computer simulation of product size distribution of a laboratory ball mill
TRANSCRIPT
This article was downloaded by: [The University Of Melbourne Libraries]On: 17 March 2013, At: 04:22Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Particulate Science and Technology: AnInternational JournalPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/upst20
Computer Simulation of Product SizeDistribution of a Laboratory Ball MillVedat Denız a
a Department of Chemical Engineering, Hitit University, Çorum,TurkeyVersion of record first published: 26 Oct 2011.
To cite this article: Vedat Denız (2011): Computer Simulation of Product Size Distribution of aLaboratory Ball Mill, Particulate Science and Technology: An International Journal, 29:6, 541-553
To link to this article: http://dx.doi.org/10.1080/02726351.2010.536303
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.
Computer Simulation of Product Size Distributionof a Laboratory Ball Mill
VEDAT DENIZ
Department of Chemical Engineering, Hitit University, Corum, Turkey
In this study, a computer simulation of a suitable matrix notation of the Broadbentand Callcott grinding model was written. First, a chromite sample was ground inthe laboratory type batch ball mill at different grinding times, with different particlesize distributions and different capacities. Second, elements of the matrix for com-puter simulation were found and a ‘‘milling matrix’’ was formed. Third, particle sizedistributions were predicted by computer simulation for the whole test. Finally, arelationship between the computer simulation predicted by dates and experimentalresults was established, with a good correlation within the limits of mean error values.
Keywords chromite, grinding, particulate, simulation, size distribution
Introduction
Comminution is one of the most important unit operations in mineral processingand several chemical industries. The comminution process fully represents thecomplexity where many variables interact (ore type, grain size, grinding media,percent solids, particle size distribution, etc.).
In recent years, there have been considerable improvements in comminutionefficiency, both due to the development of machines with the ability to enhanceenergy utilization, and also due to optimal design of grinding systems and operatingvariables that enable more efficient use of existing machines (Oner 1999). The studyof the comminution process and the understanding of different parameters that affectit have increasingly become of interest to many researchers in the field of mineralprocessing. Milling kinetics (Klimpel and Austin 1977), mill power (Yildirim et al.1998), and load behavior (Lameck et al. 2006) have been studied as functions ofmedia size, media shape (Deniz 2003; Shi 2004; Ipek 2006; Lameck et al. 2006;Lameck and Moys 2006), mills filled with balls as well as powder (Shoji et al.1982; Deniz and Onur 2002; Deniz 2003, 2010a, 2010b), mill diameter (Austin1973; Malghan and Fuerstenau 1976; Gupta et al. 1985), mill speed (Deniz 2004a),and other variables (Austin et al. 1990; Yekeler et al. 2001; Deniz 2004b; Bozkurtand Ozgur 2007).
The comminution of solids is a highly energy-consuming process. So, in thesecond half of the 19th century, the first theories aimed at establishing the relation-ship between solid comminution effect and energy used during comminution were
Address correspondence to Vedat Denız, Department of Chemical Engineering, HititUniversity, Corum, Turkey. E-mail: [email protected]
Particulate Science and Technology, 29: 541–553, 2011Copyright # Taylor & Francis Group, LLCISSN: 0272-6351 print=1548-0046 onlineDOI: 10.1080/02726351.2010.536303
541
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
formed. These theories, also called hypotheses, are known by the surnames of theirauthors, such as: Rittinger, Kick, Bond, and Charles (Otwinowski 2006).
The main task of research into loose material comminution is to determinegeneral laws concerning the evolution of the particle size distribution during commi-nution. Mathematical models predicting the particle size distributions of the commi-nution product are called stochastic or statistic models. The models, based on thepopulation balance, are numerous and represented by a highly developed group ofstochastic models. Two probability functions, namely breakage and selection, areintroduced in stochastic modelling. Epstein (1948), Broadbent and Callcott (1956),and Gardner and Austin (1962) are regarded as the creators of the stochastic modelof comminution. Broadbent and Callcott (1956) divided the complete crushingmechanism into breakage, selection, and classifying functions to establish relationsbetween the feed and product distribution. Additionally, the classifying functionin the ball mill did not conform to the functions that occurred in othercomminution mechanics (Lynch 1977; Pitchumani and Venkateswarlu 1980).
A more recent study to represent grinding has used a completely differentapproach: the kinetic or size-mass balance models. One of the earliest attempts todescribe grinding using an approach other than energy consumption was done byEpstein (1948), who described grinding, in terms of two basic concepts: the prob-ability of breakage of the material of a given size and the size distribution of the pro-duct in a breakage event. In 1956, Broadbent and Callcott, for the first time applied,a matrix notation to describe grinding, using the concepts introduced earlier byEpstein (1948).
The fundamental aspect of all comminution models is the description of theproduct of a single breakage event. This is called the breakage function, a functionthat has been very difficult to determine experimentally because there is nonon-destructive testing technique that will give information about the inherentbreakage properties of minerals.
Breakage of particles from one size to a smaller one will be described usingbreakage functions Si for the specific rate of breakage out of size i and bij for theprobability that a particle that is broken out of size j will be distributed to size i.The breakage functions Si and bij depends on the physicomechanical properties ofthe material and the grinding conditions (Park et al. 1998). Studies by Austin et al.(1984) and Deniz (2003) show that assuming first-order breakage for ball milling isoften an excellent approximation.
A Rosin-Rammler distribution is used to describe particle size distribution aftersize reduction of a material, that is,
Bðx;x0Þ ¼ 1� e�ðx=x0Þu ð1Þ
where x’ is a size such that a very significant portion of the product which is less thanthis size was produced by fine breakage. In the calculation, this distribution is alteredso that the material predicted above the size of the original particle appears afterbreakage at the original particle size. This amount is generally small (Lynch 1977).
Broadbent and Callcott (1956) postulated that a modification of theRosin-Rammler equation written as:
Bðx; yÞ ¼ 1� e� x=yð Þu� �
1� e�1� ��
ð2Þ
542 V. Deniz
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
gave a convenient form for the distribution of particles after breakage and theexponent u is an adjustable parameter, taken as 1.0 for ball milling. The functionB(x,y) represented the proportion of particles initially of size y, which appeared insize ranges smaller than x after breakage. bi,j was defined fractionated to elementsas B(x,y), where bi,j meant the fraction of the material of interval j, fell to a lowerinterval i due to breakage (Lynch 1977; Salopek et al. 1986; Austin et al. 1984).
The selection function is the breakage velocity parameter and is represented bySi..Particles of all sizes that enter a grinding process have some probability of beingbroken and this probability may change as the size of the particles change (Lynch1977).
To apply the mass balance model equations for scale up, the dependence of thekinetic parameters (Si, bi,j) on the mill dimensions and operating conditions must beknown (Herbst 1987).
The Broadbent and Callcott (1956) model equation, following the matrixnotation of the laboratory grinding process for n steps of breakage is:
P ¼ Dn � F ð3Þ
where, F and P represented the feed and product vectors, respectively, and n is num-ber of steps in breakage. D is the milling matrix and can be written for ball milling as:
D ¼ B � S þ I � S ð4Þ
where B is the breakage matrix, S is the selection matrix, and I is the unit matrix. Themilling matrix equation is shown in Figure 1.
The Broadbent and Callcott (1956) model makes it possible to predict theproduct size distribution, if the feed size distribution, capacity, and grinding timeare known.
In this study, a computer simulation of a suitable matrix notation of theBroadbent and Callcott grinding model was written. The results of the predictedand experimental studies were compared with each other and comments werepresented.
Experimental Studies
Material and Method
Chromite samples taken from deposits belonging to the Burdur Mining Co. in Budur(the Yesilova deposit), Turkey, were used as the experimental materials. The charac-terization of the raw material included chemical analysis with an x-ray fluorescence
Figure 1. Notation of breakage, selection and unit matrix.
Computer Simulation of Product Size Distribution 543
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
(XRF) spectrometer to determine its chemical composition. Chemical properties ofchromite samples used experimentally are presented in Table 1.
Mineralogical investigations were conducted both using the polarized micro-scope and transmitted light microscopy. Results indicate that samples prove to bemixtures of chromite [FeO �Cr2O3] and serpentine (magnesium iron silicate hydrox-ide) minerals [(Mg,Fe)3 � Si2O5(OH)4]. Chromite crystals were seen as subhedrall,unhedrall crystals, and different particle size. Wall rock is fully serpentinized ultra-bazic rock. Addition, this sample was seen fine band opaque minerals to think overmagnetite mineral. The ore contains approximately 40–50% of chromite, and about50–60% of serpentine. Relative density of chromite and serpentine are 4.6 and 2.6 g=cm3, respectively. Relative density of ore samples is approximately 3.4 g=cm3.
In this study, chromite ore was crushed in the laboratory with a jaw crusher to asize less than 2.8mm and samples were grouped for tests.
In the grinding experiments, a batch laboratory mill (20� 20 cm) was used. Inevery experiment 10 kg iron balls containing 2 cm (5 kg), 1 cm (3 kg), and 0.5 cm(2 kg) has been charged (ball filling is J¼ 0.35). Revolution of the mill was constantat 66 rev=min.
After the grinding process, as series of sieves of �2.8þ 1.7mm, �1.7þ 1.0mm,�1.0þ 0.600mm, �0.600þ 0.355mm, �0.355þ 0.212mm, �0.212þ 0.125mm, and�.125mm in a screen ratio of 1.666, were used and dry sieving was done by hand.Experimental processes have been carried out in four tests and these were definedherewith.
Tests for Measuring Grinding Times
The tests of grinding time measurements with 1200 g chromite ore (powder filling isfc¼ 0.056) were made in four stages of 2.5, 5, 7.5, and 10min. Following this, sieveanalysis of the products at each grinding stage and feed were made.
From the above tests, particle size distribution of the first stage ground productand sample of the chromite feed was used as the base-forming milling matrix.
Tests on Different Feed Size Distribution
In this test, a 1200 g sample (powder filling is fc¼ 0.056) of feed with different sizedistribution was ground at four stages and the products were sieved.
Table 1. Chemical composites of chromite sampleusing in experiments
Oxides (%)
Cr2O3 30.81Fe2O3 15.84Al2O3 8.36SiO2 11.13MgO 25.27CaO 0.26Loss on ignition 5.70Cr=Fe 1.90
544 V. Deniz
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
Tests on Capacity
In these capacity tests, an additional 1600 g (160 g=min) (powder filling is fc¼ 0.075)and 1920 g (192 g=min) (powder filling is fc¼ 0.090) samples were tested undersimilar conditions and the products were sieved.
Tests of Different Feed Size Distribution and Capacity
In this test, an additional 1600 g (160 g=min) (powder filling is fc¼ 0.075) of feedsamples having a different size distribution was ground under similar conditionsand the resulting products were sieved.
Figure 2. Flow chart of the computer simulation.
Computer Simulation of Product Size Distribution 545
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
Definition of a Computer Program
The result of the experimental data on the studies of the sieve analysis of productswas put in to computer simulation. First, experimental errors were removed fromthe program by using Rosin-Rammler-Bennet distribution approach.
In addition, the milling matrix was formed by finding the breakage and selectionparameters from the resulting particle size distribution in the feed and the first stageof the grinding test.
Figure 3. Screen view of the computer program. (Figure available in color online.)
546 V. Deniz
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
Input of the simulation consisted of
. a series of sieves,
. a number of sieves,
. the ratio of the sieves,
. a grinding stage number,
. grinding stage capacities,
. whether substages were or were not required, and
. feed and product size distributions.
Output of the simulation consisted of:
. product size distribution at different stages,
. product size distribution at different capacities,
. the size distribution of products obtained at different feed size distributions,
. the mean square error of the predicted and experimental results, and
. the correlation coefficients of the predicted and experimental results.
Simulation was in the Basic language and used in every type of P.C. computer. Aflow chart of this simulation is given in Figure 2. Additionally, a screen view of thecomputer simulation is given in Figure 3.
Result and Discussion
Results of Grinding at Different Times
Sieve analysis of the feed, product and inputs of different grinding times are given inTable 2. Additionally, in Figure 4, the predicted and experimental, cumulative undersize values are compared.
Table 2. Comparison of experimental and predicted size distributions at differentgrinding time
2.5min 5.0min 7.5min 10min
Particle size(mm)
Feed%
Exp.%
Predict.%.
Exp.%
Predict.%
Exp.%
Predict.%
Exp.%
Predict.%
�2800þ 1700 11.11 7.05 7.00 3.42 4.41 2.20 2.78 1.56 1.75�1700þ 1000 12.69 9.35 9.34 5.98 6.75 4.36 4.80 3.25 3.38�1000þ 600 16.36 13.63 13.67 10.40 11.09 8.35 8.79 6.62 6.84�600þ 355 15.28 14.34 14.44 12.67 13.02 11.08 11.36 9.38 9.66�355þ 212 13.16 13.72 13.73 13.67 13.53 12.87 12.81 11.62 11.77�212þ 125 9.85 11.29 11.47 12.40 12.32 12.46 12.56 11.94 12.33�125þ 000 21.58 30.63 30.36 41.46 38.88 48.68 46.90 55.62 54.27Correlationcoefficient
0.99991 0.99984 0.99998 0.99995
Mean error values 0.130 1.122 0.755 0.559
Computer Simulation of Product Size Distribution 547
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
Grinding Results of Feed Having Different Size Distribution
Same samples in different size distributions ground at four stages, and the experi-mental and predicted values of the products are given in Table 3. In Figure 5, thepredicted and experimental, cumulative under size values are compared.
Figure 4. Comparison of experimental and predicted, cumulative under size values at differentgrinding times.
Table 3. Comparison of experimental and predicted size distributions at differentfeed size distribution
2.5min 5.0min 7.5min 10min
Particle sizemm
Feed%
Exp.%
Predict.%.
Exp.%
Predict.%
Exp.%
Predict.%
Exp.%
Predict.%
�2800þ 1700 17.88 10.17 11.26 6.70 7.10 4.94 4.47 3.75 2.82�1700þ 1000 19.10 13.54 14.19 10.15 10.32 7.76 7.39 6.19 5.22�1000þ 600 20.42 17.77 17.80 15.05 14.86 12.41 12.03 10.51 9.51�600þ 355 15.46 16.33 15.97 15.56 15.23 13.97 13.81 12.62 12.08�355þ 212 10.85 13.59 12.93 14.39 13.81 14.07 13.81 13.52 13.20�212þ 125 6.69 9.75 9.49 11.34 11.35 12.02 12.39 12.23 12.77�125þ 000 9.60 18.85 18.36 26.81 27.33 34.83 36.11 41.18 44.41Correlationcoefficient
0.98473 0.99813 0.99966 0.99948
Mean error values 0.599 0.367 0.583 1.409
548 V. Deniz
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
Grinding Results of Different Capacities
The samples with similar size distribution and at different capacities were ground,and the experimental and predicted size distributions of the products are given inTable 4. In Figure 6, the predicted and experimental, cumulative under size valuesare compared.
Figure 5. Comparison of experimental and predicted, cumulative under size values at differentfeed size distribution.
Table 4. Comparison of experimental and predicted size distributions at differentcapacity
120 g=min 160 g=min 192 g=min
Particle sizemm
Feed%
Exp.%
Predicted%
Exp.%
Predicted%
Exp.%
Predicted%
�2800þ 1700 11.11 1.56 1.75 1.80 2.78 1.75 3.59�1700þ 1000 12.69 3.25 3.38 4.43 4.80 4.73 5.78�1000þ 600 16.36 6.62 6.84 9.13 8.79 9.94 9.94�600þ 355 15.28 9.38 9.66 12.36 11.36 13.86 12.19�355þ 212 13.16 11.62 11.77 14.19 12.81 15.00 13.17�212þ 125 9.85 11.94 12.33 13.33 12.56 13.68 12.44�125þ 000 21.58 55.62 54.27 44.75 46.90 41.54 42.89Correlationcoefficient
0.99995 0.9960 0.99390
Mean error values 0.559 1.17 1.34
Computer Simulation of Product Size Distribution 549
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
Grinding Results of Different Capacities and Size Distribution
The results of the tests of samples in different capacities and different size distribu-tions are given in Table 5. In Figure 7, the predicted and experimental, cumulativeunder size values are compared.
Conclusions
According to experimental and predicted results, the grinding matrix model can beused in the predicted size distribution of grinding in laboratory mills. Similarly,
Figure 6. Comparison of experimental and predicted, cumulative under size values at differentcapacities.
Table 5. Comparison of experimental and predicted size distributions at differentcapacity and different size distribution
Particle size Feed 1120 g=min
Feed 2160 g=min
mm % Exp. % Predicted % % Exp. Predicted %
�2800þ 1700 11.11 1.56 1.75 11.14 2.75 2.79�1700þ 1000 12.69 3.25 3.38 19.14 6.63 6.50�1000þ 600 16.36 6.62 6.84 23.21 11.89 11.99�600þ 355 15.28 9.38 9.66 17.99 14.61 14.35�355þ 212 13.16 11.62 11.77 12.29 15.29 14.52�212þ 125 9.85 11.94 12.33 7.21 13.18 12.94�125þ 000 21.58 55.62 54.27 9.02 36.91 35.94Correlation coefficient 0.99995 0.99916Mean error values 0.559 0.492
550 V. Deniz
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
the matrix model developed for industrial size mills should be used by industries.This model was very simple to use and reliable.
The high reliability of this model was indicated by not only the high correlationcoefficient and but also mean square error values generally close to zero or notexceeding 1.4. The results of the v2 tests were given 94% reliability.
By this method provides many advantages as follow:
1. a reduction in number of experiments,2. a reduction of comminution costs,3. the effects of capacity changes can be investigated,4. the grinding circuit performance can be determined,5. the effect of ore flow changes that occur in the mill can be investigated, and6. the investigation of the effect of grinding parameters can be conducted easily.
The reliability of this model depended on the initial milling matrix determinationby regression analysis using the Least Square, Rosin-Rammler-Bennet,Gaudin-Schuman-Gates, etc.
Nomenclature
i size class number of productJ size class number of feedSi specific rate of breakage (min�1)bi,j breakage distribution function, part of interval j falling into interval iB(x,y) dimensionless function of two variablesn number of steps in breakageu an adjustable parameter, taken as 1.0 for ball milling
Figure 7. Comparison of experimental and predicted, cumulative under size values at differentcapacities and different feed size variations.
Computer Simulation of Product Size Distribution 551
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
F feed vectorsP product vectorsD milling matrixI unit matrixS diagonal matrix of selectionB triangular matrix of breakageJ ball fillingfc powder filling
References
Austin, L. G. 1973. Understanding ball mill sizing. Inds. Eng. Chem., Process Des. Dev. 12:121–129.
Austin, L. G., R. R. Klimpel, and P. T. Luckie. 1984. Process engineering of size reduction:Ball milling. New York: AIME, SME.
Austin, L. G., M. Yekeler, T. F. Dumn, and R. Hogg. 1990. The kinetics and shape factors ofultrafine dry grinding in laboratory ball mill. Part. Part. Syst. Charact. 7: 224–224.
Bozkurt, V., and I. Ozgur. 2007. Dry grinding kinetics of colemanite. Powder Technol. 176:88–92.
Broadbent, S. R., and T. G. Callcott. 1956. A matrix analysis of processes involving particleassemblies. Philos. Tran. Royal Soc. London A 249: 99–123.
Deniz, V. 2003. A study on the specific rate of breakage of cement materials in a laboratoryball. Cem. Concr. Res. 33: 439–445.
Deniz, V. 2004a. The effect of mill speed on kinetic breakage parameters of clinker andlimestone. Cem. Concr. Res. 34: 1365–1371.
Deniz, V. 2004b. Relationships between Bond’s grindability and breakage parameters ofgrinding kinetic on limestone. Powder Technol. 139: 208–213.
Deniz, V. 2011a. Influence of interstitial filling on breakage kinetics of gypsum in ball mill.Adv. Powder Technol. 22: 512–517. DOI:10.1016=j.apt.2010.07.004
Deniz, V. 2011b. Comparison with some porous materials and the effects of powder filling onbreakage parameters of diatomite in dry ball milling. Part. Sci. Technol. 29: 428–440.
Deniz, V., and T. Onur. 2002. Investigation of the breakage kinetics of pumice samples asdependent on powder filling in a ball mill. Int. J. Miner. Process 67: 71–78.
Epstein, B. 1948. The material description of certain breakage mechanisms leading to thelogarithmic-normal distribution. J. Franklin Inst. 244(12): 471–477.
Gardner, R. P., and L. G. Austin. 1962. The use of a radioactive tracer technique and acomputer in the study of the batch grinding of coal. J. Inst. Fuel London 35: 173–177.
Gupta, V. K., H. Zouit, and D. Hodouin. 1985. The effect of ball and mill diameters ongrinding rate parameters in dry grinding operation. Powder Technol. 42: 199–208.
Herbst, J. A., Y. C. Lo, and J. E. Bohler. 1987. Increasing the capacity of a phosphate grindingcircuit with the aid of computer simulation. Miner. Metal. Process 4: 155–160.
Ipek, H. 2006. The effects of grinding media shape on breakage rate. Miner. Eng. 19: 91–93.Klimpel, R. R., and L. G. Austin. 1977. The back-calculation of specific rates of breakage and
non-normalized breakage distribution parameters from batch grinding data. Int J. Miner.Process 4: 7–32.
Lameck, N. S., K. K. Kiangi, and M. H. Moys. 2006. Effects of grinding media shapes on loadbehavior and mill power in a dry ball mill. Miner. Eng. 19: 1357–1361.
Lameck, N. S., and M. H. Moys. 2006. Effects of media shapes on milling kinetics. Miner.Eng. 19: 1377–1379.
Lynch, A. J. 1977. Mineral crushing and grinding circuits: Their simulation, optimization, designand control. New York: Elsevier.
552 V. Deniz
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013
Malghan, S. G., and D. W. Fuerstenau. 1976. Scale-up of ball mills and using populationbalance models and specific power input. Proceedings of the 4th European Symposium,613–630, Weinheim, Germany.
Oner, M. 1999. Ball size rationing affects clinker grinding. World Cem. Res. 30: 101–106.Otwinowski, H. 2006. Energy and population balances in comminution process modelling
based on the informational entropy. Powder Technol. 167: 33–44.Park, J. K., Y. Jeong, J. I. Yang, and M. Y. Jung. 1998. Grinding time for control of the size
fraction of products in the attrition milling. Korean J. Chem. Eng. 15: 375–380.Pitchumani, B., and D. Venkateswarlu. 1980. Matrix representation of batch and continuous
size reduction processes. In Fine particles processing, ed. P. Samasundaran, 148–163.New York: SME, AIMMM.
Salopek, B., S. Peaff, and M. Garapic. 1986. Determination of milling matrix elementsexemplified by laboratory ball mill tests. I. Int. Miner. Process. Symp. 24–34.
Shi, F. 2004. Comparison of grinding media-cylpebs versus balls. Miner. Eng. 17: 1259–1268.Shoji, K., L. G. Austin, F. Smaila, K. Brame, and P. T. Luckie. 1982. Further studies of ball
and powder filling effects in ball milling. Powder Technol. 31: 121–126.Yekeler, M., A. Ozkan, and L. G. Austin. 2001. Kinetics of fine wet grinding in a laboratory
ball mill. Powder Technol. 114: 224–228.Yildirim, K., L. G. Austin, and H. Cho. 1998. Mill power for smooth-lined mills as a function
of media type and shape. In Fine Powder Processing Technology, eds. R. Hogg, R. G.Cornwall, and C. C. Huary, 69–79. State College, PA: Pennsylvania State University.
Computer Simulation of Product Size Distribution 553
Dow
nloa
ded
by [
The
Uni
vers
ity O
f M
elbo
urne
Lib
rari
es]
at 0
4:22
17
Mar
ch 2
013