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ELSEVIER Powder Technology 80 (1994) 253-263
Energy-size reduction laws for ultrasonic fragmentation
Karl A. Kustersa, Sotiris E. Pratsinisap*, Steven G. Thornab, Douglas M. Smithb
‘Depatient of ChemrcaI Engmeenng, Unrversrty of Cmcmnatr, Cmcmnatr, OH 45221-0171, USA
bUNMINSF Center for Micro-Engmeered Ceramics, Unrversrty of New Merrco, Albuquerque, NM 87131, USA
Recewed 28 December 1993, m revised form 13 May 1994
Abstract
The energy requirement 1s a key criterion for the selection and use of a grmdmg process. Ultrasomc drspersron 1s extensively
used to disperse submicron agglomerated powders m hqurd suspensions Suspensrons of srhca agglomerates were ground wrth
solids concentration up to 50% by weight The fragmentation or grmdmg rate 1s inversely proportronal to suspensron volume
Starting from a semrempirrcal expression that relates fragmentatron rate to partrcle srze, suspensron volume and ultrasonic
power, energy consumptron laws for both eroding and non-eroding powders are developed Experrmental results supportmg
the energy consumption laws are given Lower power input for ultrasomcatron favors efficient energy use. For eroding powders
(e g silica, zrrcorna) the energy expendrture per unit powder mass (specrfic energy) by ultrasomc grmdmg IS lower than that
of conventronal grinding techniques. In contrast, it IS slightly higher than ball mrlhng for non-erodmg powders (e g. trtama)
Kqwordr Ultrasomc fragmentation, Grmdmg energy; Ceramic powders
1. Introduction
Ultrasonicatron of liquid suspensions of ceramic pow-
ders provides an effectrve tool for the elimination of
agglomerates [l] that cause problems in postprocessing
and degrade product quality [2]. Aoki et al. [3] showed
that ultrasomcally induced cavitatron 1s necessary for
deagglomeration to take place. Most likely the intense
pressures generated in the vicinity of imploding cavi-
tation bubbles [4] are the primary means of particle
degradation. These pressures have been measured with
the use of hollow glass bubbles [5,6] and were found
to be strong enough to fragment agglomerated powders
by cracking [7]. It was shown that agglomerated powders
could be well dispersed into their constituent submicron
primary particles under the action of these cavitation
pressures [7-91.
The fragmentation can occur either by fracture or
erosion. Erosion refers to particle size reduction due
to the loss of primary particles from the surface of the
agglomerate, whereas fracture is the partitioning of the
original agglomerate mto several smaller agglomerates.
Which breakage mechanism dominates may depend on
*Correspondmg author
the apphed ultrasonic intensity, but it is certamly a
function of material properties. For erosion to take
place, the primary particles have to be freed from the
surface of the agglomerates. This means that the cav-
itation pressure must be larger than the cohesive strength
with which the surface primary particles are bound
together. Fracture results from cracking of the ag-
glomerate compact [lo]. Stresses exerted on the ag-
glomerate initiate and propagate cracks from flaws on
the surface. Resistance agamst fracture (agglomerate
strength) depends on the srze of the surface flaws and
the fracture toughness of the particle assembly.
Recently a population balance model has been de-
veloped describmg quantitatively the ultrasonic frag-
mentation of agglomerate powders [ll]. This model
predicts the evolution of the size distribution of ag-
glomerated partrcles during ultrasonication With the
model, the required processing times for desired degrees
of dispersion can be calculated. The important param-
eters in the model are the fragmentation rate and the
breakage distributron function. The fragmentation rate
is defined as the frequency of break-up events, and
results from the physical phenomena governing the
disintegration of the agglomerate particles. 1.e their
interactrons with the collapsing cavities The breakage
distribution function is defined as the fragment size
0032-5910/94/$07 00 0 1994 Elsevler Science S A All rights reserved
SSDI 0379-6779(94)02852-F
254 K.A Kusters et al / Powder Technology 80 (1994) 2.53-263
drstrlbution resulting from each break-up event, and
is a distmct powder characteristic.
A theoretical expression for the fragmentation rate
as a function of particle size and power input was
derived and evaluated by examining experimental data
on the ultrasonic dispersion of silica and tnama ag-
glomerated powders [ll]. The breakage distributron
functions were determined from the self-preserving size
distributions these powders developed after a short
period of ultrasonication.
In the present article, energy consumption laws are
derived for ultrasonic dispersion. In order to obtain
an energy-size reduction law for non-eroding powders,
the similarity solution of the grinding equation has to
be combined with the above-mentioned relationship
between fragmentation rate and power input [12]. With
eroding powders, the energy consumption must be
related to the production of fines. Furthermore, to
calculate the specific energy for a given size reduction
of agglomerated powder, rt is necessary to consider the
effects of particle concentration and irradiated sus-
pension volume on the fragmentation rate. Experiments
were conducted to elucidate these effects on the energy
consumption. Finally, a comparison is presented be-
tween the energy efficiency of ultrasomc dispersion and
other grinding techniques such as ball milling.
2. Theory
2.1. Fragmentation rate expremon
The fragmentation rate, S(V), represents the relative
change m number or volume concentration of particles
of volume v per unit time by the disruptive action of
collapsmg cavities. These cavities are generated in a
restricted cavitation zone around the ultrasonic probe.
Mechanical mixing ensures that all the particles in the
suspension are exposed to the ultrasonic forces in the
cavitation zone.
In the beginmng of the ultrasonic treatment, the
cavities originate from gas nuclei in the liquid as well
as on the particle surfaces. These cavities expand in
the dilation half-cycle of the pressure sound wave and
collapse violently when the pressure enters the compres-
sion half-cycle. Upon collapse the cavities break up
into tmy bubbles that can act as new cavitation nuclei.
This chain reaction stops when the nuclei become too
small for expansion under the applied acoustic pressure.
Finally a constant level for the total number of cavities
collapsmg per unit time, NC, is reached m about ten
pressure cycles [13] The final number of collapsing
cavities per unit time is independent of the initial
number of gas nuclei in the suspension, but rt is related
to the power input, E [ll]:
(1)
5
7 7rrb3Ph
where r, is the cavity radius at maximum expansion,
P, is the hydrostatic pressure m the suspension and K
denotes the cavitation efficiency, defined as the ratio
of energy spent for the creation of cavities to the total
ultrasonic energy employed [14]. Only particles m the
vicinity of the collapsing cavity, 1.e. those close to or
at the cavity wall, are exposed to the intense pressures
generated by its implosion. Neglecting cavity interaction,
the number of agglomerates of radius r, touching the
cavity wall at maximum expansion (rbx=-rJ, i.e. prior
to collapse, N,, is given by:
N,(v, t) = 4mb2r,(v)n(v, t) (2)
where n(v, t) is the number concentration of agglom-
erates of volume 2, at time t. The total number of
agglomerates that are fragmented per unit time in the
cavitation zone is given by the product of Eqs. (1) and
(2). Dividing this product by the total number of particles
of size v in the suspension [ =rr(~, t) x V,,,] yields for
the fragmentation rate:
3KE ra(v)
S(v)- --
V,,, Phrb
For sphencally shaped agglomerates with sohds density
4, the relationship between the agglomerate radius, r,,
and volume, V, is:
v = $ ra(v)3+ (4)
With r,- d” [13,15], the expression for the fragmen-
tation rate reduces to:
The dependency on particle size has been verrfied by
experimental data on the ultrasonic dispersion of ag-
glomerated silica and trtania particles [ll]. The effect
of power input on the fragmentatron rate was found
to be slightly less than predrcted:
S(V) = 2.0 x 10-3eo ‘“V’“lV,,, (6)
The fragmentatron rate is also a function of the sohds
density, as follows from Eq. (5). The solids density C$J
ranged from 0.4 to 0.67 for the employed powders, a
variation that was insufficient to significantly affect the
fragmentatron rate. Eq. (6) can be also written as a
power law [12]:
S(d) =A@ (7)
where d represents the equivalent solids volume
diameter of the agglomerates. For the particular case
KA Kusters et al I Powder Technology 80 (1994) 253-263 255
of ultrasonic dispersion, A = 1.6 x 10-3~o 38/Vlot and
a = 1.
It may seem odd that the agglomerate strength does
not enter the fragmentation rate expression. Basically
this means that Eq. (6) applies only to those cases
where the ultrasonic forces acting on the particles are
always strong enough to induce breakage of the particles.
It has been observed that for strong materials there
is a certain induction period or time lag before fracture
occurs [ll]. Apparently, during the time lag, flaws or
cracks are created on the particles and grow until they
reach the critical crack size, above which fracture
contmues uninhibited [16] and is described by Eq. (6).
Hence after this time period, the original agglomerates
and the resulting fragments are too weak to resist
ultrasonic fragmentation. For eroding agglomerates that
exhibit no time lag, Eq. (6) can be utilized from t= 0,
but a different breakage distribution function should
be implemented in the population balance model. Ero-
sion results in the formation of prunary particles and
coarse fragments comparable in size with the original
particle, i.e. a bunodal breakage distribution function.
The extent of erosion is proportional to the external
surface area of the agglomerates [ll]. The breakage
distribution function resulting from fracture depends
on the cracking pattern. Flaws inside the particle de-
termine the final fragment size distrrbution by fracture,
because they give rise to multiple branching of the
propagating crack [17].
2.2. Energy-size reduction law
The fragmentation expression, Eq. (6), is used in the
general grinding equation [18] to describe the ultrasonic
fragmentation of agglomerate particles:
m
WV, 0
- = -S(v)q(v, t) +
af s
p(v, v’)S(v’)q(v’, t) dv’ (8)
v
Here q(v, t) is the volume concentration of particles
of volume V, and p(v, w’) is the breakage drstribution
functron. For non-eroding particulate systems, B(v, v’)
is usually normalized with respect to parent agglomerate
size, i.e.
P(v, 2)‘) = P(V/V’) (9)
This may be the result of the similarity of the fracture
pattern in particles of different sizes [19]. For such a
normalized breakage distrrbution, a self-preserving so-
lution for Eq. (8) has been formulated by Kapur [20].
The self-preserving particle size distribution is usually
obtained after a short period of grinding. In the self-
preserving limit, the particle size distribution does not
change its form relative to the mass mean diameter,
d,. An expression for the reduction in average particle
size is available [20]. This expression will be utilized
to derive an energy-size reduction law for non-eroding
powders. For eroding powders, the breakage distribution
function cannot be normalized because the size of the
resulting primary particles is invariant with respect to
parent agglomerate size. Here, however, the energy
consumption can be related to the production of fines.
2.2.1. Non-eroding powders
In the self-preserving limit, the average particle size
changes with time according to Kapur [20]:
d&SO
- = -AKd,(ty + l
dt
where K is a constant determined by the functional
forms of the particle size dependency of the fragmen-
tation rate and breakage distribution functions. If the
breakage distribution function does not change with
power input, K is also invariant to changes in E. For
the particular case in which the cumulative fragment
size distribution, B(v, v’), is represented by the following
power law expression:
b (11)
and a = 1, K equals 1/3b.
Here we introduce the reduction ratio R and drmen-
sionless time 7 as:
R_ d”do)
440 (12)
T= S[d,(O)]t = Sot (13)
Substitution of Eqs. (12) and (13) into Eq. (10) and
setting a = 1 yields:
dR
-=
dr -K (14)
Integration of Eq. (14) over T in the self-preserving
limit gives:
R(T) =K(T- 7,) + R(q) (15)
where rs denotes the time to reach the self-preserving
state. For most fragmentation processes, the self-pre-
serving state is attained very raprdly, so T,+O [20]. In
that case Eq. (15) can be approximated by:
R(T) =KT+ 1 (16)
This grinding law in time domain can be transformed
to the energy domain by recognizing that:
E(f) = Et (17)
where E is the consumed ultrasonic energy.
Combining Eqs. (7), (12), (13), (16) and (17) gives:
E(t) = 620 Kd,(o)
VtotEO [R(t) _ 11 (18)
256 K.A Kusters et al I Powder Technology 80 (1994) 253-263
Divrding Eq. (18) by the total particle mass ( = p,C, V,,,)
gives for the energy consumption per umt particle mass,
E,:
E,,,(t) = 620 KPP;;m(o) P(t)- 11
=KJ 1 1
d,(t) - -
L(O) (1%
The specific surface area (surface area per unit particle
mass), a,, is inversely proportional to average particle
size, i.e. uv~dmml. Hence Eq. (19) can be rewritten
as:
EO 62
Em(t) = 620 &&d,,,(O)
(20)
Eq. (20) implies that for a constant ultrasonic power
input, the specific surface area of the powder increases
proportionally to energy expense. Furthermore it shows
that more energy is needed to obtain the same increment
in surface area at a higher power input. This is in
contrast with conventional grinding techniques like ball
milhng, where the size reduction is only dependent on
the energy expense but not on the applied power [21].
2.2.2. Eroding powders
In the case of erosion, the breakage distribution
function is expressed as [22,23]:
P(V, 2)‘) =x(z)‘)w(2)) + [l -x(v’)]n(v, V’) (21)
where x(v’) 1s the mass fraction of produced fines, w(v)
represents the fine fragment size distribution and Qv,
v’) the size distribution of the coarse fragments. The
coarse fragments may result from additional fracture.
Hence the coarse part of the breakage drstributron
function may again be normalized. With this provision,
a particle size reduction law similar to Eq (14) can
be derived for eroding powders. However, the particle
size reduction becomes a function of the amount of
erosion product, which also changes with time. This
prevents a simple relationship between energy con-
sumption and size reduction.
With erosion, however, it is more appropriate to
define the energy consumption with respect to the
produced fine fragments rather than the degree of size
reduction. If the size distributions of the fine and coarse
fragments do not overlap, the increase in the amount
of erosion product, C,, is given by:
x(v’)S(v’)q(v’, t) dv’ (22)
where x(21’) denotes the volume percentage (fraction)
of fine fragments resulting from break-up of agglom-
erates of size v’, and vp,max represents the upper
boundary for the fine fragment size drstribution. Erosion
has been shown to be proportional to external ag-
glomerate surface area [ll]:
x(z)‘)= --& or x(d’) = g (23)
Substitution of Eqs. (7) and (23) mto (22) yields:
q(v’, t) dv’ (24)
vp, msx
The integral on the right-hand side equals (1 -f,) C,,
where f, denotes the erosion product fraction. Since
f,= C,/C,, Eq. (24) reduces to:
y =AH(l -f,(t))
Integration of Eq. (25) with f,(O) =0 yields:
1 -&(t) = exp( --AS) (26)
Combmmg Eqs. (7) (13), (23) and (26) gives:
1 -f,(~-) = exp( --x07) (27)
where xo=x[d,,,(0)]. Combining Eqs. (7), (17) and (26)
gives, for the energy-agglomerate fraction relationship
for eroding powders:
E(t) = - 620 !L?$Y ln(l -_f,) (28)
Hence the energy consumption per unit particle mass,
E,, is given by
E,,,(t) = - 620 & ln(1 -f,) (29)
P s
3. Experimental
The above equations are based on the inherent
assumption that the fragmentation rate 1s independent
of particle concentration This assumption, however, is
expected to break down when the particle concentration
approaches unity. It is important to establish the particle
concentration range for which Eqs. (18) and (28) are
applicable. Furthermore, the dependency on total sus-
pension volume stems from theory but has not been
thoroughly investigated experimentally. Experiments
were conducted to elucidate the effects of these process
variables on fragmentation.
The experimental data comprise the ultrasonic frag-
mentation of silica and titania powders made by wet
chemistry techniques [8,9]. These powders were sub-
K.A Kusters et al. / Powder Technology 80 (1994) 253-263 257
sequently heat-treated and ground in a mortar until
they passed a 105 pm sieve. The silica prunary particles
were made by the Stober process [24]. The titania
primary particles were produced by thermal hydrolysis
or by base precipitation from a tltania solution (1.0 M
TiOz and 3.8 M H,SO,) and will be termed hydrolyzed
and base-precipitated titania, respectively. Experiments
were also conducted with a commercially available
zu-conia powder. Sample suspensions were prepared
from these powders in a total volume of 40 and 120
cm3 of 0.05 wt.% sodiummetaphosphate-water solution.
Ultrasonic fragmentation was accomplished with a 20
kHz Tekmar TSD500 Sonic Disruptor equipped with
a VlA horn with OS-inch tip in a water-jacketed glass
sonication vessel. Thoma et al. [6] calibrated the ul-
trasonic force field with hollow glass bubbles of known
compression strength. This calibration gave the effective
break-up pressure of the collapsing cavities as a function
of power input. The applied cavitation pressure varied
from approximately 20 to 800 bar between the lowest
(2.5 W) and the highest (100 W) power setting of the
ultrasonic device. The temperature of the suspension
was maintamed at 25 “C with the use of a water bath.
The agglomerates were kept m suspension by magnetic
stirring
The evolution of the particle size distribution during
ultrasonic treatment was measured using a Microm-
critics Sedlgraph 5100 particle size analyzer. Sedimen-
tation of the particles in the sample suspension 1s
measured by X-ray transmission. This particle size
analyzer yields mass distributions, Q(r), over 250 geo-
metrical sections in a user-selected size range, usually
from 0.1 to 100 pm.
4. Results
4.1. Effect of solrds concentration
To examine the effect of solids concentration, sus-
pensions of 5, 25, 50 and 75 wt.% solids were treated
at a power input of 2.5 W for 5, 10 and 30 min in a
40 cm3 container. The powder that was used consisted
of agglomerates (d,= 19.4 pm) of 1 pm silica spheres
that were produced by heat treatment at 1100 “C for
5 h. With p,=2000 kg/m3 181, the applied solids con-
centrations correspond to 2.5, 14, 33 and 60 vol.%,
respectively.
Fig. 1 presents the cumulative mass distribution curves
for each of the suspensions after 30 min of ultrason-
ication. The steep rise in the curves at a particle size
of 1 pm represents the fine fragment peak, whereas
the moderate rise starting at approximately 10 ,um
corresponds to the agglomerate fraction of the size
distribution. There is already a fan amount (ca. 40%)
of primary particles present in the ongmal sample which
100 r
g?
- 80
OL
o untreated (t - 0)
05 1 10 100
Parttcle diameter (urn)
Fig 1. Cumulative mass dlstrlbutlons of sihca powder (prmary particle
size= 1 pm), ultrasomcally treated for 30 mm at 2 5 W m 40 cm3
suspensions of variable sohds concentration The size dlstrlbutlon of
the ongmal powder IS also mdlcated
was produced by the grinding process. For solids con-
centrations of 5, 25 and 50 wt.%, there 1s an identical
increase in the percentage of primary particles, namely
up to 60%, by ultrasonic treatment. At the same time,
the agglomerate size distribution shifts to lower particle
size values. Only the result at a solids concentration
of 75 wt.% deviates from this behavior. The 75 wt.%
suspension was very viscous, and magnetic mixing was
insufficient to wet the ultrasonic probe. A small volume
around the probe was de-watered, thereby making
impossible the proper transmission of ultrasonic waves
into the suspension. This explains why no fragmentation
was observed with the 75 wt.% solids suspension.
At the lower concentrations the suspension is non-
viscous and mixing is unhindered. As a result, the
breakage rate is independent of solids concentration
up to 50% by weight, in agreement with theory.
4.2. Effect of suspension volume
Experunents were conducted to elucidate the effect
of suspension volume on the ultrasonic fragmentation
rate. Base-precipitated titania agglomerates ([9],
d,= 22.4 pm) were ultrasonically treated at 2.5 W in
40 and 120 cm3 suspensions. Fig. 2(a) shows that the
reduction ratio increases with time faster at the small
than at the large suspension volume for equal power
input, as predicted by Eq. (6). When these data are
plotted as a function of dimensionless time they collapse
on a single line, as predicted by Eq. (16), providing
further validation of the proposed model.
From this analysis we may conclude that the course
of the fragmentation process is unaffected by the scale
of operation at equal values of the power input per
umt volume of suspension. For a constant solids con-
centration, the power input per unit mass of solids may
be used as a scaling parameter
258 IL4 Kusters et al I Powder Technology 80 (1994) 253-263
0 40 ml
7 120 ml
0 100 200 300 400
Grinding Time (mm)
I? I I
p20
Ii i
4 h 1.5 @I
1.0 q
0 2 Dimensionles~Tlme, 4 &s, 10
t
Fig 2. Reductzon factor, R, versus (a) time, t, m mmutes and (b)
dlmenslonless time, T, for ultrasomc treatment of base-preapltated
tltarua powder at 2 5 W m 40 and 120 cm3 suspensions. C, = 2 5
vol %
4.3. Energy-sue reduction laws
4.3.1. Non-eroding powders
Experimental data on non-eroding thermally hydro-
lyzed trtama powder [9] were also analyzed. The cu-
mulative fragment size distribution of these powders
(d,,,(O) = 12.3 Frn) is normalized and can be described
by Eq. (11) with the power law exponent b= 1 [ll].
The size spectra of this powder become self-preserving
after a short period of ultrasonic fragmentation. The
self-preservmg drstribution is independent not only of
the initial size distribution but also of the applied power
input.
Fig. 3 shows the energy consumption, E, versus size
reduction, R, at an ultrasonic power input of ~=2.5
and 5 W. A linear relationship between E and R is
obtained, as predicted by Eq. (18). The slopes of the
lines in Fig. 3 amount to 9.3 and 14.0 kJ for E= 2 5
and 5 W, respectively. According to Eq. (18) these
values are obtained with K=0.37, i.e. a power law
exponent, b, equal to 0.9, in close agreement with the
previous experimental result. Eq. (18) is only valid if
attainment of the self-preserving state is fast. Analysis
of the evolution of the measured size distributions
showed that this is indeed the case for the hydrolyzed
titania [ll].
J
10 Reckon ratio, it/d) 25
Rg. 3 Energy consumptton, E, versus size reduction, R for ultrasomc
fragmentation of hydrolyzed tltama powder at 25 and 5 W m 40
cm3 suspension of 2.5 ~01%
The hydrolyzed titania powder displayed a distinct
time lag prior to fragmentation [ll]. This time lag
results m a non-zero abscissa with the vertical axis at
R=l in Fig. 3.
4.3.2. Eroding powders
Experimental data from fragmentatron of eroding
silica [8] and zirconia powders were used to evaluate
the functional form of the relationship between the
energy consumption and the fraction of produced fines.
The silica agglomerates (d,=31.0 pm) consisted of
monodisperse spherical primary particles of about 0 3
pm. Fig. 4(a) is a representative scanning electron
micrograph SEM of a zirconia agglomerate. The debris
on the surface of the agglomerate is the fines that were
already present in the untreated powder. They may
have resulted from grinding steps in the process of the
zirconia powder production. Fig. 4(b) 1s a larger mag-
nification of these particles, which were also of the
order of 0.3 ,um but irregular in shape.
Fig. 5 shows the evolution of the size distribution
of the silica powder at an ultrasonic power mput of
20 W. Ultrasonic treatment results m the formation of
a bimodal distribution with a distinct separation between
the agglomerate and fine fragment size modes. With
increasing processing time, the agglomerate size dis-
tribution shifts to lower particle sizes and shrinks in
magnitude. The erosion of agglomerates results m the
production of fine fragments, as reflected by the increase
in the second mode of the size distribution around a
particle diameter of 0.3 pm and the final extinction of
the initial agglomerate mode. The fraction of primary
particles (fine fragments), f,, was calculated by:
vp milx
s q(v, t)dv
fpW= $
J
q(v t)dv (30)
-0
K.A Kmters et al / Powder Technology 80 (1994) 253-263
0 5 25
Frg 6 Decrease m fraction of agglomerates, 1 -f,, m sihca powder
as a functron of dtmensronless trme, r, for vartous power settmgs of
ultrasomc devrce
z 24 ot- 0
z20
5
6 1 6
2
212
;
7J
< 0.6
5 0.4
Z
‘00 01 1
Partlcle diameter (pm)
Frg 7 Evolutron of partrcle srze drstrrbutton, Q(L), of zucoma powder
at E= 20 W m 40 cm3 suspensron of 2 5 vol % Partrcle srze range
m Sedrgraph IS from 0 15 to 1013 Frn
Ftg 4. Scanmng electron mrcrographs of (a) zrrcoma agglomerate
and (b) prrmary parttcles of employed zucoma powder
0.1 1 IO
Partlcle diameter (pm) 100
Frg 5 Evolutron of parttcle size dtstrrbutron, Q(r), of srhca powder
(prtmary parttcle srze = 0 3 pm) at e=20 W in 40 cm3 suspension
of 25 ~01%. (-o-) t=O, (-+-) t=15 mm, (u) t=30 mm.,
(-V-) t =45 mm, (-A-) t=60 mm , (t) t =90 mm
The upper boundary of the silica fine fragment srze
drstribution, d,, ,_, was taken as 1.0 pm, i.e. v,, max=
(7r/6) pm3. Fig. 6 shows the corresponding decrease in
the fraction of agglomerates, 1 -f,, with processing
time, together with the experimental data for other
ultrasonic power inputs: E = 2.5,5, 20 and 55 W. These
data are in excellent agreement with the present theory,
Eq. (27) for x,=0.1.
Fig. 7 shows the change in the particle srze distribution
of the zirconia powder (d,(O) = 4.5 pm) under ultrasonic
treatment at l =20 W. A behavior similar to that of
the sihca powder is observed. The agglomerate and
fine fragment modes, however, start to overlap, resulting
in a less accurate determination of the fraction of
produced fines. As the particle size corresponding to
the mmrmum between the two modes seems to be
independent of processing time, this value (1.0 pm)
was used as d p, max to determine the fraction of produced
fines. Fig. 8 shows the calculated fraction as a function
of dimensionless time, 7, for various ultrasonic power
inputs. Again, an exponential decrease in the fraction
of agglomerates, 1 -f,, is obtained, similar to Eq. (27)
with x0 = 0.2.
As follows from the constant values for x0 of both
powders, the erosion amount is independent of applied
ultrasonic power input. The parameter x,, appears to
be merely a powder characteristic.
260 KA Kusters et al I Powder Technology 80 (1994) 253-263
0 1 D:mens~onles4s Tlrrk, A.t 7
Fig. 8 Decrease m fractton of agglomerates, 1 -fr, m zucoma powder
as a functton of dtmenstonless trme, r, for vartous ultrasontc power
Inputs
5. Comparison with conventional grinding
In the previous sections, the application of the size
reduction laws, Eqs. (16) and (27) for ultrasonic frag-
mentation of non-eroding and eroding powders, re-
spectively, was demonstrated. Furthermore, the frag-
mentation rate proved to be independent of solids
concentration, validating the use of Eqs. (19) and (29)
for the calculation of the energy consumption per unit
mass for any given solids concentration (up to at least
50 wt.% or 33 vol.%). Here, these energy consumptton
laws for ultrasonic fragmentation are compared to
Rittinger’s law, which applies to the conventional fine
grinding of brittle materials [25] such as employed m
the ultrasonic dispersion experiments. The fact that
the size distributions of the coarse fragments* resulting
from the employed powders are normalized [ll], i.e.
self-similar, provides direct justification for utilizing
Rittmger’s law in the comparison. A normalized frag-
ment size distribution is indicative of a Weibull uni-
formity coefficient close to 6 [17]**. For a Weibull
uniformity coefficient of about 6, Rittinger’s law follows
directly from the energy equation that was derived by
Oka and Majima [26] starting from fracture mechanics.
Rumpf [19] also arrived at Rittinger’s law for grinding
of geometrically similar particles by means of dtmen-
sional analysis. In terms of the mass mean diameter,
Rittmger’s law can be written as:
E,(t)=KR 1 1
- - -
4&) L(O) 1
= -& [W- 11 (31)
m
Fig. 9 shows the comparison between Rittinger’s law
for conventional fine grinding and the energy-size re-
*The term ‘coarse’ IS used here to dtstmgmsh these fragments
from prrmary partrcles resultmg from the erodrng powders The
produced fragments, however, may be small compared to the orrgmal
parttcle stze
**Wetchert [17] defines a Wetbull exponent, Z, whrch 1s related
to the Werbull umfornuty coeffictent, m, by z=(2+m)/5
- Ball Mllmg (Rtttmger)
Ultrasomc Grinding
L ----- titania
5
:
--._
--._
-..-.
slllca
5 1000 _-<-.._. l -._ --., --- zwconla
-~~,-“-..-..,. - . . . .
2 N.- . . .
z --\
15 1 \ --‘~~q.q~ . . .
.
ci 100 ’ \, ,,,,, . .
z
b 01 1 10
co Product Powder Diameter, a (pm)
Rg. 9 Theorettcal spectfic energy consumptton, Em, versus mass
mean diameter, d,,,, accordmg to Rittmger’s law for conventtonal
fine grmdmg (KR = lC? kWh/ton pm-‘) and for ultrasomc fragmen-
tatton of the powders discussed m section 4 3
duction curves for ultrasonic fragmentation. Rittinger’s
law has been calculated for d,,,(t) <<d,,,(O) and with
KR = lo3 kWh/ton pm [ 19,271 which is a characteristic
value for ceramic powders. The ultrasonic energy con-
sumption curves have been calculated for a power input
of 2.5 W and a solids concentration of 15 vol.%. The
energy consumption of the non-eroding titania powder
was calculated using Eq. (19) and the analogy with
Rittinger’s law 1s evident.
The corresponding curves for the silica and zirconia
powders are obtained by relating the amount of pro-
duced fines to the mass mean diameter. Because of
erosion, these curves do not display the linear de-
pendency between energy consumption and increase
in specific surface area as given by Rittinger’s law. With
decreasing particle size, the increment in energy con-
sumption decreases because a larger volumetric part
of the agglomerates is directly reduced to the fine
fragment size. Initially, less energy is needed for size
reduction of the zirconia powder due to the smaller
original particle size. The energy expenditure, however,
rises more rapidly with the zirconia powder than with
the silica powder because the zirconia erosion is less
intense, and also the large fragments that are formed
are coarser than with the silica powder. For complete
dispersion, i.e. reduction of the agglomerate size down
to the fine fragment size (-0.3 pm), almost equal
amounts of energy are required.
Eq. (19) for non-eroding powders differs from Rit-
tmger’s law mainly in the sense that the parameter,
Ku, increases with ultrasonic power input. In conven-
tional grinding, KR is independent of applied power
input [20]. In Fig. 10 the increase in energy consumption
with power input is shown for the eroding silica powder.
Note that initially the specific energy consumption at
a power input of 2.5 W is higher than at ~=5 W. This
is the result of the time lag that occurred at E= 2.5
W but was not observed at higher power inputs [ll].
KA titers et al / Powder Technology 80 (1994) 253-263 261
w
: _____ 5 w : :
- 25W i,‘;
:
.
10
w
c% 01 1 10 100
Product Powder Diameter, d (pm)
Fig 10. TheoretIcal energy consumption, E,, versus mass mean
diameter, d,, of sdlca powder (pnmary particle slze=O 3 pm) for
various ultrasomc power Inputs
The increase in energy consumption with power mput
is attributed to the indirect fragmentation mechanism
of ultrasonic dispersion. In conventional grmdmg, there
1s a direct transport of energy to the particles by the
grinding media. With ultrasonication, the supplied en-
ergy 1s first used for the generation of cavitation bubbles.
The energy released by the collapse of these cavities
IS dissipated in the surrounding liquid. If agglomerates
are m the vicinity of the collapsmg cavrtres, some of
this energy may be directed to the creation of new
surface area of particles by fragmentation. This energy
transfer, however, cannot increase linearly with the
power input, as it is locahzed m a small area around
the collapsmg cavities To mmrmrze the energy re-
quirement, operation at low power inputs is recom-
mended.
The localized fragmentation also explains the re-
duction m the energy requirement with increasing solids
concentratron A higher concentration increases the
number of agglomerates around the collapsing cavrtres,
so more cavitation energy is directed to ultrasonic
dispersion. The energy consumed can be reduced by
a factor of 2 by increasing the solids concentration
from 15 to 30 vol.%.
The ultrasonic size reduction laws are only applicable
in the case of pure fragmentation. The occurrence of
simultaneous agglomeration and fragmentation [28] 1s
favored primarily by high power inputs [ll] and sec-
ondarily, by high solids concentrations. As agglomeration
counteracts fragmentation, rt is undesirable in ultrasonic
dispersion processes. Lowering the power input there-
fore helps prevent agglomeration and reduces the cost
of srze reduction. Maximizing the solids concentration
is only an applied means of saving energy m the absence
of agglomeration.
These results indicate that the ultrasonic dispersion
technique can be as efficient as conventional grinding
techniques. The mdustnal practice of ultrasonic dis-
persion (e.g. larger probes, contmuous throughput of
suspension) may alter these results somewhat, but over-
all it is expected that the specific energy consumption
1s not the reason for the selection of this comminutron
technique but rather its ability to produce extremely
fine (submrcron) particles.
6. Conclusions
Recently, Kusters et al. [ll] have presented a semi-
emprrrcal expression for the ultrasonic fragmentation
rate as a function of particle srze, suspension volume
and applied ultrasonic power. Here, it was experi-
mentally shown that the fragmentation rate is inversely
proportional to the total suspension volume in agree-
ment with that theory. Furthermore, the fragmentation
rate was found to be independent of solids concentration
(up to 33 vol.%), giving ample evidence that this type
of fragmentation results from particle-cavity rather than
particle-particle interactions.
Energy consumption laws for size reduction by ul-
trasonic fragmentation were derived also. For non-
eroding agglomerated powders, an energy-srze reduc-
tion relationship was obtained by combining the frag-
mentation rate with a self-preserving solutron to the
population balance equation for grinding. With erosion
proportional to the agglomerate specific surface area
and independent of ultrasonic power input, the energy
consumption for eroding powders was lmked to the
amount of produced fines. These energy equations have
been verified by experimental data on the ultrasonic
fragmentatron of agglomerated submicron titama, srhca
and zirconia powders.
The specific energy consumption as a function of
initial particle size is determined by the material, power
input and solids concentration of the ultrasonicated
suspension. The fragmentation mechanism (fracture or
erosion) as well as the fragment size distribution depend
on the material properties. For non-eroding powders,
the energy expenditure follows Rittmger’s law. With
erodmg powders, energy consumption diminishes with
decreasing particle size because a larger portion of the
particles is directly converted to primary particles. Less
energy per unit mass of solids is required by operating
ultrasonic dispersion at low power inputs and high
solids concentrations. When optimum conditions are
used, ultrasonic fragmentation is shown to be equally
efficient as conventional grinding.
Acknowledgements
This research was supported by the National Science
Foundation grant GTS-8957042. K.A Kusters gratefully
262 KA kksters et al I Powder Technology 80 (1994) 253-263
acknowledges financial support by the NATO Science
Fellowship awarded by the Netherlands Organization
for Scientific Research.
List of symbols
A
a
L?(z), 21’)
b
H
K
KZ
Ku
Na(v, 0
NC
n(v, 0
p.4
pa
pll
Q,(t), Q(i)
q(v7 4
R
rb
ra
WJ)
&I
pre-exponential constant in Eq. (7)
power law coefficient m Eq. (7) for break-
up rate
specific surface area (m-l)
cumulative breakage distribution func-
tion
Greek letters
Ph v’>
E
K
power law coefficient in Eq. (11) for
fragment size distribution
volume concentration of erosion product
total volume concentratron of solids
particle diameter (m)
mass mean diameter (m)
energy consumptron (J)
specific energy consumption (J/kg)
erosion product volume fraction of par-
ticle size distnbutlon
proportionality constant for erosion
amount as a function of inverse of ag-
glomerate size v, see Eq. (23)
proportionality constant for erosion
amount as a function of inverse of par-
ticle diameter, see Eq. (23)
constant in Eq. (10)
constant in Rittinger’s law [defined in
Eq. (31)l
constant in energy-size reduction law for
ultrasonic fragmentation [defined in Eq.
WI
number of agglomerates of srze v at
cavity wall at time t (m-‘)
number of collapsing cavities per second
(s-l)
number concentratron of particles of size
v at time t (m-“)
amplitude of acoustic pressure wave (Pa)
acoustic pressure (Pa)
hydrostatic pressure (Pa)
volume fraction in size section i at time
t
fqv, v’)
References
[II
VI
volume concentration in size range (v,
v + dv) at time t
reduction ratio [defined in Eq. (12)]
maximum radius of cavity (m)
agglomerate radius (m)
fragmentation rate of particle with vol-
ume v (s-l)
[31
r41
[51
VI
[71
PI
[91
PO1
1111
P21
1131
P41
P51
fragmentation rate correspondmg to mi- WI
tral mass mean diameter El71
t
V tot
21, 2)’
x(v)
X0
time (s)
total suspension volume (m3)
particle volume (m”)
volume fraction of fragments resultmg
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volume fraction of fragments resulting
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breakage distribution function
ultrasonic power input (W)
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ultrasonic energy
particle density (kg m-“)
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(v <v’)
size distribution of large fragments re-
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