an algorithm for determination of the size-dependent breakage frequency of droplets, flocs and...

315 Part. Part. Syst. Charact. 8 (1991) 315-322

An Algorithm for Determination of the Size-Dependent Breakage Frequency of Droplets, Flocs and Aggregates

Yu-Chen Chang, Richard V. Calabrese, James W. Gentry * (Received: 17 January 1991; resubmitted: 21 May 1991)

Abstract

A method, based on the use of discrete size classes, is developed to extract size dependent breakage frequencies from sequcntial measurements of the size distribution. In order to obtain good resolution and allow for several breakage modes, interlaced Fibonacci series are used to define the size classes. Both binary and ternary breakage are considered. An approach based on Kernel Discriminant Analysis is used to focus the breakage at

the discrete size classes. The algorithm is tested using simulated distribution data. The sensitivity of the retrieved breakage frequencies to the assumed progeny distribution and to the number of breakage events between samples is considered. The numerical experiments show that the method is effective even when the breakage mechanism is not well understood.

1 Introduction

The algorithm described in this paper incorporates two basic ideas-(1) the extension of the algorithm suggested by Gentry et al. [l] and Soulen et al. [2] for determination of size dependent collection efficiency to predict particle breakage frequency and (2) the use of nested Fibonacci series to define the size categories. The algorithm to extract the size dependent frequencies is related to the kernal method of discriminant analysis, especially the method of Rosenblatt [3]. The objective of the algorithm is to obtain the size dependent breakage frequency from experimental data consisting of the ratio of number concentrations and the size distributions before and after the breakage event. In typical applications these data will consist of photographs at high magnifications before and after breakage. The ratio of number concentrations is inferred from the conservation of volume. In most applications the number of drops or clusters actually sized will be relatively small (100-lOOO), so that the algorithm must be able to effi- ciently use small samples. In general there are two unknown functions in breakage and fracture processes - the breakage frequency B ( Y ) and the distribution frequency of progeny K ( I : X ) . Here they are defined such that B ( Y ) is the conditional probability that a particle of size Y will break during a discrete event (e. g. time step or experiment). K (I: X ) is the product of the number of progeny that are formed when a particle of size Y breaks and the conditional probability that a particle of size X will be formed. We assume that all particles can be assigned to a small number (20-400) of size classes and that each consists of a large number (up to lo8) of primary units. A discretized procedure for sor- ting is devised and the algorithm is designed to compute B,, the breakage frequency of drops with sizes falling in the i th class.

~~~~ ~ ~

* E C. Chang (Graduate Student), R. T.: Calabrese (Associate Pro- fessor), J. W Gentry (Professor), Department of Chemical Engineer- ing, University of Maryland, College Park, Maryland 20742 (USA).

The distribution frequency is then a matrix of elements, Ki,, corresponding to the probable fractions of particles belonging to size class j that would result from fracture of a particle belonging to size class i. In the algorithm the distribution frequency of breakage progeny must be assumed. For a particle whose size is such that it consists of M primary units, we consider two types of breakage. Binary breakage is defined by

and ternary breakage is defined by

M U ) = M ( p ) + M ( q ) + M ( r ) . (2)

M (j) is the number of primary units belonging to the j th size class and is therefore a measure of the mass or volume of the parent. With these definitions the distribution frequencies are constrained by the relations :

C Kij = 2 for all i (binary breakage)

and

C Kij = 3 for all i (ternary breakage).

In general it is not possible to separate the breakage and distribution frequencies, nor are the distribution frequencies known. Based upon analogous problems in Kernel Discriminant Analysis, there is reason to believe that the breakage frequencies obtained from the algorithm may not be sensitive to the distribution of progeny provided that the algorithm employs suitable weighting techniques. To test this conjecture we consider a series of numerical experiments in which the K (I: X ) employed to retrieve the breakage frequencies are different than those used to simulate thc breakage process. Unless the retrieved breakage frequwcies are relatively insensitive to the assumed

0 VCH Verlagsgesellschaft mhH, D-6940 Weinheim, 1991 0934-0866/91/0412-0315 $3.50 + .25/0

316 Part. Part. Sysl. Charact. 8 (1991) 315-322

K ( I : X ) , the algorithm is of limited value, for as mentioned above, the distribution of progeny are not known. In summary, the principal contributions of this work are the following :

An algorithm for computing the discretized size-dependent breakage frequencies is derived and developed. A method for assigning discrete size classes based on Fibonacci series is developed. The procedure is applied to binary and ternary breakage. Sets of interlacing Fibonacci series consisting of 3 and 6 sequences for binary breakage and 3 sequences for ternary breakage are found. The algorithm is tested for simulated measurements and found to be efficient in retrieving the breakage frequencies. Based on these simulations, several rules-of-thumb are pro- posed for processing measurements when the distribution of progeny are not known.

Derivation of Method

In this section the algorithm for determining the breakage frequency from raw data consisting of particle or drop size distributions before and after breakage is derived. The method assumes that the total volume is conserved and that the breakage distribution frequency K ( I : X) is known. In a subse- quent section the choice of K (I: X ) is discussed. Here the derivation proceeds under the assumptions that K (I: X ) is known and has non-zero values for only a finite number of X’s for each Y Consider a size class characterized by particle size X and bin width dX. For a single breakage event the material balance for particles in this size class is given by

(3)

where subscripts 1 and 2 refer to quantities before and after the breakage event, respectively. N is the total of the number of particles and F ( X ) is the number probability density of particles of mass or volume X. The term on the right hand side of Eq. (3) is the number of particles or droplets in the size class after breakage. The first term on the left hand side is the number which do not fracture, while the second term represents the pro- duction of new particles of size X from the fracture of all larger particles. Eq. (3) can be multiplied by a weighting function as described below and integrated from X = 0 to 00.

In an experiment, the data would consist of the measured distribution functions F , ( X ) and F, ( X ) and an estimate of N,/N, as determined from mass conservation (to be discussed later). For the ideal case, the sample size would be sufficiently large to acquire the true value of these quantities. F, (X) and F, (X) could be described by continuous, “smooth” functions. The inversion process would be “well behaved” and extracted values of B (Y) would be smooth and continuous in E: In reality, the sample size will be insufficient to insure such behavior. The histograms for F, (X) and F, ( X ) will be noisy, the inversion process will not be well behaved and the resulting B (Y) could behave erratically. To circumvent this problem a scheme, consistent with Kernel Discriminant Analysis, is employed to smooth the data. We choose a specific weighting function H (a, x) which, in the limit of large a has the form

H ( a , X ) = c S ( X - X,) (4) K

where the subscript on X refers to a particle’s size class and the X k include the size XI and all sizes X, > Xj which, after fracture, produce one or more particles in size class Xj. 6 (X - X,) is a Dirac function which gives all discrete size classes an equal weighting. That is, there is no smoothing or distribution of particles among adjacent size classes. When Eq. (3) is multiplied by Eq. (4) and the result integrated with respect to X over all allowable size classes, we obtain:

F,j and Pzj are the number fractions for size class j prior to and after breakage, and Bk is the unknown breakage frequency for size k. The distribution frequency Kk, assigns to size class j the progeny resulting from fracture of particles of size k. When applied to all size classes, Eq. ( 5 ) yields a set of simulta- neous linear equations of the form

where the B, are the breakage frequencies, D, = N2F2p - N I F l p , and the matrix element Cpq is defined by

(7)

C,, represents particles of size p which break to form smaller particles and Cpq represents particles of size p formed by breakage of larger particles. Because of the upper triangular structure of the matrix Cpq, the elements B, can be obtained directly by back substitution. For smooth simulated data, the functional form of H ( a , X) given in Eq. (4) can be used. For real data an expression for H(a, x) should be used that can properly smooth the data. While several useful functional forms are possible, the following one is recommended:

H ( a , X) = Exp k

As the parameter a increases, the functional form of Eq. (8) asymptotically approaches that of Eq. (4). A logarithmic form ILn (X/X,)/ was chosen instead of an algebraic form I X - X, I /X, since it allows geometrical symmetry without undue complexity. In the absence of the absolute value brackets, a reduce3 to a power law parameter. In related problems [4, 51 we found the method to be somewhat insensitive to the particular functional form of H ( a , X). This is consistent with results reported for related problems in Kernel Discriminant Analysis [6]. In summary, a critical assumption in the above derivation is that there are only a finite number of nonzero KkI for each size classj. That is, there are only a few size classes k whose progeny are of size j . Furthermore, we assume that the Kkl can be specified a priori.


3 Fibonacci Sequence

The procedure for specifying size classes is based on the assumption that drop or cluster sizes can be divided into 20 to 400 size classes. These classes must have the property that if a drop or cluster having a size of M u ) primary mass or volume units fractures, then all the fragments have sizes M ( k ) which belong to one of the defined size classes. This closure property severely restricts the way in which size classes can be chosen. We have found two types of Fibonacci sequences which satisfy this con- straint. In addition we have found that in certain cases it is possible to use sets of coupled or interlaced Fibonacci sequences. As previous discussed, we consider two types of fracture or breakage; binary breakage in which the particle splits into two parts and ternary breakage in which the progeny consist of three parts. For ternary breakage we are particularly in- terested in the case where two of the progeny (usually the smaller droplets) are the same size. Geometrical sequences are defined by the property that the number of units in two adjacent size classes are related by

MO‘ + 1) = OMO’) . (9)

For example, a size classification based on a geometrical sequence frequently used in numerical solutions of the coagula- tion equation is defined by

M(1) = 1

MO‘) = 2MO’ -1) = 2 j - l M ( l ) .

This definition results in size classes of 1, 2, 4, 8, 16, 32, . . . units. Fewer than 30 size classes are required to describe clusters with sizes up to lo7 units. Disadvantages of this classification is that for binary breakage the only practical possibility is for equally sized progeny and the size classification is coarse since the size ratio for adjacent bins is R = MO’)/M(j - 1) = 2. For other than equal sized progeny, the number of size classes that are required becomes unbounded. The Fibonacci sequences used for binary breakage are defined by

MO’) = MO’ - 1) + MO’ - 2 ) . (11)

Each number is the sum of the previous two numbers in the sequence. These sequences have the property that for large j , the size ratio R = Mfj) /MO’ - 1) = [5”* + 1]/2 = 1.618 regard- less of the values of M(1) and M(2). This means that for any two choices of starting pairs (M (I), M(2)), two Fibonacci sequences will be generated which either intertwine or are identical after a few terms. These intertwined series can be employed to obtain even finer resolution. In this work sets consisting of three interlaced Fibonacci sequences are used. These sets are given in Table 1 for binary breakage. We see that for each constituent sequence, 4 progeny pairs are possible. By interlacing the 3 constituent sequences we increase the possible number of progeny pairs from 4 to 12. Fur- thermore, in the interlaced sequence the size ratio for adjacent bins is smaller than for a single sequence, thereby improving resolution. For the interlaced sequence, all members of a given constituent sequence have the same size ratio with their adjacent member in the next constituent sequence. For the sequences of Table 1 these are 1.170 for series 2 to 1, 1.236 for series 3 to 2 and 1.118 for series 1 to 3. Since these represent

Table 1 :

Parent Sequence 1 Sequence 2 Sequence 3 Constraints

Sequence of three nested Fibonacci series (binary breakage).

M ( 3 J + 1 ) M ( 3 J + 2 ) M ( 3 J + 3 )

Initiation jM(I), 2,1 1 9 1 292

(211 Ratio’ 1.170 (2to1) 1.236 (3to2) 1.118 ( l t o3 ) Progeny 1 M ( 3 J ) M ( 3 J + 1) M ( 3 J + 2) J > 4

M ( 3 J - 13) M ( 3 J - 10) M ( 3 J - 7) 2 M ( 3 J - 1) M ( 3 4 M ( 3 J + 1 ) J > 3

M ( 3 J - 7) M ( 3 J - 7 ) M ( 3 J - 4)

M ( 3 J - 5 ) M ( 3 J - 4) M ( 3 J - 3)

M ( 3 . 1 - 4) M ( 3 J - 3) M ( 3 J - 1)

Size ratio for adjacent bins in interlaced sequence.

3 M ( 3 J - 2) M ( 3 J - 1) M ( 3 J ) J > 2

4 M(3.1 - 3 ) M ( 3 J - 2) M ( 3 J - 1) J > 2

mass or volume size ratios, the diameter ratios would be 1.054, 1.073 and 1.038 respectively. These can be compared to a diameter ratio of 1.260 for the geometrical sequence of Eq. (10). Even finer resolution can be obtained by employing more constituent series. For instance, as will be discussed below, size classes can be defined based on an interlaced sequence of six Fibonacci series. The third set of progeny in Table 1 have the property that they all belong to the same constituent sequence as the parent. Fur- thermore, after the first few terms, the series combining the three constituent sequences has the property that every third term belongs to the same constituent sequence. This means that the relation :

M ( 3 k + i ) = M(3k + i - 3) + M ( 3 k + i - 6) with i = 1, 2, or 3 (12)

corresponds to the Fibonacci relation for the original sequence. As a result Eq. (12) can be used to generate the most general set of size classes that will accommodate all possible binary breakage events of Table 1. Examining the elements in the com- bined sequence we note that

2M(3k + i - 4) 2 M(3k + i ) > 2M(3k + i - 5 ) . (13)

In the discussion below we consider simulations in which particle breakage results in progeny whose sizes are not drastically different. A combination of 3-6 breakage and 4-5 or 4-4, 4-5 breakage is employed. 3-6 breakage (see Eq. 12) occurs when both progeny are within the same constituent sequence as the parent. The latter breakage results in breakage into two particles of nearly equal size with both progeny contained in the constituent sequence adjacent to that of the parent. As previously noted, interlaced sequences consisting of more than 3 constituent sequences can be used to define the size classes. We carried out extensive numerical simulations with different combinations of Fibonacci series for binary breakage. It was found that by choosing M(2) 5 M(1) 5 N*, one could generate interlaced sequences consisting of K* = (N* + 1) N*/2 independent, constituent sequences. Not all of the sequences had the closure property necessary for completeness. The required closure property is that when any two elements of the set are added they produce an element which also belongs to the


set. Specifically, we found that for N* I 3 or K * I 6 the interlaced sequences were complete but, for N* ? 4 and N* s 9, there were some breakages which gave products outside the set. Based on these simulations we doubt if there are suitable closed sets of nested Fibonacci sequences other than the K* = 1, 3, and 6 element sets which are listed in Table 2. For N* > 3, between 1.2 and 1.5% of the possible breakage pairs are not per- missible in that they yield elements outside of the set. The precluded breakages for N* = 4 and N* = 5 are given in Table 2. The sequences are not uniformly spaced in that the limiting value of the ratio R = M ( k + I ) / M ( k ) is not cons- tant. It has the average value of 1.6181’K* with an upper bound of 1.6182K* and a lower bound of 1.618°.5K*.

Table 2 : Properties of nested Fibonacci series (binary breakage).

Number of Initiation (N*) Ratio2 Prohibited Total Series K* N * r M ( l ) z M ( 2 ) M(J+ l ) /M(J ) breakages breakages

1 1 1.618 0 1 3 2 1.174 0 12 6 3 1.084 0 48

10 4’ 1.049 2 139 15 5 1.032 4 3 26

No Closed Sequences Were Found for K* > 6 . The Precluded Breakages Were

Sequence Initiation Pairs

A. K* = 10

M(1OJ + 6 ) = M(1OJ - 8) + . . . 4, 3 , and 4, 2 M ( 1 0 J + 10) = M ( l 0 J - 4) + ... 4, 4 , and 4, 3

B. K* = 15 M(15J + 5 ) = M ( 1 5 J - 15) + ... 5 , 4 and 5 , 3 M ( 1 5 J + 7 ) = M ( 1 5 J - 15) + . . . 3 , 1 and 5 , 3 M ( 1 5 J + 9 ) = M ( 1 5 J - 10) + . , . 5 , 5 and 5 , 4 M ( 1 5 J + 15) = M ( 1 5 J - 5 ) + ... 5 , 3 and 5 , 2

Example: For N* = 4, there are 10 constitient series in the interlaced sequence and the initiation pairs are the 10 independent binary com-

Average size ratio for adjacent bins in interlaced sequence. ’ binations of 1 , 2 , 3 , and 4 .

Table 3 : Sequence of three nested Fibonacci series (ternary breakage).

Parent Sequence 1 Sequence 2 Sequence 3 M ( 3 J + 2) M ( 3 J + 3 ) M ( 3 J + 4)

Initiation 1,l 397 1,5 Ratio’ 1.25 (2to1) 1.20 (3to2) 1.33 ( l tO3) Progeny

1 M ( 3 J + 1) M ( 3 J + 2) M ( 3 J + 3 )

M ( 3 J - 8 ) 2 M ( 3 4 M ( 3 J + 1) M ( 3 J + 2)

M ( 3 J - 6 ) 2 M ( 3 J - 7 ) M ( 3 J - 6 ) M ( 3 J - 8 )

M ( 3 J - 4) 2 M ( 3 J - 4) 2 M ( 3 J - 4) M ( 3 J - 7 )

3 M ( 3 J - 1) M ( 3 4 M ( 3 J + 1)

4 M ( 3 J - 1) M ( 3 J) M ( 3 4 2 M ( 3 J - 4) 2 M ( 3 J - 3 ) 2 M ( 3 J - 2)

M ( 3 J - 2 ) M ( 3 J - 1) fM(3J - 1) M ( 3 J - 7 ) M ( 3 J - 7 ) M ( 3 J - 2)

5 M ( 3 J - 2) 2 M ( 3 J - 1) 2 M ( 3 J ) 2 M ( 3 J - 3 ) M ( 3 J - 4) M ( 3 J - 4)

Size ratio for adjacent bins in interlaced sequence.

For ternary breakage, size class selection is much more com- plicated. For an interlaced sequence, the bin spacing and allowable breakages that result depend upon how the breakage is specified for a single Fibonacci series. Here we consider the case where one relatively large and two smaller but equally sized daughters are produced. This type of fracture occurs when satellite droplets are stripped off a parent. The single Fibonacci like sequence analogous to Eq. (11) is then

The allowable breakages which result for a set consisting of 3 interlaced Fibonacci like sequences are given in Table 3. For each constituent series, 5 progeny pairs are possible. Three- fifths of the allowable progeny pairs contain two equal size particles. Although breakages within a constituent sequence must contain two equal sized daughters, the interlaced sequence allows breakages into daughters which are all different in size.

4 Numerical Simulations

In order to test the algorithm a series of numerical experiments were carried out. A size dependent breakage frequency B ( Y ) and the progeny distribution frequency function K A (I: X ) were assumed. These were used with an assumed distribution function, Fi ( X ) prior to breakage to calculate the distribution F2 (X) after breakage. A second progeny distribution frequency function KB (X X ) , which may be different from KA (I: X), was then assumed. This was employed in the inversion algorithm with the assumed F, ( X ) and calculated F2 (X) to extract the breakage frequency. The retrieved breakage frequency could then be compared with the “true” or assumed breakage frequency. In the simulations the sensitivity of the results to the sharpness of the breakage frequency and assumed progeny distribution frequency are examined. The simulations are carried out under the following assumptions : (1) The initial distribution, F, ( X ) , was simulated using

Nl = lo7 primary particles. It was assumed that the particles were log-normally distributed in number with a mean equivalent to 100 diameter units and a number standard deviation of 0.35. The number distributions were converted to volume units for use in the algorithm and then converted back for presentation of results. Since the initial distribution was well behaved, a was chosen sufficiently large so that Eq. (8) reduced to Eq. (4).

(2) The assumed breakage frequency is given by

where X,,, is the mean of the inital distribution in volume units. No assumption or restriction is imposed on the retrieved breakage frequency.

(3) The size classes are described by three interlaced Fibonacci series.

(4) The breakages are binary with there being two possible breakages : (i) 3-6 breakage where the progeny belong to the same

(ii) 4-5 breakage where the progeny are more nearly equal constituent Fibonacci series and

sized.

Part. Part. Syst. Charact. 8 (1991) 315-322 319

0.0

Therefore, the summation in Eq. (5 ) contains only a few terms. It is necessary to impose an additional condition that the number of progeny belonging to each of the constituent sequences be the same. This is necessary if the distribution after breakage is not to contain oscillations. Consider the interlaced series consisting of the three constituent sequences with initiation pairs (M(1) and M ( 2 ) ) of [2,1], [2,2] and [1,1]. Let the fraction of particles in the constituent series which experience 3-6 breakage be 8, r$ and t respectively. Then the fractions for 4-5 breakage are 1-0,l - @ and 1-7. In order to have smooth (non- oscillating) breakage the following constraints must apply :

/ PO% 50% 70% 90%

, I ,--.dC.w4, I I L 4 I I A . I I I

0 50 100 150 200

For example, the size distribution is plotted in Figure 1 for particles or droplets undergoing pure 4-5 breakage (8 = r$ = r = 0). The final distribution oscillates unrealistic- ally. For @ = 0.75 and @ and r as given by Eq. (16), the final distribution is both smooth and probably more realistic.

, '1 0.12

0.0

FINAL DISTRIBUTION

10% 50% 70% 90%

, I , + z . 4 A 1 8 , , . 14. 1 , ,

b = 5 a = 1 4

0.0

DIAMETER

Fig. 1 : breakage steps.

Effect of progeny distribution on size distribution after two

0

0 10% 50% 70% 90%

, 1 . d , 4 , I I A . 4 . $ 4 , 3 I I

The key question in using the method is whether the retrieved breakage frequencies are dependent upon being able to specify the true progeny distribution function a priori. In the simulations this quantity is written in terms of 3-6 and 4-5 breakage events and the fraction of particles undergoing each type of breakage. Therefore, given Eq. (16) a single parameter, 8, can be used to define K (X X ) . That is, if one assumes a particular value of 8, then the K (I: X ) is determined by or must be consistent with this 8. The simulation conditions are summarized in Table 4. 8, gives the true progeny distribution while 8, speci- fies that used in the retrieval process.

Table 4 : Summary of breakage modes used in simulated and retrieval.

Breakage hlode Retrieval Mode Figure 8, 0,

2 0.5 3 0.75 4 1 .o 5 0.5

0.75 0.75 0.75 1 .o

(before breakage) are designated by and arrow superimposed on the abscissa of each figure. For example the arrow indicates that 90% of the initial distribution has a size less than 152 diameter units. The values of 8, were chosen to cover the entire range of binary breakage which yields a smooth distribution.

o e, = 0.5 ; e, = 0.75

- T R U E EFFICIENCY b = 5

I t i 4 i

Fig. 2 : frequency.

Effect of assumed progeny distribution on retrieved breakage

o o, = 0.75 ; e, = 0.75

b = 5 - TRUE EFFICIENCY

1

0 50 100 1 DIAMETER

50 200

Fig. 3 : Effect of assumed progeny distribution on retrieved breakage frequency.

o e, = 1.0 ; 8 , = 0.75 - TRUE EFFICIENCY b = 5

0.5

The results of the simulations are given in Figures 2 to 5. The cumulative number fractions of the initial size distribution


o 8, = 0.5 ; e, = 1.0

- TRUE EFFICIENCY

o = l

b = 5

w a -

LL

w - 4

d

& 1.0-

8 - 0 0

g 0.5 -

L 10% 50% 70% 90%

Fig. 5 : Effect of assumed progeny distribution on retrieved breakage frequency.

The principal points that follow from this series of simulations are : (i) It is essential that the retrieval function include both 4-5

and 3-6 breakages (ie, 0.5 < 8, < 1) if the breakage frequencies are to be good estimates of the true breakage. This can be seen by comparing Figures 2 and 5.

(ii) If the progeny distribution function for the retrieval mode includes both 4 - 5 and 3-6 breakage, then the retrieved breakage is a close estimate of the true breakage for the entire range of 6.

5 Application to Experimental Data

In an experiment, the data would consist of photographs taken of the clusters at different times. In general one would have only the size distribution and one would not have a direct measure- ment of the number concentration. The number concentration would have to be estimated from

where X j is the particle volume for size class j and subscripts 1 and 2 refer to times tz and 1, respectively (tz > tJ. It is impor- tant to note that it is not necessary to know the values of N2 and N,, but only their ratio. Asspreviously mentioned the data consist of the size distributions either before and after the breakage event or as a function of time and the number ratio can be determined from the conservation of volume. However, one complication is that during the interval between measurements, several breakage events may have occured. There would be no way that this could be detected a priori. In fact the only resolution to this complication is to choose the time interval between breakage events sufficiently short that this problem is minimized. However, simulations do provide a means of quantifying the errors that one would an- ticipate from such events. Below we present a method for simulating multiple breakages, illustrate the method with an example, and briefly describe the type of simulation program necessary to obtain useful correlations. First, we assume that one can have N multiple, successive breakage events which have an equal time interval and that the

breakage frequency for a single event is B and is given by Eq. (15). Let B [N, XI be the breakage probability after N identical breakage intervals. If the sequences are identical then one has :

B [l,X] = B

B [ 2 , X ] = B + B(l - 8 ) (18)

A series of simulations were carried out for N steps and the number distributions were determined for each size class and for each intermediate breakage interval. Using these results and the initial size distribution, the overall breakage frequency was retrieved. These values are B [M, XI where M ranges from 1 to N. Once B [M, XI is known, the corresponding value of B [I, XI can be determined by inversion of Eq. (18). This method is illustrated in Table 5 where the breakage frequencies for several representative sizes are given. Column 1 is the diameter and column 2 is the theoretical or assumed breakage probability for a single event. Columns 3 , 4 and 5 give the retrieved overall breakage rates after 1, 2 and 3 events, respectively. The conditions for these simulations were similar to those for the previously discussed simulations except for the values of a and b in Eq. (15). These values were chosen here to allow breakage over a much broader range of sizes. In these numerical experiments 8, and 6, were taken to be identical. Table 6 gives values of B [l, XI or the single event breakage pro- babilities, as determined from the 2 and 3 event numerical data

Table 5 : Retrieved overall breakage frequency.

Breakage frequency Particle True 1 event 2 events 3 events Diameter frequency' (1.1052)2 (1.2165) (1.3 33 8)

274 264 246 23 3 225 209 199 191 178 169 163 152 144 139 129 123 118 110 104 101 94 89 86 80 76

0.3610 0.3453 0.3168 0.2968 0.2832 0.2588 0.2418 0.2303 0.2098 0.1956 0.1860 0.1690 0.1573 0.1494 0.1354 0.1258 0.1194 0.1081 0.1003 0.0952 0.0860 0.0798 0.0756 0.0683 0.0633

0.3610 0.3453 0.3169 0.2968 0.2832 0.2588 0.2418 0.2303 0.2098 0.1956 0.1860 0.1690 0.1572 0.1494 0.1354 0.1258 0.1194 0.1081 0.1003 0.0952 0.0860 0.0798 0.0756 0.0683 0.0633

0.5917 0.5714 0.5333 0.5268 0.5085 0.4778 0.4574 0.4380 0.4037 0.3816 0.3647 0.3349 0.3152 0.3008 0.2754 0.2583 0.2463 0.2251 0.2104 0.2005 0.1830 0.1707 0.1627 0.1484 0.1382

0.7391 0.7194 0.6812 0.7021 0.6845 0.6582 0.6444 0.6204 0.5794 0.5558 0.5337 0.4956 0.4719 0.4523 0.4184 0.3961 0.3794 0.3502 0.3300 0.3160 0.2913 0.2734 0.2619 0.2413 0.2259

u = 0.5, b = 0.1, 0 = 0.75 number ratio = N2/Nl

Part. Part. Syst. Charact. 8 (1991) 315-322 3 21

Table 6 : Retrieved single event breakage frequency. ~ ~

Breakage frequency Particle T h e 2 events 3 events 4 events Diameter frequency' (1.2165)2 (1.3338) (1.4572)

274 264 246 233 225 209 199 191 178 169 163 I52 144 139 129 123 118 110 104 101 94 89 86 80 76

0.3610 0.3453 0.3168 0.2968 0.2832 0.2588 0.2418 0.2303 0.2098 0.1956 0.1860 0.1690 0.1573 0.1494 0.1354 0.1258 0.1194 0.1081 0.1003 0.0952 0.0860 0.0798 0.0756 0.0683 0.0633

0.3610 0.3453 0.3169 0.3119 0.2987 0.2771 0.2631 0.2501 0.2276 0.2134 0.2027 0.1843 0.1723 0.1637 0.1487 0.1387 0.1317 0.1196 0.1114 0.1058 0.0961 0.0893 0.0849 0.0771 0.0716

0.3610 0.3454 0.3169 0.3304 0.3174 0.2985 0.2883 0.2733 0.2487 0.2349 0.2227 0.2025 0.1903 0.1806 0.1643 0.1538 0.1462 0.1331 0.1243 0.1183 0.1079 0.1005 0.0958 0.0875 0.0815

0.3610 0.3454 0.3169 0.3534 0.3403 0.3237 0.3184 0.3008 0.2735 0.2605 0.2464 0.2240 0.2115 0.2005 0.1825 0.1715 0.1630 0.1487 0.1393 0.1327 0.1215 0.1135 0.1085 0.0996 0.0930

' u = 0.5, b = 0.1, B = 0.75 * number ratio = N2/N,

of Table 5 and from similar 4 event results, by inverting Eq. (18). These can be compared with the theoretical or assumed values in column 2. The agreement is sufficiently close that in this case one can retrieve the single event breakage probability with con- fidence. The results of Tables 5 and 6 are independent of any initial distribution. Figure 6 shows similar results using the assumed initial log normal distribution of the previous simulations. Results are given for 1, 3 and 5 successive breakage events. The retrieved single event breakage frequency shows an upward

0.50 7' 1 EVENT 0'40b 1 3 EVENT

s 0.20 d f i "L 0.00

a = 0.5

b = 0.1

A A ~3 + j

0 50 100 150 200 DIAMETER

Fig. 6 : Effect of number of successive events on retrieved single event breakage frequency (0, = 0, = 0.75).

drift with number of events. Additionally, other factors can af- fect the accuracy of the algorithm. These include broadness of the initial distribution and the sharpness of the breakage frequency function. Of course, one must be able to specify the cor- rect number of breakage events a priori.

6 Summary

An algorithm was developed and tested by which size dependent particle breakage frequencies can be extracted from breakage event data in the form of initial and final size distribution. Discrete size classes which accomodate binary and ternary breakage were specified based upon interlaced Fibonacci series. The method was extended to accomodate data for multiple breakage events.

7 Symbols and Abbreviations

B [N XI

Kkj

constants in Eq. (15) breakage frequency for particle of size X breakage frequency for particle in size class j overall breakage frequency for particle of size X undergoing N discrete breakage events progeny distribution matrix defined by Eqs. (6) and (7) gain in particles of size p due to a single breakage event number probability density for distribution before and after breakage, respectively number fractions for size class j before and after breakage, respectively weighting function defined by Eq. (8) size class indices distribution frequency for progeny of size X resulting from breakage of particles of size Y distribution frequency for progeny in size class j resulting from breakage of particles in size class k number of interlaced Fibonacci series size of particles in size class j in number of primary mass or volume units number of successive breakage events upper bound on size of initiator pairs total number of particles before and after breakage, respectively size ratio for adjacent size classes particle mass or volume mass or volume of particle in size class k volume of particle with the mean diameter parameter in Eq. (8) breakage fractions for each constituent Fibonacci series, respectively, as in Eq. (16)


8 References [3] M. Rosenblatt: Remarks on Some Non-Parametric Estimates of a Densitv Function. Ann. Math. Statist. 27 (1956) 832-837.

[4] J. JK Gentry, R. J. Han: Determination of Size Dependent Efficien- cies from Optical and Electron Microscopy. Proc. 5th World Filtra- tion Congress, Nice, France (1990) Vol. 2, pp. 592-598.

[5] S. A . Soulen: B. S. Thesis, University of Maryland, 1990. [6] D. J. Hand: Kernel Discriminant Analysis. Research Studies Press,

J. K Gentry, K. R. Spumy, .l Schormann: Collection Efficiency of Ultrafine Asbestos Fibers Experiment and Theory. Aerosol Sci. Technol. I1 (1989) 184-195. S. A. Soulen, Z G. Cleary, R. J. Han, X C. Chang, J. W Gentry: Size Dependent Penetrations from Microscopy : Experiment and Theory. GAeF Annual Meeting, Vienna, 1989. John Wiley & Sons Ltd., New York, 1982.

an algorithm for determination of the size-dependent breakage frequency of droplets, flocs and...

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