Changes in fractal dimension during aggregation

$Page 1: Changes in fractal dimension during aggregation$
Water Research 37 (2003) 873–883

Changes in fractal dimension during aggregation

Rajat K. Chakrabortia,*, Kevin H. Gardnerb, Joseph F. Atkinsona,John E. Van Benschotena

aDepartment of Civil, Structural and Environmental Engineering, State University of New York at Buffalo, 207 Jarvis Hall, Buffalo,

New York 14260, USAbDepartment of Civil Engineering, University of New Hampshire, Durham NH 03824, USA

Abstract

Experiments were performed to evaluate temporal changes in the fractal dimension of aggregates formed during

flocculation of an initially monodisperse suspension of latex microspheres. Particle size distributions and aggregate

geometrical information at different mixing times were obtained using a non-intrusive optical sampling and digital

image analysis technique, under variable conditions of mixing speed, coagulant (alum) dose and particle concentration.

Pixel resolution required to determine aggregate size and geometric measures including the fractal dimension is

discussed and a quantitative measure of accuracy is developed. The two-dimensional fractal dimension was found to

range from 1.94 to 1.48, corresponding to aggregates that are either relatively compact or loosely structured,

respectively. Changes in fractal dimension are explained using a conceptual model, which describes changes in fractal

dimension associated with aggregate growth and changes in aggregate structure. For aggregation of an initially

monodisperse suspension, the fractal dimension was found to decrease over time in the initial stages of floc formation.

r 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Coagulation–flocculation; Fractal dimension; Particle size distribution; Shear rate; Image analysis; Pixel resolution; Alum

1. Introduction

Coagulation is a common process used in water and

wastewater treatment plants. It also affects the behavior

of particles in natural systems. Factors affecting this

process include coagulant type and dose, solution pH,

mixing intensity and particle concentration [1–3]. One of

the most important of these factors is mixing intensity

(fluid shear), which promotes interactions between

aggregates and particles, and affects overall settling

characteristics [4,5]. However, if mixing is too intense,

particles may not flocculate efficiently [6]. Spicer and

Pratsinis [7] studied the effect of various shear rates on

coagulation and found that aggregates reach an

equilibrium, steady-state aggregate structure and floc

size distribution relatively quickly at higher shear rates.

They concluded that floc breakage was the main process

responsible for maintaining a stable particle size and

limiting further growth.

During coagulation, characteristics of the particles

change as they interact with the coagulant and with each

other. The collision frequency and growth of aggregates

depend on the relative size of the colliding particles, their

surface charge and roughness, local shear forces and the

suspending electrolyte [2,3,8]. Particle concentration also

affects the rate at which particles and aggregates collide

and is therefore important in determining changes in

aggregate size.

Many models have been developed to simulate

particle aggregation. Traditional approaches, based on

the Smoluchowski equation, consider the aggregates as

impermeable spheres (e.g., [9,10]). However, recent

studies have shown that, in reality, flocs consist of

multi-branched structures that are not consistent with

the floc structure described by classical Euclidean

geometry [11,12]. Fractal concepts have provided a

new way of describing floc geometry and various

physical properties such as density, porosity and settling*Corresponding author. Fax: +1-716-645-3667.

E-mail address: [email protected] (R.K. Chakraborti).

0043-1354/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 4 3 - 1 3 5 4 ( 0 2 ) 0 0 3 7 9 - 2

velocity [12–16]. However, if the application of fractal

concepts is to be useful and provide an improvement in

aggregation modeling, then simple and reliable methods

of measuring fractal dimensions, as well as methods to

understand the relationship between fractal dimensions

and aggregation processes, must be developed. The main

goal of the present study was to demonstrate such a

measurement procedure and to develop a descriptive

model that can illustrate expected changes in fractal

dimension during an aggregation experiment.

2. Aggregate structure and fractal dimension

Various characteristics of aggregates derived using

fractal geometry should be useful in improving our

understanding of aggregation processes and modeling.

In fact, several studies have already shown how collision

frequency functions used in particle aggregation models

are strongly dependent on the structure of the aggregate

and the corresponding fractal dimensions [8,11,13].

There are several physical interpretations of this result,

including the fact that a fractal aggregate is spread out

over a larger space than a single sphere of equal mass,

and that flow patterns around and through an aggregate

are significantly different than flow around a sphere. It is

possible that temporal changes in collision efficiency

also may be affected by floc size and structure, with a

corresponding dependence on fractal dimension, though

this relationship has not yet been investigated.

Fractal aggregation theory provides a means of

expressing the degree to which primary particles fill the

space within the nominal volume occupied by an

aggregate [16]. Fractal dimensions may be defined in

linear, planar or volumetric terms, resulting in so-called

one-, two- or three-dimensional values, respectively. For

example, considering the planar aggregate shown in

Fig. 1a, where each solid circle represents the mass of a

primary particle, it is clear that circles of different radii

drawn about the center of the aggregate enclose different

masses. The two-dimensional fractal dimension, D2; isdefined in terms of the relationship between the increase

in radius and the corresponding increase in mass

contained within the circle or, in geometrical terms,

AprD2 ð1Þ

where A is the sum of areas of all primary particles

contained within a circle of radius r: Thus, D2 may be

found as the slope of a plot of (log A) versus (log r).

Note that in Euclidean geometry, for solid circles, D2 ¼2: One- and three-dimensional fractal dimensions may

be defined in an analogous manner, but the present

discussion will focus on D2 since in this study area was

evaluated directly, using an image analysis technique. As

aggregates grow (Fig. 1b), porosity increases and the

number concentration of primary particles occupying

the nominal volume of the aggregate decreases, resulting

in a decrease in the fractal dimension.

In fractal simulation models, aggregation is usually an

irreversible process, so that once two particles stick to

each other, they do not subsequently come apart. This is

a critical limitation, when compared with naturally

occurring processes, where aggregate size is limited by

local physical-chemical conditions. In reality, there is a

restructuring of primary particles within an aggregate

due to break-up and re-formation that occurs in

response to ambient shear or other conditions [6,17].

This process is illustrated in Fig. 1b. Over time, this can

lead to stronger and more compact aggregates, with an

associated higher fractal dimension. In other words,

fractal dimension increases when the primary particles in

an aggregate are arranged more compactly. This change

in fractal dimension appears in the data of Kaye et al.

[18] and Spicer and Pratsinis [7], for example.

Another consideration for development of fractal-

based descriptions of aggregation is the variation of

fractal dimensions among aggregates in a given suspen-

sion. Based partly on this variation, as well as noting

that theoretical work has largely ignored the issue of

rearrangement of primary particles, Meakin [19] sug-

gested that traditional fractal models (e.g., diffusion-

limited aggregation or its various derivatives—see

below) can be used to develop theoretical values for

aggregate fractal dimensions, but that natural aggre-

gates should be looked at only in a composite, or

statistical sense. This is because individual aggregates

are not necessarily expected to exhibit the theoretical

Fig. 1. (a) Variation of contained mass as a function of

increasing radius. (b) Effect of breakup and reattachment,

leading eventually to stronger and more compact aggregate

structure, with associated higher fractal dimension.

R.K. Chakraborti et al. / Water Research 37 (2003) 873–883874

scaling of an ideal fractal. The fractal dimensions for a

sample of particles can be developed following an

analogous approach as for the single particle illustrated

in Fig. 1a. In this case, however, different aggregates in a

sample are analyzed, each with its own respective length

and area. Since aggregates are not circular (or spherical),

the longest dimension, l; is usually used in place of r

[11,15,20], so

AplD2 ð2Þ

and D2 is then the slope of a plot of (log A) versus (log l).

This is the approach followed in the present study.

3. Materials and methods

A non-intrusive optical sampling technique was used

to obtain digital images of particles, which were then

analyzed to develop particle size distributions, geome-

trical properties, and calculations of the fractal dimen-

sion. The basic procedures were adapted from the recent

work of Chakraborti et al. [11], who studied aggregate

characteristics produced after mixing suspensions with

different coagulant (alum) doses. A major advantage of

their method is that it requires no sample handling, so

there is no concern for disturbing the floc characteristics

during measurement. The main difference between their

study and the present work is that in the present

experiments the images were obtained while mixing was

still taking place. A similar measurement technique was

used by Kramer and Clark [21] to observe dynamic (time

dependent) particle size behavior for particles suspended

in a turbulent flow.

The experimental apparatus consisted of a stirred 2L

jar (Phipps and Bird, Richmond, VA, USA) with

variable mixing speed, an automated stroboscopic lamp

to illuminate suspended particles in the jar and a digital

charge coupled device (CCD) camera (Kodak Megaplus

model 1.4) to capture particle images. The camera was

placed on the opposite side of the mixing jar from the

lamp, so that backlit shadows of particles were

produced. A PC served to control the camera and

provided storage for the particle images. Further details

of the experimental setup can be found in Chakraborti

et al. [12].

Monosized polystyrene latex microspheres with a

density of 1.05 gm/mL (Duke Scientific Corporation,

Palo Alto, CA, USA) were used as the primary particles

for the experiments. The nominal particle diameter was

9.975mm, with a standard deviation of 70.061mm.Particles were taken from a 15mL sample of aqueous

suspension with 0.2% solids content (manufacturer’s

specification) and used without any pre-treatment. The

number concentration of particles in the concentrated

suspension was 3.66� 106 particles/mL (710%). Ali-

quots of 0.06 or 0.1mL of the suspension were added to

the mixing jar along with 1L of deionized water,

resulting in initial number concentrations of 220 and

366 particles/mL, respectively.

4. Procedures

Images were taken to examine aggregate geometry

and size distributions at different times during each

experiment. The images were analyzed to track changes

in aggregate morphology for a given experiment, as well

as differences between experiments resulting from

varying coagulant (alum) dose, particle concentration

and mixing speed, or shear rate. Two levels for each of

these variables were tested, resulting in a total of eight

experiments, as summarized in Table 1. The analyses are

reported in terms of the fractal dimension and asso-

ciated particle size distribution.

A stock solution of alum was prepared by dissolving

Al2(SO4)3 � 18H2O (Fisher Scientific, Pittsburgh, PA,

USA) in deionized water to a concentration of 0.1M

(0.2M as aluminum). Alum concentrations of 3.33 and

5.33mg/L were used in the experiments. For each test,

after an initial rapid mixing period (with G ¼ 100 S�1,

where G is the velocity gradient) for 1min, the mixing

speed was reduced to either G ¼ 20 or 80 S�1 and

continued until the end of the experiment. These values

are within the normal range used for treatment

processes. All tests were conducted at room temperature

(20–231C) and a constant pH of 6.5 was maintained by

adding acid or base as required. Measurements were

taken at 10, 20 and 30min of flocculation.

The images were recorded through an imaging

window of size 2mm� 2mm. The shutter speed was

synchronized with the strobe pulses using the camera

control software of a standard particle image veloci-

meter (PIV) system (TSI Inc., St. Paul, MN, USA). A

public domain software package, NIH-Image (National

Institutes of Health, Bethesda, MD, USA), was used for

image analysis, and pixel values recorded in the image

were calibrated to the appropriate length scale using a

stage micrometer (in this study, 540 pixels=1mm).

Further details of the image acquisition and processing

techniques are described by Chakraborti et al. [12].

Before processing a particular image, image thresh-

olding was applied, where pixels were classified into two

categories, either background or particle. This step

effectively filters out random noise, as well as particles

that are not well focused. The overall image is rendered

binary, where particles with sharp edges are identified

and separated from the background. To extract the

greatest amount of information, this step requires good

quality images with the least distortion of the edge of the

particles [22,23]. A number of tests were performed with

the NIH-Image software to determine the best settings

for capturing images with good resolution, by adjusting

R.K. Chakraborti et al. / Water Research 37 (2003) 873–883 875

the position and frequency of the strobe light and the

camera shutter speed, depending on the particular

sample conditions. Using similar settings Cheng [22]

found that particles as small as 6mm could be resolved.

A well-focused image permits analysis of more

particles in one threshold operation, whereas an image

with relatively poor resolution requires a higher thresh-

old level to provide contrast between the image and the

background, resulting in a possible loss of information.

To avoid loss of data, several images were taken at each

measurement time by moving the camera in one plane so

that different locations of the jar could be captured. This

was done with a light-camera assembly that maintained

constant distance between the camera, light and jar. This

procedure provided guidance for choosing the best light

and camera shutter speed, for the particular residual

turbidity of a given test, and images that provided the

largest number of particles at each measurement time

were selected for further analysis.

Fig. 2 shows the aggregate parameters obtained from

the NIH-Image software following application of

thresholding. An ellipse was fitted to each aggregate in

such a way that the area of the ellipse was equal to the

area of the aggregate and the major axis of the ellipse

was used as an estimate for l (Eq. (2)).

5. Results and discussion

5.1. Pixel resolution

The ability of the system to accurately resolve the

smallest particles expected in a sample must be

determined in order to evaluate the scale to which

Table 1

Physical-chemical parameters used for the experiments and resulting two-dimensional fractal dimensions (D2) and particle size

distribution (PSD); n is number of aggregates

Experiment

number

Alum

dose

(mg/L)

Shear

rate

G (S�1)

Particle

conc.

(#/mL)

Observation

times

(min)

Mode in

PSD

(mm)

Fractal dimensions n

D2 R2

1 5.33 20 366 10 14 1.8370.10 0.89 86

20 18 1.6870.08 0.91 77

30 22 1.5170.09 0.89 65

2 5.33 80 366 10 14 1.8270.10 0.87 97

20 16 1.7070.08 0.91 95

30 25 1.6470.06 0.90 84

3 3.33 20 220 10 14 1.9170.21 0.72 62

20 18 1.8270.18 0.81 51

30 18 1.7370.16 0.85 54

4 3.33 80 220 10 14 1.9470.18 0.85 47

20 18 1.8870.22 0.77 45

30 18 1.8270.19 0.83 43

5 5.33 20 220 10 18 1.7270.08 0.92 80

20 24 1.6270.08 0.91 76

30 26 1.4870.08 0.91 66

6 5.33 80 220 10 18 1.7570.18 0.78 62

20 20 1.6470.12 0.84 60

30 22 1.6070.11 0.82 52

7 3.33 20 366 10 12 1.8470.09 0.81 91

20 16 1.7770.08 0.88 72

30 18 1.7070.09 0.86 58

8 3.33 80 366 10 16 1.9170.10 0.72 83

20 18 1.8670.09 0.84 76

30 20 1.7570.09 0.85 66


geometrical properties such as the fractal dimension may

be determined. Depending on pixel size, relative to

particle size, the area of a small particle or aggregate

may be overestimated, which could result in a smaller

calculated value for D2 (i.e., if areas are overestimated

for the smallest aggregates, the resulting slope of the line

correlating area with length could decrease). In order to

illustrate this issue, consider first a plain circle,

approximated by different numbers of pixels (Fig. 3a).

When the pixel length is equivalent to the circle diameter

(first panel), then depending on the placement of the

circle relative to the pixels, from one to four pixels may

be used to ‘‘register’’ the circle. If the circle diameter is

taken to be one unit, then the ratio of the registered pixel

area (Ap) to the actual area of the circle (A0) is between

1.27 and 5.09. As the pixel resolution increases, i.e.,

more pixels are used across the circle diameter, the ratio

(Ap=A0) approaches one and its variability decreases,

since the ratio becomes less dependent on the placement

of the circle relative to the pixel locations. Fig. 4

illustrates this variability, where N is defined as the

ratio of circle diameter to pixel ‘‘length’’. In other words,

N is the number of pixels required to traverse one circle

diameter. Note that in Fig. 4 there are several area

values plotted for each resolution. These areas corre-

spond to different placement of the object, relative to the

pixel boundaries, as in the first panel of Fig. 3.

The issue of required pixel resolution was explored

further by considering a simple aggregate consisting of

three primary ‘‘particles’’, each represented by a circle as

shown in Fig. 3b. Following a similar procedure as for

the single circle, the area ratio for this case is also plotted

in Fig. 4, where N is the number of pixels required to

traverse the diameter of one of the primary circles. For

(a)

(b)

Fig. 3. (a) Variation of pixel resolution for a single circular particle; the first panel shows three different possibilities for measuring a

single primary particle, resulting in three different area values; (b) variation of pixel resolution for a simple aggregate.

Fig. 2. Parameters measured for an individual particle by image processing software.


both cases shown in Fig. 4, the area ratio is less than

1.02 when N ¼ 20: For the resolution of the present

study, N ¼ 5:4; so that a single primary particle is

represented by approximately 29 pixels and the average

area ratio is 1.3. This implies the area measurement for a

single primary particle may be overestimated by about

30%. However, the error decreases rapidly with

increasing resolution (Fig. 4), and it also should be

noted that most aggregates analyzed were larger than

the primary particles, so the average error in area

measurement is reduced.

To further investigate the potential effect of pixel

resolution on calculations of D2; the data for aggregate

length and area for a sample were divided into sub-

ranges, based on length, and D2 was calculated for each

sub-range and compared with the values obtained for

the sample taken as a whole. These calculations are

shown in Table 2, for experiments 1 and 2, where it can

be seen that there is negligible difference in D2 values

calculated over the different size ranges, although there

is a significant change in D2 for the different times. This

comparison indicates that potential problems with area

calculations for the smaller aggregates did not influence

D2 values for the sample as a whole, although it is

possible that D2 values are slightly underestimated for

the smaller size sub-range (note that an overestimation

in area for the smaller aggregates would tend to result

in smaller slopes, and therefore smaller D2; for

the regression of Eq. (2)). Even smaller divisions in size

ranges were not considered, due to limited data available

within each sub-range (see Table 2).

5.2. Images

Images captured at the three different times show the

evolution of aggregate structure during flocculation.

After 10min of mixing, the initially monodispersed latex

particles formed flocs of variable shapes and sizes

(Fig. 5). With additional time, larger flocs were formed

and examples of some of these flocs are shown in

Figs. 5b and c. Compared with images taken after

10min, the images at later stages were composed of

fewer but larger particles. The larger aggregates in the

image taken after 30min (Fig. 5c) appear to be more

porous and spread out than the smaller aggregates

observed at earlier times and should be associated with

lower fractal dimensions, as previously noted.

5.3. Particle size distribution

Fig. 6 shows the evolution of the particle size

distribution for experiments 1 and 2, using the major

axis of the ellipse fitted to the aggregates as a

characteristic length (l). Similar characteristics of the

distributions were observed for the other experiments

and are not shown here. The same alum dose and

particle concentration were used in these two experi-

ments, with only the mixing speed varied (see Table 1).

For both experiments, there is a clear movement of the

peak of the size distribution (the mode) towards larger

size classes with time. This movement is rather regular

for experiment 1 (Fig. 6a), whereas the peak moved

more at the 30min measurement for experiment 2,

relative to earlier times (Fig. 6b). At 30min, the mode

for experiment 2 is slightly larger than for experiment 1.

However, a greater portion of the distribution in

experiment 1 is in larger sizes, suggesting that the lower

G condition was more favorable for aggregation.

Plots of particle size distribution were generated for

all the experiments and they all showed the occurrence

of a broad range of particle sizes, particularly at later

times when both large and small aggregates were

present. A heterogeneous suspension is believed to lead

to higher aggregation rates [4,8,16]. This appears to be

consistent with the changes in size distribution and

gradual movement of the peak of the size distribution

plots. In other words, on average there is a slightly

greater change in the position of the peak size between

0

0.5

1

1.5

2

2.5

3

3.5

4

0 10 0 30 0 50

N

Ap/

Ao

single circle

cluster

Fig. 4. Dependence of area ratio on pixel resolution.


20 and 30min, relative to the change between 10 and

20min (see Table 1).

5.4. Fractal dimension

Calculated values of D2 for the eight experiments are

shown in Table 1. The mean values and 95% confidence

levels (7values in Table 1) were calculated from

regression analyses based on Eq. (2). The correlation

coefficients (R2) for all regressions range between 0.72

and 0.92, indicating a good correlation between logA

and log l: In addition, values for D2 compare well with

previous studies examining fractal dimensions for flocs

produced using alum. For example, Gorczyca and

Ganczarczyk [20] reported D2 values for alum flocs

formed in inorganic clay and mineral suspensions to be

between 1.71 and 1.97 (70.05) and Li and Ganczarczyk

[24] calculated fractal dimensions of alum flocs to be

between 1.59 and 1.97 from the measurements carried

out by Tambo and Watanabe [25]. Except for the 30-min

measurements of experiments 1 and 5, the present values

fall within the ranges of these earlier studies. It also is

interesting to note that these values are consistent with

cluster–cluster modeling results obtained by Meakin

[26]. This similarity may be worth further investigation

in future studies.

When the alum dose is low, as in experiments 3 and 4,

the initial (10min) D2 values are close to the Euclidean

dimension (D2 ¼ 2), since there was apparently insuffi-

cient coagulant to cause significant growth of aggregates

at that time. A sharp decrease in the measured D2 values

is observed for experiments 5 and 6, particularly for

experiment 5. Although the early fractal dimensions (at

10min) for these two experiments were almost identical,

the different D2 values at later stages apparently resulted

from the lower shear rate applied for experiment 5.

When comparing measurements at any corresponding

time, experiment 5 produced the lowest D2 and largest l

values. This effect is also seen in comparing experiments

3 and 4, where the lower mixing speed of experiment 3

resulted in smaller values for D2; although the modes

were identical at the three different times. Apparently,

the slower mixing speed allows less dense aggregates to

form.

From Table 1, it can be seen that experiments 1 and 5

produced the lowest D2 values. These experiments both

had higher alum dose and lower shear rate. In addition,

upon comparing paired experiments (1 and 2, 3 and 4, 5

and 6, or 7 and 8), the lower shear rate (the first

experiment in each of these pairs) always produced a

lower value of D2; particularly for the 30-min measure-

ments. It appears then, that the lower mixing intensity

was more favorable for producing less compact aggre-

gate structures (lower D2), though alum dose and

particle concentration also affected D2: Similarly, anexamination of data in Table 1 shows that higher alumT

able2

Calculationof

D2basedonsub-ranges

foraggregatesize,forexperiments1and2;

n=number

ofparticlesin

each

sub-range

10min

20min

30min

Majoraxisrange(l)(mm)

Sample


Sample


Sample

10–18

18–25

10–18

18–25

25–45

10–18

18–25

25–45

>45

Ex

per

imen

t1

D2

1.8270.12

1.8370.15

1.8370.10

1.6870.30

1.6770.20

1.6870.30

1.6870.10

1.5570.40

1.5270.30

1.5170.30

1.5270.30

1.5170.10

R2

0.80

0.60

0.89

0.60

0.70

0.52

0.91

0.80

0.42

0.80

0.63

0.89

N50

36

86

27

32

18

77

630

10

19

65

10min

20min

30min


Sample


Sample


Sample

10–18

18–25

10–18

18–25

11–25

25–50

>50

Ex

per

imen

t2

D2

1.8270.12

1.8170.20

1.8270.10

1.7070.09

1.6970.30

1.7070.10

1.6670.15

1.6370.26

1.6570.50

1.6470.10

R2

0.64

0.72

0.87

0.81

0.53

0.91

0.71

0.58

0.54

0.90

n73

24

97

72

23

95

48

27

984


dose generally produced lower D2 values (and larger

aggregates). This is true for both the higher (experiments

2 and 8 or 4 and 6) and lower (experiments 1 and 7

or 3 and 5) shear rates. There is no significant difference

in results for the different particle concentrations

used, but that is likely because only a relatively

small variation in particle concentration was tested

here.

Fig. 5. Images from the mixing jar for multiple aggregates after (a) 10 min, (b) 20 min, and (c) 30 min.

0

5

10

15

20

25

30

35

0.50 1.00 1.50 2.00

Log l

Rel

ativ

e F

requ

ency

(%)

10 min

20 min

30 min

0

5

10

15

20

25

30

35

0.5 1 1.5 2 2.5

Log l

Rel

ativ

e F

requ

ency

(%

)

10 min

20 min30 min

(a)

(b)

Fig. 6. Particle size distributions at three sampling times for experiments (a) 1, and (b) 2.


In order to develop relationships between fractal

dimensions and aggregate growth, it should first be

noted that aggregates in natural systems do not

generally follow the theoretical scaling laws developed

for ideal fractals, as discussed previously. In theory,

fractal dimensions are scale-invariant; that is, they do

not change as a function of aggregate size. However, this

concept is expected to hold only when aggregates

become sufficiently large relative to the primary particle

[19]. This idea is illustrated by considering the basic

approach of the diffusion-limited aggregation (DLA)

model [27] or one of its several variations (reaction-

limited, cluster-particle, cluster–cluster, etc.), which has

been used to develop theoretical values of fractal

dimensions for aggregates. In these models, aggregation

occurs as primary particles enter a sample space and

follow a random walk until they join a collector, which

is another particle or cluster of particles. The primary

particles are assumed to be solid, so their fractal

dimensions correspond to Euclidean values.

Now consider the changes in size and fractal dimen-

sion that may be expected in an aggregation experiment

starting with a monodisperse suspension of spherical

primary particles, as in the present experiments. In this

case the aggregates grow over time, at least initially, and

increasing time corresponds with increasing size. Chemi-

cal conditions and mixing speed are assumed to be held

constant. Initially, there are primary particles only and

D2 ¼ 2: As particles and clusters start to join, average

aggregate size grows and fractal dimensions decrease, as

illustrated in Fig. 7. Also shown in Fig. 7 are average

values for D2 and mode of particle size distribution

obtained at three different times in the experiments.

Eventually, D2 values are expected to level off (point A in

Fig. 7), or may start increasing slightly with longer times,

as restructuring and rearrangement result in more

compact and stable floc structures.

In the present analysis, values for D2 are seen to vary

with different experimental conditions and with time

(Table 1 and Fig. 7). In particular, D2 decreases over

time for each of the eight experiments, in some cases

more dramatically than in others. A statistical analysis of

the combined results from all experiments indicated that

D2 values were statistically different (Po0:05) for the

three measurement times. Following the above argu-

ments, a faster rate of decrease in D2 should imply that

the process is farther away from the semi-equilibrium

point (A) identified in Fig. 7. It is not clear at this time

why D2 should decrease with time, but not with size, at

least for aggregates that are not much larger than the

primary particles (Table 2). It is possible that aggregates

are simply joined differently (an aggregate growth

resulting from collision and rearrangement of particles)

after different mixing times due to restructuring effects.

Furthermore, the radius of the whole aggregate also

changes with restructuring and growth and thereby, the

mass contained in the aggregate also changes according

to the aggregate structure (see Fig. 1b). As more and

more particles are collected, the aggregate reaches a stage

where further addition of particles or clusters does not

result in further changes in D2:

5.5. Effect of shear rate on coagulation

Upon evaluating the effects of different parameters

used in this study, it was found that shear rate had the

most significant effect on aggregate growth and asso-

ciated fractal dimension. Fig. 8 illustrates this effect by

comparing the results for experiments 1 and 2, in which

a constant and relatively higher alum dose and particle

concentration were used (Table 1). It can be seen that

the correlation curves for these data points were

significantly separated (slope7standard error in the

regression line for cumulative distributions in experi-

ments 1 and 2 are 1.6270.02 and 1.7570.03, respec-

tively). This separation was the most pronounced,

relative to other comparisons (e.g., high or low alum

or particle concentration) used in this study. Fig. 8 also

shows that a lower shear rate produced a smaller slope

(i.e., a smaller D2 value), whereas the higher shear rate

produced relatively dense flocs with a correspondingly

higher D2 value, closer to D2 ¼ 2:As previously reviewed, existing studies of fractal

aggregation relate various properties of aggregates to

fixed values of fractal dimensions. In other words, given

particular values of D2 and corresponding one- and

three- dimensional fractal dimensions, it is possible to

calculate features such as porosity, settling velocity and

collision frequency. As shown here, however, it is clear

that changes in fractal dimensions, along with changes

in aggregate size, need to be taken into consideration

when describing a dynamic aggregation process. This

will be especially important for aggregation modeling,

where temporal changes in particle sizes and distribution

are simulated. Li and Logan [4] and Chakraborti et al.

[28], for example, showed that collision frequencies

Fig. 7. Temporal changes in aggregate size and fractal dimen-

sion; data points represent ranges of values and averages

obtained from the present experiments.


calculated according to fractal theory could be orders of

magnitude greater than standard values based on a

spherical particle assumption. This suggests that sig-

nificant differences may be obtained using a model that

assumes fractal aggregates, compared with one based on

impermeable spherical particles. In addition, the present

results show that changes in fractal dimensions should

be incorporated into aggregation modeling in order to

simulate the aggregate growth process as realistically as

possible. Models based on the Smoluchowski equation

would require significant modification in order to

incorporate fractal dimension and alternate modeling

approaches may be worth pursuing.

6. Conclusions

In this study, the effects of different process condi-

tions for aggregation have been examined. Results were

reported in terms of particle size distributions and floc

morphology, as represented by the two-dimensional

fractal dimension. An in situ, non-invasive optical

technique was used to obtain the required measure-

ments. The optical sampling and digital image analysis

technique was found to be useful for obtaining and

evaluating time-varying in situ aggregate data. In

combination with the NIH-Image software, the proce-

dure provides a convenient means of obtaining data to

calculate fractal dimensions and size distributions. An

important aspect of the image processing technique is

resolution, and an analysis was conducted to evaluate

the smallest particle size that could be accurately

resolved, based on relative pixel size. A conceptual

framework was also developed to describe changes in

fractal dimension with aggregate growth over time. The

measurements and discussion in this study apply to

relatively early stages of coagulation, at least prior to the

semi-equilibrium state indicated by point A in Fig. 7.

According to the pixel resolution analysis, the area of

a single primary particle in the present experiments may

be overestimated by about 30%. However, the expected

error decreases rapidly for increasing size and most

aggregates analyzed were larger than the primary

particles. In addition, analysis of the fractal dimensions

of the sample divided into several different size ranges

indicated that a possible error in area measurement did

not have an impact on the calculated D2 values.

As may be expected, higher alum dose generally

resulted in larger flocs, while lower shear rate produced

flocs with lower values of D2: There also was

a statistically significant change in D2 values over

time (i.e., as larger flocs were produced). Particle

concentration had only a minor effect, but further

studies should be conducted over a larger range

of concentrations before firmer conclusions can be

made.

The observations reveal a trend in decreasing D2

and increasing l values over time, for coagulation of an

initially monodisperse suspension of homogeneous

spherical particles. Furthermore, the decrease in D2 is

associated with an increase in particle size. Additional

experiments should be conducted to monitor changes

over longer periods of time, which are more closely

representative of natural conditions, to evaluate the

possible effect of restructuring of primary particles.

As a final note, aggregation modeling can likely be

improved by incorporating fractal dimensions directly in

the modeling framework. This is particularly evident in

the calculation of collision frequencies, which are very

different when calculated using fractal theory as

opposed to standard techniques based on the assump-

tion that interacting particles are solid spheres. Fractal

dimension is difficult to incorporate directly in the

traditional Smoluchowski equation, so alternative ap-

proaches may be needed.

Acknowledgements

Mr. Chakraborti was supported as a Sea Grant

Scholar by the New York Sea Grant Institute during

this study. Comments and suggestions from anonymous

reviewers are greatly acknowledged.

References

[1] Gregory J. Fundamentals of flocculation. CRC Crit Rev

Environ Control 1989;19(3):185–230.

[2] Amirtharajah A, O’Melia CR. Coagulation processes:

destabilization, mixing, and flocculation. In: Water quality

and treatment. New York: McGraw-Hill, 1990.

[3] O’Melia CR. Coagulation and sedimentation in lakes,

reservoirs and water treatment plants. Water Sci Technol

1998;37(2):129–35.

[4] Li X, Logan BE. Collision frequencies between fractal

aggregates and small particles in a turbulently sheared

fluid. Environ Sci Technol 1997;31(4):1237–42.

Fig. 8. Change in fractal dimension as a response to two levels

of shear rate.


[5] Gregory J. The role of floc density in solid-liquid

separation. Filtr Sep 1998;35(4):367–71.

[6] Oles V. Shear-induced aggregation and breakup of

polystyrene latex particles. Colloid Interface Sci 1992;

154:51–358.

[7] Spicer PT, Pratsinis SE. Shear-induced flocculation: the

evolution of floc structure and the shape of the size

distribution at steady state. Water Res 1996;30(5):1049–56.

[8] Wiesner MR. Kinetics of aggregate formation in rapid

mix. Water Res 1992;26(3):379–87.

[9] O’Melia CR, Tiller CL. Physicochemical aggregation and

deposition in aquatic environments. In: Buffle J, Leewen

van HP, editors. Environmental particles, vol. 2. Boca

Raton, FL: Lewis Publishers, 1993. p. 353–86 (Chapter 8).

[10] Elimelech M, Gregory J, Jia X, Williams RA. Particle

deposition and aggregation—measurement, modeling and

simulation. Butterworth-Heinemann Ltd., London, UK,

1995.

[11] Logan BE. Environmental transport processes. New York:

Wiley Inc., 1999.

[12] Chakraborti RK, Atkinson JF, Van Benschoten JE.

Characterization of alum floc by image processing.

Environ Sci Technol 2000;34(18):3969–76.

[13] Jiang Q, Logan BE. Fractal dimensions of aggregates

determined from steady state size distributions. Environ

Sci Technol 1991;25(12):2031–8.

[14] Jiang Q, Logan BE. Fractal dimensions of aggregates

from shear devices. J Am Water Works Assoc

1996;88(2):100–13.

[15] Logan BE, Kilps JR. Fractal dimensions of aggregates

formed in different fluid mechanical environments. Water

Res 1995;29(2):443–53.

[16] Gardner KH. Kinetic models of colloid aggregation. In:

Hsu JP, editor. Interfacial forces and fields: theory and

applications. New York: Marcel Dekker, Inc., 1999.

p. 509–50.

[17] Clark MM, Flora JRV. Floc restructuring in varied

turbulent mixing. J Colloid Interface Sci 1991;

147(2):407–21.

[18] Kaye BH, Leblanc JE, Abbot P. Fractal description of ht

structures of fresh and eroded aluminum shot fine

particles. Part Charact 1985;2:56–61.

[19] Meakin P. Fractal aggregates. Adv Colloid Interface Sci

1988;28:249–331.

[20] Gorczyca B, Ganczarczyk JJ. Image analysis of alum

coagulated mineral suspensions. Environ Technol

1996;17:1361–9.

[21] Kramer TA, Clark MM. The measurement of particles

suspended in a stirred vessel using microphotography

and digital image analysis. Part System Charact 1996;13:

3–9.

[22] Miller BV, Lines RW. Recent advances in particle size

measurements: a critical review. CRC Crit Rev Anal Chem

1988;20(2):75–116.

[23] Cheng CY. Application and development of particle image

technology to study turbulence and particle aggregation.

Ph.D. Dissertation, SUNY, Buffalo, New York, 1997.

[24] Li D, Ganczarczyk JJ. Fractal geometry of particle

aggregates generated in water and wastewater treatment

processes. Environ Sci Technol 1989;23(11):1385–9.

[25] Tambo N, Watanabe Y. Physical characteristic of flocs: I.

The floc density function and aluminum floc. Water Res

1979;13:409–19.

[26] Meakin P. A historical introduction to computer models

for fractal aggregates. J Sol–Gel Sci Technol 1999;15:

97–117.

[27] Witten TA, Sander LM. Diffusion-limited aggregation.

Phys Rev B 1983;27(9):5686–97.

[28] Chakraborti RK, Atkinson JF, Van Benschoten JE. Non-

intrusive characterization of fractal aggregates formed by

alum. AWWA Annual Meeting, Universities’ Forum,

Denver, June, 2000.


Changes in fractal dimension during aggregation

Documents