KONA No.21 (2003) 121

INTRODUCTION

The optimal design and control of comminution cir-cuits require a mathematical model capable of depict-ing the size reduction behavior of every size fractionfor grinding conditions of technological importance.This is particularly important for modeling closed-cir-cuit comminution systems in which oversize materialfrom a classifier or screen is recycled back to thecomminution device. Comminution kinetics did notreally have any practical application until a mathemati-cal model of batch grinding was introduced � amodel that incorporated both disappearance and pro-duction kinetics into one mathematical construct. Arigorous mathematical approach to comminution waspublished in 1954 by Bass [1] but the catalyst that ledto worldwide utilization of this so-called populationbalance model (or batch grinding model) to the analy-sis of comminution in tumbling mills was perhaps the1962 paper presented by Gardner and Austin [2] atthe First European Comminution Symposium.

The batch-grinding model entails the formulation ofa mathematical model which is phenomenological innature in that it lumps together the entire spectrum ofstress-application events which prevail in the systemunder a given set of operating conditions. The appro-priately defined average of these individual events isconsidered to characterize the over-all breakage prop-erties of the device and material. A single parameteris assumed to represent the resistance of particles ofthat size (or size fraction) to fracture, given the aver-age grinding environment which exists in the mill.The isolation of such a parameter and a related setof quantities which constitute the breakage productsize distribution for the average event in this size frac-tion allows the formulation of physically meaningfuldescriptive equations capable of yielding detailedinformation for simulation. The continuous-time, size-discretized solution of the batch grinding equation byReid [3] provides a valuable practical simplificationfor data treatment since size distributions of commi-nuted products are usually determined in terms of aseries of finite size intervals by sieving.

In dry batch ball milling, grinding kinetics follow alinear model. Linear breakage kinetics are said to pre-vail in a mill when neither the probability of breakage

D. W. FuerstenauDepartment of Materials Science andEngineering, University of California*and P. C. KapurDepartment of Metallurgical Engineering,Indian Institute of Technology**A. DeABB. Inc

Modeling Breakage Kinetics in Various Dry Comminution Systems†

Abstract

In modeling comminution systems, breakage rate (selection) functions are generally inf luencedmore by the comminution machine than are breakage distribution functions, which are controlled bymaterial properties. Linear grinding kinetics can be expressed either in terms of grinding time orspecific energy consumption. Nonlinearities are caused by energy transfer mechanisms in the com-minution machine whereby coarser particles might be ground preferentially or are protected by fines,by energy dissipation through interparticle friction in compressed bed comminution, and sometimesfrom heterogeneities produced in the feed particles. This paper discusses modifications of breakagerate functions in the grinding model for a number of situations and compares simulated and experi-mental results.

* Berkeley, CA 94720-1760, USA**Kanpur, India† Accepted: September 9, 2003

of a particle (as measured by the breakage rate func-tion) nor the distribution of fragments resulting fromthe primary breakage of that particle (as measuredby the breakage distribution function) is inf luencedby the size consist in the mill. However, the rate ofparticle breakage may deviate from linearity due tochanges in the nature of the comminution events orfrom changes in the heterogeneity of the particlesbeing ground. For example, in some cases conditionsmay exist where the larger particles are ground pref-erentially or in other cases the larger particles maybecome protected after time. Under wet grinding con-ditions, at lower pulp densities suspension of the fineparticles in the slurry may lead to an increase in theprobability of coarser particles being ground, or athigh pulp densities the slurry may become so viscousthat grinding is retarded unless a grinding aid isadded. In confined particle bed grinding, the productparticles can be weakened through the generationof f laws and microcracks and consequently exhibitwidely distributed strength behavior, which leads tononlinearities in the regrinding of such particles. Thispaper discusses approaches taken to modify themodel to describe that behavior.

THE LINEAR BATCH GRINDING KINETICMODEL

Consider the size range of a particulate assembly,characterized by a maximum size, x1, and a minimumsize xn�1 to be subdivided into n intervals (with nsieves). The i th size fraction is the interval boundedby xi above and xi�1 below and denote the mass frac-tion of material in this size interval at time t by mi(t).If r is the geometric sieve ratio (which is generally M2for a series of sieves such as the Tyler sieve series),then x l�rxi�1.

The Reid solution to the size-discretized integrodif-ferential equation of grinding kinetics yields a massbalance for the material in the i th size interval in abatch ball mill at any time t:

��ki(t)m i(t)�∑i�1

j�1kjbi�jmj(t); i�1, 2, .... (1)

where the parameter k i is the breakage rate function(also called the selection function) for the solids inthe i-th size interval which gives the fraction of mate-rial in size fraction i that is broken out in time t tot�dt, and bi�j is the breakage distribution functionand gives the fraction of progeny particles reportingto size i when unit mass fraction of particles of size j isbroken.

dmi(t)dt

If we consider the top size, first fraction only, then

��k1m1(t) (2)

for first-order grinding kinetics where the breakagerate function is independent of time. Under these con-ditions, the behavior of the top fraction is predicted bythe following relationship:

m1(t)�m1(0) exp (�k1t) (3)

Several different methods have been used to deter-mine breakage rate and breakage distribution func-tions experimentally. For example, individual interiorsize fractions have been replaced by radioactively-labeled size fractions of the same feed material, acomplicated technique that provides the greatestamount of information. In a simpler way, an insertfraction of a different material with some easily identi-fiable physical or chemical property has been used,such as using a quartz insert in a limestone bulk feed.Herbst and Fuerstenau [4] showed the mathematicaland experimental basis for obtaining these parame-ters by batch grinding single-sized feed particles. Thebreakage distribution function is normalizable andcan be considered invariant and will not be discussedfurther in this paper. However, there is a strongpower law dependence of the breakage rate functionon particle size, x:

k(x)�Axα (4)

where A is a constant depending on the material prop-erties. In many of their simulations, Kelsall et al. [5]assumed α�1.0. However, Herbst and Fuerstenau [4]showed that the value of α has the same value as thedistribution modulus of the Gaudin-Schuhmann sizedistribution of the comminuted product. To obtain thenecessary data, a series of batch grinds should be car-ried out in a test mill to quantify not only the break-age rate but also the breakage distribution functions.

Using a 254-mm diameter instrumented ball mill, aseries of batch grinding experiments was carried outdry by Herbst and Fuerstenau [4,5] with 7�9 mesh(2.8�2.0 mm) dolomite feed for various operatingconditions: mill speeds N*, ball loads Mb

*, and particleloads Mp

*. Details of the instrumented torque mill andexperimental procedures can be found in the thesepapers [4,6]. According to Eq. 3, a semilog plot of thefraction of feed material remaining in the top size vs.time in batch experiment will result in a straight linewhose slope is proportional to the breakage rate func-tion. Figure 1, which shows plots of batch dataobtained for selected mill speeds, ball loads, and parti-

dm1(t)dt

122 KONA No.21 (2003)

cle loads (for two different batches of dolomite feed)illustrates the appropriateness of Eq. 2 for the rangeof operating variables tested. This figure also illus-trates the strong dependence which feed disappear-ance kinetics have on mill operating conditions.

Herbst and Fuerstenau [4] experimentally grounda range of monosized dolomite feeds to obtain thedependence of breakage rate functions on particlesize (and also found the cumulative breakage distribu-tion function to be normalizable) and proved theirpredicted relation to zero order production kineticphenomena. Figure 7 given later, illustrates the de-pendence of breakage rate functions on particle sizefor dry ball mill grinding.

Using the values of breakage rate functions forparticles of different size fractions and the normal-ized breakage distribution functions, the Reid solu-tion to the batch grinding equation was programmedfor a digital computer and the grinding behavior wassimulated. Figure 2 presents a comparison of thecomputed size distributions with the experimentallyobserved distributions. The simulation based on thelinear batch grinding model indeed predicts the prod-uct size distribution over a wide range of grind times.

The very extensive experimental work carried outduring the last four decades has proven the validity ofthe linear grinding model and the size-dependence ofthe breakge rate functions and the normalizability of

breakage distribution functions. With confidence it,therefore, is possible to back calculate these parame-ters for simulation purposes.

THE NONLINEAR KINETIC MODEL FORROD MILL GRINDING

Rod mill comminution represents a nonlinear grind-ing system, one in which the order is less than unity.In Eq. 1, k1(t) represents the grinding rate functionfor monosized feed disappearance. There are twoforms of k1(t) which have physical meaning. The firstis

k1(t)m1(t)�k1(0)m1(0) (5)

which represents a “zero order” feed disappearancelaw where the feed grinding rate is constant for alltime. This would occur if the grinding zones were sat-urated with respect to feed-size material and all com-minution events are applied to feed-size particles. Thesecond form of the breakage rate function is

k1(t)m1(t)�k1(0)m1(t) (6)

KONA No.21 (2003) 123

MILL OPERATING CONDITIONS

FR

AC

TIO

N O

F F

EE

D S

IZE

PA

RT

ICL

ES

RE

MA

ININ

G

0 1 2 3GRINDING TIME, MINUTES

4 5 6

1.00

0.50

0.20

0.10

0.05

0.02

0.01

N* M*B M*

P

0.53 0.50 1.00.60 0.50 1.00.70 0.50 1.0

BATCH 2

0.60 0.35 1.00.60 0.50 1.00.60 0.50 1.6 BATCH 1

��

Fig. 1 The inf luence of mill operating variables on the kinetics ofdisappearance of 7�9 mesh dolomite in dry batch ballmilling.

7�9 MESH DOLOMITEN* �0.60M*

B�0.50EXPERIMENTAL

GRIND TIME,MIN.

CU

MU

LA

TIV

E F

RA

CT

ION

FIN

ER

TH

AN

ST

AT

ED

SIZ

E0 100

400 200 100 48 28 14

1000PARTICLE SIZE, MICRONS

PARTICLE SIZE, MESH

1.0

0.10

0.01

0.001

0.5 1 2 4 6 816

PREDICTED

Fig. 2 Simulation of the comminution of 7�9 mesh dolomite feedusing the Batch grinding equation and the experimentallydetermined values of the breakage rate and breakage dis-tribution functions.

or k1(t)�constant. This “first order” feed disappear-ance occurs when statistically independent commi-nution events are applied to an infinite particlepopulation in a completely mixed environment, asexemplified by dry batch ball mill grinding. Non-inte-ger order grinding does not have physical meaning,except possibly for systems that can be representedby an arbitrary linear combination of zero and firstorder kinetics. Such is the case for rod milling wherebridging of coarse particles between the rods de-creases the proportion of comminution events appliedto the fines.

Grandy and Fuerstenau [7] proposed that for com-minution systems which have apparent grinding or-ders less than one, the system can be modeled by aconvex linear combination of zero and first orderkinetics:

�φ1k1(0)m1(t)�(1�φ1)k1(0)m1(t) (7)

where φ1 and (1�φ1) give the respective fraction offirst and zero order kinetics assumed, For rod millgrinding with kinetics that follow the foregoing rela-tionship, the behavior of the top size material is pre-dicted by Eq. 8:

m1(t)� � (8)

functions are environment-dependent. Since thegrinding order is between zero and one, it is appropri-ate to fit Eq. 8 to the data. A semilog plot of [m1(t)�m1(0)(1�φ1)/φ1] vs. time should give a straight linewith slope equal to �φ1 k1(0). For the foregoing rodmill grinding data, it was found by trial and error thata value of 0.91 for φ1 gave the best straight line forfeed loads.

To carry out computer simulations of these rod millgrinding tests, the mathematical representation of theenvironment dependence is given by

ki(t2)�ki(t2)φ i�(1�φ i)� � where t1�t 2 (9)

Using the technique of Herbst and Fuerstenau [4] toestimate the size dependence of all the breakage rateparameters and the fraction of first-order grindinggiven in Figure 4, the dry batch rod milling of the7�9 mesh dolomite was accurately simulated. SinceThe simulated plots and experimental date fit exactly[7], those results are not presented here.

In a detailed investigation of the wet grinding ofdolomite in the batch ball mill, Yang [8] found thatplots of feed size breakage kinetics had the same

m1(t1)m1(t2)

(1�φ1)m1(0)φ1

m1(0) exp[�φ1k1(0)t]φ1

dm1(t)dt

shape as those shown in Figure 3 for rod mill grind-ing. This means that under the conditions of Yang’sexperiments (60% solids), the fine product particlestended to be suspended in the slurry inside the mill

124 KONA No.21 (2003)

7�9 MESH FEED

CHARGE WEIGHTgrams, dry

MA

SS F

RA

CT

ION

OF

FEE

D R

EM

AIN

ING

0 1 2 3 4 5 6 7

GRIND TIME, minutes

1.0

0.1

0.01

0.001

1320198033005280

Fig. 3 First order feed disappearance plot for rod mill grinding atseveral different feed loads in the batch mill.

M1(

t)�

p 1

0 1 2 3 4 5 6 7 8

GRIND TIME, minutes

0.1

1.0

� 91% FIRST ORDER 9% ZERO ORDER

7�9 MESH FEEDCHARGE WEIGHT

grams, dry1320198033005280

Fig. 4 Linear combination of first order and zero feed disappear-ance kinetics in dry rod milling.

p1�� �M1(0)�0.11�φ1

φ1

with the result that there was some preferential grind-ing of the coarser particles under those conditions.On the other hand, Klimpel [8] found that the disap-pearance kinetics plots for the fine wet grinding ofcoal at 57% solids density were first order but at 73%solids density became strongly concave as grindingprogressed. Under these conditions the slurry insidethe mill becomes increasingly viscous. As grindingprogressed the rate of disappearance of feed size coalwas significantly retarded. But by adding a polymericgrinding aid, which reduced the slurry viscositymarkedly, Klimpel [8] was able to return the grindingkinetics to first order.

LINEAR GRINDING KINETICS EXPRESSEDIN TERMS OF SPECIFIC ENERGY

In conducting their batch grinding experimentswith dolomite feed under a wide range of operatingconditions, Herbst and Fuerstenau [6] also accuratelymeasured the specific energy consumed by their mill.An anaysis of the grinding kinetics in the dry ball millrevealed that the size-discretized breakage rate func-tions are proportional to the specific energy input tothe mill and that the breakage distribution functionscan be taken as invariant. Taking the experimentalresults used to prepare the plots given in Figure 1for a range of mill speeds, mass of solids in the mill,and the mass of grinding balls in the mill, the resultsare replotted and presented in Figure 5 using spe-

cific energy as the independent variable instead oftime.

Based on that extensive experimentation, Herbstand Fuerstenau [6] showed that over a fairly widerange of conditions the feed size breakage rate func-tion can be approximated by

k1�kE1 � � (10)

where P is the power input to the mill, Mp is the massof feed material in the mill, and kE

1 is a constant. FromEq. 9, we can write the batch response for the firstinterval:

m1(t)�m1(0) exp ��kE1 � � t� (11)

Since the product of specific power and time is equalto the net specfic energy input to the mill, E

�, Eq. 10

can be expressed alternatively as

m1(E�

)�m1(0) exp (�kE1 E�

) (12)

The preceding analysis leads to expressing the batchgrinding equation in terms of specific energy ex-pended rather than in terms of grinding time as fol-lows

��kEi m i(E

�)�∑

i�1

j�1kE

i bi�jmj(E�

) (13)

The fact that breakage kinetics can be accurately ana-lyzed in terms of specific energy instead of time, hasbecome very useful, with regard to mill scale-up andthe analysis of other types of comminution systems,such as the roll mill.

Malghan and Fuerstenau [9] conducted a detailedstudy of the scale-up of ball mills using the populationbalance grinding model normalized through specificpower input to the mills. In their investigation, theyconstructed three instrumented and scaled batch ballmills : 5-inch (12.7 cm) diameter, 10-inch (25.4 cm)diameter, and 20-inch (50.8 cm) diameter. They dryground 8�10 mesh (2.4�1.7 mm) limestone. Theyfound that the feed size breakage rate functions for allthree different mill speeds, ball loads and feed loadsin each mill fell on a single line when the feed size dis-appearance was plotted as a function of expendedenergy. Likewise, the breakage distribution functionfor all of the grinds were self-similar, completely inde-pendent of mill size and mill operating conditions. Toillustrate the utility of the specific energy reducedbreakage rate function concept, Figure 6 presentsthe experimental size distributions obtained fromgrinding the 8�10 mesh limestone in the 20-inch mill

dm i(E�

)dE

PMp

PMp

KONA No.21 (2003) 125

FRA

CT

ION

OF

FEE

D S

IZE

PA

RT

ICLE

S R

EM

AIN

ING

0 0.5 1.0 1.5 2.0SPECIFIC ENERGY INPUT, KWH/TON

1.00

0.20

0.05

0.10

0.20

0.50

0.01

LEAST SQUARES FIT TO ALL THE DATA

MILL OPERATING CONDITIONS N* M*

B M*P

0.53 0.50 1.00.60 0.50 1.00.70 0.50 1.0 BATCH 20.90 0.50 1.00.60 0.35 1.00.60 0.50 0.800.60 0.50 1.00 BATCH 10.60 0.50 1.60

��

Fig. 5 Feed disappearance kinetics normalized with respent tospecific energy in kWh per ton of feed material.

and the simulation of grinding using the breakageparameters determined from grinds carried out in the5-inch mill.

Malghan and Fuerstenau [9] also showed that if thebreakage rate function were expressed in time ratherthan specific energy, for the range of mill speeds, ballloads, and feed loads, it would be proportional to D0.56

and proportional to (D�db) where D is the mill diam-eter and db is the ball diameter. The reduced break-age rate function simplifies analysis and designconsiderably since there would be no necessity forachieving kinematic and loading similarity. Scalingwith time would require that the mills be compared atthe same fraction of critical rotational speed.

Expressing grinding kinetics in terms of specificenergy instead of grinding time also clarifies the com-plex behavior observed in wet grinding at high solidscontent [10].

MODELING NONLINEAR HIGH-PRESSUREROLL MILL COMMINUTION

After single-particle breakage, the next most effi-cient method of comminution is particle-bed com-minution [11,12]. In this mode, comminution occursprimarily by very high localized interparticle stressesgenerated within the particle bed. No separate carrieris employed for the transport of energy to the solids,unlike in tumbling mills. Particle bed comminution is

carried out continuously in a device comprised of twocounter-rotating rolls. As the feed particles passthrough the roll gap, the particle bed is compressedand the coarser particles undergo an isostatic-likecompression by the fine particles in which thecoarser ones are embedded. Energy is lost in thehigh-pressure roll mill due to friction between the par-ticles as they pass through the roll gap and due to theineffectiveness of the isostatic loading phenomenon.

There really is no explicit running grinding time inthe high-pressure roll mill, only a fixed time of pas-sage of solids through the grinding zone of the rollsin a more or less plug f low manner. Therefore, inorder to simulate roll-mill grinding, it is necessary toformulate the population balance model in terms ofenergy input to the mill. In addition, as the particlebed or column passes down through the rolls, it iscompacted and densified and compacted more andmore with an increasing rate of energy dissipationdue to interparticle friction and incipient visco-plasticf low. Consequently, the energy component that actu-ally goes into stressing the particle to fracture is pro-gressively reduced. Kapur et al. [13] proposed thatthe increase in retardation of the breakage rate withenergy input can be incorporated into the populationbalance model by defining a rescaled energy:

E′� E1�y where 0�y�1 (14)

For high-pressure roll mill grinding, the populationbalance equations for grinding kinetics can be formu-lated in terms on cumulative energy input, taking intoaccount energy dissipation in accordance with Eq. 13,to yield the following relation:

�� m i(E)�∑i�1

j�1bi�jmj(E) (15)

Fuerstenau et al. [13] determined the product sizedistributions for grinding quartz, limestone anddolomite at various energy expenditures in a labora-tory high-pressure roll mill (roll diameter of 200 mm)and from the results estimated the breakage rate andbreakage distribution parameters. For each of thesematerials, the breakage distribution functions haveessentially the same shape as those found for othercomminution systems, such as the ball mill and rodmill. However, the breakage rate functions all remainquite high at all particle sizes, unlike in a ball millwhere the specific rate constants drop sharply withparticle size, especially in the medium and fine parti-cle size range (the slope of a log-log plot of k-vs-sizebeing the distribution modulus). Figure 7 compares

koj

Ey

ko1

Ey

dmi(E)dE

11�y

126 KONA No.21 (2003)

CU

MU

LAT

IVE

WE

IGH

T F

RA

CT

ION

FIN

ER

TH

AN

ST

AT

ED

SIZ

E

10 20 50

400 200 100 48 28 14 8

PARTICLE SIZE, MICRONS

0.25

SIMULATED USING5 IN. MILL k i

E AND Bi1

0.500.740.99

1.491.98

PARTICLE SIZE, MESH

1.0

0.5

0.2

0.1

0.05

0.02

0.01100 200 500 1000 2000 4000

DRY GRINDING SIMULATION20 IN. MILLN*�0.5, M*

B�0.4, M*

P�1.0

�E (KWH/T)

Fig. 6 Comparison of experimentally determined size distribu-tions and those Predicted for grinding 8�10 mesh lime-stone in a 20-inch (50.8 cm) diameter mill, using breakagerate and breakage distribution functions obtained fromgrinding the same material in the 5-inch (12.7-cm) mill.

the effect of particle size on the breakage rate func-tion for grinding dolomite in the high-pressure rollmill with grinding the material in a dry ball mill. As aconsequence of this, the pressurized roll mill can per-form size reduction much more efficient energy-wisethan the ball mill. This probably explains the greaterenergy efficiency of the high-pressure roll mill atlower reduction ratios. The reason that the breakagerate functions for the high pressure roll mill appear tohave only a small particle size effect must result fromthe energy transfer mechanism in the roll mill,namely that the high compression stresses are trans-ferred from one particle to another as they passthrough the roll gap, in contrast to the probabilisticnature of stress transfer in a ball mill. This apparentlyis the crucial difference between the two kinds ofgrinding mills. As a consequence, the pressurized rollmill can perform size reduction tasks much more effi-ciently energy-wise than the ball mill (at lower reduc-tion ratios).

With the breakage rate parameters estimated fromthe high-pressure roll milling experiments, the sizedistributions of dolomite, limestone, and quartz weresimulated with the population-balance grinding equa-tion that had been suitably modified to account for

energy dissipation in the roll gap (Eq. 15). Figure 8shows the simulation of grinding quartz in the high-pressure roll mill at three different energy levels [13].The simulations of the size distributions of the rollmill products are in good agreement with the experi-mentally determined size distributions.

MODELING THE KINETICS OF GRINDINGDAMAGED PARTICLES

Significant energy savings can be gained by usinga two-stage grinding system which utilizes the effi-ciency of the high-pressure roll mill at low reductionratios and the higher efficiency of ball mills at highreduction ratios [14]. As discussed in the foregoingsection, particle breakage in the roll mill occursthrough interparticle loading of the feed as it passesthrough the roll gap. Because the particles are notloaded by impact, as in ball milling, but by directtransmission of stresses from one particle to another,high-pressure roll mill grinding results in progenyparticles that are broken, cracked, damaged or other-wise weakened. Consequently, the particles are dis-tributed in strength which implies that the rate atwhich these particles will be ground in a ball mill willalso be distributed. As already discussed, the linearpopulation balance model used routinely for the simu-

KONA No.21 (2003) 127

NO

RM

ALI

ZED

BR

EA

KA

GE

RA

TE

FU

NC

TIO

N, k

i/k

1

SIZE INDEX, i

1

0.1

0.01

0.001

DOLOMITE2.36�3.35 mm

Ball MillHP Roll Mill

13 11 9 7 5 3 1

Fig. 7 The effect of particle size on the estimated breakage ratefunctions for dolomite being ground in the high-presssureroll mill and a ball mill (size index 1�8�10 mesh here).

Cum

ulat

ive

Fine

r, %

12 11 10 9 8 7 6 5 4 3 2 1 0Size Index, i

100

1098765

20

30

40

5060708090

4

Quartz(8�10 mesh feed)

SimulatedFunctional formFine tuned

8 mesh

Expt’l Energy(kWht�1) 3.11 2.19 1.17

Fig. 8 Experimental and simulated size distributions of quartzground in a high-pressure roll mill at different energy lev-els.

lation of ball mill grinding of ordinary feeds assumesa single-valued rate parameter for particles of a givensize fraction.

In order to successfully use the population balanceapproach to model the ball mill grinding of materialthat has first been ground in the high-pressure roll,we need to account for the pronounced heterogeneityin strength (or ease of grinding) of the high-pressureroll mill product particles [15]. We have noted earlierthat the parameter A in Eq. 4 is a function of the mate-rial properties. In the simulation of batch ball millingof primary particles, A could be assumed to be con-stant, without affecting the quality of simulation. Wecan account for the heterogeneity in the strength ofhigh-pressure roll-milled particles by letting A be vari-able. It is reasonable to say that A is a measure of theease with which the particles could be broken, that is,a measure of the grindability. The larger the value ofA, the easier it would be to break the particle. Whileprimary particles are expected to be relatively uni-form in strength, the particles in the high-pressureroll mill product are damaged and weakened to dif-ferent extents, that is, that are distributed in A. Weassume, however, that the distribution is independentof the particle size. In our analysis, we consider thatthe distribution in the grinding rate parameter, k, ofthe high-pressure roll mill product can be describedby a modified gamma function:

m(A)� exp[�λ(A�Ao)]; A o�A� (16)

where A o is the minimum grinding rate constant en-countered in the ball mill. And β and λ are the shapeand scale parameters of the distribution. By incorpo-rating Eq. 16 into Eq. 12, and integrating, we obtainthe expression for the mass fraction retained on size xafter an energy expenditure of Eb in ball milling thehigh-pressure roll mill product:

R(x,Eb)�R(x,0) (17)

The shape and scale parameters are functions of theenergy expenditure in the high-pressure roll millingstage.

These concepts were tested through a detailedstudy of the hybrid comminution of bituminous Pitts-burgh No. 8 coal, that is for the open-circuit ballmilling of coal that had first been ground in the high-pressure roll mill. Figure 9 presents the experimen-tal results of the mass fraction of ball mill productretained on a 200-mesh sieve (74 µm) as a function ofthe energy expended for ball mill grinding the pres-

exp(�AoxαEb)(1�xαEb/λ)β

λβ(A�Ao)β�1

Γ(β)

surized roll mill product, together with the computedresults. In these experiments, the coal was firstground in the high-pressure roll mill (HPRM) at fourdifferent energy levels. Figure 10 presents the ex-perimental and computed size distributions of theground coal after it had been comminuted in the ballmill at different expenditures of energy in the ball millstep after first being roll-milled at an energy expendi-ture of 2 kWh/t. The simulated results given in Fig-ure 10 are in excellent agreement with experiment.

The distribution in the grindability of coal particles

128 KONA No.21 (2003)

FRA

CT

ION

RE

TAIN

ED

AT

200

ME

SH

0 1 2 3 4 5 6

GRINDING ENERGY IN THE BALL MILL, kWh/t

HYBRID BATCH GRINDINGPITTSBURGH NO. 8

1.00.9

0.8

0.7

0.6

0.5

0.4

0.3

EHPRM

1.14 kWh/t2.00 kWh/t3.16 kWh/t3.74 kWh/t

Fig. 9 For the hybrid grinding of Pittsburgh No. 8 coal, the frac-tion retained on a 200-mesh sieve (74 µm) sieve as a func-tion of the energy expended in the ball mill on materialthat had been comminuted in the high-pressure roll mill atdifferent HPRM energy levels: experimental data pointsand simulated curves.

CU

MU

LAT

IVE

FIN

ER

, per

cent

10 100 1000

PARTICLE SIZE, microns

computedOPEN-LOOP HYBRID GRINDING

PITTSBURGH NO. 8

100

10

EbmkWh/t

0.330.651.302.605.20

EHPRM�2.00 kWh/t

Fig. 10 Experimental and computed size distributions of Pitts-burgh No. 8 coal produced by open-circuit hybrid ballmill grinding of material that had first been ground at anenergy expenditure of 2 kWh/t in the HPRM.

in the high-pressure roll mill product was assessed byfitting the mathematical model to the experimentaldata. Figure 11 was constructed to show plots of thecalculated density functions m(k) representing thedistribution of breakage rate functions of 200-meshparticles of Pittsburgh No. 8 coal produced at energyconsumptions of 1.14, 2.0, 3.16 and 3.74 kWh/t in theHPRM. Although the coal is discharged from theHPRM in a briquetted form, the individual particlesare easily separated by stirring in aqueous methanolto determine their size distribution, if desired [15].The fact that the mean of the distributions shiftstowards higher grinding rate values with increasingenergy input to the high-pressure roll mill stronglysuggest that these results and concepts are phenome-nologically meaningful.

OPTIMIZATION OF OPEN-CIRCUIT HYBRIDGRINDING SYSTEMS

The non-linearity of the kinetics of grinding pri-mary particles in the high-pressure roll mill and thesubsequent ball milling of the roll mill product pre-sent us with an interesting optimization problem. Foropen-circuit hybrid grinding, our goal would be tofind the optimum energy expenditure in the high-pressure roll mill that would result in maximum parti-cle damage at minimum energy dissipation, and theenergy expenditure range in the ball mill where theenergy is primarily spent in breaking the damaged

particles, that is the range where the ball mill grind-ing of damaged particles is essentially non-linear. Fig-ure 12 presents the simulated production of minus200-mesh product and the specific energy expendi-ture per ton of product, as a function of the energyinput in the high-pressure roll mill and the ball millper tonne of feed for the open-circuit grinding of Pitts-burgh No. 8 coal.

The specific grinding energy contours are shown assolid lines, and the contours of percent fines pro-duced are presented as dashed lines. The simulationresults indicate that the percentage of minus 200-mesh produced increases with the increase in grind-ing energy input in the ball mill for a given energyexpenditure in the high-pressure roll mill. However,for a fixed energy input in the ball mill, the percentageof minus 200-mesh material produced goes through amaximum with increasing energy input in the high-pressure roll mill. As the production of minus 200-mesh particles increases, the high-pressure roll millenergy at which the maximum occurs shifts to higherenergy values. The specific energy required to pro-duce a unit weight of minus 200-mesh product, on theother hand, increases with the increase in energyinput in both the ball mill and the high-pressure rollmill (except in the region defined by high-pressureroll mill energy between 1.75 kWh/t and 2.5 kWh/tand ball mill energy greater than 3.5 kWh/t). At agiven specific energy consumption, particularly athigher percentage of fines production, an optimumpartition exists between the energy input in the high-

KONA No.21 (2003) 129

DIS

TR

IBU

TIO

N F

UN

CT

ION

, m(k

)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

GRINDING RATE CONSTANT, k

HP ROLL MILL PRODUCTPITTSBURGH NO. 8

4

0

8

12

16

20

Ehp�1.14

Ehp�3.16

ko�0.093

Ehp�3.74 kWh/t

Ehp�2.0

Fig. 11 Density functions for the distribution of the specific ballmill grinding rates of Pittsburgh No. 8 coal produced inthe high-pressure roll mill at different energy inputs.

HP

Rol

l Mill

Ene

rgy,

kW

h/t

0 1 2 3 4 5

Ball Mill Energy, kWh/t

1

1.5

2

2.5

3

3.5

4 10% 20% 30% 40%

22 16

13

13.5

50%

15

14 kWh/t

12

10% 20% 30% 40%

14 kWh/t

Fig. 12 Relationship between specific grinding energy (solidlines) and the percentage of minus 200 mesh (74 µm)fines produced (dashed lines) at different energy inputsin the high-pressure roll mill and the ball mill in open-cir-cuit hybrid grinding of Pittsburgh No. 8 bituminous coal.

pressure roll mill and the ball mill in order to achievea maximum percentage of minus 200-mesh product.For example, at a specific energy consumption of13.25 kWh/t of minus 200-mesh product, the energyinput should be 2.5 kWh/t of feed in the high-pressureroll mill and 3.2 kWh/t in the ball mill in order to pro-duce a comminuted product with 43 percent of minus200-mesh fines.

Because of the visco-plastic nature of coal, it is pos-sible that the potential energy saving resulting fromthe fracturing and weakening of the particles as theyare stressed in the HPRM could be more than offsetby the increased briquetting at the high compressivestresses at higher energy expenditures in the pressur-ized roll mill step. Thus, a maximum benefit mightbe expected from the hybrid grinding of coal at anoptimal energy expenditure in the high-pressure rollmill. In Figure 13, the percentage of energy savedthrough the hybrid grinding mode relative to the en-ergy expended in ball mill grinding alone for the pro-duction of a fixed amount of fines is plotted againstthe energy expended in the HPRM step. These plotsclearly show that not only is there an optimum for theenergy input to the high-pressure roll mill but alsothe energy saving is negated if the energy expendi-ture in the high-pressure roll mill is higher than athreshold value, which would depend on the nature ofthe coal and the percentage of fines produced. Thereis a similar optimum for the hybrid grinding of miner-als than can deform plastically, such as calcite anddolomite [14].

SUMMARY

Industrial comminution processes are inherentlynonlinear. The extent of nonlinearity is governed bythe mill characteristics, comminution environment,and the material properties. However, over a limitedrange of time or energy input, the processes could betreated as linear and modeled accordingly. For in-stance, in the initial stages of dry batch ball milling,grinding kinetics follow a linear model. Linear break-age kinetics prevail in the mill as neither the probabil-ity of breakage of a particle (as measured by thebreakage rate function) nor the distribution of frag-ments resulting from the primary breakage of thatparticle (as measured by the breakage distributionfunction) is significantly inf luenced by the size con-sist in the mill. In the rod mill, on the other hand,where the larger particles are ground preferentially,the initial grinding behavior is significantly nonlinear.The nonlinearity can be modeled in such cases bysuitably modifying the breakage probability function.

The change in the grinding environment in themedia mills, over time, also leads to nonlinear com-minution kinetics in these mills. This is primarilybecause of reduced energy utilization at larger grind-ing times due to particle shielding under dry grindingconditions, or increased slurry viscosity in wet grind-ing. Such nonlinearities can be easily resolved byrecasting the grinding model equations in terms ofenergy input, instead of the grinding time. Expressinggrinding kinetics in terms of energy instead of timealso leads to simplifications in grinding mill scale up.

For confined particle bed grinding, as in high-pressure roll milling, due to the energy dissipationthrough interparticle friction and isostatic stresses athigher confining pressures, can only be modeled as anonlinear process. A modified population balancemodel, that explicitly handles the retardation in en-ergy utilization, has been presented. The modifiedmodel results in fairly accurate simulation of high-pressure roll mill grinding.

High pressure roll milling results in product parti-cles that may become can be weakened though thegeneration of f laws and microcracks and conse-quently exhibit widely distributed strength behavior.This leads to nonlinearities in the regrinding of suchparticles in media mills. We have incorporated thisdistribution in particle strength into the batch ballmill grinding model and successfully simulated theopen-circuit hybrid grinding of coal, where coal parti-cles are first ground in the high-pressure roll mill andthen in the ball mill. Experimental as well as simula-

130 KONA No.21 (2003)

TO

TAL

SPE

CIF

IC G

RIN

DIN

G E

NE

RG

Y, k

Wh/

t

0 1 2 3 4

SPECIFIC ENERGY EXPENDED IN HP ROLL MILL, kWh/t

HPRM/BM Hybrid GrindingPittsburgh No. 8

2

0

4

6

8

10

40% minus 200-mesh

30%

20%

Fig. 13 Relationship between the total specific comminution en-ergy and the energy expenditure in the high-pressure rollmill for the production of minus 200-mesh fines in thehybrid grinding of Pittsburgh No. 8 coal.

tion results indicate that optimum specific energy perton of product of desired fineness could be achievedby operating each of the mills in appropriate regimes.The optimal operating conditions and the energy par-titioning would be dictated by the material propertiesand fineness of product.

REFERENCES

[1] L. Bass (1954), Zeits. Angew. Math. Phys., 5, 283-292.[2] R. P. Gardner and L. G. Austin (1962), Symposium

Zerkleinern, 1st European Symposium on Comminu-tion, ed. H. Rumpf and D. Behrens, Verlag Chemie,Weinheim, 217-248.

[3] K. J. Reid (1965), Chemical Engineering Science, 20,953-963.

[4] J. A. Herbst and D. W. Fuerstenau (1968), Trans.SME/AIME, 241, 528-549.

[5] D. F. Kelsall, K. J. Reid, and C. J. Stewart (1969-1970),Powder Tech., 3, 170-179.

[6] J. A. Herbst and D. W. Fuerstenau (1973), Trans.

SME/AIME, 254, 343-347.[7] G. A. Grandy and D. W. Fuerstenau (1970), Trans.

SME/AIME, 247, 348-354.[8] R. R. Klimpel (1982), Powder Technology, 32, 267-277.[9] S. Malghan and D. W. Fuerstenau (1976), in Zer-

kleinern, Dechema-Monographien, Verlag Chemie,GMBH, 613-630.

[10] D. W. Fuerstenau, K. S. Venkataraman and B. V.Velamakanni (1985), International Journal of MineralProcessing, 15, 251-267.

[11] K Schoenert (1988), International Journal of MineralProcessing, 22, 401-412.

[12] D. W. Fuerstenau and P. C. Kapur (1995) PowderTechnology, 82, 51-57.

[13] D. W. Fuerstenau, A. Shukla and P. C. Kapur (1991),International Journal of Mineral Processing, 32, 59-79.

[14] A. De (1995), Modeling and Optimization of FineGrinding of Minerals in High-Pressure Roll Mill/BallMill Hybrid Comminution Circuits, Ph.D. Thesis, Uni-versity of California, Berkeley.

[15] D. W. Fuerstenau and J. Vazquez-Favela (1997), Miner-als and Metallurgical Processing, 13, 41-48.

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Author’s short biography

D. W. Fuerstenau

D. W. Fuerstenau received his Sc.D. degree in metallurgy (mineral engineering)from MIT in 1953. After a period in industry, he joined the faculty in the Depart-ment of Materials Science and Engineering of the University of California at Berke-ley in 1959, where he continues as Professor in the Graduate School. He has beenactively involved over the years in research on the processing of minerals and par-ticulate materials, including extensive research on interfacial phenomena in thesesystems. Areas in which he has worked extensively include comminution, agglom-eration, mixing, f lotation and applied surface and colloid chemistry.

P. C. Kapur

P. C. Kapur received his M. S. degree in 1964 and his Ph.D. degree in materials sci-ence and engineering in 1968 from the University of California at Berkeley. Afterworking at the Colorado School of Mines Research Foundation, he returned toIndia as Professor of Metallurgical Engineering at the Indian Institute of Technol-ogy at Kanpur. Since retiring from IIT Kanpur in 1995, he has been on the staff ofTata Research Development and Design Centre as Consulting Advisor. He iswidely recognized for his extensive research on the processing of particulate mate-rials, mainly agglomeration, comminution and size distributions, and particularlythe mathematical modeling and computer simulation of processing operations.

A. De

A. De received his B. Tech. degree in 1981 and M. Tech. degree in 1983 in metal-lurgical engineering from the Indian Institute of Technology at Kanpur. In 1994,he received his Ph.D. in materials science and engineering from the Universityof California at Berkeley. At Berkeley his major research efforts were directedtowards the comminution of coal, including experimental investigations and math-ematical modeling. He currently is Senior Software Engineer with ABB, Inc.