J. Appl. Phys. 119, 223302 (2016); h ps://doi.o g/10.1063/1.4953649 119, 223302
© 2016 Au ho (s).
Elec ic cha ge limi s on se led powde s
Ci e as: J. Appl. Phys. 119, 223302 (2016); h ps://doi.o g/10.1063/1.4953649
Submi ed: 17 Ma ch 2016 . Accep ed: 28 May 2016 . Published Online: 13 June 2016
J. Pé ez-Vaque o, M. A. S. Quin anilla , and A. Cas ellanos
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Elec ic cha ge limi s on se led powde s
J. P
e ez-Vaque o, M. A. S. Quin anilla,
a)
and A. Cas ellanos
Depa men o Elec onics and Elec omagne ism, Facul y o Physics, Uni e sidad de Se illa, Se illa, Spain
(Recei ed 17 Ma ch 2016; accep ed 28 May 2016; published online 13 June 2016)
In lows o d y pa icula e sys ems, elec ic cha ge is gene a ed on pa icle su aces by hei
collision wi h walls and wi h o he pa icles. Cha ge build-up on single pa icles can yield local
cha ge alues high enough o su pass he limi ing elec ic ield o co ona discha ge in o he su -
ounding gas. Then, local cha ge is dec eased o a lowe alue ha becomes s abilized when lows
s op and pa icles deposi in a con aine . In his pape , we ha e used a Fa aday pail sys em o mea-
su e he esidual pa icle cha ge a e using wo di e en de ices— ibocha ge s— o pa icle
cha ging. One o he ibocha ge s allowed us o di ec ly measu e he o al cha ge ha was ans-
e ed om he walls o he pa icles, and his was compa ed o he inal alues in he bulk powde
once i was collec ed in he Fa aday pail. The esul s show ha he elec ic cha ge o pa icles dis-
pe sed in gas is limi ed by co ona discha ge and depends mainly on he pa icle size. In addi ion,
we p esen a simple model o he discha ge o he collec ed powde based on elec os a ic conside -
a ions. I he powde e ec i e conduc i i y and he elec ic cha ge o he se ling pa icles a e
known, he model p edic s he empo al e olu ion o he o al cha ge o he collec ed powde and
he spa ial dis ibu ion o he elec ic cha ge and elec ic ield. Published by AIP Publishing.
[h p://dx.doi.o g/10.1063/1.4953649]
I. INTRODUCTION
D y powde handling ope a ions a e usually accompa-
nied by elec ic cha ge buil -up due o iboelec i ica ion,
some imes also called con ac cha ging. Pa icle con ac
cha ging can be bene icial in some cases, such as elec o-
s a ic p ecipi a ion and xe og aphic p in ing, o de imen al,
o example, in pneuma ic con eying. The physical p oc-
esses ha de e mine he magni ude o he cha ge ans e ed
be ween wo su aces ha e a ac ed in e es o a long ime,
and while he physics o iboelec i ica ion be ween he con-
duc i e su aces is well unde s ood, he e is no well es ab-
lished heo y ha explains iboelec i ica ion be ween non-
conduc i e su aces. While o conduc i e ma e ials he
cha ge ans e ed be ween su aces in con ac esul s om
elec on ans e d i en by he di e ences in Fe mi le els;
1,2
when a leas one o he su aces in con ac is no conduc i e,
he esul ing cha ge ans e has been explained using an
e ec i e wo k unc ion o he non-conduc i e solid,
3,4
as a
consequence o he exchange o mobile ions
5
o he exis-
ence o dono s and accep o cen e s ep esen ing localized
elec on si es on he su aces o he solids in con ac .
6,7
In
any case, a dis inc ion has o be made be ween he cha ge
ans e ed be ween wo su aces when hey a e in con ac
and he cha ge emaining in he su aces a e sepa a ion.
When wo su aces cha ged wi h di e en pola i ies a e sep-
a a ed, he elec ic ield be ween bo h su aces inc eases wi h
inc easing sepa a ion, up o he poin when some cha ge can
be back ans e ed o i s o iginal su ace.
8
The p ac ical con-
sequence o his ac o powde handling is ha he maxi-
mum cha ge han an isola ed pa icle can hold mus be
limi ed by he elec ical b eakdown ield in i s su ounding
gas, a he han by he physical mechanisms ha caused he
cha ge ans e be ween he pa icle and he solid su ace
om whe e he cha ge was picked by he pa icle while in
con ac . Indeed, i is known ha he cha ge ans e ed
be ween non-conduc i e polyme s in acuum is much la ge
han in he a mosphe e.
9
Howe e , in mos powde handling
p ocesses, pa icles canno be conside ed as isola ed and he
o al cha ge p esen in a powde is also limi ed by spa ial
cha ge e ec s. In pneuma ic anspo , spa ial cha ge e ec s
a e no ele an unless he olume ac ion o dispe sed pa -
icles is g ea e han a ce ain limi ha depends on pa icle
size and pipe diame e ,
10
bu when he pa icles se le o
o m a packed powde , he concen a ion o cha ge can c e-
a e elec ic ields la ge enough o igge elec ical dis-
cha ges. The ype o discha ge ha occu s depends on he
dep h o he packed powde . Fo dep hs o he o de o a me-
e , he o al cha ge accumula ed can igge a ype o elec i-
cal discha ges known as cone discha ges
11
in which he
elec ical b eakdown o he su ounding ai occu s h ough
he o ma ion o a conduc i e channel ha ca ies he cha ge
om he powde o i s su oundings: ypically, he walls o
he silo holding he powde .
12
Howe e , in he ini ial s ages
o he o ma ion o heap o powde , he e is no enough
cha ge accumula ed o o m a conduc i e channel and i is
expec ed ha a co ona discha ge om he su ace o he
powde o he su ounding ai akes place, as i happens in
some si ua ions in elec os a ic powde coa ing.
13
The cha ge
emaining in he powde would hen be limi ed o he amoun
necessa y o c ea e he elec ic ield o main ain he co ona
discha ge. In his wo k, we ha e measu ed bo h he cha ge
acqui ed by he powde du ing pneuma ic anspo and he
elec ical cha ge emaining in he powde once collec ed.
a)
Au ho o whom co espondence should be add essed. Elec onic mail:
[email p o ec ed]
0021-8979/2016/119(22)/223302/9/$30.00 Published by AIP Publishing.119, 223302-1
JOURNAL OF APPLIED PHYSICS 119, 223302 (2016)
II. EXPERIMENTAL SETUP
In ou expe imen s, powde s a e i s dispe sed in a gas
s eam and acqui e elec ical cha ge h ough collisions wi h
a ibocha ge . Once hey come om a ibocha ge , he
powde is collec ed in a Fa aday pail whe e he mass and
cha ge o he collec ed powde a e measu ed as a unc ion
o ime. A pho o o he se up is shown in Fig. 1.Thedispe -
sion p ocedu e depends on whe he he powde is luidiz-
able o no ( he lis o powde es ed and hei p ope ies
a e gi en a he end o his sec ion). Fo ee- lowing ma e-
ials, a small silo wi h an adjus able hole on i s bo om is
used o g a i y unloading o he powde in o he suc ion
po o a en u i. Fo non ee- lowing ma e ials, he silo is
eplaced by a closed cell wi h a po ous pla e subjec ed o
mechanical ib a ion and blown wi h a cons an gas low.
The pa icles which a e elu ia ed om he sample go in o
he suc ion po o he en u i. In bo h cases, he mass low
a e is kep as cons an as possible. The ypical alue o gas
low a es exi ing he en u i yields inpu gas eloci ies in
he ibocha ge s ha ange om 10 o 20 m/s o all he
measu emen s we made.
We ha e made expe imen al uns wi h wo di e en
ibocha ge s: a nylon cyclone (350 mm in leng h, 74 mm
in e nal diame e ) and a s eel ube (leng h 540 mm and in-
e nal diame e 21 mm). In bo h cases, he inle po o he
ibocha ge is angen o i s walls o imp o e he collision
a e. Cyclone ibocha ge s ha e been used by o he
au ho s
14
due o hei abili y o sepa a e he pa icles om
he gas s eam, al hough pa icle-gas sepa a ion is inc eas-
ingly ine icien o pa icles wi h sizes close o o below
10 lm. The s eel ube ibocha ge does no sepa a e he
pa icles om he gas bu , being made o a conduc i e
ma e ial, i allows us o measu e he elec ic cha ge Q
d
ans e ed o he pa icles while hey a e dispe sed in o he
gas s eam. To his aim, a p og ammable elec ome e
(Kei hley 6512) wo king in he amme e mode connec s
g ound o he ibocha ge and measu es he elec ic cu en
o he s eel pipe. Q
d
is ob ained by nume ical in eg a ion o
he egis e ed cu en . In o de o educe he noise picked-
up by capaci y coupling, an elec ically g ounded me allic
meshco e s hepipe.Themeshand hepipea einsula ed
o m each o he and g ounded by sepa a e connec ions. The
use o one ibocha ge made o conduc i e ma e ial and
ano he ibocha ge o non-conduc i e ma e ial allows us
o p obe i he mic oscopic mechanism o he cha ge ans-
e be ween he pa icles and he walls o he ibocha ge
has any e ec on he cha ge acqui ed by pa icles dispe sed
in he gas s eam.
The powde ha exi s he ibocha ge se les by g a i y
in o a cylind ical cell whose walls a e made o insula ing
ma e ial (a me hac yla e ube), and i is closed a i s bo om
wi h a me allic il e o help sepa a e he pa icles om he
gas. The il e inside cell is elec ically insula ed o he ou -
side o he cell. The cell is loca ed inside he Fa aday pail
consis ing o an inne and an ou e cage made o conduc i e
ma e ial, he o me being connec ed o g ound h ough a
picoamme e (Kei hley 6485). The picoamme e measu es
he cu en lowing om g ound o he inne cage o he
Fa aday pail as he collec ing cell is illed by he ma e ial
exi ing he ibocha ge . In eg a ion o his cu en yields he
cha ge Q
s
( ) in he se led powde as a unc ion o ime. The
ou e cage is g ounded o ac as a shield o ex e nal in e e -
ences, and in some uns, i was supplemen ed by a g ounded
me allic g id a ound all he Fa aday pail ac ing as a second
shield. No e ha we make a dis inc ion be ween he elec ic
cha ge Q
d
acqui ed by he powde when dispe sed and he
elec ical cha ge Q
s
emaining in he pa icles when se led,
because he cha ges Q
d
and Q
s
a e no necessa ily he same.
The Fa aday pail sys em es s on a mass balance (Me le
Toledo AB204-S o P ecisa XT se ies, depending on he
mass in ol ed) ha measu es he collec ed powde mass
m( ) as a unc ion o ime.
Bo h mass and elec ic in ensi y da a a e egis e ed in a
PC (pe sonal compu e ) a a a e o 4–10 samples pe second.
Unce ain ies in he alues o he cha ge a e calcula ed om
he oo mean squa e ampli ude o he noise ol age in he ana-
log ou pu channels o he picoamme e and he elec ome e .
FIG. 1. The expe imen al se up o measu ing iboelec i ica ion le els,
wi h he s eel ube ibocha ge in place and he nylon cyclone ibocha ge
in he inse , shown wi hou elec ic shielding. The Fa aday pail es ing on
he balance collec s he powde coming om he ibocha ge . The elec ic
cu en lowing in o he Fa aday pail is measu ed by he picoamme e and
eco ded by he PC. When he s eel pipe is used as ibocha ge , he elec-
ome e depic ed in he igu e is connec ed o i and i s da a a e also egis-
e ed by he PC. The me allic mesh a ound all he se up is g ounded o
educe elec ic noise om he es o he lab. The dispe sion uni s do no
appea in he igu e.
223302-2 P
e ez-Vaque o, Quin anilla, and Cas ellanos J. Appl. Phys. 119, 223302 (2016)
III. MATERIALS
Di e en ma e ials ha e been used o co e a ange o
pa icle sizes as la ge as possible: poly-(me hyl me hac yla e)
(PMMA) beads (Polysciences, Inc.), 5–50 lm glass mic o-
sphe es (Duke Scien i ic Co p.), 70–100 lm and 90–150 lm
glass beads (Sigmund-Lindne GmbH), comme cial suga ,
comme cial semoline, and co ns a ch (Maizena
V
R
). Pa icle
size analysis o e e y ma e ial bu PMMA beads (whose a e -
age size was aken om he manu ac u e da a shee ) was pe -
o med wi h a lase di ac ion analyze (Mas e size Sci occo
2000), by d y dispe sion in ai (1 ba p essu e) acco ding o
ISO 13320. The mean su ace (Sau e mean) diame e o hese
ma e ials is lis ed in Table I. The condi ions and he esul s
o he es s on each ma e ial a e lis ed in Table I.Insome
cases, he sample es ed was s o ed a con olled ambien con-
di ions a wo di e en ela i e humidi ies (30% and 60%
RH) be o e each es o in es iga e he e ec o s o age
humidi y on he elec ical cha ge picked by he powde .
IV. RESULTS
A. Cha ge pe pa icle in suspension
The cha ge acqui ed by he pa icles in he ibocha ge
while hey a e dispe sed in he gas s eam q
d
can only be
e alua ed o he s eel ube ibocha ge , o which he o al
cha ge Q
d
gi en o he powde can be measu ed. An example
o such measu emen is shown in Fig. 2, whe e Q
d
is p e-
sen ed as a unc ion o he mass mcollec ed in he Fa aday
pail. In all expe imen s pe o med, he plo o Q
d
agains m
has a linea dependence wi h he mass m, as shown in Fig. 2.
This ac indica es ha , assuming he sample is monodis-
pe se, he pa icles acqui e a cons an cha ge q
d
¼Q
d
m
p
/m
om he s eel ube o he du a ion o he expe imen , whe e
m
p
is he pa icle mass. Fig. 3shows he alue o q
d
as a
unc ion o he pa icle adius
p
(hal he su ace mean diam-
e e lis ed in Table I) o he expe imen s using he s eel i-
bocha ge . Since he e is always some powde ha emains
s uck o he inne walls o he ibocha ge , using he col-
lec ed mass msomewha o e es ima es he alue o q
d
. The
ypical mass loss anges om 19% o 69% o he o al mass
o dispe sed powde o 5–50 lm glass beads and 1%–20%
in 90–150 lm. In gene al, he mass loss dec eases wi h la ge
pa icles and highe s o age humidi y.
Such o e es ima ion in cha ged mass has impac on wo
p ocess a iables, q
d
and olume ac ion o powde , w,as
TABLE I. Ma e ials used in he expe imen s p esen ed in his pape . Thei
mean pa icle size (su ace-mean diame e ) was measu ed in a Mas esize
2000 using a d y dispe sion module (ai , dispe sion p essu e 1 ba ).
Ma e ial Su ace mean diame e (lm)
Co ns a ch 7.3
5–50 glass beads 33.9
70–110 glass beads 92.0
90–150 glass beads 125.4
PMMA beads 200
Suga 720
Semoline 796
FIG. 2. Elec ic cha ge Q
d
ceded by he s eel ibocha ge o PMMA beads
(s o age humidi y no con olled) as a unc ion o he collec ed mass min
he Fa aday pail. The linea dependence o Q
d
wi h mindica es ha he pa -
icles a e cha ged uni o mly when he pass h ough he ibocha ged.
FIG. 3. Resul s o absolu e alues o ans e ed cha ge q
d
om he s eel
pipe ibocha ge o pa icles o di e en ma e ials. Plo ed lines a e he he-
o e ical alues o maximum cha ge o pa icles limi ed by plana co ona
discha ge (do ed line) and size dependen co ona discha ge (solid line), Eq.
(2). The dispe sing gas and s o age condi ions o he ma e ial a each expe i-
men al poin a e gi en acco ding o he ollowing code: 1: Dispe sed in N
2
,
s o ed a 10% RH; 2: Dispe sed in N
2
, s o ed a 60% RH; 3: Dispe sed in
ai , s o ed a 60% RH; 4: Dispe sed in ai , s o ed a 30% RH; and 5:
Dispe sed in ai , no humidi y con ol du ing s o age. Fo expe imen s in
which he collec ed mass m
d
was a ailable ( illed symbols), he pa icle
cha ge q
d
was calcula ed using he dispe sed mass m
d
a he han he col-
lec ed mass m. E o ba s indica e he ypical unce ain y o each size ange.
Fo suga pa icles, e o ba and symbol a e app oxima ely he same size.
Subplo s in linea scale zoom in o a eas o in e es o he main plo .
223302-3 P
e ez-Vaque o, Quin anilla, and Cas ellanos J. Appl. Phys. 119, 223302 (2016)
shown in Figu es 3,4,and8. In hese igu es, g ey colou ed
ba s by each g oup o symbols indica e he unce ain y in
ei he q
d
,w, o bo h, when mass losses be ween 19% and 69%
a e conside ed o he 5–50 lm glass beads, and 20% o he
90–150 lm glass beads. The ba co esponding o a mass loss
o 1% is neglec ed since i s e ec is no isually no iceable.
In Fig. 3, we ha e d awn lines o he maximum pa icle
cha ge q
d,max
, assuming he alue o q
d,max
is gi en by he
condi ion ha he elec ic ield on he pa icle su ace equals
he b eakdown ield o co ona discha ge as sugges ed in
Re . 8. We ha e plo ed wo lines: one (dashed line in Fig. 3)
assuming his ield is equal o E
c
¼310
6
V/m i espec i e
o pa icle size, and ano he line (solid line in Fig. 3) assum-
ing i s alue is pa icle size dependen , as gi en in Re . 1
Ec¼9:29 105 0:3
p;(1)
whe e 
o
is he elec ic pe mi i i y o he gas su ounding
he pa icle, which we assume is equal o he pe mi i i y o
acuum (o¼8:85 1012F=m). Acco ding o his, he
maximum cha ge o an isola ed pa icle in ai , assuming a
size-dependen co ona discha ge, is gi en by
qd;max ¼1:03 104 1:7
p;(2)
whe e q
d,max
is measu ed in coulombs and
p
in me e s. All
he expe imen al da a lie e y close o he co ona discha ge
line which conside s a cons an c i ical elec ic ield. Thus,
he da a suppo he ac ha , when dispe sed, he pa icles
cha ge up o hei maximum a ainable alue.
Fo each ma e ial p esen ed in Fig. 3, se e al RH alues
a s o age ha e been used and, in he case o he ines
glass beads, di e en dispe sion gases (ai and d y N
2
).
The condi ions on humidi y and dispe sing gas o each
expe imen al poin a e gi en in he igu e cap ion. F om he
esul s o Fig. 3, changing he s o age humidi y om 60% o
30% does no g ea ly in luence he cha ge acqui ed by he
pa icles in dispe sion q
d
. The only a ailable da a wi h d y
N
2
and 10% RH we e ob ained wi h 5–50 lm glass beads.
Finally, he ac ha he expe imen al alues o he pa icle
cha ge q
d
lay close o he limi imposed by co ona discha ge,
i espec i e o he na u e o he pa icle ma e ial, indica es
ha he de ails o he con ac cha ging be ween he pa icles
and he s eel ibocha ge a e e ased by he co ona discha ge
o pa icles o he su ounding gas, as p oposed by Re . 10.
An isola ed pa icle, howe e , ep esen s an idealized
case since dispe sed pa icles in he gas s eam a e no
imme sed in an in ini e olume o gas bu su ounded by
o he pa icles con ained in he same olume and simila ly
cha ged. Inside a conduc i e ube, he maximum cha ge o e
a pa icle is lowe han o he case o an isola ed pa icle
due o he in luence o he cha ges in neighbou ing pa icles
and he p esence o image cha ges in he ube. Acco ding o
Re . 10, he maximum cha ge q
d,max
o pa icles o adius
p
dispe sed in he gas s eam wi h olume ac ion uin a con-
duc i e ube o diame e Dis
qd;max ¼8:80 104 3
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2D2þ136:8 1:5
p

2
q;(3)
whe e q
d,max
is gi en in coulombs and
p
in me e s. Fo an
isola ed pa icle, u!0 and Eq. (3) yields
qd;max ¼1:82 105 1:5
p;(4)
which in he ange o pa icle adius
p
be ween10and100lm
yields alues o q
p,max
simila o hose gi en by Eq. (2).
In Fig. 4, we ha e plo ed he alues o he elec ic q
d
acqui ed om he s eel ube ibocha ge by pa icles in sus-
pension as a unc ion o he olume ac ion uo he pa -
icles, calcula ed om he gas low a e in he ou side po o
he en u i G(in sccm/s), he a e age mass low a e m/
exp
(in g/s) a which he powde is collec ed in he Fa aday pail
(
exp
is he du a ion o he expe imen ) and he pa icle den-
si y q
p
(g/cm
3
). When da a a e a ailable, he mass low a e
in o he en u i has been subs i u ed in m/
exp
u¼1
Gqp
m
exp
:(5)
In Fig. 4, he alues o he pa icle cha ge q
d
a e unde
he limi gi en by Eq. (3) because he olume ac ion uo
he pa icles in he dispe sing gas s eam is no la ge enough
o allow he elec ic cha ge on each pa icles in luence signi i-
can ly he elec ic cha ge on neighbo ing pa icles. Fo he
olume ac ions a which we ha e dispe sed he powde s,
Eq. (3) p edic s ha he e should be an e ec o neighbo ing
pa icles on he maximum elec ic cha ge o pa icles o
diame e less han 10 lm. All he powde s cha ged wi h he
s eel ube ibocha ge ha e pa icle sizes la ge han 10 lm,
and he e o e o he condi ions es ed, hei pa icle cha ges
a e no a ec ed by he p esence o neighbo ing pa icles, bu
hese may be impo an issues when es ing ine ma e ials.
FIG. 4. T ans e ed cha ge pe pa icle q
d
in he s eel ibocha ge as a unc-
ion o he olume ac ion o he powde in he dispe sing gas s eam u.As
a e e ence, he lines ep esen he maximum cha ge pe pa icle when he
e ec o he cha ge o each pa icle on i s neighbo s is aken in o accoun
(Eq. (4)) o d
p
¼10 lm and d
p
¼100 lm pa icle size.
10
All he expe imen-
al poin s a e abo e he limi se by Eq. (4) o hei pa icle size. Fo expe i-
men s in which he collec ed mass m
d
was a ailable ( illed symbols), he
pa icle cha ge q
d
was calcula ed using he dispe sed mass m
d
a he han he
collec ed mass m.
223302-4 P
e ez-Vaque o, Quin anilla, and Cas ellanos J. Appl. Phys. 119, 223302 (2016)

Fo one o he ma e ials es ed, (5–50 lm) glass beads,
we ha e eco ded he pa icles ajec o ies as hey exi he
s eel ube ibocha ge using a high-speed came a (Phan om
Mi o310). Fo eco ding hese images, he s eel ube ibo-
cha ge was connec ed o a ibe glass ex ension wi h glass
windows ha allowed o see in he di ec ion pe pendicula o
he pa icle laden gas low. A high speed came a was i ed
wi h magni ica ion op ics yielding an image scale o app oxi-
ma ely 7 lm/pixel. Back illumina ion was p o ided by a
high in ensi y LED du ing he eco ding ime. In he ibe -
glass ex ension, wo pa allel elec odes connec ed o a unc-
ion gene a o and a high ol age ampli ie (model 20/20 A,
T ek, Inc.) c ea ed an oscilla ing elec ic ield pe pendicula
o bo h he gas low and he iewing di ec ion. The elec o-
des we e loca ed in he ou side o he ibe glass ex ension, so
he pa icles could no ha e di ec con ac wi h he elec o-
des. The came a eco ding is synch onized wi h he elec ic
ield so he phase o he ield is known in he eco dings. Fo
eco ding he images, a sho gas pulse (0.3–1 s) is used o
d i e he pa icles h ough he ibocha ge . Sho pulses a e
necessa y o allow he pa icles o slow a he exi o he i-
bocha ge down o a speed ha allows ajec o ies o be
acked in he images a he used ame a e (4000 ames pe
second).
The elec ic cha ge o each pa icle is ob ained om he
analysis o pa icle ajec o ies when he pa icles pass
h ough a egion occupied by he al e na ing elec ic ield.
Cha ged pa icles oscilla e in he ield, and he alue o hei
cha ge and i s sign can be ob ained om he oscilla ion am-
pli ude and phase lag wi h he ield using he p ocedu e
desc ibed in Re . 15. The esul s o a o al o 330 ajec o-
ies a e shown in Fig. 5. In his igu e, we p esen he pa icle
su ace cha ge densi y as a unc ion o pa icle adius Ras
well as he pa icle size dis ibu ion o he ajec o ies
eco ded and he pola i y o he pa icle cha ges. Each
pa icle adius was measu ed om he pa icles ha a e in
he eco ded images.
Mos pa icles cha ged nega i ely, i.e., wi h he same
size o he dispe sed powde a e lea ing he ibocha ge ,
bu in e es ingly, some o he smalle pa icles a e posi i ely
cha ged. The plo o he su ace cha ge densi y o he pa -
icles as a unc ion o he pa icle adius shows ha is
mo e o less cons an o he la ge pa icles o he sample,
as i would be i hese pa icles would ha e a ained hei
maximum cha ge, bu de ia es o he smalle pa icles. The
de ia ion in he alue o o he smalle pa icles may be
explained i hese pa icles, being able o ollow he gas low
lines be e han he la ge pa icles, expe ienced ewe colli-
sions wi h he ibocha ge walls and s ill main ained, a leas
pa ially, he elec ic cha ge dis ibu ion hey had be o e he
sample was dispe sed, which is expec ed o expand bo h
signs as i is he esul o cha ge ans e be ween pa icles.
16
B. Cha ge pe pa icle in se led powde
We ha e seen ha he cha ge ans e p ocess in he s eel
ube ibocha ge p oduces he same cha ge q
d
in all he pa -
icles i we assume he sample is monodispe se. I he pa -
icles did no discha ge du ing collec ion o he sample in o a
se led powde , he cha ge Q
s
in a se led sample would equal
he o al cha ge Q
d
ans e ed o i s pa icles in he s eel ube
ibocha ge . Howe e , his is no so. In ac , i we plo Q
s
as
a unc ion o he collec ed mass mas we ha e done in Fig. 6
o he same expe imen , as shown in Fig. 2, in all cases,
Q
s
<Q
d
. Mo eo e , he plo o Q
s
e sus collec ed mass m
always shows a sublinea dependence o Q
s
on m. Since he
cha ge o mass a io qm o he collec ed sample is he de i a-
i e o he cu e o Q
s
s. m, his ac means ha he las pa s
o he sample o be collec ed e ain less cha ge han he i s
pa s. In he ollowing pa s o he manusc ip , whene e we
e e o he speci ic cha ge o he collec ed sample qm , we
FIG. 5. Dependence o he su ace cha ge densi y
q
on he pa icle adius
o 5–50 lm glass beads cha ged in he s eel ube ibocha ge . The elec ic
cha ge and he adius o indi idual pa icles ha e been measu ed om he
analysis o he pa icle ajec o ies eco ded by a high-speed came a as he
pa icles lea e he ibocha ge . The inse displays he pa icle size dis ibu-
ion o he imaged pa icles and he sign o he cha ge ca ied by he
pa icles.
FIG. 6. Elec ic cha ge Q
s
in he collec ed sample as a unc ion o he col-
lec ed mass o he same expe imen as in Fig. 4. The sublinea dependence
o Q
s
wi h he mass mindica es ha he collec ed powde is no cha ged uni-
o mly. No e ha he collec ed cha ge Q
s
is abou 1/100 o he cha ge Q
d
acqui ed by he pa icles in he ibocha ge .
223302-5 P
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will e e o i s a e age alue ob ained by di iding he inal
alue o Q
s
agains he o al collec ed mass m.
Fig. 7illus a es he speci ic cha ge qm o he collec ed
powde agains he collec ed mass m o all he a ailable
expe imen s, ha is, using he s eel ube and he nylon cyclone
ibocha ge s. Fo a gi en ma e ial, he speci ic cha ge qm
ends o dec ease when mo e mass is collec ed in ag eemen
wi h he sublinea dependence obse ed o he cha ge Q
s
in
he collec ed mass mshown in Fig. 6. Al hough he da a poin s
p esen ed in Fig. 7ha e a la ge sca e , he e is a isible end-
ency o dec easing speci ic cha ge qm wi h sample mass m.
Mo eo e , da a ob ained wi h di e en ibocha ge s wi h he
same ma e ial ollow he same end, indica ing ha he mech-
anisms ha limi he elec ic cha ge in he dispe sed pa icles
and he collec ed sample a e he same o bo h ypes o ibo-
cha ge s, al hough he o al cha ge Q
d
ans e ed o he pa -
icles when suspended in he gas s eam canno be measu ed
o he nylon cyclone.
The a e age elec ic cha ge pe pa icle in he collec ed
sample can be calcula ed om qs¼Qsmp=m. Fo he expe i-
men s wi h he s eel ube ibocha ge , q
s
can be compa ed
wi h he elec ic cha ge pe pa icle when he sample was
dispe sed in he gas s eam q
d
o e alua e he amoun o
cha ge los du ing se ling. Such a compa ison is done in
Fig. 8and shows ha in mos cases q
s
is be ween 1/100 and
1/10 o he cha ge q
d
acqui ed om he ibocha ge .
V. DISCUSSION
To explain he discha ge o he pa icles when hey se -
le, we assume ha he e is a co ona discha ge om he su -
ace o he se led sample o he su ounding ai and ha he
emaining cha ge is de e mined by he condi ion ha he
elec ic ield a he su ace o he sample mus equal he elec-
ic ield o co ona discha ge in ai , which we ake o sim-
plici y as E
c
¼310
6
V/m, al hough i should ha e some
dependence on he sample size.
1
Since he calcula ion o he
elec ic ield c ea ed by a cylind ical heap o powde is no
s aigh o wa d, o he sake o gaining insigh on he p ob-
lem i s , we will discuss a simpli ied p oblem in which he
sample is assumed o be an in ini e laye in he XY plane ha
g ows in he posi i e Zdi ec ion by uni o m addi ion o pa -
icles o i s su ace. Bo h he eal geome y o ou se up and
he geome y o he simpli ied p oblem a e depic ed in Fig. 9.
The laye es s on a conduc i e pla e ep esen ing he me allic
il e o he collec ing cell. The mass low a e o new pa -
icles pe uni a ea is F
m
, so i he mass densi y o he powde
laye is q
m
, he heigh Ho he powde laye g ows as
H¼Fm
qm
;(6)
un il a ¼
he sample collec ion s ops and he powde
laye a ains i s inal dep h H
. Modeling he de ails o he co-
ona discha ge om he powde is complica ed and we will
FIG. 7. Speci ic cha ge qm (cha ge o mass a io) o he collec ed sample as
a unc ion o he collec ed mass m. Da a o all he expe imen al uns a e
included. The oid symbols ep esen da a om expe imen s using he s eel
ube ibocha ge . Da a wi h illed symbols ep esen da a om he expe i-
men s using he nylon cyclone ibocha ge . The solid line ep esen s he
esul o he model p esen ed in Sec. V o PMMA.
FIG. 8. Elec ic cha ge pe pa icle q
s
in he collec ed sample e sus he
elec ic cha ge pe pa icle q
d
in he dispe sed sample o he samples es ed
using a s eel ube ibocha ge . Fo expe imen s in which he collec ed mass
m
d
was a ailable ( illed symbols), he pa icle cha ge q
d
was calcula ed
using he dispe sed mass m
d
a he han he collec ed mass m. Lines a e he
cases when q
s
is equal o 1, 0.1, and 0.01 imes q
d
.
FIG. 9. COMSOL model o a cha ged powde laye inside he collec ing
cell and Fa aday cage used in ou expe imen s. Coo dina es in he diag am
a e gi en in me e s. To he igh , he simpli ied model discussed in Sec ion
V. A ows in he COMSOL model indica e he magni ude and he di ec ion
o he elec ic ield displacemen i he e we e no discha ge in he powde .
223302-6 P
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e-so o he simpli ying assump ion ha he powde laye
has an elec ical conduc i i y . This elec ical conduc i i y
ep esen s he abili y o he elec ic cha ge inside he powde
laye o mo e once he co ona discha ge is ini ia ed. In igo -
ous e ms, should only ha e a nonze o alue once he co-
ona discha ge is igge ed, bu in his analysis, we will
assume as a cons an alue independen o ime. The em-
po al e olu ion o he cha ge pe uni olume qðz; Þ, he
elec ic displacemen Dð~z; Þinside he laye as well as he
su ace cha ge densi y on he conduc i e pla e pð Þa e
gi en by he solu ion o he se o equa ions
~
D¼qþ pdðzÞ;(7)
@q
@ þ ~
j¼qod zqm
Fm

;(8)
d p
d ¼jz0;
ðÞ
¼
Dz0;
ðÞ
;(9)
whe e dð zqm=FmÞis he Di ac’s del a unc ion and q
o
ep esen s he cha ge pe uni olume on a newly deposi ed
laye o powde . Due o he geome y o he p oblem, he
spa ial de i a i es a e gi en by ¼~
uz@=ð@zÞ, he elec ic
displacemen by ~
D¼Dz~
uz, and he cu en densi y by
~
j¼ Dz=~
uz. As he powde losses i s cha ge, he cha ge
pe uni olume qðz; Þdec eases om he alue q
o
:apa
o he cha ge is los o he su ounding ai by co ona dis-
cha ge and a pa mig a es o he me allic il e . As ini ial
condi ions, we ake ha he ini ial heigh o he powde
laye o be ze o and ha he pla e ep esen ing he il e is
discha ged.
The solu ion o Eq. (7) is
qz;
ðÞ
¼qoexp 
 qm
Fm
z

H qm
Fm
z

 o
Dzz;
ðÞ
¼qo

Fm
qm
exp 
 qm
Fm
z

exp 
2

()
p
ðÞ¼qo

Fm
qm
1þexp 


2 exp 
2

> 0
Dzz;
ðÞ
¼qo

Fm
qm
exp 
 qm
Fm
z

exp 
  o
ðÞ

þexp 
2  o
ðÞ

exp 
2

( )
p
ðÞ¼qo

Fm
qm
2 exp 
2  o
ðÞ

1exp 
2 o

þexp 
  o
ðÞ

exp 
 o

1
 :(10)
The empo al e olu ion o he cha ge densi y and he elec ic
ield displacemen a e shown in Fig. 10. As he powde laye
g ows, he deepe laye s discha ge while he uppe laye s
emain cha ged wi h he esul ha only he po ions o he
powde laye close o he su ace emain cha ged. The dep h
o he cha ge-holding laye depends bo h on he e ec i e
conduc i i y o he powde and he speed a which he pow-
de is ed o he laye . The elec ic ield displacemen (which
is p opo ional o he elec ic ield) is nonze o only in he
egion whe e he powde emains cha ged and abo e he
powde su ace. The same pa e n o cha ge and ield dis i-
bu ion has ound o happen du ing loading o silos.
17
The
o al cha ge pe uni a ea in he laye o powde is gi en by
Q
A¼ðH
ðÞ
o
dzqz;
ðÞ
)
Q
A¼qo

Fm
qm
1exp 


;  o
Q
A¼qo

Fm
qm
exp 
  o
ðÞ

1exp 
 o

;  o:
(11)
While he powde laye is g owing = , he elec ic dis-
placemen on he su ace o he powde laye ends o he
alue
DzH;
ðÞ
!qo

Fm
qm
¼Dz;lim:(12)
FIG. 10. Tempo al e olu ion o he elec ic ield displacemen D
z
inside he
powde laye and he olume cha ge densi y qas a unc ion o posi ion
inside he powde laye zand he elapsed ime since he s a o he deposi-
ion o he laye . The ime is measu ed in uni s o s¼/ and he zcoo di-
na e in uni s o he heigh L
z
o powde deposi ed du ing a ime s.
223302-7 P
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The elec ic ield ou side he powde laye is gi en by
Ez¼DzðH; Þ=o. I we iden i y he alue o he elec ic ield
ob ained om Eq. (12) wi h he elec ic ield E
c
¼30 kV/cm
o co ona discha ge on ai , we ge
Ec¼q0

o
1
Fm
qm
;(13)
whe e he conduc i i y o he laye and he ini ial cha ge
densi y on a newly deposi ed powde laye q
o
mus be e al-
ua ed om expe imen al da a, p o ided we use can w i e
ano he equa ion in which bo h quan i ies a e in ol ed. We
ge a second equa ion om he empo al e olu ion o he
o al cha ge in he Fa aday pail Q=Aþ p. While he powde
laye is g owing, he o al cha ge pe uni a ea is gi en by
Q
Aþ p¼2qo

Fm
qm
1exp 
2

;(14)
and once no mo e new powde is deposi ed on he powde
laye , he o al cha ge decays as
Q
Aþ p¼Q
Aþ p

o
exp 
2  o
ðÞ

;(15)
whe e he alue o ðQ=Aþ pÞ ois ob ained by subs i u ing
o
in Eq. (14). The empo al e olu ion o he o al cha ge
inside he Fa aday pail is ep esen ed in Fig. 11. Du ing he
ini ial s ages o g ow h o he powde laye , he o al cha ge
g ows sublinea ly wi h ime, esembling he dependence o
he collec ed cha ge on collec ed mass depic ed in Fig. 6,
since in ou expe imen s he mass is collec ed a a cons an
a e and hus i is p opo ional o ime.
Once he powde laye s ops g owing, he o al cha ge in
he Fa aday pail decays as expð =ð2ÞÞ (see Eq. (15)).
This means we can es ima e he alue o s¼2= ,i we
measu e he a e o dissipa ion o he cha ge in ou samples.
We ha e done his o PMMA beads (see Fig. 12) wi h he
esul ha s¼61617 s. In o de o ge an es ima e o
he e ec i e laye conduc i i y , we need a alue o he
elec ic pe mi i i y o he powde . The dielec ic cons an
k¼=oo PMMA anges be ween 2.8 and 4. Acco ding
o he B uggeman mixing o mula
18
o a solid ac ion /¼0:6
o he powde , he alues ha co espond o he limi s o
he ange a e k¼1.94 and k ¼2.50 so we ake a alue o
k¼2.22 60.28 o he dielec ic cons an o he laye ; so,
¼2k
o
/s¼(5.5 61.6) 10
9
S/m. The densi y o solid
PMMA is q
P
¼1.15–1.19 g/cm
3
. Assuming a solid ac ion
o /¼0:6, his yields q
m
’0.70 g/cm
3
. The a e o dis-
cha ge o he powde in he expe imen s wi h PMMA is
abou 0.20 g/s. Fo a cell o 4 cm in diame e yields F
m
; so,
we can es ima e he cha ge pe uni olume o a newly
deposi ed laye o powde as
qo¼Ec
k
qm
Fm
;(16)
which yields q
o
¼3.3 nC/cm
3
which o q
m
¼0.70 g/cm
3
is
equi alen o qm ¼4.7 nC/g, which is o he same o de o
magni ude han he esul s ob ained o he speci ic cha ge in
he collec ed sample (qm ¼10–20 nC/g).
I he alues ob ained o q
o
, ,and a e subs i u ed in
he equa ion o he o al cha ge pe uni a ea in he Fa aday
pail as a unc ion o ime (Eq. (14)) and we use ha he sec-
ion o he collec ing cell is 12.56 cm
2
, we can ob ain a p edic-
ion o he speci ic cha ge qm as a unc ion o he collec ed
mass m. The esul ing cu e is plo ed in Fig. 7, whe e i can
be compa ed wi h he expe imen al da a o PMMA beads.
The model seems o gi e he co ec end in he da a o spe-
ci ic cha ge agains collec ed mass, al hough he alues o he
speci ic cha ge a e abou an o de o magni ude smalle han
he expe imen al alues. The di e ence be ween he p edic-
ions o he model and he expe imen al esul s a ises om
wo easons. The i s one is ha we ha e assumed ha he
e ec i e elec ical conduc i i y o he powde is nonze o
since he s a o powde collec ion, whe eas in he eal pow-
de , i migh be ze o un il he elec ic ield on he su ace o
he powde has isen o b eakdown ield in ai , and hus, in
he ini ial s ages o powde collec ion, he cha ge does no dis-
sipa e. The second eason is ha he plana geome y o he
FIG. 11. Tempo al e olu ion o he o al cha ge in he Fa aday pail.
FIG. 12. Tempo al e olu ion o he o al cha ge in a sample o PMMA a e
collec ion. The s aigh line is a i o an exponen ial decay.
223302-8 P
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