Ž .Powder Technology 110 2000 246–252www.elsevier.comrlocaterpowtec

Monte Carlo simulation of particle breakage process during grinding

B.K. MishraDepartment of Materials and Metallurgical Engineering, Indian Institute of Technology, Kanpur, India

Received 1 January 1999; received in revised form 1 August 1999; accepted 7 December 1999

Abstract

The Monte Carlo method is quite useful in the modeling of particulate systems. It is used here to simulate the particle brekage processduring grinding that can be represented by a population balance equation. The simulation technique is free from discretization of time orsize. The results of simulation under restricted conditions of grinding compare very well with the available analytical solution of thepopulation balance equation. The procedure is extended to simulate the grinding process in its entirety. This method provides analternative to the modeling of the grinding process where the governing population balance equation cannot be readily solved. q 2000Elsevier Science S.A. All rights reserved.

Keywords: Monte Carlo simulation; Grinding; Population balance

1. Introduction

Mathematical models of comminution processes pro-vide a quantitative description of the size reduction phe-nomena. These models are used for the purpose of simula-tion in order to improve the performance of the millthrough proper design and control. These also allow pre-diction of the size spectra of ground particles duringcomminution. A general form of the population balanceequation for grinding is as follows:

dw x w x w x w xn d ,t s Inflow y Outflow q Birth y DeathŽ .

d t1Ž .

Ž .Here n d,t is the number concentration of particles be-tween size d and dqdd at time t. The birth and deathterms account for increase or decrease in the number ofparticles due to agglomeration, attrition, brekage etc. In-corporating all the sub-processes of grinding into the aboveequation results in an integro-differential equation, whichin its most general form is highly intractable, and in someinstances insoluble. A review of the solution methods ofthe population balance equation is given by Ramakrishnaw x1 . The methods that are known to work on the continuouspopulation balance equation include: method of momentsw x w x2 , method of weighted residuals 3 , similarity solution

w x w x4 , and Monte Carlo method 5 . The purpose of this paperis to solve the most fundamental grinding equation incontinuous-time and continuous-size by using the MonteCarlo method. This numerical approach is conceptuallysimpler and computationally efficient compared to manyother available solution methods.

The Monte Carlo method is a numerical simulationtechnique for solving problems by means of random sam-pling. This technique is generally applied to analyze physi-cal systems where direct experimentation is impossible ormathematical problems that cannot be solved by directmeans. It has been successfully applied to study diverseproblems in the area of semiconductor to spread of choleraepidemic.

The Monte Carlo method can be directly applied insimulation of particulate systems. Mathematical descrip-tion of particulate systems involves integro-differentialequations resulting from population balance whose solu-

w xtion is complex and time-consuming. Shah et al. 5 usedthe Monte Carlo method to solve problems involving

w xparticulate systems. Rod and Misek 6 have also used thismethod to simulate the dispersion and formation of dropsin agitated liquid–liquid systems. The difference in theirapproaches lies in the manner in which the Monte Carlomethod is implemented. In the former case it is the‘‘event-driven’’ approach, and in the latter, ‘‘time-driven’’approach is used to simulate the particulate systems. A

0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved.Ž .PII: S0032-5910 99 00281-8

( )B.K. MishrarPowder Technology 110 2000 246–252 247

connection between these two approaches is established byw xRajamani et al. 7 by considering the dispersion of bub-

bles in liquid–liquid systems. The two approaches differ inthe manner in which the random waiting times are decided.In the time-driven approach the time step involved in thesimulation is pre-specified. The time step in the event-driven approach is decided by considering a random wait-ing time, which is generated by a known distributionfunction. A detailed mathematical treatment of quiescentinterval distributions for mono-particle, bi-particle, and

w xmulti-particle events is described by Shah et al. 5 . An-other application of Monte Carlo method in the area ofparticulate system is size sorting and particle segregation.Recently, this technique is successfully used by Rosato et

w x w xal. 8 and further extended by Castier et al. 9 to simulateparticle segregation due to shaking and vibration whichhas immense industrial application.

Industrial grinding in ball mill also involves particulatesystems, and therefore, it becomes amenable to MonteCarlo analysis. In a ball mill particles are constantlybroken within a size class and reappear in another sizeclass. The population balance models can describe thischange of particle size with time during milling. However,except for some special cases, analytical solution of thegoverning population balance equation is not feasible.Therefore, numerical techniques are used to arrive at asolution. In this work, the time-driven Monte Carlo ap-proach is used to simulate the particle breakage process ina ball mill. The algorithmic detail of the numerical methodand its computer implementation is discussed. Computa-tional results of the simulations are compared with theanalytical solution for a very special case of the grindingprocess where the analytical solutions of the populationbalance model is available.

2. Theory of ore grinding

In the population balance approach, the average behav-ior of the particles during grinding is considered. Twobasic descriptive functions such as the specific rate ofbreakage S and the breakage distribution function B areused to describe the average behavior of many particles.The equation governing ore grinding leading to breakage isan integro-differential equation of the following form

`EP d ,tŽ .X X X Xs S d B d ,d p d ,t dd 2Ž . Ž . Ž . Ž .H

Et d

Ž .where P d,t is the cumulative size distribution functionwhich is the fraction of particles in the population by mass

Ž X .whose size is less than d at time t, p d ,t is the corre-Ž X .sponding density function, B d,d is the breakage distri-

bution function, which is the cumulative amount of parti-cles by mass less than size d broken out of the parent size

X Ž .d ,S d is the specific rate of breakage of particle of sizeclass d. By definition,

dXXb d ,d dds1 3Ž . Ž .H

0

and

dX XB d ,d s b d ,d dd 4Ž . Ž . Ž .H0

Ž X.where b d,d dd is the fraction of particles in the popula-tion of size d and dqdd produced due to breakage ofparticles of size dX.

There is no straightforward analytical solution of thepopulation balance equation. However, assuming certainfunctional forms for the parameters involved in the one

w xcan arrive at the solution 10 . The standard functionalŽ X. Ž .forms of B d,d and S d for any ore subjected to

grinding, are generally assumed as follows

a1 a2X X XB d ,d sQ drd q 1yQ drd 5Ž . Ž . Ž . Ž . Ž .and

cS d sk drd 6Ž . Ž . Ž .max

where a1, a2, c, and Q are constants whose values dependon the physical properties of the ore, d is the largestmax

diameter of the particle in the population. For the specialcase, where it is assumed that

S dX b d ,dX sa k day1 7Ž . Ž . Ž .0

Ž .the analytical solution of Eq. 1 can be arrived at by usingŽ . Ž .Eqs. 2 and 3 as

aP d ,t s1y 1yP d ,0 exp k d t 8Ž . Ž . Ž .0

A general close-form analytical solution of the integro-differential equation of grinding is apparently an insolubletask. Therefore, there arises a need for discretization inorder to make the model equation less intractable. Thediscretized equation is arrived by assuming the mass frac-tion m in the ith size class asi

dim s p d ,t dd 9Ž . Ž .Hi

diq1

and integrating both sides of the continuous model equa-tion between d and d with further assumption that theiq1 i

size interval is small enough so that

p d ,t sp t 10Ž . Ž . Ž .i i

one arrives at the discretized solution as

iy1Em m md di i ji isy S d d d qŽ . Ž . ÝH HEt d yd d ydd di iq1 j jq1iq1 iq1js1

=dj X X XS d b d ,d d d d d 11Ž . Ž . Ž . Ž . Ž .H

djq1

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Defining the size discretized selection and breakage func-tion as

1 diS s S d d dŽ . Ž .Hi d yd di iq1 iq1

1 d di j X X Xb S s b d ,d S d d d d dŽ . Ž . Ž . Ž .H Hi j j d yd d di iq1 iq1 jq1

12Ž .

the most familiar form of the size discretized grindingequation is obtained. Thus, the size discretized grindingequation is

iy1dmisyS m q b S m 13Ž .Ýi i i j j jd t js1

This equation can be solved by several techniques that arew xwell documented in the literature 1 . In this paper, an

attempt is made to solve the size-continuous and time-con-tinuous population balance grinding equation by using theMonte Carlo method.

3. The Monte Carlo method

The goal of the Monte Carlo method in the grindingcontext is to simulate the evolution of the size spectra byrandom sampling from known functions that describe thegrinding process. These are selection and breakage func-tions. The ideal method for selecting a particle for break-age and subsequently determining the daughter size is

Ž .sketched in Fig. 1. A uniform random number R 0,1 isgenerated, and the abscissa corresponding to the distribu-tion function value of R is read, which gives the size orsize class of the particle to be selected. This approach turnsout to be more time consuming, as always the cumulativedistribution function needs to be updated after every calcu-lation cycle. An alternative to using the cumulative distri-

w xbution function is to use the acceptance–rejection 6method which is computationally much more efficient.

In the acceptance–rejection method, given a densityŽ . Ž .function f d breakage or selection function the limiting

Fig. 1. Drawing random sample from the cumulative distribution func-tion.

values of the argument and the maximum value of thefunction f are determined. Here, size d is considered asM

the argument, which is bounded between 0 and d . Amax

pair of random numbers is generated and the followinginequality is examined

ŽIf the above is fulfilled, then the corresponding size dmax.=R is accepted since it is going to follow the density1

Ž .function f d . In the opposite case, the trial is rejected andthe procedure is repeated.

4. Computational procedure

In order to apply the Monte Carlo technique to simulatethe grinding process certain assumptions are made. Firstmonosize particles are used as the starting material. Whenthese particles break, progeny particles are generated thatappear in lower size classes. It is assumed that at any giventime a single particle can only break. Furthermore, afterany breakage event, the entire mass of the parent particle isshifted from its size class to a size class below, which israndomly chosen within the framework of Monte Carlomethod. A schematic of this breakage process is shown inFig. 2. In addition to above, functional forms for S and bi i j

are assumed which are treated as probability density func-tions. A time-driven Monte Carlo technique is applied tocompute the size distribution of the particles during grind-ing. The algorithmic details of this technique are describedin the following steps.

4.1. Start

Initially, all the breakage frequencies are summed overall the particles to obtain S . The critical time step for theb

simulation, D t, is calculated as

D ts1rrSb

where r is an arbitrary constant termed as tuning factor.The physical significance of the above equation is thatlarger the particle size, smaller is the value of D t. In otherwords, larger particles break at a faster rate.

4.2. EÕent selection

In the time driven Monte Carlo approach, at any time itis essential to ascertain whether a breakage event occurred.This is done by generating a random number, R, such that

if RFS =D t: breakage occurs, calculation proceedsb

with increment of timeelse: simply increment the simulation time

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4.3. Identification of particle for breakage

A particle is chosen for breakage by using anotherrandom number and the selection of this particle is biasedtowards the larger particle size. This particle is selectedbased on the following condition

if RFS rS : accept the particle for breakagej max

else: reject the particle for breakagewhere S is the breakage rate for a particle in the jth sizej

class and S breakage rate of the largest particle. Themax

above inequality physically means that the probability ofbreakage of larger particle is more than that of the smallerparticle.

4.4. Identification of daughter particle

Once the particle is selected for breakage, the size ofthe daughter particle is determined by using a pair ofrandom numbers and the breakage density function in thefollowing way

Let, d sR=di j

if RFb rb : accept the size class as daughter sizei max

classelse: choose another size class

4.5. Update data

The total number particles in each size class and theoverall breakage frequency S is updatedb

N

S s SÝb iis1

and the calculation cycle is repeated from step 2.In step 3, the particle to be broken is identified. The

mass of this particle is added to the size class of thedaughter particle as determined through step 4 and accord-ingly the mass of the parent size class is updated. At theend of a calculation cycle, the maximum size of theparticles present and the amount of material in each size

Fig. 2. Schematic of the breakage process for a single particle.

Fig. 3. Flow chart of the Monte Carlo method.

class are updated. Another cycle of breakage events isinitiated and this process is repeated until the total timeexceeds the desired time of simulation. Breakage of verysmall particles is ignored. These particles are thought ofthose that would remain in the smallest size class. A flowdiagram of the calculation cycle is given in Fig. 3.

5. Numerical results

The simulation program is written in a modular fashionin structured C language. The program is implemented in a

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Hp 735 workstation. It uses the intrinsic pseudo randomnumber generator available in the computer: drand48. Thetwo important components of the program that deal withthe selection and the breakage of particles merit a detaileddiscussion. The algorithms of these components are shownin the flow diagram within the dotted rectangular boxes.The top rectangular box shows the selection algorithm. Itdeals with the rate at which particles are broken and it

Ž .factors as S d in the grinding equation. The kinetics ofthe grinding process is such that the breakage of monosizeparticles is similar to one that is obeyed by the moleculesparticipating in a first order chemical reaction. Thus, thefirst-order hypothesis as applied to the top size material inbatch grinding reduces the grinding equation to

dw tŽ .1syS w t 15Ž . Ž .1 1d t

whose solution is

w t S tŽ . Ž .1 1log sy 16Ž .ž /ž /w 0 2.3Ž .

Ž .where w t is the mass fraction of the total material in the1Ž .top size interval at time t, w 0 is the initial mass, and S1

is the breakage rate constant. It is always interesting tofind out whether the grinding follows the first-order break-age hypothesis.

The selection algorithm is tested to check the first-orderbreakage hypothesis. It is argued that if there are Nmonosize particles, all having the same rate constant orselection constant S then an event is selected in thefollowing way:

P N™Ny1 sSN t D tŽ . Ž .P null s1ySN t D t 17Ž . Ž . Ž .

Ž .It is clear from the above that as time progresses N tdecreases and the selection becomes slower. Above algo-rithm is tested using 100 particles. The results of simula-tion are shown in Fig. 4 for two breakage rate constants:1.0rmin and 0.5rmin. Fig. 4 shows an excellent correla-tion between the analytical and simulated results.

Fig. 4. First-order kinetics behaviour during grinding: A comparisonbetween Monte Carlo and analytical results.

Fig. 5. Breakage characteristics of single particles. A comparison betweenMonte Carlo and analytical results.

In a real situation, particles from different size classesare to be selected for breakage. It can be easily done oncethe probability density function for the selection process isknown. There are algebraic equation established for theprocess that relates the breakage rate to size. This algebraicequation can be used to draw samples to accumulate theoverall selection behavior in a multi-size particle environ-ment. It may be realized that during the simulation theselection function in the cumulative form has to be up-dated each time the particle population is changed. Thisincreases the number of calculation involved in selecting aparticle for breakage. Alternatively, the acceptance–rejec-tion method, as outlined earlier, is the most appropriatealgorithm for drawing random sample that results in re-duced computation time.

The next most important component of the main pro-gram is the breakage module. This module is tested usinghundred monosize particles as before. It is assumed thatthe single particle breakage follows the relationship as

Ž .given in Eq. 5 , which is termed as the cumulativebreakage distribution function. The corresponding densityfunction is

a =da1y1 a =da2y11 2Xb d ,d sQ= q 1yQ 18Ž . Ž . Ž .X aXa1 2dd

which is used in the Monte Carlo simulation to drawstatistical samples for the evolution of the size spectra. Inthis test, the particles are allowed to break one at a time.Each breakage event results in a daughter fragment whosesize is decided by the breakage density function accordingto the algorithm shown in the lower rectangular dotted boxof Fig. 3. This algorithm embeds in itself the acceptance–rejection method. The results of simulation are presentedin Fig. 5. Again, excellent agreement is obtained betweenthe simulated and analytical result.

Finally, the entire grinding process is simulated incor-porating the selection and breakage parameters. These twoparameters can be represented through their correspondingdensity functions. Now, it becomes a matter of sequen-tially drawing of samples using these density functions to

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describe the evolution of the overall system. At first, thegrinding process is simulated considering the functionalforms for the breakage and selection parameters for whichanalytical solution of the grinding equation is available.The solution to the grinding equation under these condi-

Ž .tions is given in Eq. 8 . This allows a comparison be-tween the analytical and that of the Monte Carlo results.

It is possible to carry out the Monte Carlo simulationeither by counting the change in the number of particles ineach size class or by mass. The latter approach is consid-ered here due to computational simplicity. Moreover, evenwith one hundred starting particles the sheer number ofparticles towards the end of the simulation, particularly inthe lower size classes, become overwhelmingly large andunmanageable. The grinding process is simulated usinghundred monosize particles of 2000-micron nominal diam-

Ž .eter. Other parameters such as k and a of Eq. 8 , are0

1=10y7 and 2.0 respectively. Using these values, theprocess was simulated for five minutes. Fig. 6 shows acomparison of results at a time interval of one minute. It isobserved that the analytical results agree very well withthe Monte Carlo simulation results.

Finally, the above simulation is extended to incorporatethe complete functional forms for both selection and break-

Ž . Ž .age function as stated in Eqs. 5 and 6 . These functionsinclude certain parameters whose values for the purpose ofsimulation are chosen as: Ks0.63; cs0.62, als0.598,a2s2.21, Qs0.336. As mentioned earlier, there is noavailable analytical solution to the grinding equation whenthese functional forms are imposed. However, the MonteCarlo simulation poses no such problem. Using the re-quired breakage and selection functions, Monte Carlo sim-ulation was carried up to 10 minutes and the result ofsimulation are shown at 1, 5, and 10 min in Fig. 7. Thisfigure shows at a glance whether the grinding systemconfirms to the typical grinding behavior of ores estab-lished through numerous experimental data available in theliterature. These data could not be compared with anyother results, as there is no analytical solution availableand other numerical methods only attempt to solve the

Fig. 6. A comparison between Monte Carlo and analytical solution.

Fig. 7. Simulation result showing change in the size distribution withtime.

discretized grinding equation of the form shown in Eq.Ž .13 .

6. Computational efficiency

The Monte Carlo method is a computationally efficientmethod compared to many other available numerical meth-ods for solving population balance equations. Consider the

Ž .batch grinding equation as stated earlier in Eq. 13 . For aset of n discrete size classes, the discretized grindingequation simply represents n simultaneous differentialequations. It can be conveniently represented as a singlematrix equation as

d m t� 4Ž .s IyB S m t 19Ž . Ž .

d t

The solution of this equation in the matrix form is

y1m t sTJ t T m 0 20Ž . Ž . Ž . Ž .which has become quite useful in batch mill simulation.Here, T , J and T matrices contain the numerical values ofbreakage and selection function. The mathematical details

w xof the above solution are given by Grandy et al. 11 .While arriving at the solution by using the matrix tech-niques, the determination of any one unknown requiresknowledge of all remaining unknowns. On the other hand,in the Monte Carlo method it is not necessary. Here, thearithmetic operation is simply proportional to the numberof equations and not to the cube of the number as othermatrix methods. More importantly, it is not required to theexact form of the governing differential equation in orderto simulate the process.

7. Conclusions

The analytical solution of the time-continuous andsize-continuous batch grinding equation has not been solvedincorporating the most appropriate functional forms for thebreakage and selection function. It can be solved by com-

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plicated numerical techniques that make the very modelunsuitable for simulation purpose. However, it can readilybe solved by using the Monte Carlo method. The results ofthe Monte Carlo simulation agree quite well with knownsolutions of the governing batch grinding equation wheresimplified forms of the breakage and selection functionsare assumed. Incorporating the true functional forms ofbreakage and selection function, the Monte Carlo tech-nique is employed to simulate the grinding process withoutany computational difficulty.

The Monte Carlo method is found conceptually simplerand computationally efficient compared to many otheravailable numerical methods for solving population bal-ance equations. It enables one to investigate the behaviorof the grinding equation in its most complete form forwhich no analytical solution exists and the respectivenumerical solutions are tedious. This sort of investigationallows a comparison between the discrete continuous batchgrinding equations with a view to correctly establish theparameters such that the solutions of the discretized aswell as the continuous equations match with one another.A step in this direction is successfully pursued by Hill and

w x w xNg 12 and Eyre et al. 13 .Several other studies in the area of particulate system

are easily accessible via the Monte Carlo method. Onesuch study that currently being pursued is to analyze thecrystallite size of particles during mechanical alloying. Inthis process metal powders are put in a ball mill andground until alloying takes place. During the process ofalloying, size enlargement of particles takes place due tothe ductile nature of metallic particles before the particlesbreak. A mathematical model of the process based onpopulation balance can be developed but the solution isquite tedious. However, using the Monte Carlo method,both the processes of size enlargement and reduction can

be treated separately and the governing population balanceequation can be solved easily. This analysis should permitthe formulation of a clearer definition of critical particlesize for rupture and subsequent alloying.

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