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
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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
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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
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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
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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.
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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.
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