VOLUME 82, NUMBER 6 PHYSICAL REVIEW LETTERS 8FEBRUARY 1999
Flow Regimes in Fine Cohesi e Powde s
An onio Cas ellanos,1,* José Manuel Val e de,1Albe o T. Pé ez,1An onio Ramos,1and P. Kei h Wa son2
1Depa amen o de Elec ónica y Elec omagne ismo, Uni e sidad de Se illa, 41012 Se ille, Spain
2Xe ox Co po a ion, Wilson Resea ch Cen e , Webs e , New Yo k
(Recei ed 20 No embe 1997; e ised manusc ip ecei ed 2 June 1998)
G anula ma e ials exhibi se e al egimes o beha io : plas ic, ine ial, luidized, and en ained low,
bu no all ma e ials can pass h ough all o hese s a es. Ou conce n is wi h he c i e ia ha de e mine
he ansi ion om one egime o ano he and wi h he bounda ies o he a ious low egimes ha hese
c i e ia de ine. Expe imen ally we ha e ocused on ine, cohesi e powde s, whe e he in e pa icle
cohesi e o ce domina es o e g a i a ional o ce and whe e en ained ai can cause mo ing powde o
become luidized. [S0031-9007(98)08339-2]
PACS numbe s: 45.70.Mg, 81.20.E , 83.70.Fn
The pas decade has wi nessed a s ong in e es in
g anula ma e ials by he physics communi y, bu he e
is an impo an class o g anula ma e ials ha has been
la gely igno ed in spi e o i s comme cial impo ance;
hese ma e ials can be classi ied as ine cohesi e powde s.
In hese powde s, wi h pa icle diame e s less han abou
30 31026m, in e pa icle cohesi e e ec s a e dominan ,
and ambien gas plays an impo an ole in he beha io o
he powde . G anula ma e ials display ou di e en low
egimes: plas ic beha io , ine ial low, luidized low, and
en ained low. Pa icle size, pa icle densi y, cohesi i y,
and gas low de e mine which o hese ypes o beha io
occu .
(i) The plas ic egime is cha ac e ized by a small spacing
be ween neighbo ing pa icles. Veloci ies a e small o ze o
and he s esses a e independen o eloci y o simple
geome ies. Plas ic beha io de e mines he s abili y o
heaps and slopes and he e is an ex ensi e li e a u e on he
subjec because o i s impo ance in ci il enginee ing.
(ii) In he ine ial egime he s esses a e due o he
anspo o momen um by in e pa icle collisions. The
spacing be ween pa icles is much smalle han he pa icle
size bu g ea e han in he plas ic egime. In e e yday
li e g anula ma e ials such as sand, suga , and g ound
co ee exhibi he ansi ion om plas ic solid o ine ial
low when he limi o plas ic s abili y is eached. We no e
ha he in e s i ial luid plays no pa in ine ial low.
(iii) Powde s a e capable o being luidized by gas
low p o ided hei cohesi i y is no oo g ea . In his
egime he in e pa icle dis ance is o he same o de o
magni ude as he pa icle size. The in e s i ial luid is
he agen o ans e o momen um be ween pa icles, and
luid eloci y de e mines he s esses in he ma e ial. The
bes known example o his si ua ion is he luidized bed,
in which gas is o ced h ough a bed o pa icles and he
gas low causes a p essu e d op ac oss he powde . When
he p essu e d op is su icien o suppo he weigh o he
powde and o o e come he in e pa icle cohesi e o ces,
he bed expands and becomes luidized. The powde hen
akes on many o he p ope ies o liquid, i s uppe su ace
emaining ho izon al when he con aine is il ed.
(i ) A ou h egime is ha o en ainmen , o suspen-
sion o he pa icles by he gas. In his case he dis ance
be ween pa icles is much g ea e han he size o he pa -
icles, he mean eloci y o he ma e ial is close o he
luid eloci y, and he in e ac ion be ween pa icles is neg-
ligible. This is he case in a dus s o m o a sand s o m.
Fo each egime o g anula beha io he e is a domi-
nan mechanism ha de e mines he o de o magni ude
o he s esses in he bulk, and he ansi ions be ween he
a ious low egimes can hen be ob ained by compa ing
he magni ude o hese s esses. Fo he plas ic egime,
in he absence o ex e nal s esses, he dominan s ess
is gh ( is he bulk densi y o he powde , gis
he accele a ion due o g a i y, and his he e ical
leng h scale o he sample). In ine cohesi e powde s
he in e pa icle cohesion may be much g ea e han he
pa icle weigh , and i is cohesion ha de e mines he
o de o magni ude o s esses; ha is why cohesi e
ma e ials some imes exhibi slopes s eepe han 90±
. Two
mechanical p ope ies o cohesi e powde s ha can be
measu ed a e (i) he shea s ess ha will cause he
powde o ac u e in he absence o comp essi e s ess
and (ii) he ensile s eng h, i.e., he no mal s ess ha
will ac u e he powde in he absence o shea . Fo
mos ine powde s bo h quan i ies a e o he same o de
o magni ude.
Fo low consolida ion s a es [1,2], he ensile s eng h
s inc eases linea ly wi h he consolida ion s ess sc,so
ha s asc1s
0and adec eases as he ma e ial
ha dness inc eases. Fo uncha ged, d y, ine pa icles
he ensile s eng h a ze o consolida ion s 0scales as
s 0bdad22
p, whe e dais he size o he aspe i ies, dp
is he pa icle diame e , and bdepends on he ma e ial
p ope ies and bed oidage [2–4]. This exp ession is
consis en wi h he obse a ion ha he cohesion inc eases
as dainc eases and dpdec eases. This in u n explains
he echnique o educing in e pa icle cohesi i y by he
addi ion o low con ol addi i es [5] such as Ae osil
(Degussa, Ge many) o Cab-o-Sil (Cabo Co p., Wal ham,
MA). The addi i es consis o submic on agg ega es
o umed silica nanopa icles, which a e dispe sed on
1156 0031-9007y99y82(6)y1156(4)$15.00 © 1999 The Ame ican Physical Socie y
VOLUME 82, NUMBER 6 PHYSICAL REVIEW LETTERS 8FEBRUARY 1999
he powde pa icle su aces. The addi ion o hese
nanopa icles esul s in a educ ion in abecause 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
size o he con ac s, hus educing s 0.
A use ul echnique o es ima ing he cohesi i y is o
luidize he powde . In o de o luidize a cohesi e powde
he gas low has o o e come no only he weigh o
he powde , bu also i s ensile s eng h. The p essu e
d op ac oss he powde a he poin o luidiza ion is
hen gi en by DP gh 1s
. This p essu e d op is
also gi en by Ca man’s equa ion [6] DPyhEhUs12
ed2e23d22
p, whe e his he gas iscosi y, Uis he gas
eloci y, eis he oid ac ion o he bed, and E,180.
Equa ing hese ela ions we ob ain he gas eloci y needed
o luidiza ion. Fo noncohesi e powde s ss .0d he
minimum eloci y o luidiza ion is p opo ional o he
squa e o he pa icle diame e . When pa icles a e e y
ine he in e pa icle cohesi e o ces become dominan
and he minimum eloci y o luidiza ion becomes less
dependen on he pa icle diame e . Re e ing now o
Fig. 1, line Ain he diag am ep esen s he minimum
eloci y o luidiza ion as a unc ion o pa icle diame e
(ob iously he exac loca ion o his line depends on he
cohesi i y o he pa icles). The ho izon al con inua ion
o Aon he le side o he igu e ep esen s he case o
e y cohesi e powde s, and he loca ion o his bounda y
depends on he ha dness o he pa icles and on he size o
he aspe i ies. The oid ac ion echanges wi h he a io
o pa icle weigh o in e pa icle cohesi e o ce. Pa icles
abo e 331025m in diame e pack nea he andom close
packing limi se.0.45dand he e o e he slope o line A
does no depend on he diame e o hese la ge pa icles.
Below his alue einc eases o a maximum o abou 0.8
(nea he andom ballis ic agg ega ion limi ). This will
change sligh ly he shape o line A o e y ine pa icles.
The uppe limi o he luidized zone, deno ed by line B,
is gi en by S okes d ag U pgd2
py18h, whe e p
is he pa icle densi y (gas densi y has been neglec ed).
This alue o gas eloci y, U, is he minimum equi ed
o le i a e a pa icle, and excep o e y ine powde s
sdp#1026md his eloci y is g ea e han he minimum
eloci y o luidiza ion. Thus line B ep esen s he mini-
mum eloci y o pa icle en ainmen o suspension. As
may be seen, he e is a c i ical alue o pa icle diame e
o which he suspension eloci y equals minimum lu-
idiza ion eloci y. The e o e, i he powde consis s o
pa icles o diame e less han his c i ical size, luidiza-
ion by gas low is impossible as he pa icles will become
en ained by he low, a he han luidized.
Conside ing now he ine ial egime, acco ding o
Bagnold [7] he s esses in g anula low a e p opo ional
o he squa e o he eloci y g adien , , pd2
psUydd2,
whe e dis he ex en o he shea laye and Uis he
eloci y dec emen o he ma e ial ac oss he shea laye
(we assume ha in he ine ial case he gas is swep
FIG. 1. Phase diag am de e mining he ansi ion be-
ween low egimes as a unc ion o pa icle diame e .
p1000 kgm23,h231025kgm21s21,E180,
g10 ms22,d8d
p,P1Pa (we assume a ee mo ing
slab, hickness ,1mm, o one ), h10 mm, and e0.5.
Values o a0.26,b0.1, and da131027m ha e
been aken om expe imen al measu emen s on a xe og aphic
one [1].
along by he as -mo ing pa icles so ha U ep esen s
bo h gas and pa icle eloci ies). Acco ding o Sa age
and Hu e [8] he ansi ion be ween plas ic and ine ial
egimes is gi en by pd2
pU2ysPd2d$0.1, whe e Pis
he o al no mal s ess; he shea laye hickness dscales
wi h dpso his ansi ion is ep esen ed by a ho izon al
line such as Cin Fig. 1. We see om his pa o
he low egime diag am ha o pa icles abo e 1024m
diame e , as low eloci y is inc eased, he ansi ion om
plas ic o ine ial beha io is ollowed a highe lows by
a ansi ion o he luidized egime. In con as o his,
pa icles be ween 1025and 1024m do no exhibi ine ial
beha io bu unde go a di ec ansi ion o luidiza ion.
Below 1025m luidiza ion becomes inc easingly sensi i e
o in e pa icle cohesion; i also becomes e y sensi i e o
ini ializa ion condi ions, i.e., o he uni o mi y and deg ee
o consolida ion o he powde (see below). Fo hese
easons he low egime diag am has o be unde s ood in a
semiquan i a i e way, bu he gene al ends o he lines
indica e he app oxima e ela ionships be ween pa icle
size and eloci y o he a ious egimes and ansi ions.
We ha e used he low diag am o analyze wo ypes
o expe imen s: (i) he ac u e and low o powde in
a il ed bed and (ii) he low o powde in a o a ing
d um. In ou il ed bed s udies [9], we ind ha a ypical,
ine cohesi e powde con ains egions o widely di e ing
deg ees o consolida ion, and his implies ha he yield
s ess a ies om poin o poin wi hin he sample. I ,
howe e , he powde is luidized, he dispe sed ma e ial
is qui e uni o m, and when he luidized gas is u ned
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VOLUME 82, NUMBER 6 PHYSICAL REVIEW LETTERS 8FEBRUARY 1999
o , and as he bed collapses unde i s own weigh , i
condenses in o a ep oducibly consolida ed s a e. We
ind ha his echnique, ep esen ed by line ain Fig. 1
p o ides a eliable, con enien me hod o ini ializing
cohesi e powde s.
Ha ing ini ialized he powde , we hen slowly il he
bed and obse e he sample su ace while main aining
he condi ion o ze o gas low. As he angle o il
inc eases, a shea s ess is gene a ed in he powde laye ,
and he e comes a poin a which he sample ails in
shea and a laye o powde slumps o one side o
he bed. Expe imen s o his ype ha e been ca ied
ou using a comme cially a ailable xe og aphic one
(Canon CLC 500 one ) and a se ies o Xe ox one s
o di e ing cohesi i y [9]. Analyzing he b eaking and
sliding p ocess by ideo came a and eco de indica es
ha i he eloci y o he laye exceeds a ce ain c i ical
alue he mo ing powde becomes luidized; his c i ical
eloci y depends on he powde cohesi i y and is o he
o de 0.2–0.4 mys. When he luidized powde comes o
es i se les wi h i s op su ace ho izon al, and hen as
he en ained ai escapes he powde e u ns o he s a ic,
plas ic s a e. This p og ession om plas ic o luidized
s a e and back again is simila o he ini ializa ion p ocess
and is ep esen ed by line ain Fig. 1. This beha io
con as s wi h ha o a coa se g anula ma e ial such as
sand, which b eaks in a succession o ine ial, su ace
a alanches when il ed beyond i s angle o epose. This
p og ession om plas ic o ine ial s a e, and back again,
is ep esen ed by a line such as line bin Fig. 1.
Ou o a ing d um expe imen s also illus a e he use
o he low egime diag am and make clea he ole o
pa icle-gas in e ac ion in he beha io o ine powde s.
The equipmen consis s o a cylind ical Plexiglas d um,
designed o o a e a ound i s axis, which is ho izon al.
The dimensions a e 21 mm in e nal adius and 49 mm
leng h. One o he ends o he cylinde has a po ous
il e and a connec ion ha allows he ex ac ion o
FIG. 2. (a) P o ile o sand in a o a ing d um. (b) P o ile
o one RT (0.4% Ae osil) in he o a ing d um a he
same o a ion eloci y. Double headed a ow indica es small
oscilla ions o he one ho izon al su ace.
ai om he d um ye e ains he pa icles. The d um
is d i en by a mo o ha allows a maximum angula
eloci y o 225 pm. A ideo came a is used o eco d
he mo ion, and he came a is connec ed o a compu e
o image p ocessing. Th ee ypes o g anula ma e ials
ha e been es ed in he o a ing d um: d y sand, 1.8
o 3.5 31024m diame e ; xe og aphic one , pa icle
diame e a ound 131025m; and polyme beads, 3 o
531024m diame e , o he same esin as he one .
When he d um is pa ly illed wi h d y sand and is o a ed
a low eloci ies s,4 pmd he sand su ace is plana and
makes a cons an angle wi h he ho izon al. A hin laye
o ma e ial, a ew g ains in hickness, cascades o e he
su ace con inuously ollowing a phase ajec o y such as
line bin Fig. 1. The slope o he su ace inc eases wi h
he angula eloci y and a 20 pm, he su ace akes he
o m o il ed “S.” The slope is minimum a he lowe pa
(25±
) and maximum a he uppe pa (64±
) o he p o ile
[see Fig. 2(a)]. This ype o beha io is widely epo ed
[10–13] o coa se, g anula ma e ials sdp.1024md.
Wha we obse e is ha he maximum angle o he slope
is a con inuously inc easing unc ion o he eloci y, as
shown in Fig. 3. The same measu emen s ha e been
epea ed wi h he chambe e acua ed o 1024a m, and he
esul s a e he same as a a mosphe ic p essu e. The same
ype o ine ial beha io is also ound when he s y ene
bu adiene beads a e es ed in he o a ing d um.
We ha e ca ied ou he same ype o expe imen on
ine, cohesi e powde s, and as one would p edic om ou
discussion o Fig. 1, hei beha io is qui e di e en om
ha o sand. The powde s we ha e s udied a e xe og aphic
one s: Canon CLC 500 one , pa icle diame e 8.5 3
1026m, and wo model ma e ials consis ing o pigmen ed
polys y ene bu adiene, pa icle diame e 12.7 31026m,
whose cohesi i y has been educed by adding 0.2% and
0.4%, espec i ely, by weigh o he low con ol addi i e
Ae osil. Bo h o hese one s luidize a a supe icial gas
eloci y o a ew mm pe sec.
FIG. 3. Maximum angle o he slope o sand and beads (same
esin as Xe ox one s) and a e age angle o Canon CLC 500
and model Xe ox one s (wi h 0.4% silica and 0.2% silica) a
ac u e as a unc ion o o a ion a e in a o a ing d um a
a mosphe ic p essu e.
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VOLUME 82, NUMBER 6 PHYSICAL REVIEW LETTERS 8FEBRUARY 1999
FIG. 4. Maximum angle o he slope o model Xe ox one s
(wi h 0.4% and 0.2% silica) and o Canon CLC 500 as a
unc ion o o a ion a e in he o a ing d um a ai p essu e
1024a m.
A s iking ea u e o he one beha io in he o a ing
d um is ha he mean angle o he slope a ac u e is a
dec easing unc ion o he angula eloci y (see Fig. 3).
The p o iles a low speed a e simila o hose o sand, bu
as speed inc eases a egion appea s ha is ho izon al in
he lowe pa o he p o ile [see Fig. 2(b)]. The ex en
o his ho izon al egion inc eases wi h eloci y and, a
he same ime, he powde expands, inc easing i s olume
conside ably o eloci ies abo e 50 pm. The eason
o his beha io is ha he ai becomes en ained in
he powde wi h each e olu ion and pa ially luidizes
he powde ; he amoun o en ained ai inc eases wi h
o a ional eloci y, and so he luidized egion inc eases
wi h eloci y. As we would an icipa e om Fig. 1,
he one does no pass h ough he ine ial egime bu
passes di ec ly om he plas ic o he luidized condi ion,
ollowing line ain Fig. 1. This luidized egion beha es
like a liquid and p esen s a negligible ic ion o he wall;
he e o e, he a e age angle o he slope dec eases as he
luidized egion inc eases wi h he speed o o a ion.
To make clea he ole o he ai -pa icle o ces in
he one low p ocess we ha e un he o a ing d um
while e acua ing he chambe wi h a acuum pump. The
expe imen was conduc ed a 1024a m. Unde hese
condi ions he one canno be luidized by he esidual ai ,
and as a consequence he e is a ansi ion om he plas ic
egime o a sys em o a alanches, in a simila way o ha
o sand. Mo eo e , as shown in Fig. 4, he maximum
angle o he slope is simila o ha o sand. Clea ly,
luidized low equi es an ambien gas, and a low gas
p essu e he luidiza ion p ocess is supp essed. Rie ema
[3,14] also pe o med g anula low expe imen s in a
o a ing d um while a ying gas p essu e and obse ed
ha he angle o he slope dec eases as he p essu e
inc eased. This is a clea indica ion ha he e is a
p og essi e luidiza ion o he powde as he p essu e
inc eases due o he inc ease o he gas e ec i e iscosi y.
He asc ibed he dec ease in slope o he en apped gas in
he powde .
In conclusion, we no e ha he low egime diag am
we ha e p esen ed p o ides a use ul way o in e p e ing
he low p ope ies o bo h ine, cohesi e powde s and
coa se g anula ma e ials. In gene al he mo ion o
coa se g anula ma e ial is cha ac e ized by ansi ion
om plas ic o ine ial low, whe eas ine pa icle mo ion
a a mosphe ic p essu e is cha ac e ized by he ansi ion
om plas ic o luidized low. Fluidized low, howe e ,
equi es an ambien gas and a low gas p essu e he
luidiza ion p ocess is supp essed.
We a e indeb ed o Mike Mo gan and F ank Gen-
o ese o hei help and o Paul Julien o p o iding us
he one s used in his s udy. This esea ch has been
suppo ed by he Xe ox Founda ion, Xe ox Co po a ion,
and by he Spanish Go e nmen Agency Di ección Gen-
e al de Ciencia y Tecnologı
´a (DGES) unde Con ac
No. PB96-1375.
*Co esponding au ho .
Email add ess: [email p o ec ed]
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Wa son, in Powde s & G ains 97, edi ed by R. Beh inge
and J.T. Jenkins (Balkema, Ro e dam, 1997), pp. 163–
166.
[2] J.M. Val e de, A. Ramos, A. Cas ellanos, and P.K.
Wa son, Powde Technol. 97, 237–245 (1998).
[3] K. Rie ema, The Dynamics o Fine Powde s (Else ie ,
London, New Yo k, 1991), pp. 233–250.
[4] P.K. Wa son, H. Mizes, A. Cas ellanos, and A.T. Pé ez,
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[9] J.M. Val e de, A. Ramos, A. Cas ellanos, and P.K.
Wa son, in Powde s & G ains 97 (Re . [1]), pp. 135–138.
[10] J. Rajchenbach, Phys. Re . Le . 65, 2221 (1990).
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1879 (1995).
[12] G.H. Ris ow, Eu ophys. Le . 34, 263–268 (1996).
[13] A.A. Boa eng and P.V. Ba , J. Fluid Mech. 330, 233–
249 (1997).
[14] K. Rie ema and W. Co a , Powde Technol. 50, 147–154
(1987).
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