Fine cohesive powders in rotating drums: Transition from rigid-plastic flow to gas-fluidized regime

Fine cohesi e powde s in o a ing d ums: T ansi ion om igid-plas ic low o gas- luidized egime A. Cas ellanos, J. M. Val e de, 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 28 No embe 2001; published 12 June 2002兲 We in es iga e he dynamics o ine cohesi e powde s inside o a ing d ums. We show ha hese powde s may be luidized due o en apmen o ambien gas, and we de e mine he onse o luidiza ion. Expe imen al measu emen s on he bed expansion as a unc ion o he o a ion eloci y ha e been pe o med. D ums o di e en diame e s and ine powde s o a ying cohesi eness ha e been es ed. We show ha 共i兲 ine powde s ansi di ec ly om a igid-plas ic s a e o a gas- luidized s a e in acco dance wi h he low egime bounda ies p edic ed elsewhe e 关A. Cas ellanos e al., Phys. Re . Le . 82, 1156 共1999兲兴,共ii兲 he onse o luidiza ion in he o a ing d um is de e mined by he a io o he powde kine ic ene gy pe uni olume o i s ensile s eng h, and 共iii兲once he powde is comple ely luidized he a e age in e s i ial gas eloci y inc eases p opo ionally o he o a ion eloci y. The las wo esul s imply ha he equi ed eloci y o luidize a powde , ␻ R共 ␻ angula eloci y, R adius o he d um兲, mus inc ease as he squa e oo o i s ensile s eng h, and his has been con i med by independen measu emen s and es ima ions. DOI: 10.1103/PhysRe E.65.061301 PACS numbe 共s兲: 45.70.⫺n, 47.55.K , 47.55.Mh I. INTRODUCTION P oblems wi h powde lowabili y a e e y impo an in indus ial applica ions in ol ing powde anspo a ion and mixing. Powde s in plas ic low exhibi low lowabili y and poo mixing, and i is common p ac ice in indus y o low gas h ough hese powde s in o de o enhance he abo e men ioned p ope ies. Fo una ely, in many indus ial de ices ha use mechanical mo ing pa s he ambien gas may be en apped in he powde hus acili a ing i s anspo and mixing. Due o his, he beha io o ine powde s in o a ing d ums displays qui e di e en ea u es om noncohesi e g ains, as al eady obse ed a long ime ago by Rie ema 关1兴. The pa icle size, anging om less han 1 ␮ m o 1000 ␮ m o mo e, cons i u es a c i ical pa ame e in he low beha io o g anula ma e ials 关2兴. A low s esses g anula assemblies exhibi he solid-plas ic egime, esis ing shea by unde going plas ic de o ma ions. When he limi o plas ic s abili y is eached, noncohesi e g anula ma e ials such as d y sand 共pa icle diame e ⲏ100 ␮ m), display an ine ial egime, whe e s esses a e mainly ca ied by in e pa - icle collisions. This high shea a e low has been o en in- es iga ed in a ho izon al o a ing d um pa ially illed wi h he g anula sample 关3兴. In such de ice in e mi en quasip- e iodic a alanches o g ains a e obse ed o low o a ion speeds 关4兴. As he o a ion eloci y ␻ is inc eased he e- quency o he a alanches inc eases un il a con inuously cas- cading supe icial laye o ⬃10 g ains is o med, eeded by he es o he g ains ha unde go igid-body o a ion 关5兴. The slope o he su ace inc eases wi h ␻ due o ine ial o ces and a highe eloci ies, he s eady p o ile o he ee su ace akes he o m o a il ed ‘‘S.’’ On he con a y, ine powde s 共pa icle diame e dpbe- ween ⬃1 ␮ m and ⬃100 ␮ m) do no exhibi he ine ial egime bu expe ience a di ec ansi ion om he solid- plas ic o he luidized egime due o he s ong in e ac ion be ween pa icles and in e s i ial ai 关2兴. A e y low o a ion a es quasipe iodic a alanches may be obse ed, bu as op- posed o noncohesi e g ains, hese a alanches a e o a ypi- cal dep h co ela ed o powde cohesi eness and bounda y condi ions and much la ge han pa icle size 关6兴. In his igid-plas ic egime hese ine d y powde s a e akin o we sand mo e han o d y sand. Howe e , hey di e om we sand in one impo an espec . As we inc ease he angula eloci y he a e o a alanching inc eases. Du ing he a a- lanching p ocess he su ounding gas is en ained in he bulk. In addi ion he alling ma e ial wedge may be pa ially lu- idized upon impac on he d um wall. As a esul he su - ounding gas is mo e and mo e en ained in he bulk as we inc ease he angula eloci y and adius o he d um. The en ained gas akes a ime o lea e he luidized powde ha may be e en ually la ge han he ime in e al be ween suc- cessi e a alanches o high enough o a ion eloci ies. Then a pe manen luidized egion appea s a he lowe pa o he slope, whe eas he es o he powde emains in he solid- plas ic egime. The luidized egion g ows wi h he o a ion eloci y un il he whole ma e ial becomes luidized. A his s age he ee su ace becomes ho izon al esembling he be- ha io o a low iscosi y liquid 共see Fig. 1兲. I he o a ion eloci y is u he inc eased beyond he onse o ull luidi- za ion he powde is u he expanded. The objec i e o his pape is o de e mine he pa ame e s on which his ansi ion depends in o de o gain an unde s anding o he unde lying physical p ocess. In a i s pa o he p esen wo k he e ec s o powde cohesion and d um diame e on he onse o lu- idiza ion o ine pa icles in o a ing d ums a e in es iga ed. Ano he de ice used o in es iga e he p ope ies o pow- de s has been desc ibed p e iously 关7兴, a schema ic o which is shown in Fig. 2. I consis s o a e ical essel closed a he bo om by a po ous pla e. A con olled gas low is sup- plied om below o a bed o powde pa icles pa ially illing he essel and he gas p essu e d op ac oss he powde is measu ed by a di e en ial p essu e ansduce . The bed heigh is ead om an ul asonic senso and om his da a he solid olume ac ion is calcula ed. Wi h su icien ly high gas low he upwa d d ag o ce exe ed by he gas on he pa icles coun e ac s g a i y and in e pa icle cohesi e o ces, allowing us o measu e he ensile s eng h o he PHYSICAL REVIEW E, VOLUME 65, 061301 1063-651X/2002/65共6兲/061301共7兲/$20.00 ©2002 The Ame ican Physical Socie y65 061301-1 sample 共see 关7兴 o de ails on he measu emen p ocess兲. Fu - he inc ease in he low causes he bed o expand homoge- neously and s able luidiza ion occu s. Using such a de ice i is possible o analyze he ela ionship be ween he luidizing gas eloci y and he solid olume ac ion o he luidized bed as a unc ion o powde cohesi eness 关8兴. A second pa o he p esen wo k will consis o ela ing he a e age in e - s i ial gas eloci y in he o a ing d um o he o a ion speed. To his end, da a om he powde bed echnique will be employed. II. EXPERIMENTAL SETUP In he expe imen s we used a compu e con olled s ep mo o o o a e polyca bona e d ums o se e al diame e s (D⫽42 mm, 50 mm, and 90 mm in e nal diame e s, and 40 mm leng h兲. The d um was placed on an ai able and connec ed o he mo o by an elas ic ca dan o isola e i om any ex e nal ib a ion 共see Fig. 3兲. To eco d he expe imen a cha ge-coupled de ice 共CCD兲came a was in e aced o a compu e and con olled wi h an image p ocessing so wa e. In all ou measu emen s we begin by ini ializing he sample, which we do by luidizing he powde . To luidize he powde he d um is d i en o an angula eloci y a ound 200 pm. Once he powde eaches a s a iona y luidized s a e he d um is suddenly s opped and he powde is allowed o col- lapse unde g a i y. In his way a e y ep oducible consoli- da ion s a e o he powde is ob ained. Then he o a ion speed is slowly inc eased om ze o o 200 pm and he ol- ume ac ion o he d um occupied by he powde 共 兲and leng h o he ee su ace o he luidized egion (d, see Fig. 1兲a e compu ed using an image so wa e. III. MATERIALS Measu emen s ha e been pe o med on a ange o xe o- g aphic one s made om polyme 共pa icle densi y ␳ p ⯝1g/cm 3). Powde s o a ying pa icle size ha e been es ed 共4.4, 7.2, and 12.7 ␮ m) blended wi h di e en amoun s o su ace addi i es such as Ae osilRand Cab-o-silR 共nanopa icles o umed silica兲, well known o hei abili y o diminish in e pa icle adhesion o ces. Expe imen s ha e been also ca ied ou using a low cohesi eness comme cially a ailable one 共Canon CLC 700兲. an de Waals a ac i e o ces a e dominan be ween ou uncha ged and d y ine pa - icles. The addi ion o nanopa icles esul s in a educ ion in he powde ensile s eng h because he addi i es a e made o a ha d ma e ial and he e o e hey inc ease he ha dness o he con ac s; hey also educe he powde ensile s eng h by educing he size o he con ac s. Ano he impo an pa am- e e a ec ing he ensile s eng h h ough i s e ec on he packing ac ion and he con ac a ea be ween pa icles is he consolida ion s ess 共weigh pe uni a ea兲. As a gene al ule he ensile s eng h inc eases wi h he consolida ion s ess. Fu he mo e he a e o inc ease is a dec easing unc ion o he amoun o addi i es 共see 关9兴and e e ences he ein兲. The ensile s eng h may inc ease d ama ically wi h he consoli- da ion ime, indica ing iscoplas ic de o ma ion o he pa - icle con ac a ea 关10兴. To ule ou he in luence o his e ec , FIG. 1. Expe imen al p o iles o he low o a xe og aphic one in a o a ing d um a inc easing alues o he o a ion eloci y 共10 pm, 45 pm, and 100 pm兲. FIG. 2. Expe imen al se up o he powde bed echnique. FIG. 3. Expe imen al se up o he o a ing d um expe imen . CASTELLANOS, VALVERDE, AND QUINTANILLA PHYSICAL REVIEW E 65 061301 061301-2 all es s ha e been pe o med wi hin he i s 5 m a e se - ling. A summa y o he ele an physical p ope ies o ou pu poses o he samples es ed in his wo k is gi en in Table I. IV. RESULTS Figu es 4 and 5 display / 0共whe e 0is he ini ial ol- ume ac ion兲and d/Das a unc ion o ␻ . As an icipa ed, bo h / 0and d/Dinc ease wi h he o a ion speed, bu a e also s ong unc ions o he powde cohesi eness and d um diame e . Indeed, he end ollowed by / 0and d/Dwi h ␴ and Dmigh be expec ed. On one hand, dec easing he ensile s eng h dec eases he minimum in e s i ial gas eloc- i y o luidiza ion and his acili a es luidiza ion and he consequen u he expansion o he bed. This is e iden om he di e en beha io o samples a-d,b-e, and c- -g,in Figs. 4 and 5; hese samples di e only in hei ensile s eng h, and we may see ha he smalle he ensile s eng h he as e is he ansi ion o he luidized s a e. On he o he hand, he in e s i ial gas eloci y would also be inc eased by he inc ease in he angen ial eloci y when using la ge d ums, hus acili a ing luidiza ion. Again his can be e i- ied by looking a samples a-b,d-e,h-i,j-k, and l-m. Each pai consis o he same powde , and di e only in he diam- e e o he o a ing d um, which is 9 cm o he i s membe o he pai and 5 cm o he second membe . As i is appa en he i s membe o he pai shows a as e inc ease in bo h / 0and d/D. V. DISCUSSION Le us now sea ch o he main pa ame e s go e ning lu- idiza ion in he o a ing d um. F om he h ee a iables in- ol ed in he p oblem, namely, he powde ensile s eng h ␴ , d um adius R⫽D/2, and o a ion eloci y ␻ , wo non- dimensional pa ame e s may be de i ed. A simple choice a e he F oude numbe 共 a io o cen i ugal accele a ion o g a - i y accele a ion兲,F ⫽ ␻ 2R/g, and he a io o ␴ o he ypi- cal po en ial ene gy densi y ␳ gR, whe e ␳ is he powde densi y, Co⫽ ␴ / ␳ gR, named he ea e cohesion numbe . Thus, bo h / 0and d/Dmus be ce ain unc ions o F and Co, / 0⫽G共F ␣ ,Co ␤ 兲,共1兲 d/D⫽H共F ␦ ,Co ␥ 兲.共2兲 In he limi ␻ →0, i mus be → 0and d→0, and his leads us o hypo hesize he simple dependence / 0⫽1⫹AF ␣ Co ␤ ,共3兲 d/D⫽BF ␦ Co ␥ ,共4兲 FIG. 4. Ra io o he olume ac ion o he d um occupied by he powde o he ini ial olume ac ion as a unc ion o he angula o a ion eloci y. FIG. 5. Leng h o he ho izon al pa o he ee su ace o he powde as a unc ion o he o a ion eloci y. TABLE I. Physical p ope ies o xe og aphic one s used in he expe imen s wi h d ums o di e en diame e s. The concen a ion o addi i e is gi en in w . %, dpis he a e age pa icle size, Dis he d um in e nal diame e . ␾ 0is he solid olume ac ion o he powde , pa ially and uni o mly illing he s a ic d um, and ␴ is he co esponding ensile s eng h, measu ed by means o he powde bed echnique 关7兴. Sample no. dp( ␮ m) Addi i e 共%兲D共cm兲 ␾ 0 ␴ (Pa) a12.7 0.4 9 0.4 25 b12.7 0.4 5 0.4 28 c12.7 0.4 4.2 0.38 16 d12.7 0.02 9 0.31 58 e12.7 0.02 5 0.32 70 12.7 0.2 4.2 0.33 26 g12.7 0.1 4.2 0.33 48 h8.5 9 0.34 20 i8.5 5 0.34 20 j7.2 2 9 0.41 36 k7.2 2 5 0.4 30 l4.4 2 9 0.39 25 m4.4 2 5 0.37 16 FINE COHESIVE POWDERS IN ROTATING DRUMS:... PHYSICAL REVIEW E 65 061301 061301-3 whe e ␣ ⬎0 and ␦ ⬎0, and Aand Ba e p opo ionali y con- s an s. Expe imen al da a o / 0 o he whole se o pow- de s in es iga ed i p e y well o he p oposed law o ␣ ⫽0.97⫾0.27 and ␤ ⫽⫺1.07⫾0.28, which a e he bes i pa ame e s om a eg ession analysis 共see Fig. 6兲. App oxi- ma ing ␣ ⯝1 and ␤ ⯝⫺1 we ha e / 0⯝1⫹AF /Co⫽1 ⫹A⌰, whe e a new nondimensional pa ame e ⌰ ⫽ ␳␻ 2R2/ ␴ has a isen. Since d/Dis a measu e o he lon- gi udinal ex en o he luidized egion and / 0is a measu e o he a ea 共 he d um heigh is held cons an 兲, we migh an icipa e d/D⬀ 冑 ⌰. Figu e 7 shows d/Das a unc ion o 冑 ⌰. I can be seen ha d/Dinc eases linea ly wi h 冑 ⌰as expec ed. Mo eo e , he condi ion o ull luidiza ion can be es ablished as d⬇D, and, om Fig. 7, his happens o ⌰1/2⬃1. Al e na i ely, he expansion o he powde can be measu ed by he dec ease o i s solid olume ac ion ␾ e- la ed o he olume ac ion by / 0⫽ ␾ 0/ ␾ , ␾ 0being he ini ial solid olume ac ion o he powde . The e o e, he ela i e dec ease o he solid olume ac ion ( ␾ 0⫺ ␾ )/ ␾ is p opo ional o ⌰. We conclude ha luidiza ion o he pow- de in he o a ing d um is mainly uled by he nondimen- sional numbe ⌰. We now u n o in es iga e he a e age in e s i ial eloci y 具 i 典 o he luidizing gas ac oss he powde . To his end we will make use o he powde bed echnique desc ibed in 关8兴. The powde is homogeneously luidized by lowing a con- olled gas low h ough he bed and he ela ion be ween he solid olume ac ion and he in e s i ial gas eloci y is p e- cisely measu ed. Fine pa icles in he luidized s a e a e ag- g ega ed due o s ong in e pa icle adhesi e o ces 关11兴. Ag- g ega es we e modeled as e ec i e pa icles wi h a adius o gy a ion equal o i s hyd odynamic adius and cha ac e ized by a numbe o agg ega ed pa icles Nand a ac al dimen- sion D 关12兴. By means o an ex ended Richa dson-Zaki equa ion, he se ling eloci y scan be ela ed o he solid olume ac ion ( ␾ ) o he luidized powde by means o he equa ion 关12兴 s a ⫽共1⫺ ␾ e 兲n,共5兲 whe e ais he S okes se ling eloci y o a single agg ega e ( a⫽ p0N1⫺1/D , being p0 he se ling eloci y o a single pa icle兲, ␾ e is he olume ac ion occupied by he agg e- ga es ( ␾ e ⫽ ␾ N⫺1⫹3/D ), and na pa ame e o o de 5. Se - ling expe imen s se ed us o es ablish ha he se ling e- loci y o a ini ially luidized bed o pa icles a e suddenly s opping he gas low is equal o he supe icial gas eloci y gin he luidized s a e. The in e s i ial gas eloci y in he luidized bed is hen gi en by i( ␾ )⫽ s/(1⫺ ␾ e ). To ind Nand D o each powde in es iga ed in his s udy expe imen s on sedimen a ion ha e been pe o med. Resul s o Nand D a e shown in Table II. As we ob ained o o he ine powde s, D app oaches closely o he alue gi en by he di usion limi ed agg ega ion model 关13兴in 3D (D ⫽2.5) as a consequence o he la ge in e pa icle adhe- si e o ces as compa ed o pa icle weigh 关12兴. Le us as- sume ha he a e age in e s i ial gas eloci y in he o a ing d um expe imen 具 i 典 is o he o de o iin he luidized bed when he powde is luidized in bo h expe imen s wi h he same alue o ␾ , i.e., 具 i 典 ( ␾ )⯝ i( ␾ ). Then Eq. 共5兲may se e us o es ima e 具 i 典 ( ␾ ) and co ela e i wi h he o a ion TABLE II. Numbe o agg ega ed pa icles and ac al dimen- sion o agg ega es de i ed om se ling expe imen s 关12兴 o he di e en powde s used in his s udy and cha ac e ized by hei a - e age pa icle size (dp) and w . % o low addi i es. dp( ␮ m) % ND 12.7 0.2 20 2.64 12.7 0.4 20 2.56 12.7 0.1 22 2.66 12.7 0.02 46 2.5 7.2 2 133 2.7 4.4 2 1300 2.73 8.53共canon兲96 2.62 FIG. 6. Ra io o he olume ac ion o he d um occupied by he powde o he ini ial olume ac ion as a unc ion o he a io o kine ic ene gy densi y o powde ensile s eng h. FIG. 7. Leng h o he ho izon al pa o he ee su ace o he powde as a unc ion o he squa e oo o he a io o he kine ic ene gy densi y o he ensile s eng h. CASTELLANOS, VALVERDE, AND QUINTANILLA PHYSICAL REVIEW E 65 061301 061301-4 speed o he d um. In Fig. 8 we ha e plo ed he a io o 具 i 典 , es ima ed in his way, o he angen ial eloci y o he d um ␻ R. As can be obse ed, 具 i 典 /( ␻ R)⬃10⫺2in he egion o comple e luidiza ion o all he powde s in es iga ed. On he o he hand om Fig. 7 we in e ed ⌰⬅ ␳␻ 2R2/ ␴ ⬃1 o comple e luidiza ion. The e o e, and because ␳ ⫽ ␳ p ␾ does no change app eciably o ou powde s, he in e s i ial gas eloci y a luidiza ion 具 i 典 should scale as he squa e oo o he powde ensile s eng h ␴ . To check his beha io we ha e measu ed he in e s i ial gas eloci y o a homoge- neous luidiza ion s a e 共wi h ␾ ⫽0.15兲 o se e al powde s o same mass and pa icle size, only di e ing in he amoun o su ace addi i es. Resul s a e plo ed in Fig. 9. I is ob- se ed ha he in e s i ial gas eloci y can be well i ed by he law i⬀ 冑 ␴ , in ag eemen wi h ou expec a ion. On he o he hand, since he luidized s a e is a dilu ed s a e, acco d- ing o Eq. 5 he in e s i ial gas eloci y can be oughly es i- ma ed as he agg ega e se ling eloci y i⬃ a⫽N1⫺1/D . Acco ding o a powe law i o he da a p esen ed in 关12兴, N1⫺1/D ⬃Bo0.43⯝Bo0.5, whe e Bo is he a io o he in e - pa icle adhesion o ce o he pa icle weigh and labeled Bo because o i s simila i y o he Bond numbe in luid mechan- ics. Fo he ensile s eng h is expec ed o ela e linea ly wi h he in e pa icle adhesion o ce 关12兴, i leads us again o he conclusion ha he in e s i ial gas eloci y mus scale as he squa e oo o he powde ensile s eng h o he packed bed. Ano he cha ac e is ic gas eloci y is he minimum luidi- za ion eloci y sm , de ined as he supe icial gas eloci y a which he p essu e d op balances he weigh o he powde bed pe uni a ea ( ␴ c) plus i s ensile s eng h. The p essu e d op ac oss he powde a he poin o incipien luidiza ion is hen gi en by ⌬Pm⫽ ␴ c⫹ ␴ . A low Reynolds numbe s he gas low h ough he packed bed is lamina and he p es- su e d op is a linea unc ion o he supe icial gas eloci y 共Ca man-Kozeny law 关14兴兲: ⌬P h⫽E ␮ dp 2 ␾ p 2 共1⫺ ␾ p兲3 s,共6兲 whe e ␮ is he dynamic gas iscosi y ( ␮ ⫽1.89 ⫻10⫺5Pas a ambien empe a u e兲, sis he supe icial gas eloci y, ␾ pis he solid olume ac ion o he packed bed, E⬃200 is an empi ical cons an , and his he powde bed heigh ha is ela ed o ␾ p关h⫽ ␴ c/( ␳ pg ␾ p)兴. The gas eloci y needed o luidiza ion is de i ed by imposing he condi ion ⌬P( sm)⫽⌬Pm, sm⫽ ␳ pgdp 2 E ␮ 共1⫺ ␾ p兲3 ␾ p 冋 1⫹ ␴ ␴ c 册 ,共7兲 and he in e s i ial gas eloci y a incipien luidiza ion is simply gi en by im⫽ sm /(1⫺ ␾ p). Resul s o im ob- ained om he measu ed alues o ␾ pand ␴ a e ep e- sen ed in Fig. 10. Again, he end ound o he in e s i ial gas eloci y is qui e close o a squa e oo law o he powde ensile s eng h. Figu es 9 and 10 also indica e ha he in e - s i ial gas eloci y o a solid olume ac ion o 0.15, close o he bubbling egime, is abou one o de o magni ude la ge han he minimum gas eloci y needed o luidiza ion, and his esul has been con i med by di ec measu emen s o bo h cha ac e is ic eloci ies in samples o di e en powde s. FIG. 8. Ra io o he es ima ed a e age in e s i ial gas eloci y in he d um o d um angen ial eloci y as a unc ion o he a io o kine ic ene gy densi y o powde ensile s eng h. FIG. 9. Measu ed in e s i ial gas eloci y o a homogeneous luidiza ion s a e ( ␾ ⫽0.15) as a unc ion o he squa e oo o he ensile s eng h o he packed bed. The con inuous line is a linea i o he da a. FIG. 10. Compu ed in e s i ial gas eloci y a incipien luidiza- ion om expe imen al measu emen s o he packed solid olume ac ion and ensile s eng h as a unc ion o he squa e oo o he ensile s eng h. FINE COHESIVE POWDERS IN ROTATING DRUMS:... PHYSICAL REVIEW E 65 061301 061301-5 A. Physical model We a e now in a posi ion o pos ula e a physical model o he luidiza ion and de luidiza ion p ocesses in he o a ing d um ha p o ides us wi h a a ionale o unde s anding he physical basis o he abo e nondimensional numbe s. Figu e 11 ep esen s h ee ypical s ages o he luidiza ion p ocess as we inc ease he angula eloci y. In Fig. 11共a兲, which co - esponds o Fig. 1共a兲, he wedge o powde , consis ing o pa s 2 and 3, a alanches. As a esul o he impac on he wall pa 2 o he wedge is luidized, bu be o e he nex a alanche akes place he en apped ai has ime o escape and he ma e ial ends up in a plas ic s a e wi h a ho izon al su ace. This in e p e a ion is backed by independen mea- su emen s o he quasis a ic a alanching p ocess made in a il ed bed o ec angula shape 关15兴. In hese expe imen s he ma e ial is i s ini ialized by luidiza ion. A e shu ing o he gas supply he ma e ial collapses unde i s own weigh wi h a ho izon al ee su ace. Once he ma e ial has col- lapsed, he bed is il ed slowly un il an a alanche is igge ed and a slice o powde slides down he slope o med by he es o he ma e ial. I he ma e ial in he slice is luidized du ing he a alanche, i will beha e like a liquid as i comes o es wi h a ho izon al ee su ace. Reco ding he expe i- men wi h he CCD came a makes i possible o measu e he a e age eloci y o he slice du ing he a alanche ( w). As we know om o he expe imen s he ensile s eng h o he ma e ials used in his expe imen , we know o each a a- lanche he ensile s eng h o he ma e ial in he slice and i s eloci y and we can make a plo like he one in Fig. 12. This igu e shows ha in o de o become luidized du ing an a alanche, he eloci y o he slice mus o e come a ce ain h eshold ha depends on he ensile s eng h o he ma e ial in he slice. In Fig. 11共b兲 he model co esponds o he si ua ion o Fig. 1共b兲. A alanching is now a p ac ically con inuous p o- cess. The alling mass du ing a ime in e al ⌬ scales as ⌬m ⬃d2 ␻ R⌬ . Since he p ocess is s a iona y, he same amoun o mass is de luidized in he same ime in e al and se les in pa 4. The se ling eloci y can be es ima ed as he gas eloci y and hus he de luidized mass scales as ⌬md ⬃d1 i⌬ ⬃⌬m . Since he in e s i ial gas eloci y in he luidized egion 共pa 1兲o he bed g ows as he squa e oo o he ensile s eng h, we ob ain d1/d2⬃ ␻ R/ 冑 ␴ ⬀ 冑 ⌰. Thus, i u ns ou ha he pa ame e ⌰is a na u al pa ame e o desc ibe he e olu ion o he powde . Finally, Fig. 11共c兲co esponds o Fig. 1共c兲, and shows he onse o luidiza ion. The powde ha sedimen s in one o a- ion pe iod 共pa 4兲is he powde ha comes ou a he igh side pul e ized. The sedimen hickness is p opo ional o R i⌬ , and i is ejec ed wi h eloci y ␻ Rdu ing he ime ⌬ , h ough a leng h d2⫽ ␣ R共 om Fig. 1 we see ha ␣ mus be o o de 10⫺1o smalle 兲. The e o e, a he onse o luidiza ion imus be p opo ional o ␻ R. Due o he ela- ion be ween iand ␴ and he negligible a ia ion o he ini ial densi y o ou powde s, we may choose again ⌰as a sui able pa ame e o cha ac e ize his ansi ion. FIG. 11. Physical model on he luidiza ion-de luidiza ion p o- cess. FIG. 12. A alanching wedge eloci y as a unc ion o he a e - age wedge ensile s eng h om il ed bed expe imen s and o di - e en samples o powde . Fo solid symbols he wedge luidizes upon impac on he wall, whe eas o open symbols i emains in he plas ic egime. CASTELLANOS, VALVERDE, AND QUINTANILLA PHYSICAL REVIEW E 65 061301 061301-6 VI. CONCLUSIONS In conclusion, we ha e ound ha o a se o powde s o a ying cohesi eness and o di e en geome ies, luidiza- ion is uni e sally uled by he a io o he kine ic ene gy densi y o he powde ensile s eng h. Mo eo e he es i- ma ed a e age in e s i ial gas eloci y scales wi h he angen- ial o a ion eloci y. This scaling implies ha he in e s i ial gas eloci y should inc ease as he squa e oo o he powde ensile s eng h, and his has been con i med by independen measu emen s using a powde bed echnique. As a inal ema k we wan o emphasize ha o he pow- de s used in his expe imen he onse o luidiza ion, de ined as he lowes angula eloci y a which he ee su ace o he ma e ial becomes ho izon al (d/D⬃1), occu s a a alue ⌰⬃1共Fig. 7兲. Gi en he di e en pa icle sizes, cohesi e- ness, and d um diame e s, we may expec ha a use ul de- sign ule o indus y applica ions in ol ing luidiza ion o ine cohesi e powde s in o a ing d ums should be ⌰⬃1. ACKNOWLEDGMENTS This esea ch was suppo ed by he Xe ox Founda ion, Spanish Go e nmen Agency Minis e io de Ciencia y Tecno- logia 共DGES兲unde Con ac No. BMF2000-1056, and NATO G an No. LINKAGE PST.CLG.976575. 关1兴K. Rie ema, The Dynamics o Fine Powde s,共Else ie , Am- s e dam, 1991兲, pp. 233–237. 关2兴A. Cas ellanos, J.M. Val e de, A.T. Pe ´ ez, A. Ramos, and P.K. Wa son, Phys. 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