www.elsevier.com/locate/powtec

Powder Technology 143–144 (2004) 174–178

The breakage matrix approach to inadvertent particulate degradation:

dealing with intra-mixture interactions

John Baxter*, Azlina Abu-Nahar, Ugur Tuzun

Chemical and Process Engineering, School of Engineering, University of Surrey, Guildford, Surrey GU2 7XH, UK

Available online 19 June 2004

Abstract

The description of particulate degradation using a matrix equation is a potentially useful tool in modelling degradation processes. One of

the main constraints in adopting this approach is measuring the relevant breakage matrix, especially where degradation of particles in a given

size class is influenced by the presence of particles in other size classes. This paper describes a technique for quantifying such intra-mixture

interactions in degradation processes, hence calculating the full breakage matrix for interacting mixtures, for a relatively modest increase in

necessary observational effort. The result indicates that the breakage matrix approach may be more useful and versatile than previously

realised.

D 2004 Elsevier B.V. All rights reserved.

Keywords: Breakage matrix; Particulate degradation; Intra-mixture interaction

1. Introduction lems. Transport and handling of intermediates in sugar

Particulate degradation is a major issue in many process

industries concerned with the transport, handling and

storage of bulk solids. In some cases, including milling

and pulverisation, such degradation is intentional. Howev-

er, degradation also arises as an (often-undesirable) side

effect of normal transport and handling operations. Degra-

dation can give rise to handling difficulties—for example,

generation of fines can inhibit the aeration and hence

handleability of bulk solids. More generally, the particle

size distribution (PSD) of bulk solids is often an important

quality factor in its own right, and changes to the PSD

owing to degradation can lead to reduction in product

functionality, appearance or other factors affecting its

quality and saleability. Inadvertent degradation has em-

erged as a significant problem in many processing scenar-

ios, partly because degradation considerations are often

taken as secondary in design (or not considered at all)

compared to other factors such as the energy consumption

required for transport. Sugar processing provides an illus-

trative industrial example of inadvertent degradation prob-

0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.powtec.2004.04.011

* Corresponding author. Tel.: +44-1483-686-594; fax: +44-1483-689-

510.

E-mail address: j.baxter@surrey.ac.uk (J. Baxter).

refining processes generates a substantial quantity of fine

particles, which affect the saleability (value) of the finished

product in a number of ways. The result is necessary

recycling and reprocessing of considerable quantities of

such ‘‘dust’’ and hence considerable additional costs to the

refiner.

Therefore, it is clearly important to be able to model and

predict inadvertent particulate degradation in process oper-

ations, with an eventual view to avoiding or even ‘‘design-

ing out’’ such effects as far as is possible. Whilst

significant research into inadvertent particle degradation

is a relatively recent development, studies of intentional

degradation (milling and pulverisation) have a long history

of a century or more. It emerges that much of this

knowledge may be of significant interest for inadvertent

degradation studies.

In this paper, we describe a compact mathematical

formulation for describing degradation effects in the form

of a matrix equation. We discuss the perceived limitations

of the so-called breakage matrix approach, and offer a

simple technique by which many of these limitations can

be overcome. We demonstrate that the breakage matrix

technique is a promising tool in an overall strategy towards

better understanding and hence modelling of inadvertent

particulate degradation.

Table 1

Breakage matrix for pneumatic conveying process

Input size class

850–1180 600–850 425–600 300–425 212–300

Output 850–1180 0.838 0.000 0.000 0.000 0.000

size 600–850 0.113 0.878 0.000 0.000 0.000

class 425–600 0.023 0.097 0.959 0.000 0.000

300–425 0.010 0.011 0.033 0.966 0.000

212–300 0.005 0.005 0.004 0.027 0.982

< 212 0.010 0.009 0.004 0.006 0.018

J. Baxter et al. / Powder Technology 143–144 (2004) 174–178 175

2. The breakage matrix approach

Definite assertions as to the ‘‘quality’’ of particulate solids

are often made in terms of the PSD. For example, the median

particle size of a distribution may be set within certain

tolerances, or limits on composition above or below a certain

target size might be specified. From the standpoint of break-

age modelling, many unit operations such as milling and

crushing may be susceptible to analysis as continuous pro-

cesses, even though the operations themselves are batch or

semi-batch in nature. This approach is based on calculating

time-dependent rates of breakage (i.e., breakage functions)

and hence the evolution of PSDs over (residence) time. The

result is a population-balance approach to modelling. Most of

the literature on breakage modelling over time (for example,

Refs. [3,4]) has been done on this basis. However, a key

practical issue with such an approach is that it requires

continual sampling at suitable points within the process for

model calibration/validation. A common practical problem is

that such intra-process sampling is difficult or essentially

impossible.

A possible alternative, which can yield useful data and

generate predictive modelling capability, is to treat an entire

process (or series of consecutive processes) as a single

breakage ‘‘event’’, thereby capturing the overall breakage

behaviour in such an event. This implies capturing overall

(time-averaged, time-invariant) analogues of the breakage

functions mentioned above, to give the overall relationship

between input and output PSDs. It is useful to express this

relationship in a compact mathematical form that has been

widely used in describing milling processes, namely the

breakage matrix approach [1,2].

The technique yields rather less information than popula-

tion-balance type approaches, but lends itself well to practical

implementation. The ‘‘boundaries’’ of a single breakage

event can be specified according to the availability of data

from sampling. For example, a conveying system followed

by a receiving hopper may be regarded together as a single

‘‘breakage event’’ if sampling is only possible upstream of

the conveyor and at the hopper outlet, and intermediate

sampling between these points is not possible. In such a case,

the breakage matrix gives the mathematical relationship

between (overall, time-averaged) input and output PSDs via

the matrix equation:

I � M ¼ O ð1Þ

where I andO are column vectors of the PSD in the input and

output streams, and M is the breakage matrix. The elements

ofM give the probability of breakage of given-sized particles

in the input stream, and the (statistical average) size distri-

bution of debris from broken particles. In general, the

breakage matrix formulation can be applied to any degrada-

tion process or series of processes.

The breakage matrix is effectively a (time-averaged)

combination of two better-known mathematical tools: the

so-called selection function [3] which gives the proportion

of particles of various sizes that break during the process,

and the breakage function [4] which gives the (statistical

average) fragment size distribution (FSD) from any break-

age event. The selection function is encompassed in the

diagonal elements of M, and the breakage function in the

off-diagonal elements. Table 1 gives an example breakage

matrix for the degradation of an industrial material in a

pneumatic conveying system.

Each column of the matrix tells us the ‘‘fate’’ of particles

in each size class in the input stream on a mass basis. For

example, 83.8% by mass of 850–1180 Am particles in the

input stream remain in that size class in the output. 11.3%

break into fragments of size 600–850 Am and so on.

Several features of the matrix are immediately apparent.

All of the elements one side of the diagonal are zero; this

follows if there is only breakage (size reduction) in the

process of interest, and no agglomeration (size enlargement)

of the fragments. It follows that the elements on the

diagonal are the effective ‘‘survival rates’’ of particles in

each size class—effectively, the ‘‘selection function’’ often

referred to in the literature. By also giving the FSDs for

each input size class via the off-diagonal elements (aka the

breakage functions), the matrix contains a wealth of infor-

mation about the degradation event in a compact mathe-

matical form.

To illustrate the terminology, taking the breakage matrix

in the form of Table 1 (assuming for the moment that this

can be measured appropriately) a typical degradation pro-

cess might be represented as:

0:06

0:51

0:25

0:13

0:05

0:00

2666666664

3777777775

0:838 0 0 0 0 0

0:113 0:878 0 0 0 0

0:023 0:097 0:959 0 0 0

0:010 0:011 0:033 0:966 0 0

0:005 0:005 0:004 0:027 0:982 0

0:010 0:009 0:004 0:006 0:018 1:000

2666666664

3777777775¼

0:05

0:45

0:45

0:29

0:14

0:06

0:01

2666666666664

3777777777775ð2Þ

where the two column vectors represent the input and output

PSDs. Moving down the columns represents decreasing

size—so the input has 6% by mass in the > 850 Am size

range (taking the size ranges from Table 1), 51% in the range

600–850 Am and so on. Note that the input has no material in

J. Baxter et al. / Powder Technology 143–144 (2004) 174–178176

the ‘‘pan’’ of the sieve stack ( < 212 Am) used to measure the

PSDs whereas a small fraction (1%) of the debris appears in

this size class.

3. Measuring breakage matrices with no intra-mixture

interactions

The depth of information contained within the breakage

matrix gives rise to the major perceived problem of the

approach. It has been seen as being a somewhat unwieldy

and detailed description of the undoubtedly complicated

breakage process for a given set of conditions, making

transfer of the findings to other sets of conditions somewhat

difficult [5]. In addition, it has been noted that the breakage

matrix is mathematically indeterminable from a simple

knowledge of the input and output size distributions I and

O; there are, at least theoretically, an infinite number of

solutions. Numerical techniques can sometimes be used in

an attempt to suggest possible solutions and ‘‘optimise’’ the

values in the breakage matrix. The technique described in

this paper means that such optimisation is no longer

necessary.

The problems with the breakage matrix approach largely

depend upon the difficulty of measurement. If effective

means of measuring breakage matrices can be found, it

promises to be a relatively simple practical tool for charac-

terising and predicting degradation.

If the degradation behaviour of different size classes in

a mixture is unaffected by the presence of other size

classes in the mixture, measuring breakage matrices is

relatively straightforward, and involves N experimental

measurements for N size classes in the input mixture.

The technique involves pre-separating the material into

different size classes, and testing the degradation behaviour

of each separately. In effect, this builds up the breakage

matrix a column at a time. This approach has been

demonstrated successfully for degradation of wheat in a

roll mill [5,6].

However, the assumption of no intra-mixture interac-

tion is fairly restrictive and is not reasonable in many

degradation scenarios. The remainder of this paper dem-

onstrates how with a minimal increase in necessary

experimental effort (from N to N + 1 measurements)

and a simple assumption, the breakage matrix can be

found for scenarios where intra-mixture interaction is

significant.

4. Measuring breakage matrices with intra-mixture

interactions

We are concerned with calculating the full breakage

matrix for a N-component mixture containing known frac-

tions by mass of various size fractions given by a, b, c and

so on. Suppose that the completely general form of the

breakage matrix for a given process with significant intra-

mixture interaction is as follows:

a

b

c

::

::

::

::

2666666666664

3777777777775

M11 0 0 0 0 :: ::

M21 M22 0 0 0 :: ::

M31 M32 M33 0 0 :: ::

M41 M42 M43 M44 0 :: ::

M51 M52 M53 M54 M55 :: ::

:: :: :: :: :: :: ::

:: :: :: :: :: :: ::

2666666666664

3777777777775

¼

A

B

C

D

E: : :

: : :

2666666666664

3777777777775

ð3ÞEq. (4) below illustrates the procedure. We begin by

performing a degradation test on the entire mixture—

effectively measuring the output PSD in Eq. (3) above

but not, as yet, saying anything about the breakage matrix,

M. We then take unit mass samples of each of the

individual size classes in the mixture, and subject them to

degradation tests under the same conditions. We repeat the

tests illustrated in Eq. 4(b)–(d) for as many size classes as

we have in the mixture.

a

b

c

::

::

26666664

37777775!

A

B

C

::

::

26666664

37777775

4ðaÞ

1

0

0

0

::

26666664

37777775!

a1

a2

a3

a4: : :

26666664

37777775

4ðbÞ

0

1

0

0

: : :

26666664

37777775!

0

b2

b3

b4: : :

26666664

37777775

4ðcÞ

0

0

1

0

: : :

26666664

37777775!

0

0

c3

c4: : :

26666664

37777775

4ðdÞ

ð4Þ

If there is no intra-mixture interaction, we simply com-

bine the results of the tests given by Eq. 4(b) and onwards,

giving a breakage matrix as:

abc::::

266664

377775

a1 0 0 : : : : : :

a2 b2 0 : : : : : :

a3 b3 c3 : : : : : :

a4 b4 c4 : : : : : :: : : : : : : : : : : : : : :

266664

377775 ¼

AVBVCVDV: : :

266664

377775 ð5Þ

where the output size distribution in Eq. (5) is a predicted

distribution in the absence of intra-mixture interaction. So

if AV=A, BV =B, and so on, we can say with confidence

that interaction is absent, and that the breakage matrix in

Eq. (5) is the actual breakage matrix for the process (that

in Eq. (3)).

J. Baxter et al. / Powder Technology 143–144 (2004) 174–178 177

If intra-mixture interaction is in fact significant, the

breakage matrix is not, in general, tractable. However, we

can calculate the breakage matrix by invoking a simple

assumption regarding the size distribution of fragments

from a given size class. We assume that whilst intra-mixture

interaction may affect the extent of degradation in any given

size class, that the fractional distribution of fragment sizes is

unaffected. Thus, for example, if intra-mixture interaction

serves to suppress degradation of mother particles in a

given size class, it is assumed that fragment production in

each of the smaller size classes is similarly suppressed. The

proportional production of fragments is established through

the individual size class tests (those effectively assuming no

intra-mixture interaction) and hence no further experimental

observation is necessary to calculate the full breakage

matrix.

This assumption effectively divorces the selection func-

tion (i.e., diagonal matrix elements) from the breakage

function (the off diagonal elements). On this basis, we

proceed by stating the above assumption symbolically; if

fragment size distribution is unaffected by intra-mixture

interaction, then considering degradation of the coarsest

fraction in the mixture yields:

M21XNi¼2

Mi1

¼ a2XNi¼2

ai

ð6Þ

or, more simply:

M21

1�M11

¼ a2

1� a1ð7Þ

since the sum of each column in the breakage matrices, both

for the overall mixture test and for the individual size

fraction tests, must be unity. Similar equations can be

written for the other elements in the first column of the

general breakage matrix. For example:

M31XNi¼2

Mi1

¼ a3XNi¼2

ai

ð6bÞ

or

M31

1�M11

¼ a3

1� a1ð7bÞ

and we can continue writing analogous equations for every

element in the first column of the matrix irrespective of the

matrix size.

Now, it emerges that the only unknown in Eq. (7) is M21,

since all the terms on the right-hand side are extracted from

the individual size class test, and we already know (or can

calculate) M11 from the general matrix Eq. (3), i.e.,

aM11 ¼ A hence M11 ¼ A=a ð8Þ

Physically, this means that debris particles in the a size

class can only originate from mother particles in the same

size class (since there is no agglomeration of fragments,

only breakage of mother particles). Combining Eqs. (7) and

(8) gives:

M21 ¼a2 1� A

a

�

1� a1ð9Þ

and similarly for the next element in the first column

(combining Eqs. (7b) and (8)):

M31 ¼a3 1� A

a

�

1� a1ð10Þ

and so on, for each subsequent element in the first column.

We can show that calculations for subsequent columns of

the breakage matrix depend only on data that is already

known or has previously been calculated. Thus, by induc-

tion we can extend the analysis to a matrix of any size. For

the second column of the matrix, by analogy with Eq. (7) we

have:

M32

1�M22

¼ b3

1� b2ð11Þ

where, once again, everything on the right-hand side is

already known. We do not, at present, know the survival rate

of particles initially in the second size class (M22). However,

this is readily calculable from available data; by definition,

from the original matrix equation, we have:

aM21 þ bM22 ¼ B ð12Þ

and hence:

M22 ¼B� aM21

bð13Þ

where everything on the right-hand side is known. Then,

from Eq. (11) we can calculate the other elements in the

second column of the matrix:

M32 ¼b3ð1�M22Þ

1� b2ð14Þ

and also:

M42 ¼b4ð1�M22Þ

1� b2¼ b4

b3M32 ð15Þ

J. Baxter et al. / Powder Technology 143–144 (2004) 174–178178

We then continue for all other size classes in the mixture in

the same way. For the general matrix element Mqp ( q>p) we

begin with:

aMp1 þ bMp2 þ cMp3 þ : : :pMpp ¼ P ð16Þ

where P is the proportion by mass appearing in the p-th size

fraction in the overall breakage test, and everything on the

left-hand side of Eq. (16) apart from Mpp is either known or

calculated from previous columns of the matrix. Then:

Mqp ¼pqð1�MppÞ

1� ppð17Þ

where the coefficients pq and pp are measured from the

individual size fraction test on the p-th size class.

5. Discussion and conclusions

The breakage matrix technique is a potentially powerful

tool in the modelling of particulate degradation. It contains a

wealth of information on a complex process, yet (with

certain restrictions) can be measured with relative ease.

This work has demonstrated a technique whereby one of

the most important restrictions on use of the breakage

matrix, namely the issue of intra-mixture interactions, can

be relaxed.

We have shown that for a N-component mixture, the full

breakage matrix can be calculated for scenarios where intra-

mixture interaction is significant by employing N + 1 exper-

imental observations (only N observations are necessary

when intra-mixture interaction can be ignored). Thus, we

can effectively determine the nature and extent of intra-

mixture interaction—this is highly significant. Intra-mixture

interaction is one of the main problems restricting the use of

the breakage matrix concept in modelling degradation. The

technique offers the possibility of a systematic and quanti-

tative assessment of intra-mixture interaction in degradation,

which to date has apparently been lacking.

Assuming the essential similarity between the breakage

function for interacting and non-interacting cases is clearly

something of an idealisation. However, divorcing the selec-

tion and breakage functions for this purpose definitely

makes the assumption less restrictive than it otherwise

would have been. To some extent, it implies that the

mechanisms for breakage are similar whether there is

intra-mixture interaction or not—but it is perhaps reasonable

to envisage that if particles in a given size class break at all,

that they will break in a certain way. The breakage matrix

can always be calculated given some means of classifying

particles in a given size class of the debris (i.e., unbroken

‘‘virgin’’ particles or fragments from the breakage of larger

particles)—our approach offers an alternative if such a

distinction is not possible experimentally.

The (implicit) alliance of the breakage matrix technique

with the use of simple sieving to measure PSDs is undoubt-

edly something of a simplification. Effectively, the above

technique refers to ‘‘catastrophic’’ degradation; particle

breakage that does not result in a change in size class is

effectively no breakage at all. As has been discussed

elsewhere [7,8], there are numerous mechanisms for particle

degradation, by no means all of which will result in

sufficient particle size change to give a change in size class.

Indeed, particle size in itself is probably too simplistic a

basis for a full theoretical understanding of the complicated

processes and mechanisms of breakage. However, particle

size (and PSD) is nevertheless a parameter of primary

importance, especially in the industrial environment, and

very definite assertions as to product quality can often be

made in quite simplistic terms on the basis of PSD.

To date, our experimental observations have largely been

on model systems designed to be analogous to lean-phase

pneumatic conveying scenarios, where intra-mixture inter-

action is in fact rather unlikely. It is easy to show if there is

no intra-mixture interaction, simply by pre-separating the

input mixture, measuring the breakage matrix as if there

were no interaction, and matching the breakage matrix

predictions to the experimental output for degradation of

the mixture. However, we are now in a position to make

definite assertions on the intra-mixture interaction. Ongoing

work is extending this analysis to experimental scenarios

where intra-mixture interaction is very much more likely,

such as the degradation in dense-phase assemblies of

particles owing to compression and/or shearing.

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