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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

from erosion of agglomerates of the ini-

tial mass mean diameter

breakage distribution function

ultrasonic power input (W)

coefficient of cavitation utihzation of

ultrasonic energy

particle density (kg m-“)

drmensionless time [defined in Eq. (13)]

agglomerate solids density

size distribution of fines resulting from

erosion of agglomerates of size v’ where

(v <v’)

size distribution of large fragments re-

sulting from break-up of agglomerates

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