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Short communication

Application of the RosinRammler and

GatesGaudinSchuhmann models to the particle

size distribution analysis of agglomerated cork

A. Macas-Garcaa,*, Eduardo M. Cuerda-Correab, M.A. Daz-Deza

aEscuela de Ingenieras Industriales, Area de Ciencia de los Materiales e Ingeniera Metalurgica, Universidad de Extremadura,

Avda de Elvas S/N. E-06071 Badajoz, SpainbDepartamento de Qumica Inorganica, Facultad de Ciencias. Universidad de Extremadura, Avda de Elvas, S/N. E-06071 Badajoz, Spain

Received 6 November 2003; accepted 30 April 2004

Abstract

In the present, work samples prepared from cork waste and low quality cork have been analyzed from the standpoint of their

particle size distribution (PSD). The distribution functionF(/) (mass fraction) and density function f(/) (number of particles

binned between two given mesh sizes) of the agglomerated samples have been obtained by applying two widely-used

mathematical models, namely those proposed by RosinRammler (RR) and GatesGaudinSchuhmann (GGS). RR model

provides excellent results when applied to the samples here studied, which leads to a more accurate separation of the differentparticle sizes in order to obtain a better industrial profit of the material.

D 2004 Elsevier Inc. All rights reserved.

Keywords: Particle size distribution; Modeling; Agglomerated cork

1. Introduction

Many methods of varying complexity have been

developed to determine the size distribution of partic-

ulates [17]. Particle size is probably the most im-

portant single physical characteristic of solids. It

influences the combustion efficiency of pulverized

coal, the setting time of cements, the flow character-

istics of granular materials, the compacting and sinter-

ing behavior of metallurgical powders, and the

masking power of paint pigment[8].These examples

illustrate the intimate involvement of particle size in

energy generation, industrial processes, resource uti-

lization, and many other phenomena.

The current demand for cork far outstrips its

annual production. There have consequently been

several studies published aimed at optimizing the

use of natural cork slabs and cork waste, some of

which are unfit for industrial use [912]. Control of

the cork particle sizes is an important factor be-

cause it allows one to make better use of the

material and to select more efficiently the PSDs

according to their potential application.

1044-5803/$ - see front matterD 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.matchar.2004.04.007

* Corresponding author. Tel.: +34-24-289604; fax: +34-24-

289601.

E-mail address:[email protected]

(A. Macas-Garca).

Materials Characterization 52 (2004) 159 164


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In the process of milling cork, one obtains a PSD

that may be treated cumulatively or differentially. The

determination and subsequent treatment of this distri-

bution requires the use of accurate methods of anal-ysis of particle sizes, i.e., the use of methods of PSD

analysis or characterization. Correct information in

this sense will be the basis of not only the subsequent

design of the milling circuits, but also of the particle

concentration circuits and control of the operation

when the plant is running.

A number of methods aimed at determining

PSDs (i.e., sieving, cycloning, microscopy, etc.)

have been described in the literature [13,14]. Using

different characterization techniques for the PSD

analysis of a material, one may obtain quite radi-

cally different information [1517]. Hence, which

analysis technique is used will depend on the

ultimate goal of the characterization.

The results of a PSD analysis may be expressed in

different forms: binning by particle diameter indicating

the nominal mesh sizes, or by PSD, in grams, in

percentage by weight of each fraction (differential

distribution, as the cumulative percentage of sizes

below a given valueundersizeand as the cumula-

tive percentage of size above a given valueoversize)

[18,19].

The aim of this study is to obtain the distributionF(/) (mass fraction) and density f(/) functions

(number of particles binned between two given mesh

sizes) of a cork waste sample by applying the RR

and GGS mathematical models to the PSD data

obtained by sieving through different size meshes

[20].

2. Materials and experimental methods

Samples were prepared from cork waste and low-

quality cork unfit for industrial use. This material

was ground in a star or tooth mill to yield a

granulate of suitable particle size and to make a

preliminary elimination of impurities (sand, dust,

etc.). The product was passed through a hammer

mill for further reduction of particle size and

separation of impurities. The purified granulate

was then passed through a blade mill to yield the

final different PSDs.

After milling, the cork particles were separated

into different particle sizes using a 200-mm-diam-

eter sifting column whose internal diameters corre-

spond to those set out in the UNE Norm 7-050Part II. This column was placed on a vibrating

table at 100 vibrations/min for 12 min, rotating at

1 turn/min.

The resulting values allow one to obtain the

experimental PSD curves. These represent the percen-

tages by weight versus particle size.

Several mathematical models have been utilized

to obtain the distribution and density functions from

experimental PSD curves. The mostcommonly used

are those of RR and GGS [2123].

The RR distribution function has long been used

to describe the PSD of powders of various types

and sizes. The function is particularly suited to

represent powders made by grinding, milling, and

crushing operations. The general expression of the

RR model is:

F/ 1 exp /

l

m 1

where F(/) is the distribution function, and / is theparticle size (mm), l is the mean particle size (mm),

and m is a measure of the spread of particle sizes;

l and m are adjustable parameters characteristic of

the distribution. This expression may be rewritten

as:

lnfln1F/g mln/ mlnl 2

A plot of the first term of this expression versus thenatural logarithm of/ will result in a straight line of

slope m if the behavior of the material fits the RR

model.

The application of the function to a specific

distribution and the calculation of its parameters

are often done via linear regression of data, repre-

sented as ln{ ln[1 F(/)]} versus ln/, indicativeof the applicability of the RR distribution function to

the PSD. Often a least-squares regression analysis is

used to fit a line to the data point. The correlation

A. Macas-Garca et al. / Materials Characterization 52 (2004) 159164160


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coefficient may be used as the parameter for good-

ness of fit. The density function in the RR model

will be:

f/ m

lm /m1exp

/

l

m 3

One relatively uncomplicated method that has

found favor in the metalliferous mining industry since

1940 is the GGS equation, defined by:

F/ /

/max

m4

whereF(/) is the fraction of the sample finer than size

/, / is the particle diameter, /max is the maximum

particle diameter of the distribution (size modulus),

and m (distribution modulus) is an adjustable param-

eter. If the logarithm of the F(/) is plotted versus the

logarithm of particle size /, a relatively straight line is

often obtained, with a slope equal to m, obtaining the

expression:

logF/ mlog/ mlog/max 5

Hence, a plot of the logarithm of the distribution

function versus the logarithm of the particle diameter

will give a straight line if the PSD curve fits the

Fig. 1. PSD curve obtained by sieving.

Fig. 2. Plot of the distribution function vs. particle size.

Fig. 3. Plot of the density function vs. particle size.

Table 1

PSD analysis of milled cork

Range

of sizes

(mm)

Mesh

size

(mm)

Fraction

(g)

Fraction

(%)

Cumulative

% weight

(under)

Cumulative

% weight

(over)

< 0.71 0.71 1.00 1.39 1.39 98.61

0.71 1 1.0 1.70 2.36 3.75 96.25

1 1.4 1.4 8.60 11.94 15.69 84.31

1.4 2 2.0 7.20 10.00 25.69 74.31

2 2.8 2.8 17.70 24.58 50.27 49.73

2.8 4 4.0 20.50 28.47 78.74 21.26

4 5.6 5.6 11.40 15.83 94.57 5.43

5.6 8 8.0 2.30 3.19 97.76 2.24

8 11.2 11.2 1.60 2.22 100 0

A. Macas-Garca et al. / Materials Characterization 52 (2004) 159164 161


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GGS model. The density function in this model will

be:

f/ m/m1

/mmax6

3. Results and discussion

The values of the weights of the different particle

sizes obtained in the sieving operations are listed in

Table 1, together with the cumulative percentages by

weight. Fig. 1 depicts the corresponding PSD curve.

Fig. 2shows the distribution function, F(/), obtained

from fitting the experimental results of Fig. 1 toEq. (1). This function may represent the fraction by

volume, by mass, or by number of particles. The value

of the function at a given point is the fraction of the

number of particles (mass or volume) that is below a

given size.

On the other hand, the area under the curve between

two sizes (i.e., /1 and /2) indicates the number of

particles (expressed as particle mass or volume) whose

diameters are comprised in that interval:

F/2 F/1

Z /2/1

f/d/ 7

Finally, the slope of the distribution function (plot-

ted vs. the particle diameter, /) at each point gives the

density function,f(/), defined by Eq. (3) and plotted in

Fig. 3. This function represents the differential curve

corresponding to the percentage of particles of a certain

size. Many materials present PSD curves like thatshown in Fig. 1 and are therefore well suited to be

analyzed by using this type of model.

We applied the two above-described models to the

experimental results given in Table 1andFig. 1. The

different parameters to be fitted are listed in Table 2.

Table 2

Fits to the RR and GGS models

Cumulative % weight (under) F(/) / (mm) f(/) log F(/) Log / ln{ ln[1F(/)]} ln /

1.39 0.014 0.71 0.020 1.857 0.15 4.269 0.343.75 0.038 1.00 0.038 1.426 0 3.264 015.69 0.157 1.40 0.112 0.804 0.15 1.768 0.3425.69 0.257 2.00 0.128 0.590 0.30 1.214 0.6950.27 0.503 2.80 0.180 0.299 0.45 0.359 1.0378.74 0.787 4.00 0.197 0.104 0.60 0.437 1.3994.57 0.946 5.60 0.169 0.024 0.75 1.069 1.7297.76 0.978 5.68 0.122 0.010 0.90 1.335 2.08100 1 11.20 0.089 0 1.05 2.42

Fig. 4. Fit to the RR model. Fig. 5. Fit to the GGS model.



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Figs. 4 and 5show the fits of experimental data to RR

and GGS models, respectively.

From the observation of the two figures and the

corresponding linear correlation coefficient, onededuces that the RR model provides a better fit to

the experimental PSD curve than GGS does. The

resulting distribution and density functions obtained

by application of RR model are given by the

following expressions:

F/ 1 exp /

1:33

1:90" # 8

f/ 1:11/0:90exp /

1:33

1:

90" #

9

The application of these expressions to the PSD

curve allows one to extrapolate (for estimation pur-

pose only) the percentage of material smaller than a

certain particle diameter (/) at points that do not

correspond to the sieving classification system used,

thus obtaining information at the extremes of the

particle size diagrams.

The use of the RR model may provide valuable

help to carry out the modeling during the design

phase of milling circuits. Moreover, it facilitates

making correct use of the particle sizes to obtain

more homogeneous cork agglomerate samples in the

cork industry.

4. Conclusions

Particle size is probably the most important single

physical characteristic of solids. In addition, it may

be easily determined by using low-cost methods. Acorrect determination of particle size of a cork

agglomerate is necessary prior its industrial utiliza-

tion in different fields, such as wine production,

acoustic, and thermal isolation, etc.

Two mathematical models widely used to study the

PSD of solids have been applied to a cork granulate.

The RR model provides excellent results when applied

to the sample here studied, which leads to a more

accurate separation of the different particle sizes to a

better industrial profit of the material.

On the other hand, the GGS model does not

properly fit the experimental data. Nevertheless, fur-

ther investigations are being carried out for some

other materials with hopeful results.

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