Theoretical kinetic model for particulates disintegration processes that result in bimodal size distributions

Powder Technology, 70 (1992) 21-30

21

Theoretical kinetic model for particulates that result in bimodal size distributions L. M. Popplewell”

disintegration

processes

and M. Peleg

Department of Food Engineering, Universiy of Massachusetts, Amherst, UA 01003 (USA) (Received January 15, 1991; in revised form October 1, 1991)

Abstract Disintegration is described in terms of a breakage matrix, mass balance on the various size fractions, and expressions that relate the breakage frequency, and thus its rate, to the fraction’s size. The main differences between the described and previously published models of this kind are that a large particle from which fines have been detached is allowed to remain, at least temporarily, in its original size slot, and that the breakage frequency of the particle population is described by a bimodal /?-distribution. The advantage of these modifications is a more consistent and realistic account of disintegration in general and erosion-dominated processes in particular. The model capabilities are demonstrated in computer-simulated size distributions during disintegration processes dominated by shattering and erosion, and its usefulness is shown in the comparison of the behavior of agglomerated and freeze-dried instant coffees undergoing mechanical attrition.

Introduction The

fundamental

concepts

upon

which

the mathe-

of disintegration is based were first codified by Epstein [l]. They may be stated as: (1) Disintegration may be viewed as a series of discrete breakage events. (2) There is some specific probability that a particle will be broken. (3) There is a characteristic size distribution of progeny resulting from the single breakage of a particle under specific conditions. Thus the rate of a disintegration procesi is mainly defined by (2) above, while the size distribution of the resulting population is related to (3). Combination of the concepts above with the empirical observations of a number of investigators [2] show that in many cases breakage is a first-order process that can be described by the following. Consider a particle population divided into it discrete size intervals (smallest size numbered n, largest size numbered l), with the weight fraction of material in each interval at time t denoted as Wi(t). Then, assuming first-order kinetics for a batch system, a mass balance on fraction i may be written as: matical

description

(1) *Present address: McCormick & Co., Technical Resource Center, 202 Wight Avenue, Hunt Valley, MD 21031, USA.

0032-5910/92/$5.00

where bi, j is defined as the fraction of the breakage products of size interval j that are broken into size interval i (breakage distribution parameter), and SDj is defined as the probability of breakage of a particle in size interval j per unit time (specific rate of breakage). The first term of eqn. (1) thus represents material size interval i accumulating due to the breakage of larger sizes. The second term represents the disappearance of material of size i due to the breakage of that size. Note also that for all j:

k

bisj=l

i-j+1

i.e. mass is conserved within size intervals smaller than i.

If the particle size distribution is viewed as a whole, then a convenient matrix form of the first-order grinding model is observed [3]:

$ [W(t)] = -

[I-B]S,W(t)

(3)

where W(t) is the column vector of fractional weights, B the upper triangular breakage parameter matrix, and S, is the diagonal specific rate of breakage matrix (see below). It should be noted that other forms of the first-order grinding equation may be written, e.g. instead of discrete size intervals with time continuous as in eqns. (1) and (3), both time and size may be made continuous [4].

Q 1992 - Elsevier Sequoia. All rights reserved

22

The analytical solution to eqn. (1) assuming S,i and bi, i invariant with time, is [5]: w;:(t)= i

a,, i exp( -SD&

l-&CC”

m-l

where for m = i: i-1 S,b,

a,, i= I: j-m

S, = a * (xi/xO)”

SDi-SDm

or for m#i: i-1 ai, i= K(O) - 2 am, i

(6)

m-l

Numerical solutions of eqns. (1) and (3) are also possible. Perhaps the simplest of these employs Euler’s method. In that case, for a time step b solutions for eqns. (1) and (3) are given by: i-l j-l

(7) and W(t)=[I-B&s,‘-&‘]W(f--b)

(8)

where SD’=S,At. Thus if q short time intervals are applied (Meloy and Bergstrom, 1965 as reported in [2]) we obtain for the matrix form of eqn. (3) i.e.: - q At)

(9

Breakage distribution and breakage rate firactions

A variety of continuous functions to represent the breakage distribution function or specific rate of breakage have been suggested over the years [2]. The socalled breakage distribution function is always given in a cumulative form, i.e. as the fraction of the breakage products of a particular size that is smaller than a specified size. An example of this type of function is 131: 1 - exp( -x/y) B(x,Y) = 1-exp(-1)

(10)

where B(x, y) is the fraction of the products of size y that are smaller than size X. More recently the following expression has been proposed [6]: Bi,j=~(Xi/xi+l)y+(l-aj)(Xi/Xj+l)B

(11)

where 3, y and /3 are constants for a given material and conditions, xi is the upper size of interval i, and _X~+~is the lower size interval j (i>j). Using these functional forms, the breakage distribution parameters are calculated by: bi,j=Bi,j-Bi+l,j

(13)

where C is the probability of survival of a particle of unit size. Alternatively [2]:

ia,,i

W(t) = [I- BS,’ - S,‘]vv(t

The specific rate of breakage in an early work [3] was assumed not to vary with size. Later, it was suggested for coal [7] that:

(12)

(14)

where a* and (Yare constants for a given material and conditions (0 0), and x0 is a reference size. If the connection is made between rate of breakage (proportional to probability of breakage) and strength of a particle, other expressions may be surmised. For example, for approximately spherical agglomerates [8]: Strength = kd’a 1 - SDi

(15)

where k and r are constants, and d is agglomerate diameter. In all cases mentioned above, change in either breakage distribution parameters or specific rate of breakage with time has not been allowed for. Correlation of the first-order model of particle disintegration with breakage mechanisms The first-order disintegration model has been widely

applied, yet specific correlations of the model to the mechanisms of particle disintegration are rare. The model has been used [9] to describe erosion (termed abrasion) and shattering behavior in catalysts. The authors showed, for a system consisting of five size intervals, how transitions between particle sizes might take place if each mechanism were operating, and presented matrices of allowable and forbidden transitions between these size intervals. It is readily seen from eqn. (l), however, that an inherent limitation on the first-order model of disintegration is that a particle may not break and still remain in its original size interval. In other words, all material that is broken in size j is removed from the interval. Clearly this is not mechanistically the general case, being especially untrue when erosion is dominant. In such a case the production of relatively tine daughter particles has practically no effect on the size of the parent particle, especially when particle shape and measurement variability are taken into account. Thus, to make the model more generally applicable to disintegration processes, and more mechanistically accurate, the possibility of this type of behavior must be accounted for. This may be done simply by changing the summation term of eqn. (1) to include j =i:

(16)

23 BREAKAGE

PARAMETER

MATRICES

EROSION DOMINANT

a.)

a.) EROSION DOMINANT

FROM

nl

SIZE

bfx)/\ b(x’Qd+-

n

““-,“-2”-3”-4

PARAMETERS TIVELY HIGH AREAS

b.1SHATTERINGDOMINANT FROM S,ZE

,

2

3

.

. .

“_.7”_,”

”

z

“-2

. . . . . . .

.

. .

. . .

.

. .

.

.

. . . .

.

5

4

3

2

,

.

c.)

n-2

:

n-1

”

n-1

. . . . . . . . .

:

:

b(xlLEl

2 i 3 2

n-1

”

1

n-2

SIZE

n-3

n-4

*

SIZES n,n-1, n-2 ARE DESIGNATED AS "FINES"

Fig. 1. Breakage parameter matrices (b,,j values in eqn. (1)) for erosion- and shattering-dominant cases.

This equation may be solved using the Euler method as before, resulting in: wi(t)=w,(t-~)[l-S,ht]+

$Albi,

jSmWj(t-At)

j-l

(17) It should be noted that extensions of the first-order model for describing erosion-type behavior have been developed [lo, 111, primarily for autogenous ore breakage, where abrasion produces a very fine powder from large particles. The extension described here assumes that the fines produced are somewhat larger, and is thus applicable to a rather different type of erosion process. It is also evident that biiSDi occurs as a product when j=i. It would seem, therefore, that each value cannot be determined individually. However, it must be recalled that breakage behavior in each size interval is part of an overall pattern. Thus, the time variant behavior in other size intervals should aid in separating bii from S, in the ith interval. The solution method will, of course, be somewhat more difficult, and development of manageable methods is a significant challenge.

Fig. 2. Characteristic shapes of the continuous b(x) (breakage parameter) function for the erosion-dominant case. The curves are given such that (a) is for the largest initial particle size, (b) is for a size smaller than the largest possible, and (c) is for an initial size particle near the fines range (compare Fig. 3).

The breakage function The breakage parameter matrix is a very convenient way to visualize the disintegration mechanisms. If we consider that a particle population is undergoing an erosion process, the breakage matrix would take on values as shown in Fig. la, with only fine progeny being produced from larger particles. For dominant shattering, the matrix would appear like that in Fig. lb, where larger particles result from a breakage event. Note that for breakage occurring in the lower size intervals the matrices are represented as identical. This is because at those small sixes any breakage mainly results in particles that are relatively large in comparison to the original particle. If we desire to represent the breakage parameters by a continuous function, we must only observe that for constant j, Bi, j is a normalized distribution function in cumulative form. Thus for a size interval: AT=Xi b

1

-Xi+

Bi. j-Bi+l.i

AZ-+0

Ax

(18) =

do =b(x)

z

(19)

so that dB = b(x)&

(20)

24 SHATTERING DOMINANT

TABLE

a.1

1. Selected

Size interval

j=l

G)

Upper size

b(x)

(xi) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

b.)

b(x)

b(x’lLzd n-2

n-1

”

n-3

xi dB=

we

s xi-cl

W

dx

obtain (recall eqn. (12)): d

Bi, j-Bi+l, j=bi, j=

s

b(x) h

xi+1

Thus any normalized frequency function (b(x)) may be used to calculate the breakage parameters. If we select a small size interval, Ax, and integrate using the approximation: ai s xi+1

b(x) dx=bxb(xi+,+Ax/2)

then we see that it is reasonable bi,j = b(x,)Ax

Lower size (xi+l)

Upper size (xi)

-

0.8409

0.7071 0.5946 0.5000 0.4204 0.3536 0.2973 0.2500 0.2102 0.1768 0.1487 0.1250 0.1000 0.0750 0.0500 0.0250 0

Lower size (xi+l)

1 0.8409 0.7071 0.5946 0.5000 0.4204 0.3536 0.2973 0.2500 0.2102 0.1768 0.1487 0.1189 0.0892 0.0595 0.0297

0.8409

0.7071 0.5946 0.5000 0.4204 0.3536 0.2973 0.2500 0.2102 0.1768 0.1487 0.1189 0.0892 0.0595 0.0297 0

n-4

Upon integrating,

j

j=2

‘Note that sizes 13 through 17 are designated as fines and that the fines size for j=2 are relatively bigger than in j= 1.

Fig. 3. Characteristic shapes of the continuous b(x) (breakage parameter) function for the shattering-dominant case. The curves are given such that (a) is for the largest initial particle size, (b) is for a size smaller than the largest possible, and (c) is for an initial size particle near the tines range (compare Fig. 2).

7 Bi+l,

1 0.8409 0.7071 0.5946 0.5000 0.4204 0.3536 0.2973 0.2500 0.2102 0.1768 0.1487 0.1250 0.1000 0.0750 0.0500 0.0250

intervals for j= 1 and j=2”

(23)

obtain the shapes shown in Figs. 2 and 3. Figure 2 shows how in the erosion-dominant case the probabilities are heavily weighted toward fines production and retention of original particle size. Conversely, Fig. 3 shows how shattering breakage probabilistically results in the creation of relatively large progeny. As indicated, the bi,j, at each size interval would have a somewhat different shape. This is because the range of sizes to which the breakage products can be reduced is different in each fraction. Note that even in the erosion-dominant case, the breakage pattern of relatively small particles in the population is expected to be of the shattering typeThis is due to the relative enlargement of the fines region in relation to the breaking particle.

Use of the bimodal size distribution The shapes of the b(x) functions necessary for all cases can be provided by the bimodal P-distribution [12]. (Models of the kind expressed by eqns. 10 and 11 cannot generally describe curves with these shapes). This results in the breakage function in the form:

to write (24)

Now, envisioning from the breakage matrix (Fig. 1) what bi,, as a continuous function would look like in erosion-dominant and shattering-dominant cases, we

0

25

6,

4

I SIZE

BREAKING

= 1

SIZE

BREAKING

= 1

SIZE

BREAKING

= 12

5

4

S

2

I

D SIZE

NORMALIZED

to

SIZE

SIZE

BREAKING

Fig. 4. Shapes of the b(x) function for the erosion-dominant case calculated using x,,,~,= 0.925, m, =25, mz= 1.5 and A =0.2 in the bimodal modified p-distribution (compare Fig. 2).

LLd

.

.4

NORMALIZED

.

.

to SIZE

BREAKING

Fig. 5. Shapes of the b(x) function for the shattering-dominant case calculated using x,,,, =0.7, m,=25, mz=4 and A=0.05 in the birnodal modified pdistribution (compare Fig. 3).

where A is the weight fraction of the fines and the as and ms are the characteristic powers of the fine and coarse fraction distributions (subscripts 1 and 2 refer to the fine and coarse fractions respectively).

+ (l-AjYY”~(l-x)“a 1 n

Results and discussion

1

w

1

(26)

dx

or alternatively:

hisj=

+ (1;-Ajx?~“~(1-xyn~ s

0

YWrnY( 1 -x)m%

~

The proposed extension of the first-order disintegration model, and the use of the proposed breakage function were demonstrated through computer simulations. This was done as follows. The size range of particles was.normalized over 0 to 1. Particles with normalized size below 0.125 were defmed as fines, which were not allowed to break. This fines region was, however, still divided into five equally sized intervals. For sizes above 0.125, the ratio, R,of lower to upper interval sizes was set to 2-‘I“, since it has been shown [13] that this ratio always results in first-order breakage kinetics. Or, in symbols: &Cl xi

=R=2-1"

(27)

TIME -25

-“I

6

3

0

0

2

4

6

-NORMALIZED

.8

SIZE -

-NORMALIZED

SIZE -

Fig. 6. Frequency profiles obtained for the erosion-dominant case starting with a relatively narrow particle size distribution (compare Figs. 7 and 8).

Fig. 7. Frequency profiles obtained for the erosion-dominant case starting with a relatively wide particle size. distribution (compare Figs. 6 and 9).

The resulting seventeen intervals are listed in Table 1. (Other interval ratios were also tried for comparison but since they had no major effect on the results they are not reported.) The breakage function used, eqn. (26), requires normalization to the upper size of the jth interval (the one being broken), and the other sizes must be kept in proper relation to this size. Thus, it is required that the adjusted normalized xi be calculated from:

like those shown qualitatively in Figs. 2 and 3. The following assumptions were also made: (a) The midpoint of the fines region was assumed to be the mode, X,11, of the fines distribution or [4]:

4

normalized xi = Ri-1 -

(28)

for all i and j. This type of normalization is also demonstrated in Table 1 for j= 2. It shows that the relative size of the fines increases as the size breaking decreases. The breakage rate assumed was that given by eqn. (14), with arbitrarily selected constants of a* = 0.3, a= 0.7 and x0= 1.0. The five constants of the breakage function, namely uy, Q, my, my and Wj, were selected to give shapes

(29) (b) The coarse mode, x,,, according to: xmy= GZ,-(xmz, -0.5)

j-l n,-1

was assumed

( 1

to vary

(30)

where n,, is the number of intervals that may break. (c) m,,, m, and Aj were assumed constant for all j. For the erosion-dominant case the selected constants were x,,, = 0.925, m, = 25, m, = 1.5 and A = 0.2. For the shattering-dominant case, the constants selected were X,*,=0.7, m1=25, m,=4 and A=0.05. Curves generated using these constants are shown for the breakage of various sizes in Figs. 4 and 5 for erosion-dominant and shattering-dominant cases respectively. The simulated size distributions were produced using eqn. (17)

27

9

6

3

0

9

6

3

0

L L

TIME = 2.5

I

TIME =25

TIME '12 5

9,

6

3

0

IL-

0

2

4

-NORMALIZED

6

SIZE

-1

TIME= 25

6

0

8

-

.2 4 0 -NORMALIZED

6

.8

SIZE

-

Fig. 8. Frequency profiles obtained for the shattering-dominant case starting with a relatively narrow particle size distribution (Compare Figs. 6 and 9).

Fig. 9. Frequency profiles obtained for the shattering-dominant case starting with a relatively wide particle size distribution (compare Figs. 7 and 8).

and the IMSL routine DCADRE for the integrals calculation. The results were converted from weight fractions ( Wi(t)) to frequency 6 (t)) using the conversion:

Frequency profiles obtained for the shattering-dominant cases with narrow and wide initial particle size distributions are shown in Figs. 8 and 9 respectively. In contrast to the erosion-dominant case, few fines are generated as the coarse mode rapidly declines. These simulated disintegration patterns demonstrate the flexibility of eqns. (17) and (26) and the modified first-order disintegration model (which allows a particle to break but still remain within its original size interval) in describing processes governed by different mechanisms. Thus, the same set of equations can be used for disintegration processes dominated by both erosion and shattering.

J(t)=

.-!5@-G--G+

1

(31)

In order to check the validity of results, three time intervals, ht= 0.025,O.l and 0.25, and three size interval ratios, R=2-ln, 2-lf4 and 2-lB, were tried in specific cases and compared. The time interval had almost no effect on the results, so a fixed At =O.l was used for all the presented simulations. Results with different size interval ratios similarly had little effect on the disintegration pattern, especially that of the smaller fractions. Thus the ratio R= 2-‘j4 was used for all subsequent simulations. Frequency profiles obtained for the erosion-dominant cases with narrow and wide initial particle size distributions are shown in Figs. 6 and 7 respectively. As expected, in each case fines are gradually accumulated while the coarse fraction, correspondingly, decreases in weight, but more slowly in mode.

Comparison of the model with instant coffee attrition data The disintegration of instant coffee particles is a good example of processes in which both shattering and erosion take place [15]. The relative dominance of the two mechanisms as well as the overall rate and intensity of the process, however, depend on the coffee type, freeze- or spray-dried, initial particle size, and

I

JIMF (mini

MEAN SIZE (mm)

Fig. 10. Dominant erosion Data from [15].

particle size distributions of freeze-dried instant coffee (-6+8

the mechanical conditions to which the particles are exposed. Typical kinds of particle size distributions of agglomerated and freeze-dried coffees undergoing mechanical attrition are given in Figs. 10-12. They show how the attrition rate as well as the relative dominance of the disintegration mechanism depend on the coffee type and the manner in which the mechanical energy is applied. In fact, one could also expect, intuitively or according to the described model, that as the attrition proceeds there will be a shift in the relative dominance of the breakage mechanism, especially as the particle size distribution is considerably altered, or in other words, once the particles are reduced to a small size they are more likely to shatter than to erode. This was indeed observed and documented in both coffee types

mesh) subjected to vertical vibration.

preciable number of the large particles can remain in their original size fraction despite the formation of a considerable amount of fines. Such a behavior is predicted and can be described by the proposed model but not by previously published models. This is because the proposed model allows for both a continuously changing bimodal breakage function and the survival of particles in their original size fraction after losing material to fines. Although only coffee examples are presented, this size reduction behavior is also typical to other fragile food pa&mates, notably breakfast cereals. It is expected however that the proposed model is not limited to dry food particles and is just as applicable to other brittle materials such as agglomerated pharmaceuticals and certain minerals.

WIFigures 10-12 also clearly show that once the coffee particles are disrupted their size distribution becomes unmistakably bimodal. They also demonstrate that, at least in the erosion-dominated case (Fig. lo), an ap-

Acknowledgements

Contribution of the Massachusetts Agricultural Experiment Station in Amherst. The work was partially

29

120

x)0

I 2,

E Lo x

1

NUMBER OF SAPPI-

I.

0

0

0

10

020 040 060 0 100 v 180 0260

MEAN SIZE

Fig. 11. Dominant from [15].

shattering

-

particle

(mm)

size distributions

of freeze-dried

instant coffee (-6+8

mesh) subjected

to tapping. Data

NUMBER OF TAPPINGS :

MEAN SIZE

Fig. 12. Mixed mechanism tapping. Data from [15].

-

particle

(mm)

size distributions

of agglomerated

spray dried

instant

coffee

(-6+8

mesh)

subjected

to

30

supported by the Particulate and Multiphase Program of the NSF (Grant No. CBT 8520370). The authors express their thanks to the sponsoring agencies for the support, and to Mr. Richard J. Grant for his graphical aid.

Greek letters

; Y @

power in power in power in coefficient

eqn. (14) eqn. (11) eqn. (11) in eqn. (11)

Subscripts of fraction i

List of symbols a a* A b B C d

? k m n nb 4

l? SD

t W x Y

a power of the p-distribution function defined in the text a coefficient in eqn. (14) weight fraction of the fines in a population with a bimodal size distribution proportion of breakage products that fall into a given size interval proportion of daughter particles that are smaller than a specified size a probability of survival in eqn. (13) agglomerate diameter (eqn. (5)) (arbitrary length units) frequency of a size interval (eqn. (31)) identity matrix a coefficient in eqn. (15) a power of the p-distribution function number of intervals number of intervals that are allowed to break number of time intervals (eqn. (9)) a power in eqn. (15) size ratio of successive intervals probability of breakage in a given time interval time (arbitrary time units) weight fraction of the material in a given size interval normalized or absolute particle size size in eqn. 10

i m

0 1 2

of of of of of

in size interval i fraction in size interval j a fraction that breaks in eqns. (4)-(6) a reference size the fine fraction in eqns. (25) and (26) the coarse fraction in eqns. (25) and (26)

References 1 B. Epstein, Ind. Eng. Chem., 40 (1948) 2289. 2 C. L. Prasher, Crushing and Grinding Process Handbook Chapter 8, Wiley, Chichester, 1987. 3 S. R. Broadbent and T. G. Callcott, J. Inst. Fuel, 29 (1956) 528. 4 G. Austin, Powder TechnoL, 5 (1971/72) 1. 5 K. J. Reid, Chem. Eng. Sci., 20 (1965) 953. 6 K. Shoji, S. Kohrasb and L. G. Austin, Powder Technol., 25 (1980) 109. 7 R. Berenbaum, J. Inst. Fuel, 35 (1962) 346. 8 C. E. Capes, Powder Techno/., 5 (1972) 119. 9 J. Wei, W. Lee and F. J. Krambeck, Chem. Eng. Sci., 32 (1977) 1211. 10 J. M. Menacho, Powder Technol., 49 (1986) 87. 11 L. G. Austin, C. A. Barahona and J. M. Menacho, Powder Technol., 46 (1986) 81. 12 L. M. Popplewell, 0. H. Campanella and M. Peleg, Chem. Eng. Prog., 8.5 (8) (1989) 56. 13 L,. G. Austin and P. T. Luckie, Powder Technol., 4 (1970/71) 109. 14 M. Peleg and M. D. Normand, AZChE J., 32 (1986) 1928. 15 J. Malave-Lopez and M. Peleg, J. Food Sci., 51 (1986) 691. 16 L. M. Popplewell and M. Peleg, Powder TechnoL, 58 (1989) 145.