E ec o pa icle size and in e pa icle o ce on he luidiza ion beha io o gas- luidized beds
J. M. Val e de,1A. Cas ellanos,1P. Mills,2and M. A. S. Quin anilla1
1Depa 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
2LPMDI, Uni e si e
´de Ma ne-la-Valle
´e, 5 boule a d DesCa es, Champs-su -Ma ne, F-77454 Ma ne-la-Valle
´e Cedex 2, F ance
共Recei ed 14 No embe 2002; published 23 May 2003兲
Gas- luidized powde s o ine pa icles display a luidlike egime in which he bed does no ha e a yield
s eng h, i expands uni o mly as he gas eloci y is inc eased and mac oscopic bubbles a e absen . In his
pape we es he ex ension o his luidlike egime as a unc ion o pa icle size and in e pa icle a ac i e
o ce. Ou esul s show ha o su icien ly la ge pa icles, bubbling ini ia es jus a e he solidlike luidized
egime as i is ob ained expe imen ally by o he wo ke s. A scaling beha io o he solid-phase p essu e in he
luidlike egime and a p edic i e c i e ion o he onse o mac oscopic bubbling a e analyzed in he ligh o
hese esul s.
DOI: 10.1103/PhysRe E.67.051305 PACS numbe 共s兲: 45.70.⫺n, 47.55.K , 47.55.Mh
I. INTRODUCTION
The beha io o ine powde s is a keys one o an inc eas-
ing numbe o high- echnology indus ial p ocesses. Fo ex-
ample, luidized beds, consis ing o a e ical column essel
pa ially illed wi h a powde es ing on a po ous il e and a
lowing luid pumped om below, a e ex ensi ely used in
applica ions ha ake ad an age o hei excellen luid-solid
mixing and anspo capabili ies.
Back in 1973 Gelda 关1兴p oposed an empi ical classi i-
ca ion o powde s acco ding o hei luidiza ion p ope ies
ha has been widely used by chemical enginee s. Gelda
summa ized his obse a ions in e ms o he pa icle size
e sus he ela i e densi y di e ence be ween he luid phase
and he solid pa icles. Acco ding o Gelda , G oup A共ae -
a able兲powde s gi e a egion o ‘‘homogeneous’’ luidiza-
ion in which he pa icles a e suppo ed by he d ag o ce o
he luid and he bed expands smoo hly as he luid eloci y
is inc eased, aking he appea ance o a low iscosi y luid.
Abo e a ce ain luid eloci y his luidlike egime is ol-
lowed by a bubbling egime in which a la ge pa o he gas
is apped om he dense phase by ising mac oscopic
bubbles and bed expansion is cu ailed. Mo e common
G oup Bpowde s consis o coa se and dense pa icles han
G oup Apowde pa icles and gi e only bubbling luidiza-
ion. This g oup includes sandlike ma e ials. G oup Cpow-
de s a e e y ine, cohesi e powde s ha a e incapable o
luidize and end o ise as a slug o solids o o o m chan-
nels h ough which he luid will escape a he han being
dis ibu ed h ough he bulk. The bo de be ween G oup A
and Bpowde s is no clea cu . Fo ins ance, i has been
ound ha he luid p essu e in luences he bubbling beha -
io . Ele a ed p essu e p oduce mo e homogeneous gas-solid
low s uc u e by in ensi ying pa icle- luid in e ac ion and
supp essing pa icle-pa icle in e ac ions, he eby ex ending
he uni o m luidiza ion egime 关2兴.
A ine powde showing an amazing beha io and used in
he xe og aphic indus y is one . In he las decades image
quali y equi emen s leaded o a educ ion o one pa icle
size in expenses o loosing he good lowabili y exhibi ed by
he old- ashion one s, o which he in e pa icle a ac i e
o ces we e compa able wi h he ine ial o ces. Nowadays
xe og aphic one s a e made o polyme pa icles wi h a ol-
ume a e age size dp⬍⬃10
␮
m. Fo such a small pa icle
size we would p edic G oup C ypical s icky beha io since
in e pa icle a ac i e o ces a e se e al o de s o magni ude
la ge han pa icle weigh . The need o educing cohesi e-
ness was app oached by means o he use o addi i es. The
addi ion o umed silica nanopa icles, which became dis-
pe sed on he su ace o he one pa icles, dec eases he
in e pa icle adhesion and hus causes a shi o well-beha ed
G oup Apowde s. Mo eo e , he small pa icle size and i s
low densi y (
␳
p⬃1 g /cm3) allowed o an ex ended in e al
o ‘‘homogeneous’’ luidiza ion in which he bed could ex-
pand uni o mly up o each solid olume ac ions as low as
0.1 关3兴. This beha io con as s wi h he beha io o G oup A
powde s usually encoun e ed in comme cial gas- luidized
sys ems, o which he in e al o nonbubbling luidiza ion is
a he sho .
Expe imen s epo ed by Rie ema and co-wo ke s 关4兴, Tsi-
non ides and Jackson 关5兴, and Menon and Du ian 关6兴on gas-
luidized pa icles o size dp⭓50
␮
m indica ed he absence
o g ain dynamics in he uni o mly luidized s a e. These
esea che s asse ed ha in he luidized s a e he pa icles
a e s a ic in di ec con ac wi h each o he , hus p o iding an
elas ic modulus o he bed ha s abilizes i agains small
pe u ba ions. This pic u e was indeed upheld by heo e ical
in es iga ions based on he kine ic heo y. Koch and Sangani
关7兴de i ed he pa icle-phase p essu e o a homogeneous
suspension o pa icles in a gas in e ac ing ia ins an aneous
ha d-sphe e collisions and hyd odynamic in e ac ions. Thei
linea s abili y analysis indica ed ha he homogeneous s a e
o a luidlike gas- luidized bed is always uns able, leading
hem o he conclusion ha bubbling can be only es ained
by a solidlike beha io o he expanded beds in ag eemen
wi h Rie ema and o he ’s expe imen s. Howe e , i con a-
dic ed empi ical obse a ions 关8兴on some pa icula gas-
luidized sys ems belonging o G oup Ain he Gelda ’s clas-
si ica ion scheme ha exhibi a luidlike ‘‘homogeneous’’
egime. Ou measu emen s on he yield s eng h and di u-
si i y o ou gas- luidized ine powde s wi h educed cohe-
si eness clea ly showed ha bo h egimes, he solidlike 共be-
low a ce ain gas eloci y cla ge han he minimum
luidiza ion eloci y m ) and he luidlike 共abo e c), migh
be encoun e ed in he so-called in e al o ‘‘homogeneous’’
luidiza ion 关9兴.
PHYSICAL REVIEW E 67, 051305 共2003兲
1063-651X/2003/67共5兲/051305共6兲/$20.00 ©2003 The Ame ican Physical Socie y67 051305-1
A deepe insigh in o he local dynamics o he luidlike
egime has ecen ly e ealed he p esence o mesoscale spa-
io empo al s uc u es and sho -li ed oidage ins abili ies
关3兴whose cha ac e is ic empo al equency g adually in-
c eases wi h gas eloci y. Well-o ganized luc ua ions 关10,11兴
and ansien oids 关12兴ha e also been disco e ed in liquid-
luidized beds. Du u and Guazelli showed ha in liquid lu-
idiza ion local oids g ew in o mac oscopic eal bubble-like
s uc u es o su icien ly high densi y beads 共pa icle
densi y/liquid densi y
␳
p/
␳
⬇8) 关12兴. Thus, in close simi-
la i y wi h obse a ions in liquid- luidized beds, we should
conclude ha ou gas- luidized beds a e ac ually uns able
h oughou he luidlike nonbubbling egime. E en hough
his ins abili y does no g ow in o ully de eloped bubbles, i
ce ainly denies he applicabili y o hyd odynamic linea o
weakly nonlinea s abili y analyses ha ha e been employed
in he pas o p edic he onse o bubbling 关13兴. S ill he
ques ion o why la ge bubbles de elop abo e ce ain luid
eloci ies emains unexplained.
Undoub edly bubbling mus be inhibi ed by some com-
plex hyd odynamic mechanism o su icien ly small pa -
icles, ye a ela i e ole o in e pa icle o ces should no be
unequi ocally dismissed. Nume ical simula ions o bubbling
beds ha e ecen ly shown ha when he a ac i e in e pa -
icle o ce was inc eased he a e age size o bubbles de-
c eased and he coalescence a e o bubbles slowed down
关14兴. I is well known ha an de Waals a ac i e o ce,
much la ge han pa icle weigh o mic ome e -sized pa -
icles, causes he agg ega ion o luidized ine pa icles 关15兴.
Independen empi ical obse a ions sugges ha he size o
agg ega es could in luence he ansi ion o he bubbly e-
gime: The homogenei y o luidiza ion has been subs an ially
imp o ed in some cases when he agg ega es could be dis-
up ed by means o ex e nal agen s such as ib a ion 关16兴o
ul asonics 关17兴. Changes occu ing du ing luidiza ion o
some cc ca alys s ha e been obse ed as he in e pa icle
a ac i e o ces we e inc eased by inc easing empe a u e
关18兴. Bubbling o beds o me allic pa icles could be sup-
p essed con olling in e pa icle o ces by means o an ex e -
nal magne ic ield 关19兴.
One o ou aims in his pape is o con ibu e o ill he
gap be ween he obse a ions on one side o a bubble- ee
luidlike egime o small pa icles and on he o he side o
jus bubbling luidiza ion o la ge pa icles. Following a
sugges ion by Jackson 关20兴, he ex ension o he luidlike
egime in e al and he onse o bubbling a e expe imen ally
in es iga ed using specially designed powde s o se e al pa -
icle sizes and di e en cohesi eness. This wo k will allow
us o elucida e a ole o in e pa icle o ces and pa icle size
on luidiza ion beha io .
II. EXPERIMENTAL POWDERS
Tone pa icles made o he andom copolyme s y ene
n-bu ylme hac yla e (
␳
p⫽1.14 g/cm3) we e p oduced by an
a i ion p ocess ollowed by size classi ica ion by means o a
cyclone sepa a o appa a us. In his way we had a ailable
ou se s o powde s wi h olume a e aged pa icle size dp
⫽19.1, 15.4, 11.8, and 7.8
␮
m. These powde s we e subse-
quen ly blended wi h nanopa icles o umed silica, which
became dispe sed on he one pa icle su ace. 32% and 8%
o su ace a ea co e age 共SAC兲we e used. The addi ion o
hese nanopa icles allowed o a educ ion o he in e pa -
icle adhesion o ce because silica is conside ably ha de
han polyme and hus he addi i e inc eases he con ac
ha dness; silica also dec eases he adhesion o ce by educ-
ing he ypical size o he su ace aspe i ies a con ac
关21,22兴. In he luidlike egime one pa icles a e agg e-
ga ed. Assuming ha in sedimen a ion agg ega es beha e
like e ec i e sphe es wi h a hyd odynamic adius equal o
hei adius o gy a ion R, we de i ed elsewhe e he a e age
numbe o pa icles pe agg ega e 共N兲and he a e age a io
o agg ega e size o pa icle size (k⬅2R/dp)关15,23兴共see
Table I兲.
III. EXPERIMENTAL SETUP
In he expe imen s epo ed in his pape a powde sample
o mass mwas held in a ⫽2.21 cm adius polyca bona e
cylinde , he base o which is a sin e ed me al il e o 5
␮
m
po e size. Fo his bed adius wall e ec s a e negligible 关24兴.
A d y ni ogen ank u nished wi h a mass low con olle
supplied adjus able gas low. All he measu emen s s a ed by
d i ing he powde in o he bubbling egime by imposing a
e y la ge gas low o he bed. Once he powde is in he
bubbling egime i had los memo y o i s p e ious his o y.
Then he gas low was dec eased down o a gi en alue.
A e he bed had eached a s a iona y s a e i s heigh hwas
ead om an ul asonic senso and he a e age solid olume
ac ion
␾
关
␾
⫽m/(
␳
p
␲
2h)兴was compu ed.
IV. RESULTS
In Fig. 1 he solid olume ac ion is plo ed as a unc ion
o he supe icial gas eloci y g o powde s wi h he same
SAC 共32%兲, i.e., same alue o in e pa icle an de Waals
o ce (F dW⯝1.6 nN 关15兴兲. In Fig. 1 we delinea e he an-
si ions be ween he di e en luidiza ion egimes. The gas
eloci y a he ansi ion o he solidlike egime cis aken
om he poin a which he de elopmen o endu ing con-
ac s cause a slow down o he se ling p ocess 关15兴. We mus
no e, howe e , ha he ansi ion solidlike- luidlike akes
place along an in e al o gas eloci ies in which ansien
TABLE I. dp, pa icle size; N, a e age numbe o pa icles pe
agg ega e in he luidized bed; k, a e age a io o agg ega e size o
pa icle size;
␾
b
exp , expe imen al alue o he solid olume ac ion
a he ini ia ion o bubbling;
␾
b
heo , heo e ical alue o he solid
olume ac ion a he ini ia ion o bubbling acco ding o Eq. 共7兲;
F , F oude numbe a he bubbling poin based on agg ega e size
共 alues co esponding o 32% SAC兲.
dp共
␮
m兲Nk
␾
b
exp
␾
b
heo F
7.8 63 5.22 0.089 0.087 0.14
11.8 23.7 3.55 0.140 0.146 0.09
15.4 12.4 2.72 0.177 0.188 0.07
19.1 9.6 2.45 0.228 0.229 0.04
VALVERDE e al. PHYSICAL REVIEW E 67, 051305 共2003兲
051305-2
luidlike egions al e na e wi h ansien solid ne wo ks. This
issue is ex ensi ely discussed elsewhe e 关3兴. In he luidlike
egime
␾
dec eases s eadily as he gas eloci y is inc eased
down o he minimum alue
␾
b ha he wo-phase sys em is
able o sus ain. I he gas eloci y is u he inc eased abo e
his c i ical poin , mac oscopic bubbles ise ac oss he bed
and
␾
inc eases, wi h oscilla ions o he ee su ace due o
gas bubbles bu s . The gas eloci y a incipien mac oscopic
bubbling bis well de e mined and we ake i om he poin
in Fig. 1 a which he solid ac ion s a s o inc ease as he
gas eloci y is inc eased.
V. DISCUSSION
A clea ou come om Fig. 1 is ha he luidlike egime
sho ens as pa icle size is inc eased. Le us conside he
in e al ⌬ g⬅ b⫺ c, which is plo ed in Fig. 2 agains he
a io o in e pa icle an de Waals o ce F dW o pa icle
weigh Wp共g anula Bond numbe Bog⬅F dW /Wp). I is
seen ha ⌬ gshows a loga i hmic inc ease wi h Bog.An
ex apola ion o he esul s gi es ⌬ g→0 o Bog⬍⬃10,
indica ing ha he luidlike egime should sh ink o ze o o
luidized beds o la ge enough pa icles. Fo ins ance, in he
case o glass beads (
␳
p⯝2.5 g/cm3) we es ima e F dW
⬅Ada/(24z0
2)⯝8nN共Hamake cons an A⯝1.5⫻10⫺19 J,
ypical aspe i y size da⯝0.2
␮
m关25兴, and minimum in e -
pa icle dis ance z0⯝4Å关26兴兲, and hus he condi ion Bog
⬍⬃10 would be me o dp⬎⬃40
␮
m. Thus, o pa icles
o size la ge han ⬃40
␮
m, mac oscopic bubbling would
ini ia e immedia ely a e he solidlike egime. This is in ac-
co dance wi h expe imen s epo ed in he li e a u e 关6兴and
wi h a luidiza ion es we made in ou lab, using a sys em o
glass beads wi h dp⯝60
␮
m. In e es ingly, a simila h esh-
old is ound o he agg ega ion o pa icles 关15,27兴, sugges -
ing ha he o ma ion o low-densi y agg ega es plays a c u-
cial ole in p omo ing he gas-solid in e ac ion and hus in
p e en ing he g ow h o la ge bubbles along an ex ended
pseudo u bulen luidlike in e al.
Figu e 3 shows he e ec o educing he su ace silica
co e age om 32% down o 8% 共 hus inc easing he an de
Waals o ce om 1.6 nN up o 3.4 nN 关15兴兲 o a gi en
pa icle size 共 he same e ec is ob ained o o he pa icle
sizes兲. We obse e ha , in he luidlike egime, he bed o
mo e cohesi e pa icles shows la ge alues o
␾
o a gi en
gas eloci y, whe eas he opposi e happens in he solidlike
egime. This beha io may be a ionalized as ollows: As he
addi i e su ace co e age is dec eased he in e pa icle a -
ac i e o ce inc eases and pa icles end o be sligh ly mo e
agg ega ed 关15兴. Thus o la ge agg ega es he gas eloci y
equi ed o each a gi en alue o he solid olume ac ion
FIG. 1. A e age solid olume ac ion
␾
o he luidized powde
as a unc ion o he supe icial gas eloci y g. Pa icle size is
indica ed in he inse . Su ace addi i e co e age is held cons an
共32%兲. The limi s be ween he solidlike, luidlike, and bubbling
egimes a e shown. E o ba s ep esen he ypical s anda d de ia-
ion o he ime signal o
␾
o a gi en gin each egime. The
a ows show he p edic ed alues o he solid olume ac ion a
incipien bubbling, using a modi ied e sion o he FG c i e ion.
FIG. 2. Ex ension o he luidlike egime as a unc ion o he
a io o in e pa icle an de Waals o ce o pa icle weigh . The
SAC% o he one pa icles is indica ed.
FIG. 3. A e age solid olume ac ion
␾
o he luidized powde
as a unc ion o he supe icial gas eloci y g o powde s wi h he
same pa icle size (15.4
␮
m) bu di e en SAC 共shown兲. The a -
ows indica e he bubbling poin . This igu e has been ecen ly e-
iewed by Sunda esan 关39兴. Please no e ha he igu e shown in
ha e iew displays he in e s i ial gas eloci y iins ead o he
supe icial gas eloci y g关 i⫽ g/共1⫺
␾
兲兴 o he powde wi h
32% SAC. This mino di e ence does no al e he discussion p e-
sen ed in he ex .
EFFECT OF PARTICLE SIZE AND INTERPARTICLE... PHYSICAL REVIEW E 67, 051305 共2003兲
051305-3
in he dilu ed luidlike egime should be highe as we see in
Fig. 3. Acco dingly he gas eloci y needed o each he bub-
bling egime inc eases wi h he educ ion o su ace addi i e.
On he o he side when he gas eloci y is dec eased down o
he poin a which he solidlike egime is eached agg ega es
oll o slide o e each o he un il hey land a s a ic posi ions
and o m pe manen ne wo ks. The highe he in e pa icle
a ac i e o ce he less e icien is he ea angemen o ag-
g ega es lea ing behind la ge oid spaces. This leads o
smalle alues o
␾
o a gi en gas eloci y as we see in Fig.
3. We ha e plo ed in Fig. 2 he ex ension o he luidlike
egime in e al o he beds wi h inc eased cohesi eness and
i can be seen ha b⫺ c ollows he same law wi h Bog.
E en hough ou expe imen s a e es ic ed o one pa -
icles o which he an de Waals o ce p o ides he main
sou ce o in e pa icle a ac ion, ecen nume ical simula-
ions as well as expe imen s wi h magne ic and we pa icles
also show ha he ole o in e pa icle o ce on luidiza ion
beha io mus scale wi h he pa icle weigh 关28兴. Fu he -
mo e, ou esul s ag ee quali a i ely wi h he esul s o
Rhodes e al. 关28兴who ob ain by means o a Disc e e Ele-
men Me hod simula ion ha he in e al o nonbubbling bed
expansion sh inks o ze o o Bog→1.
VI. A PREDICTIVE CRITERION FOR THE INITIATION
OF MACROSCOPIC BUBBLING
Mo e han hal a cen u y ago Wilhelm and Kwauk delin-
ea ed he ansi ion o he bubbling egime based on he em-
pi ical F oude numbe c i e ion, F ⫽ 2/(gdp)⬎1 whe e is
he luid eloci y and gis he g a i y ield 关29兴. Howe e ,
esea che s ound an o de o magni ude o di e ence be-
ween, o ins ance, ai - luidized glass beads 共F ⫽1.1兲and
ai - luidized lead sho 共F ⫽85兲关30兴. In addi ion, he c i e ion
ailed o p edic he obse ed ins abili y in liquid- luidized
beds, o which F ⬃0.1⬍1关12,30兴. Using he e ec i e size
o he agg ega es o ou gas- luidized powde s we es ima e
F ⬃10⫺1⫺10⫺2共Table I兲a incipien bubbling, hus deny-
ing he applicabili y o he Wilhelm and Kwauk c i e ion in
close analogy wi h liquid- luidized beds. La e in 1984, Fos-
colo and Gibila o 共FG兲关31兴 o mula ed he p edic i e c i e-
ion o
␾
b,
共gdp兲1/2⫽0.56n
␾
b
1/2 共1⫺
␾
b兲n⫺1.共1兲
He e is he S okes se ling eloci y o a single pa icle
and nis he exponen in he phenomenological Richa dson-
Zaki 共RZ兲law 关32兴, g⫽ (1⫺
␾
)n. By including he p e -
ac o 0.56 in he igh hand side o he equa ion FG admi ed
n⫽4.8 acco ding o ea lie RZ empi ical obse a ions on he
expansion and se ling o luidized beds a low pa icle Rey-
nolds numbe s 关32兴. We ind in he li e a u e a a ia ion in
he empi ical alues o he Richa dson-Zaki exponen maybe
due o wall e ec s, esidual polydispe si y e ec s and ine ial
sc eening e ec s as poin ed ou by Snab e and Mills 关33兴.As
we a e dealing wi h gas- luidized beds we gi e abo e a sim-
pli ied e sion o he FG c i e ion by neglec ing he a io o
gas densi y o pa icle densi y.
The o iginal de i a ion o Eq. 共1兲was based on he gen-
e al c i e ion p oposed by Wallis o desc ibe bubbles as con-
cen a ion shocks ha o igina e when he p opaga ion eloc-
i y o a oidage dis u bance (u
␾
) ises as e han he elas ic
esponse a e o he bed. FG used o u
␾
he well-es ablished
ela ion
u
␾
⫽⫺
␾
d g
d
␾
,共2兲
whe e gwas ela ed o
␾
by he RZ law. On he o he hand,
he elas ic wa e eloci y uemus be calcula ed as
ue⫽
冉
1
␳
p
⳵
p
⳵
␾
冊
1/2
,共3兲
whe e pis he pa icle-phase p essu e. As ea lie sugges ed
by Ve loop and Hee jes 关34兴, Foscolo and Gibila o ne-
glec ed pa icle ine ia and ela ed p o he luid-pa icle d ag
o ce in an ideal homogeneous s a e, a i ing a
p⫽1
3n
␳
pgdp
␾
2⬃
␳
pgdp
␾
2.共4兲
This is a w ong de i a ion, howe e , since he p essu e a ia-
ion canno be due only o a dissipa i e p ocess bu , as no ed
by Ba chelo 关13兴and Jackson 关35兴, momen um anspo by
pa icle luc ua ions and collisions mus be necessa ily con-
side ed. The linea s abili y analysis o Ba chelo 关13兴yields
o he pa icle p essu e p⬃
␳
p g
2
␾
F(
␾
), whe e F(
␾
)is
some unc ion o he solids ac ion. Taking in o accoun ha
when a pa icle is e ec i ely isola ed (
␾
→0) p→0 and ha
pmus be ze o again when he pa icles a e locked in he
packed assembly (
␾
→
␾
c), Ba chelo sugges ed he simple
ep esen a ion F(
␾
)⬇(
␾
/
␾
c)关1⫺(
␾
/
␾
c)兴. This ough es i-
ma e d o e him o alues o pin he dilu e limi compa able
o he FG p edic ion 关36兴. S ill he e is no a gene al consen-
sus on a co ec o mula ion o he pa icle p essu e.
I mus be ema ked ha , e en hough he bed mani es s a
smoo h appea ance o he naked eye in he luidlike egime,
he exis ence o mesoscale spa io empo al s uc u es 关3,12兴
indica es ha he luidlike egime is no uly homogeneous
a e all. In ac , as poin ed ou by Jackson 关35兴, he FG
c i e ion ails o p edic he onse o p ima y ins abili y ob-
se ed in liquid luidized beds consis ing o oidage wa es
ha do no de elop in o ecognizable bubbles. In spi e o
hese se e e d awbacks, he su i al o he FG c i e ion o
disc imina e isible bubbling along yea s o expe imen s is
ascina ing. In hei pape Foscolo and Gibila o epo ed
good ag eemen wi h expe imen al da a on liquid and gas
luidized beds a bo h lamina and u bulen condi ions 关31兴.
O e he nex yea s he FG c i e ion was success ully applied
unde sys ema ic a ia ions o ele an pa ame e s on he
bubbling poin o gas- luidized beds such as p essu e, em-
pe a u e, and addi ion o ines 关37兴. Mo eo e , measu emen s
o he collisional pa icle p essu e in liquid-solid lows
yielded, in he ange
␾
⬍⬃0.3, simila esul s o hose p e-
dic ed om he FG model, whe eas esul s o collision-based
models we e in poo co espondence wi h he expe imen s
关36兴. The e o e i seems ha he scaling p⬃
␳
pgdp
␾
2can be
aken a leas as an empi ical ac .
VALVERDE e al. PHYSICAL REVIEW E 67, 051305 共2003兲
051305-4
Le us ex end Eq. 共1兲 o ou case o agg ega ed ine pa -
icles. Acco ding o ou p e ious model 关23兴, agg ega es a e
iewed like e ec i e sphe es wi h a hyd odynamic adius
equal o hei adius o gy a ion and unde going a collisional
dynamics. The a e age numbe o agg ega ed pa icles Nand
he a io o agg ega e size o pa icle size ka e de i ed i ing
a gene alized RZ law g
*⫽
*(1⫺
␾
*)n, whe e
*⫽ N/k
关23兴is he S okes eloci y o a single agg ega e and
␾
*
⫽
␾
k3/N关23兴is he olume ac ion illed by he agg ega es,
o he esul s o se ling expe imen s 关15,23兴. We admi ha
he oidage wa e eloci y can be ob ained as
u
␾
*⫽⫺
␾
*d g
*
d
␾
*⫽k2
␾
n
冉
1⫺
␾
k3
N
冊
n⫺1
.共5兲
Following he same app oach a s aigh o wa d gene aliza-
ion o he pa icle p essu e is gi en by p*
⬃
␳
*gkdp(
␾
*)2, whe e
␳
*⫽
␳
pN/k3关23兴is he densi y o
he agg ega e. Thus, o ou e ec i e sys em o agg ega es,
he elas ic wa e eloci y can be calcula ed as
ue
*⫽
冉
1
␳
*
⳵
p*
⳵
␾
*
冊
1/2
⬃
冉
gdp
␾
k4
N
冊
1/2
.共6兲
I we admi ha isible bubbling is a consequence o a
nonlinea p ocess leading o he de elopmen o concen a-
ion shocks we can eco e he Wallis c i e ion ela ing bub-
bling o he c isis ha esul s om he balance be ween he
oidage and elas ic wa es eloci ies. Thus, equa ing u
␾
*
⬃ue
*we ob ain a p edic i e c i e ion o he solid olume
ac ion a he onse o mac oscopic bubbling
共gdp兲1/2⬃n N1/2
␾
b
1/2
冉
1⫺
␾
b
k3
N
冊
n⫺1
.共7兲
As can be seen in Table I his modi ied c i e ion yields alues
o
␾
bqui e close o he expe imen al ones. I mus be
s essed, howe e , ha we p opose an o de o magni ude o
ue
*based on he scaling o he pa icle p essu e p*
⬃
␳
*gkdp(
␾
*)2. Since agg ega es o ou cohesi e pa icles
ha e a ac al dimension (D⫽lnN/lnk) close o 2.5
共di usion-limi ed agg ega ion limi 兲关15兴we can w i e p
⬃
␳
pgdpk3/2
␾
2whe e he in luence o in e pa icle o ces is
explici in he a io o agg ega e size o pa icle size k, show-
ing ha he agg ega ion o pa icles con ibu es o ise pa -
icle p essu e and hus has a s abilizing ac ion.
VII. CONCLUSIONS
By ex ending he ange o pa icle size downwa ds we
ha e illus a ed he connec ion be ween he wo mechanisms
ha ha e been p oposed in he li e a u e as esponsible o
supp essing bubbling in luidized beds, on one hand hyd o-
dynamic o ces and on he o he in e pa icle o ces. Al-
hough bo h mechanisms seemed o be con adic o y he
p esen wo k shows ha hey a e a he complemen a y. The
ex en o he luidlike egime and hus he ela i e ole o
hyd odynamic s abiliza ion sho ens as pa icle size is in-
c eased, indica ing ha la ge bubbles a e jus es ained by
yield s esses o su icien ly la ge pa icles. Ne e heless he
inhe en ins abili y o he luidlike egime con i ms ha
bubble mac os uc u es canno be he esul o a linea de-
pa u e om a homogeneous s a e bu a consequence o a
complex nonlinea hyd odynamic in e ac ion in which in e -
pa icle o ces play a ela i e ole by inducing pa icle agg e-
ga ion. Applica ion o he Wallis c i e ion o p edic he onse
o bubbling sugges s ha he pa icle p essu e in he pseudo -
u bulen luidlike egime scales as p⬃
␳
pgdpk3/2
␾
2. This
law is in con adic ion wi h he sublinea dependence on
␾
heo e ically p edic ed o a homogeneous s a e 关7兴bu
ag ees wi h di ec measu emen s made on a liquid- luidized
bed 关36兴. P esumably mesoscale s uc u es a ec he low
cha ac e is ics p o oundly 共like is he case in ise lows
关38兴兲, and his would p o ide a s abilizing ac ion agains he
g ow h o mac oscopic bubbles. To ou knowledge we a e
he i s wo ke s able o succeed in luidizing pa icles small
enough o see an ex ended in e al o uni o m expansion
wi h hyd odynamic s abiliza ion. The complexi ies o hyd o-
dynamic in e ac ions o he pseudo u bulen wo-phase low
make a heo e ical model capable o p edic ing he scaling o
he solid phase p essu e a a he complica ed challenge ha
emains o be add essed.
ACKNOWLEDGMENTS
We a e glad o acknowledge R. Jackson o sugges ing o
us he epo ed expe imen s. This esea ch has been sup-
po ed by he Xe ox Founda ion, Spanish Go e nmen
Agency Minis e io de Ciencia y Tecnologia 共DGES兲unde
Con ac No. BMF2000-1056, and NATO G an No. LINK-
AGE PST.CLG.976575.
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