion o con olled ampli ude and equency. The esponse o he luidized bed depends essen ially on he alue o ␾ 0. Fo ␾ 0⬎0.28 he luidized bed is in a weak solidlike egime, i has a mechanical s eng h, and pa icles a e s a ic. In his egime ib a ion causes compac ion o he loosely packed bed. Fo ␾ 0⬍0.28 he mechanical s eng h anishes and s esses a e ca ied by in e s i ial gas and collisions. In his luidlike egime he luidized bed displays a di usi e dynamics and pa icles agg ega e due o he s ong in e pa icle an de Waals o ces. When ib a ion is applied he powde expands due o he pa ial dis up ion o agg ega es. Howe e a a c i ical alue o he ib a ion ampli ude A⫽Acei he su ace 共sloshing兲o low 共bubbling兲ins abili ies de elop. The nuclea ion o gas bubbles has been co ela ed o he sa u a ion in pa icle di usi i y measu ed elsewhe e. The size o he bubbles inc eases as Ais u he inc eased abo e Acand as he ib a ion equency is educed. Mo eo e , as i should be expec ed om he p edic ions o hyd odynamic models, Acis independen o cohesi i y o pa icles o he same size and densi y. DOI: 10.1103/PhysRe E.64.021302 PACS numbe 共s兲: 45.70.⫺n, 47.55.K , 47.55.Mh I. INTRODUCTION Vib a ion o g anula ma e ials plays a undamen al ole in indus y, commonly used as a means o handling, ans- po ing, mixing, o seg ega ing hese ma e ials. Beds o non- cohesi e g ains 共 ypically la ge han 100 ␮ m) display a ich and complex phenomenology unde applied ib a ions ha has been widely epo ed. When loosely packed beads a e apped o ib a ed he sys em elaxes slowly elimina ing low-densi y me as able con igu a ions owa d a mo e com- pac s a e 关1兴. Besides his, a la ge a ie y o low and su - ace ins abili ies, such as seg ega ion, dila ion, bubbling, heaping, low pa e ns, su ace wa es, and a ching 关2–13兴 ha e been obse ed in ib a ed o apped laye s o g ains. Pa icula ly in e es ing o us a e he ee su ace exci a ions ha ha e been o en epo ed o appea as a esul o he momen um ans e associa ed wi h in e g ain mul iple colli- sions 关2–4兴. The nuclea ion o ascina ing localized and s able exci a ions 共‘‘oscillons’’兲ha e been obse ed in i- b a ed deep laye s o beads 关9兴. Some o hese ins abili ies a e oo ed on he ac ion o in e s i ial gas. The slip o ce exe ed by he su ounding gas on he powde pa icles scales as Fs⬃1/a2关14兴. In ine powde s his o ce can o e come he pa icle weigh and he e o e he in e ac ion wi h he in- e s i ial gas canno be neglec ed. Powde -ai in e ac ion has been p o ed o play a c ucial ole in he o ma ion o co u- ga ed pa e ns in a apped laye s o ine powde s o pa icle size ⬃10 ␮ m关13兴. Pak e al. 关6兴also demons a ed ha he ambien gas is de e minan in gene a ing heaps in ib a ed deep beds o glass sphe es o size ⱗ1 mm. The e ec was a ibu ed o he c ea ion o a gap be ween he g ains and he base du ing he ee- ligh pa o he cycle. La e in he ib a ion cycle he apped gas in his gap is comp essed p o iding an upwa d o ce on he g ains. The applica ion o a e ical gas low was epo ed o li he hys e esis in con- ec i e ins abili ies o ho izon ally ib a ed g anula ma e i- als 关10兴. When a bed o sand g ains 共0.2 mm in diame e 兲 was shaken a qui e la ge ampli udes bubbling o ai pocke s, c ea ed nea he bo om o he bed, was obse ed by Pak and Beh inge 关5兴. The impo an e ec o su ounding ai in he bubble o ma ion was co obo a ed when bubbles disap- pea ed i he gas was e acua ed om he sandbox. Pak and Beh inge s essed he analogy o his phenomenon wi h bubble o ma ion in gas- luidized beds 关15兴. I he gas eloci y h ough a s able gas- luidized bed is u he inc eased abo e a c i ical alue he wo-phase low becomes uns able and wa es o compac egions o low pa - icle concen a ion 共gas bubbles兲mo e upwa ds h ough he bed. Hyd odynamic models ecognize pa icle di usion as he esponsible o he s abiliza ion o he luidized bed since i ends o supp ess he ampli ude g ow h o pe u ba ions on local concen a ion 关16兴. Bubbling ins abili y has been a ib- u ed o he delay in he esponse o he pa icles o a change in he local concen a ion o solids due o he la ge pa icle o gas phase ine ia 关16兴. F om he basis o a pu e hyd ody- namic in e ac ion be ween luid and pa icles, he pa icle p ope ies in ol ed in he c i e ia o ansi ion o ins abili y a e pa icle size and densi y 关16,17兴. Ou wo k will ocus on he e ec o ib a ions on s able gas- luidized beds o ine cohesi e pa icles 共pa icle size ⬃10 ␮ m). These powde s can be uni o mly luidized in a e y wide in e al o solid olume ac ions hanks o he use o low addi i es ha educe cohesi i y. By means o a con- olled gas low we a e able o ha e a p ecise con ol on he a e age solid olume ac ion ␾ 0, allowing us o explo e a ich phenomenology in he beha io o he ib a ed gas- luidized bed as a unc ion o i s bulk densi y. We ha e shown elsewhe e 关18兴 ha wo clea ly di e en ia ed egimes exis in he uni o m luidiza ion in e al. Fo solid olume ac ions la ge han a ce ain c i ical alue ␾ c he luidized bed has a mechanical s eng h. In his solidlike egime pa - icles a e s a ic, sus ained by endu ing con ac s, and he pow- PHYSICAL REVIEW E, VOLUME 64, 021302 1063-651X/2001/64共2兲/021302共8兲/$20.00 ©2001 The Ame ican Physical Socie y64 021302-1 de has a beha io qui e simila o ha o a weak solid. I he bed is il ed i s ee su ace emains s able un il i a alanches when a c i ical angle is eached. Fo ␾ 0⬍ ␾ cpa icles a e no longe in pe manen con ac wi h each o he . They agg ega e due o he domain o in e pa icle adhesion o ce o e pa - icle weigh and agg ega es a e suspended in he luidizing gas displaying a di usi e dynamics 关18兴. The powde in his luidlike egime akes on many o he p ope ies o a low iscosi y luid, i s uppe su ace emaining ho izon al when he con aine is il ed. Vib a ion expe imen s epo ed in his pape con i m he exis ence o bo h egimes since he e- sponse o he luidized bed o ib a ion depends essen ially on which egime he sys em is. When he bed is in he sol- idlike egime compac ion is obse ed. On he o he hand, when i is in he luidlike egime we obse e an ini ial ex- pansion ollowed by ei he su ace o bubbling ins abili y 共depending on he alue o ␾ 0). The c i ical ib a ion ampli- ude o he onse o ins abili y has been measu ed as a unc- ion o he ib a ion equency and solid olume ac ion. The ole o con ac o ces on he onse o ins abili y will be add essed. Finally we will show ha he ansi ion o bub- bling can be cha ac e ized as a Hop bi u ca ion o he s able expanded s a e. II. MATERIALS Measu emen s ha e been made on a comme cially a ail- able powde 共xe og aphic one Canon CLC700兲. Pa icle densi y ( ␳ p⫽1199 kg/m3) was measu ed by using an AccuPyc 1330 Pycnome e . Tone pa icles a e based on a andom copolyme and hei olume a e age diame e is dp ⫽8.53⫾2.32 ␮ m. They a e i egula in shape since hey a e o med by a g inding p ocess. Fo he mass o one used in he expe imen s he solid olume ac ion o he se led pow- de is ␾ 0⫽ ␾ s⯝0.35. In he absence o ib a ion, as he gas low h ough he bed is inc eased and he bed expands, in- s abili y in he o m o isible bubbling is obse ed o ␾ 0 ⫽ ␾ b⯝0.15. This is he minimum alue o ␾ ha he wo- phase sys em can sus ain. Acco ding o ou measu emen s 关18,19兴 he ensile s eng h o he luidized bed becomes neg- ligible o his powde o ␾ 0⬍ ␾ c⯝0.28 共see Fig. 1兲. The e- o e o ␾ 0苸(0.15,0.28) he luidized powde is in he lu- idlike egime, while o ␾ 0苸(0.28,0.35) i is in he solidlike egime. To check o he e ec o cohesi i y wo o he xe o- g aphic one s 共supplied by Xe ox Co.兲o simila pa icle size and densi y as he Canon one we e used. These one s a e iden ical in pa icle size (12.7 ␮ m diam), pa icle den- si y ( ␳ p⫽1065 kg/m3), pa icle esin ype 共s y ene- bu adiene兲, and pigmen concen a ion, bu wi h concen a- ions o 0.2% and 0.4% by weigh o su ace addi i e, espec i ely. This addi i e 共Ae osil R812兲consis s o ag- glome a es o nanopa icles o umed silica. The e ec o Ae osil on he in e pa icle con ac o ces is desc ibed in ou p e ious wo ks 关19–21兴. The addi ion o hese nanopa icles esul s in a educ ion in he in e pa icle adhesion o ce be- cause he addi i es inc ease he ha dness o he con ac s and also educe he a ea o con ac be ween g ains. In spi e o hei di e en cohesi i y and in ag eemen wi h he p edic- ions o he hyd odynamic models he h ee one s in es i- ga ed 共o simila pa icle size and densi y兲show he ansi- ion o bubbling ins abili y a app oxima ely he same alue o he solid olume ac ion ( ␾ b⯝0.15). III. EXPERIMENTAL RESULTS A. Expe imen al se up and p ocedu e In Fig. 2 a schema ic iew o he expe imen al se up is depic ed. The powde sample is held in a 2.52-cm-diame e cylinde made o polyca bona e, he base o which is a sin- e ed me al il e o 5- ␮ m po e size. D y ni ogen is supplied om a ank o comp essed gas and a gas low con olle is used o adjus he low. By using d y ni ogen we minimize he e ec o mois u e on he in e pa icle cohesion. Mo e- o e cha ge spec og aph measu emen s made a Xe ox Co. indica e ha elec os a ic o ces a e minimal. The e o e an de Waals o ces a e dominan be ween ou d y ine pa icles. All he measu emen s s a by d i ing he powde in o he eely bubbling phase. Once he powde is in he bubbling phase he luidizing gas is se down o a gi en alue below he minimum bubbling low and he powde se les o a e- p oducible and s able luidized s a e. In his way he solid olume ac ion o he sys em was con enien ly a ied by changing he gas low a e. The p essu e d op ac oss he bed is measu ed by a di e en ial p essu e ansduce . The bed heigh , which was o he o de o cen ime e s 共 e y la ge compa ed o pa icle size兲, was isualized by a ideo came a ei he wi h a di ec o a s obe ligh . F om he heigh mea- FIG. 1. Tensile s eng h o he luidized powde 共Canon CLC700 one 兲as a unc ion o he solid olume ac ion. Da a aken om 关18兴. FIG. 2. Ske ch o he appa a us. VALVERDE, CASTELLANOS, AND QUINTANILLA PHYSICAL REVIEW E 64 021302 021302-2 su ed hand he pa icle densi y ␳ pwe compu ed he a e age solid olume ac ion, gi en by ␾ 0⫽m/( ␳ pSH), whe e m ⯝6.5 g is he mass o he powde sample and Sis he a ea o he il e . In Fig. 3 ␾ 0is ep esen ed as a unc ion o he supe icial gas eloci y g. Fo g⬎0.8 cm/s 共minimum gas eloci y o bubbling兲 he mean bed heigh begins o de- c ease slowly 共i.e., ␾ 0inc eases兲, wi h ma ked oscilla ions o he ee su ace as gas bubbles bu s . The powde bed was moun ed on an elec omagne ic i- b a ion exci e d i en by a signal gene a o , which p o ided sinusoidal, e ical ib a ions o con olled equency , and ampli ude A. The equency o he oscilla ions could be con- olled o wi hin ⫾0.1 Hz in he ange om 10 o 1000 Hz. The ampli ude o ib a ions is moni o ed using a piezoelec- ic accele ome e glued on he op o he essel. Th oughou he expe imen s epo ed he e he powde was luidized o a gi en alue o he solid olume ac ion ␾ 0by a gi en gas low and hen he ampli ude o ib a ion was slowly in- c eased om ze o a a ixed equency. B. The solidlike egime When he powde is luidized a supe icial gas eloci ies below 0.1 cm/s he solid olume ac ion is abo e 0.28, pe - manen con ac s o m, and he bed beha es as a weak solid. I ib a ion is applied in his egime he bed elaxes slowly owa ds a mo e compac solid s uc u e in spi e o he p es- ence o a gas low whose supe icial eloci y is abo e he minimum luidiza ion gas eloci y 共which is abou ⬃0.01 cm/s). The compac ion p ocess is i e e sible, i.e., i ib a ion is s opped he bed does no e u n o i s o iginal densi y. The beha io o he powde bed in his egime is qui e simila o he compac ion obse ed when loose packed beads a e apped o ib a ed 关1兴. C. The luidlike egime In his egime pe manen con ac s do no exis anymo e, pa icle agg ega es a e eely suspended in he gas and s esses a e ca ied by he in e s i ial luid and by collisions. We obse e ha when ib a ion is applied o he bed in his egime he bed expands, i.e., he solid olume ac ion de- c eases. Fine pa icles a e no eely loa ing as indi iduals in he luidlike luidized bed bu a e agglome a ed due o a ac- i e an de Waals o ce 关22,23兴. On he o he hand, i has been shown ha , o highly cohesi e powde s, ib a ion o he bed imp o es he luidiza ion quali y by helping he gas low o o e come cohesi e o ces and b eak up channels ha conduc luidizing gas away om he bulk and p e en lu- idiza ion 关24兴. In ou case o homogeneously luidized agg e- ga ed pa icles he expec ed e ec o ib a ion would be o inc ease he a e o collisions be ween agg ega es, p omo ing hei dis up ion. As a esul he e would be an imp o emen o he gas-solid con ac , he slip o ce 共which is in e sely p opo ional o he squa e o he agg ega e adius兲on he educed agg ega es would be la ge , and consequen ly he s able bed would expand. Howe e , he expansion s ops when a c i ical ib a ion ampli ude Ac, which depends on he ib a ion equency, is eached. Then wo kinds o ins abili ies, depending on he alue o he solid olume ac ion, ha e been obse ed. In Fig. 4 we ha e plo ed he change in he solid olume ac- ion ⌬ ␾ ⫽ ␾ m⫺ ␾ 0un il he bed des abilizes 共o compac ion s ops i i is in he solidlike egime兲as a unc ion o he ini ial solid olume ac ion ␾ 0 o a se o uns made a di e en equencies . This igu e shows ha ⌬ ␾ depends essen ially on he ini ial alue o ␾ 0and is almos indepen- den on , excep o s a es o ␾ close o 0.15 共 o which bubbling occu s spon aneously in he absence o ib a ions兲 whose expansion is sligh ly highe a la ge equencies. Mo eo e he bounda y be ween he solidlike (⌬ ␾ ⬎0) and he luidlike (⌬ ␾ ⬍0) egimes is clea ly seen a ␾ 0⯝0.28. As we should expec , when ␾ 0app oaches 0.15, ⌬ ␾ →0. Thus ⌬ ␾ is ze o o ␾ ⫽0.15 and ␾ ⫽0.28 共bounda ies o he luidlike egime兲. The e o e ⌬ ␾ mus ha e a minimum in be ween and we ind i a ␾ 0⯝0.24, independen ly o he ib a ion equency. Fo his alue o he ini ial solid olume ac ion he maximum expansion o he bed occu s. In Fig. 5 he ini ial h0and maximum hmbed heigh s a e plo ed as a unc ion o ␾ 0. This igu e shows ha hmcan be well ap- p oxima ed by a linea unc ion o ␾ 0. Taking in o accoun ha h⫽m/( ␳ pS ␾ ) we de i e o he minimum solid olume FIG. 3. Solid olume ac ion o he luidized bed 共Canon CLC700 one 兲as a unc ion o he supe icial gas eloci y. FIG. 4. Va ia ion o he solid olume ac ion when he uni o m luidized bed is ib a ed om he ini ial s a e cha ac e ized by a solid olume ac ion ␾ 0. Each se ies o da a co esponds o a i- b a ion equency indica ed in he inse in Hz. The da a in he lu- idlike egime co esponds o he maximum dec ease in he solid olume ac ion 共jus be o e he onse o ins abili y, ei he sloshing o bubbling兲. The da a in he solidlike egime indica es he inc ease o he solid olume ac ion due o compac ion o he luidized bed. EFFECT OF VIBRATION ON THE STABILITY OF A . . . PHYSICAL REVIEW E 64 021302 021302-3 ac ion ␾ mo he expanded bed he simple exp ession 1 ␾ m⯝1 ␾ c 冉 1⫹ ␾ c⫺ ␾ 0 ␾ b 冊 ,共1兲 whe e ␾ c⯝0.28 is he c i ical solid olume ac ion o he ansi ion o he solidlike egime and ␾ b⯝0.15 is solid ol- ume ac ion o he ansi ion o bubbling in he absence o ib a ion. 1. F ee su ace ins abili y In he in e al ␾ 0苸(0.24,0.28), close o he solidlike e- gime, ee su ace ins abili y 共sloshing兲is obse ed when he ib a ion ampli ude eaches a c i ical alue Ac. The e olu- ion o he sys em a ound he c i ical ib a ion ampli ude is comple ely e e sible, i e. no hys e esis ha e been ound. This ins abili y becomes mo e appa en a low equencies ( ⬍100 Hz) o which Ac, as we will see below, is la ge . Fu he mo e a low equencies we obse e he nuclea ion o localized sloshing ha eminds o soli a y wa es 共‘‘oscil- lons’’兲obse ed in ib a ed deep beds o noncohesi e beads 关9兴. Likewise sloshing is eminiscen o he su ace ins abili- ies obse ed in ib a ed 关3兴o apped 关13兴 ine powde s. Su ace wa es induced by ib a ion we e epo ed o be spe- cially isible a e y la ge ib a ion ampli udes 关3兴. Du an 关13兴obse ed ha ine pa icles on he ee su ace we e expelled a e e y ap and a ibu ed i o he e ec o he upwa d low o he apped ai in he powde . Mo eo e he epo s on some in e es ing complex pa e ns, like w inkled olcanos o he ingbone-shaped s uc u es, when a con inu- ous ai low was passed h ough he apped powde laye . 2. Bubbling ins abili y One o he e ec s o inc easing he gas low in he luid- like egime is he dec ease o he a e age solid olume ac- ion. The o he is he inc ease o he g anula empe a u e T 关18,25兴, which is de ined in e ms o he ensemble a e age o he squa ed luc ua ion eloci y, 3T⫽ 具 2 典 ⫺ 具 典 2关18兴. The g anula empe a u e can be es ima ed by means o he e - ec i e di usion coe icien , De ⫽l 冑 T, whe e l ep esen s he mean ee pa h. We ha e p esen ed elsewhe e 关18兴mea- su emen s o De o he one Canon CLC700 in he luidlike egime. Values o De we e ob ained om measu ing he mixing a e o wo luidized samples o di e en colo . The esul s a e shown in Fig. 6. Acco ding o Fig. 6 pa icle e- loci y luc ua ions a e enhanced by he inc ease in he gas low un il ␾ is a ound 0.21. Pa icle di usi i y hen eaches a pla eau in spi e o he inc ease o gas low 共and he conse- quen educ ion in ␾ ). We expec ha he collisions a e should be a o ed by he inc ease in eloci y luc ua ions. A ␾ 0⫽0.24 when ib a ion is applied he bed expansion is maximum and ␾ eaches a minimum alue o ␾ ⯝0.21 共see Fig. 4兲. In e es ingly his alue coincides wi h he alue o ␾ o which we measu e a sa u a ion in di usi i y 共see Fig. 6兲. The p oponen s o hyd odynamic models a gue ha pa icle di usion plays a decisi e ole in s abilizing he luidized bed agains local pe u ba ions on he concen a ion o solids 关16兴. Acco ding o his, as he gas low is u he inc eased and ␾ 0is educed below 0.24 he sa u a ion in pa icle di - usi i y would educe he abili y o he sys em o supp ess dis u bances in he local concen a ion o pa icles. In o he wo ds, lowe ing ␾ below 0.21 accele a es he app oach o he low o he sys em o he c i ical s a e. F om his plau- sible a gumen i can be unde s ood ha o ␾ ⬍0.21 he pe u ba ion o he sys em caused by ib a ion p omo e low ins abili y and ha he maximum expansion o he bed de- c eases as ␾ is u he educed. Acco dingly, when ␾ 0is educed below 0.24, su ace sloshing is no he i s ins abil- i y obse ed. Ins ead he wo-phase low becomes uns able and isible bubbling appea s. Figu e 7 displays he e olu ion o he solid olume ac ion in a ypical expe imen in which bubbling ins abili y is obse ed. When he ib a ion ampli- ude app oaches a c i ical alue Ac he sys em eaches i s maximum expansion and ␾ i s minimum alue ␾ m. Then pa o he gas is apped om he dense phase by bubbles and ␾ inc eases, wi h oscilla ions o he ee su ace due o gas bubbles bu s . The ansi ion o ins abili y is no sha p. The e is an inde e minacy 共see ba plo ed in Fig. 7兲in he measu emen o Acdue o ou sensi i i y o de ec an in- c ease o ␾ 共 he e o ba o he measu ed alues o ␾ a e shown in Fig. 7兲. He ea e we ake o Ac he alue o he FIG. 5. Ini ial bed heigh h0o he luidized bed and maximum heigh hmo he expanded bed when ib a ion is applied as a unc- ion o he ini ial solid olume ac ion ␾ 0.A ␾ 0⫽ ␾ c he luidized bed en e s he solidlike egime. A ␾ 0⫽ ␾ b he non ib a ed bed en e s he bubbling egime i he gas low is u he inc eased. Only da a o ⫽300 Hz has been ep esen ed o he sake o cla i y. FIG. 6. E ec i e di usion coe icien De o he luidized bed in he luidlike egime 关 ␾ 苸(0.15,0.28)兴as a unc ion o he solid olume ac ion 共da a aken om 关18兴兲. In he solidlike egime pa - icles a e s a ic and he di usion coe icien is ze o. VALVERDE, CASTELLANOS, AND QUINTANILLA PHYSICAL REVIEW E 64 021302 021302-4 igh end o he ba in Fig. 7 since a ha alue he inc ease in ␾ is mo e ma ked and is easily de ec ed. As he ib a ion ampli ude is u he inc eased abo e Ac, ␾ keeps g owing and he oscilla ions o he ee su ace become mo e no ice- able due o he inc ease o bubbles size. We ha e obse ed a co ela ion be ween he ypical size o he ee su ace oscil- la ions and he imposed equency. As he ib a ion e- quency is inc eased he oscilla ions become less ma ked go- ing om se e al millime e s a low equencies ( ⱗ100 Hz) o less han 1 mm a high equencies ( ⲏ500 Hz). Pa allel o he bubbling ins abili y we obse e he onse o a modes con ec i e mo ion on he ee su ace ha can be ela ed o he epo ed con ec ion o he op lay- e s o shaken ine powde s 关10,11兴, o he ci cula ion o sol- ids gene a ed by bubbling 关14兴, o maybe o bo h. Indeed he e olu ion o he sys em o A⬎Ac esembles i s e olu ion in he absence o ib a ion when he gas low is inc eased abo e he minimum bubbling low 共see Fig. 3兲. The ansi ion o bubbling ins abili y does no show any hys e esis on cycling h ough he onse o bubbling. The a - e age solid olume ac ion is a con inuous e e sible unc- ion o he ib a ion ampli ude. Tennakoon e al. 关10兴 e- po ed also on he absence o hys e esis in con ec i e ins abili ies o ho izon ally ib a ed g anula beds when a gas low h ough he sys em was applied. Thus he supp es- sion o hys e esis by he use o a gas low seems o be a common ac o independen o he dynamic ins abili y in- ol ed. Tennakoon e al. conjec u ed ha he educ ion o he s a ic ic ion was he eason o emo ing hys e esis a he ansi ion o con ec i e ins abili y 关10兴. In ou expe i- men he powde in he luidlike egime has no s a ic ic ion and acco ding o he same a gumen we should no expec any hys e esis. The scena io is qui e di e en when he lu- idized powde is in he solidlike egime whe e he powde e ains a ce ain mechanical s eng h. In such a s a ic egime he compac ion obse ed when he bed is ib a ed is i e e s- ible. The o ma ion o gas bubbles in ib a ed beds o sand g ains wi hou an applied gas low was al eady obse ed by Pak and Beh inge 关5兴. An in e es ing ea u e o ha wo k, analogous o ou s in gas- luidized beds, is he epo ed in- c ease o bubble size as he ib a ion ampli ude is u he inc eased. I mus be ema ked howe e ha , as Pak and Beh inge epo 关5兴, he o ma ion o bubbles in he absence o gas low needs o exagge a ed ib a ion ampli udes, and p io o bubbling, con ec ion and su ace ins abili ies 共slosh- ing兲 ake place. Ac ually we ha e ep oduced his expe imen in ou labo a o y by using glass beads 共a e age size 50 ␮ m) and p onounced su ace con ec ion, heaping, and sloshing we e obse ed be o e he appea ance o bubbles. Howe e , when we ied o epea he expe imen wi h cohesi e pow- de , he ini ial compac ion induced by ib a ion made he powde so cohesi e ha we we e no able, wi h ou ins u- men a ion, o achie e s ong enough shaking o b eak in e - pa icle con ac s. D. C i ical ib a ion ampli ude In his sec ion we will discuss he dependence o he c i i- cal ampli ude o he onse o ins abili y 共ei he su ace o low ins abili y兲Acon he ib a ion equency and on he solid olume ac ion ␾ 0. In Fig. 8 we ha e plo ed Acas a unc ion o o h ee s a es o di e en ini ial concen a ion o solids. Acis well i ed by a po en ial law Ac ⯝ ␥ ⫺1.3, ␥ being a weakly dependen unc ion o ␾ . Howe e Acp esen s some puzzling peaks a ce ain special equencies p共100, 200, 400, and 600 Hz兲. In e es ingly, hese peaks do no depend essen ially on he solid olume ac ion 共see Fig. 8兲, no on he powde cohesi i y 共see Fig. 9兲.To ha e u he insigh on he dependence o Acon ␾ 0we p esen in Fig. 10 he c i ical ib a ion ampli ude as a unc- ion o he solid olume ac ion o a se o uns pe o med a di e en ixed equencies. As could be an icipa ed om Fig. 8, Fig. 10 does no show a s ong dependence o Acon ␾ 0.Acchanges smoo hly as he solid olume ac ion is dec eased. When ␾ 0 eaches 0.15 he bed bubbles in he absence o ib a ion and Ac alls sha ply o ze o. A low equencies Acdec eases as ␾ 0is dec eased. Howe e , his endency changes a high equencies o which Acis almos independen o ␾ 0. Measu emen s o he bubble equency Nb共de ined as he numbe o bubbles ha ise up pe uni c oss sec ion o he bed pe uni ime兲o eely bubbling non ib a ed luidized beds ha e been epo ed in he li e a u e 关26兴and he ypical FIG. 7. E olu ion o he solid olume ac ion ( ␾ ) o he luid- ized bed as he ib a ion ampli ude 共A兲is inc eased o a ypical expe imen 共gas eloci y 0.334 cm/s, ib a ion equency 200 Hz兲. The solid line is a squa e oo i . The ba size indica es he ypical inde e minacy (⌬Ac)onAc. FIG. 8. Dependence o he c i ical ib a ion ampli ude Ac o he onse o ins abili y on he ib a ion equency . Each se ies co esponds o an ini ial solid olume ac ion indica ed in he inse . EFFECT OF VIBRATION ON THE STABILITY OF A . . . PHYSICAL REVIEW E 64 021302 021302-5 alues a e below 10 Hz. P esumably he ypical equencies o he la ge pe u ba ions in he local solids concen a ion in he s able luidized bed a e o he same o de o magni ude. The magni ude o hese pe u ba ions a e expec ed o in- c ease as he solid olume ac ion dec eases un il hey be- come ini e and he bed bubbles. The na u al equency o he pe u ba ions is n⬃c/␭,cbeing he pe u ba ion cha ac- e is ic wa e speed and ␭i s wa eleng h. Thus i he s able luidized bed is ib a ed a low equencies he g ow h o he la ges pe u ba ions is expec ed o be p omo ed. Acco d- ingly we should expec a dec ease o he c i ical ampli ude o bubbling as he solid olume ac ion is educed, i.e., as local pe u ba ions inc ease in magni ude. Mo eo e we should expec , as we obse e, la ge luc ua ions on he ee su ace when bubbling is p omo ed by ib a ion. On he o he hand, i he bed is ib a ed a e y high equencies, qui e la ge han he ypical na u al equency o bubbles, he pe u ba ions magni ied o he onse o bubbling would be o la ge equency, small size, and p esumably weakly de- penden on he a e age solid olume ac ion. This would explain he obse ed independence o Acon ␾ 0and he e y small ampli ude o he oscilla ions 共 en hs o a millime e 兲o he ee su ace a la ge ib a ion equencies. No e also ha , o ⬎500 Hz, Acis smalle han he size o he pa icles, which in u n is o he o de o he a e age in e pa icle sepa a ion in he luidized s a e. Thus, a la ge equencies, ib a ions canno be ansmi ed by in e pa icle con ac s and his p o ides ano he explana ion o he independence o Ac on he a e age solids concen a ion. 1. The ole o con ac o ces F om a heo e ical poin o iew he s abiliza ion mecha- nism o uni o m luidiza ion was long ago a ibu ed o an e ec i e pa icula e-phase p essu e in he momen um equa- ion ha ends o esis a ia ions in he local solid olume ac ion and he eby has a s abilizing in luence 关27兴. Ye , he physical o igin o he pa icle-phase p essu e has been a sub- jec o s ong deba e un il e y ecen ly 关28,29兴. Rie ema 关30兴ou lined ha in e pa icle con ac o ces gi e an e ec- i e elas ic modulus o he bed ha s abilize he sys em agains small dis u bances. Tsinon ides and Jackson 关15兴 joined Rie ema’s a gumen s and s a ed ha he loss o s eng h o he assembly was esponsible o he loss o s a- bili y and he consequen onse o bubbling. Howe e his a gumen would imply ha he pa icle assembly should ha e a yield s ess h ough all he in e al o s abili y and should no beha e in a luidlike way bu a he like a weak solid. I also implies ha he ansi ion o ins abili y would be e y sensi i e o ac o s ha a ec con ac o ces, such as he su ace p ope ies in d y ine powde s, o which cohesion be ween pa icles a ise mainly om dispe sion o an de Walls o ces. S ongly opposed o Rie ema’s ideas, Foscolo and Gibila o 关17兴p oposed a s abili y mechanism o hyd o- dynamic o igin. An al e na i e s abilizing mechanism oo ed on hyd odynamic a gumen s was sugges ed by Ba chelo 关16兴. The physical o igin o his mechanism is based on he exis ence o andom luc ua ions in he pa icles eloci ies ha a e linked o he di usion o pa icles 关16,29兴. Acco d- ing o hyd odynamic models he only pa icle p ope ies ha a ec essen ially he ins abili y c i e ia a e hei size and densi y. In ou p e ious wo k 关18兴we ha e shown ha he luid- ized powde beha es like a luid 共i does no exhibi any yield s ess兲and displays a di usi e dynamics in a b oad in e al o uni o m luidiza ion p e ious o he onse o in- s abili y. Thus, acco ding o ha wo k, he sou ce o ins abil- i y canno be a ibu ed o he loss o mechanical s eng h bu a he o hyd odynamic in e ac ions. He e we ind ano he e idence o suppo he hyd odynamic o igin o bubbling ins abili y. E en hough he sou ce o bubbling in ou sys em is no he inc ease o gas low beyond a ce ain limi bu he exci a ion o he sys em by ex e nal ib a ion, we should expec ha i con ac o ces we e ele an in he ansi ion o ins abili y we would ind a ma ked dependence o he c i ical ib a ion ampli ude o bubbling on he powde cohesi i y. In Fig. 9 esul s ob ained o Acon Canon CLC700 one a e compa ed wi h some esul s ob ained on o he powde s o simila pa icle size and densi y bu wi h di e en cohesi i y. These one s a e blended wi h di e en amoun s o su ace addi i e such as Ae osil®共nanopa icles o umed silica兲, well known o i s abili y o con ol in e pa icle cohesi e o ces 关19兴. As can be seen om Fig. 9 he alues ob ained o he c i ical ib a ion ampli ude a e on he same cu e o he h ee powde s in es iga ed. This esul is in opposi ion o he p oposed s abilizing mechanism based on con ac o ces FIG. 9. C i ical ib a ion ampli ude Ac o he onse o ins abil- i y as a unc ion o he ib a ion equency o h ee powde s o simila pa icle size and densi y bu di e en pa icle su ace s uc- u e. RT one s a e model one s cha ac e ized by he weigh % o su ace addi i e. The ini ial solid olume ac ion o he h ee pow- de s es ed is ␾ 0⫽0.185. FIG. 10. C i ical ib a ion ampli ude Ac o he onse o ins a- bili y as a unc ion o he solid olume ac ion ␾ 0. Each se ies co esponds o a ixed ib a ion equency indica ed in he inse in Hz. VALVERDE, CASTELLANOS, AND QUINTANILLA PHYSICAL REVIEW E 64 021302 021302-6 关15,30兴. On he con a y hey suppo he idea ha he s abi- lizing mechanism mus ha e a hyd odynamic o igin 关16,17兴. E. The ansi ion o bubbling as a Hop bi u ca ion In a ypical expe imen , when he luidlike luidized bed is ib a ed, he alue o ␾ dec eases due o dis up ion o ag- g ega es om he ini ial alue ␾ 0down o ␾ m. Then a c i i- cal ampli ude A⫽Acis eached o which bubbling ins abil- i y occu s. As Ais u he inc eased he size o he bubbles 共as measu ed by he ee su ace oscilla ions兲and he a e age solid olume ac ion inc ease con inuously. Pak and Beh inge 关5兴 ela ed he bubbling ins abili y obse ed in a ib a ed sandbox in he absence o imposed gas low wi h a Hop bi u ca ion. On he o he hand, he ansi ion o bub- bling ins abili y in non ib a ed gas luidized beds as he gas low is inc eased has been desc ibed heo e ically also as a Hop bi u ca ion o he uni o m s a e 关31兴. Following Pak and Beh inge 关5兴we can conside he bubble size as a mea- su e o he ampli ude o he oscilla ions o he sys em. The nuclea ion o bubbles as he gas eloci y is kep cons an gi es, as a esul , a densi ica ion o he dense phase 共see Fig. 11兲. Pa o he gas low ha was ca ied h ough he dense phase is now cap u ed by he gas phase 共bubbles兲. Gas lows up h ough he bed and sho cu s be ween he bubbles wi h a high eloci y. When he gas bubbles bu s on he ee su ace he bed heigh hdec eases and oscilla es. The ising eloci y o bubbles has been co ela ed o he bubble size Rby Ub ⬀ 冑 R关32兴. Thus as bubbles g ow in size and hei eloci y inc eases he heigh dec emen h⫺hmand he size o oscil- la ions would also inc ease. The o al mass o solids in he bed mus be cons an , hence ␾ mhm⫽ ␾ h. F om his balance we ge 兩 h⫺hm 兩 /hm⫽ 兩 ␾ m⫺ ␾ 兩 / ␾ ⬅⌬ ␾ b/ ␾ . The e o e he in- c emen o he a e age solid olume ac ion ⌬ ␾ bcan be aken as a measu e o he size o bubbles ha can be in u n conside ed a measu e o he ampli ude o he oscilla ions o he sys em. F om Fig. 7 i can be seen ha once he ins abil- i y is eached he a ia ion o he solid olume ac ion can be e y well i ed by he squa e oo law ⌬ ␾ b⫽ ␣ (A ⫺Ac)1/2. In Fig. 12, ⌬ ␾ bis plo ed agains (A⫺Ac)1/2 o a se o expe imen s made a di e en alues o ␾ 0and a di e en equencies. In all he uns he expe imen al da a a e p e y well i ed by a linea law, sugges ing ha bubbling ins abili y in ib a ed gas- luidized beds can also be ela ed o a Hop bi u ca ion. IV. CONCLUSIONS We ha e in es iga ed he beha io o uni o mly luidized ine cohesi e powde s 共pa icle size ⬃10 ␮ m) subjec ed o ib a ions. These powde s show a wide in e al o s able luidiza ion 关solid olume ac ion ␾ 0苸(0.15,0.35)兴as op- posed o coa se g ains o which bubbling appea s soon a e he minimum gas eloci y o luidiza ion is eached. In he in e al o uni o m luidiza ion he powde exhibi s wo e- gimes. In he solidlike egime (0.28⬍ ␾ 0⬍0.35) pa icle con ac s a e pe manen and he expanded luidized bed has a mechanical s eng h. When he bed in his egime is ib a ed i compac s much in he same way as loose s a ic beds o noncohesi e g ains do. In he luidlike egime (0.15⬍ ␾ 0 ⬍0.28) he e a e no endu ing con ac s and agg ega es o pa icles a e eely suspended in he luid. When he bed in his egime is ib a ed i ini ially expands due o agg ega es dis up ion induced by ib a ion. Howe e a c i ical ib a ion ampli ude A⫽Acis eached a which he bed des abilizes. Two kinds o ins abili ies ha e been obse ed. Sloshing o he ee su ace occu s o ␾ 0苸(0.24,0.28). In he in e al FIG. 11. Schema ic pic u e o he nuclea ion o bubbles in i- b a ed gas- luidized beds. When he ib a ion ampli ude A eaches he c i ical alue Acpa o he gas ises in bubbles. When he bubbles bu s on he ee su ace i collapses and oscilla es acco d- ing o he bubble equency. FIG. 12. Va ia ion o he a e age solid olume ac ion (⌬ ␾ b ⫽ ␾ ⫺ ␾ m, see Fig. 7兲a e bubbling ins abili y is eached as a unc ion o he squa e oo o he bi u ca ion pa ame e (A⫺Ac). 共a兲S a es o di e en ini ial solid olume ac ion ␾ 0共indica ed in he igu e兲 ib a ed a he same equency ⫽200 Hz. 共b兲S a es o he same ini ial solid olume ac ion ␾ 0⫽0.18 ib a ed a di e en equencies 共indica ed in he igu e in Hz兲. EFFECT OF VIBRATION ON THE STABILITY OF A . . . PHYSICAL REVIEW E 64 021302 021302-7 ␾ 0苸(0.15,0.24) he wo-phase low becomes uns able and pa o he gas ises in bubbles. Then he ee su ace oscil- la es and he a e age solid olume ac ion inc eases. Ac scales as ⫺1.3 and depends weakly on ␾ . The ampli ude o he oscilla ions o he ee su ace, which a e a measu e o he size o he gas bubbles, dec ease as he ib a ion e- quency is inc eased, going om se e al millime e s a low ( ⬍100 Hz) o en hs o millime e a la ge ( ⬎500 Hz). I Ais u he inc eased abo e Ac he dec ease in he a e age solid olume ac ion scales as (A⫺Ac)1/2. Thus he ansi ion o bubbling ins abili y esembles a Hop bi u ca ion o he s able s a e. We ha e checked ha Acdoes no depend on he powde cohesi i y as pa icle size and densi y a e held cons an . This inding can be aken as a u he e idence o he in alidi y o s abili y c i e ia p o- posed in he li e a u e ha a e based on he exis ence o a mechanical s eng h in he luidized bed. ACKNOWLEDGMENTS We hank D . A. T. Pe ´ ez o ui ul discussions. 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