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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 a
Department of Civil, Structural and Environmental Engineering, State University of New York at Buffalo, 207 Jarvis Hall, Buffalo, New York 14260, USA b Department 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
*Corresponding author. Fax: +1-716-645-3667. E-mail address: [email protected] (R.K. Chakraborti).
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
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
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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 ; is defined 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 ¼ 2: 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
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.
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 fractalbased descriptions of aggregation is the variation of fractal dimensions among aggregates in a given suspension. Based partly on this variation, as well as noting that theoretical work has largely ignored the issue of rearrangement of primary particles, Meakin [19] suggested that traditional fractal models (e.g., diffusionlimited aggregation or its various derivatives—see below) can be used to develop theoretical values for aggregate fractal dimensions, but that natural aggregates should be looked at only in a composite, or statistical sense. This is because individual aggregates are not necessarily expected to exhibit the theoretical
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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 Apl
D2
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, geometrical properties, and calculations of the fractal dimension. 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 2 L 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 9.975 mm, with a standard deviation of 70.061 mm. Particles were taken from a 15 mL 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%). Aliquots of 0.06 or 0.1 mL of the suspension were added to
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the mixing jar along with 1 L 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 associated 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 0.1 M (0.2 M as aluminum). Alum concentrations of 3.33 and 5.33 mg/L were used in the experiments. For each test, after an initial rapid mixing period (with G ¼ 100 S1, where G is the velocity gradient) for 1 min, the mixing speed was reduced to either G ¼ 20 or 80 S1 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 30 min of flocculation. The images were recorded through an imaging window of size 2 mm 2 mm. The shutter speed was synchronized with the strobe pulses using the camera control software of a standard particle image velocimeter (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=1 mm). Further details of the image acquisition and processing techniques are described by Chakraborti et al. [12]. Before processing a particular image, image thresholding 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
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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 (S1)
Particle conc. (#/mL)
Observation times (min)
Mode in PSD (mm)
n
Fractal dimensions
D2
R2
1
5.33
20
366
10 20 30
14 18 22
1.8370.10 1.6870.08 1.5170.09
0.89 0.91 0.89
86 77 65
2
5.33
80
366
10 20 30
14 16 25
1.8270.10 1.7070.08 1.6470.06
0.87 0.91 0.90
97 95 84
3
3.33
20
220
10 20 30
14 18 18
1.9170.21 1.8270.18 1.7370.16
0.72 0.81 0.85
62 51 54
4
3.33
80
220
10 20 30
14 18 18
1.9470.18 1.8870.22 1.8270.19
0.85 0.77 0.83
47 45 43
5
5.33
20
220
10 20 30
18 24 26
1.7270.08 1.6270.08 1.4870.08
0.92 0.91 0.91
80 76 66
6
5.33
80
220
10 20 30
18 20 22
1.7570.18 1.6470.12 1.6070.11
0.78 0.84 0.82
62 60 52
7
3.33
20
366
10 20 30
12 16 18
1.8470.09 1.7770.08 1.7070.09
0.81 0.88 0.86
91 72 58
8
3.33
80
366
10 20 30
16 18 20
1.9170.10 1.8670.09 1.7570.09
0.72 0.84 0.85
83 76 66
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 6 mm 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 threshold 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
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Fig. 2. Parameters measured for an individual particle by image processing software.
(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.
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 correspond 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
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878
4
single circle
3.5
cluster
Ap/Ao
3 2.5 2 1.5 1 0.5 0
0
10
0
30
0
50
N Fig. 4. Dependence of area ratio on pixel resolution.
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 subranges, 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 10 min 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 10 min, the images at later stages were composed of fewer but larger particles. The larger aggregates in the image taken after 30 min (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 experiments, 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 30 min measurement for experiment 2, relative to earlier times (Fig. 6b). At 30 min, 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
1.6470.10 0.90 84 1.6570.50 0.54 9 1.6370.26 0.58 27 1.7070.10 0.91 95 1.6970.30 0.53 23 1.8170.20 0.72 24 Experiment 2 D2 1.8270.12 R2 0.64 n 73
10–18
18–25
Sample
1.8270.10 0.87 97
1.7070.09 0.81 72
Sample 18–25
Major axis range (l) (mm) Major axis range (l) (mm)
10–18
20 min 10 min
1.6670.15 0.71 48
25–50 11–25
30 min
1.6870.10 0.91 77 1.6870.30 0.52 18 1.6770.20 0.70 32 1.8370.10 0.89 86
1.6870.30 0.60 27
Major axis range (l) (mm)
>50
1.5270.30 0.42 30
Sample
1.5170.30 0.80 10
1.5270.30 0.63 19
5.4. Fractal dimension
1.8370.15 0.60 36
18–25 10–18
879
20 and 30 min, relative to the change between 10 and 20 min (see Table 1).
1.5570.40 0.80 6
18–25 10–18 25–45 18–25 10–18
Major axis range (l) (mm) Sample Major axis range (l) (mm)
Experiment 1 D2 1.8270.12 R2 0.80 N 50
Major axis range (l) (mm) Sample
30 min 20 min 10 min
Table 2 Calculation of D2 based on sub-ranges for aggregate size, for experiments 1 and 2; n=number of particles in each sub-range
25–45
>45
Sample
1.5170.10 0.89 65
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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 log A 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 (10 min) D2 values are close to the Euclidean dimension (D2 ¼ 2), since there was apparently insufficient 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 10 min) 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 measurements. It appears then, that the lower mixing intensity was more favorable for producing less compact aggregate structures (lower D2 ), though alum dose and particle concentration also affected D2 : Similarly, an examination of data in Table 1 shows that higher alum
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Fig. 5. Images from the mixing jar for multiple aggregates after (a) 10 min, (b) 20 min, and (c) 30 min.
Relative Frequency (%)
35
30 min
30 25
20 min 10 min
20 15 10 5 0 0.5
1
1.5
(a)
35 Relative Frequency (%)
2
2.5
Log l 30 min
30 25 20
20 min 10 min
15 10 5 0 0.50
1.50
1.00
(b)
2.00
Log l
Fig. 6. Particle size distributions at three sampling times for experiments (a) 1, and (b) 2.
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.
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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 (reactionlimited, 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 dimension 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. Chemical 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
Fig. 7. Temporal changes in aggregate size and fractal dimension; data points represent ranges of values and averages obtained from the present experiments.
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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 arguments, 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 associated 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 experiments 1 and 2 are 1.6270.02 and 1.7570.03, respectively). 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
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Fig. 8. Change in fractal dimension as a response to two levels of shear rate.
calculated according to fractal theory could be orders of magnitude greater than standard values based on a spherical particle assumption. This suggests that significant 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 conditions 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 measurements. 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 procedure 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 assumption that interacting particles are solid spheres. Fractal dimension is difficult to incorporate directly in the traditional Smoluchowski equation, so alternative approaches 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.
R.K. Chakraborti et al. / Water Research 37 (2003) 873–883 [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.
883
[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. Nonintrusive characterization of fractal aggregates formed by alum. AWWA Annual Meeting, Universities’ Forum, Denver, June, 2000.
Changes in fractal dimension during aggregation Rajat K. Chakrabortia,*, Kevin H. Gardnerb, Joseph F. Atkinsona, John E. Van Benschotena a
Department of Civil, Structural and Environmental Engineering, State University of New York at Buffalo, 207 Jarvis Hall, Buffalo, New York 14260, USA b Department 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
*Corresponding author. Fax: +1-716-645-3667. E-mail address: [email protected] (R.K. Chakraborti).
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
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
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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 ; is defined 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 ¼ 2: 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
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.
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 fractalbased descriptions of aggregation is the variation of fractal dimensions among aggregates in a given suspension. Based partly on this variation, as well as noting that theoretical work has largely ignored the issue of rearrangement of primary particles, Meakin [19] suggested that traditional fractal models (e.g., diffusionlimited aggregation or its various derivatives—see below) can be used to develop theoretical values for aggregate fractal dimensions, but that natural aggregates should be looked at only in a composite, or statistical sense. This is because individual aggregates are not necessarily expected to exhibit the theoretical
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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 Apl
D2
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, geometrical properties, and calculations of the fractal dimension. 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 2 L 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 9.975 mm, with a standard deviation of 70.061 mm. Particles were taken from a 15 mL 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%). Aliquots of 0.06 or 0.1 mL of the suspension were added to
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the mixing jar along with 1 L 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 associated 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 0.1 M (0.2 M as aluminum). Alum concentrations of 3.33 and 5.33 mg/L were used in the experiments. For each test, after an initial rapid mixing period (with G ¼ 100 S1, where G is the velocity gradient) for 1 min, the mixing speed was reduced to either G ¼ 20 or 80 S1 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 30 min of flocculation. The images were recorded through an imaging window of size 2 mm 2 mm. The shutter speed was synchronized with the strobe pulses using the camera control software of a standard particle image velocimeter (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=1 mm). Further details of the image acquisition and processing techniques are described by Chakraborti et al. [12]. Before processing a particular image, image thresholding 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
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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 (S1)
Particle conc. (#/mL)
Observation times (min)
Mode in PSD (mm)
n
Fractal dimensions
D2
R2
1
5.33
20
366
10 20 30
14 18 22
1.8370.10 1.6870.08 1.5170.09
0.89 0.91 0.89
86 77 65
2
5.33
80
366
10 20 30
14 16 25
1.8270.10 1.7070.08 1.6470.06
0.87 0.91 0.90
97 95 84
3
3.33
20
220
10 20 30
14 18 18
1.9170.21 1.8270.18 1.7370.16
0.72 0.81 0.85
62 51 54
4
3.33
80
220
10 20 30
14 18 18
1.9470.18 1.8870.22 1.8270.19
0.85 0.77 0.83
47 45 43
5
5.33
20
220
10 20 30
18 24 26
1.7270.08 1.6270.08 1.4870.08
0.92 0.91 0.91
80 76 66
6
5.33
80
220
10 20 30
18 20 22
1.7570.18 1.6470.12 1.6070.11
0.78 0.84 0.82
62 60 52
7
3.33
20
366
10 20 30
12 16 18
1.8470.09 1.7770.08 1.7070.09
0.81 0.88 0.86
91 72 58
8
3.33
80
366
10 20 30
16 18 20
1.9170.10 1.8670.09 1.7570.09
0.72 0.84 0.85
83 76 66
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 6 mm 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 threshold 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
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Fig. 2. Parameters measured for an individual particle by image processing software.
(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.
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 correspond 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
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4
single circle
3.5
cluster
Ap/Ao
3 2.5 2 1.5 1 0.5 0
0
10
0
30
0
50
N Fig. 4. Dependence of area ratio on pixel resolution.
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 subranges, 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 10 min 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 10 min, the images at later stages were composed of fewer but larger particles. The larger aggregates in the image taken after 30 min (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 experiments, 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 30 min measurement for experiment 2, relative to earlier times (Fig. 6b). At 30 min, 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
1.6470.10 0.90 84 1.6570.50 0.54 9 1.6370.26 0.58 27 1.7070.10 0.91 95 1.6970.30 0.53 23 1.8170.20 0.72 24 Experiment 2 D2 1.8270.12 R2 0.64 n 73
10–18
18–25
Sample
1.8270.10 0.87 97
1.7070.09 0.81 72
Sample 18–25
Major axis range (l) (mm) Major axis range (l) (mm)
10–18
20 min 10 min
1.6670.15 0.71 48
25–50 11–25
30 min
1.6870.10 0.91 77 1.6870.30 0.52 18 1.6770.20 0.70 32 1.8370.10 0.89 86
1.6870.30 0.60 27
Major axis range (l) (mm)
>50
1.5270.30 0.42 30
Sample
1.5170.30 0.80 10
1.5270.30 0.63 19
5.4. Fractal dimension
1.8370.15 0.60 36
18–25 10–18
879
20 and 30 min, relative to the change between 10 and 20 min (see Table 1).
1.5570.40 0.80 6
18–25 10–18 25–45 18–25 10–18
Major axis range (l) (mm) Sample Major axis range (l) (mm)
Experiment 1 D2 1.8270.12 R2 0.80 N 50
Major axis range (l) (mm) Sample
30 min 20 min 10 min
Table 2 Calculation of D2 based on sub-ranges for aggregate size, for experiments 1 and 2; n=number of particles in each sub-range
25–45
>45
Sample
1.5170.10 0.89 65
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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 log A 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 (10 min) D2 values are close to the Euclidean dimension (D2 ¼ 2), since there was apparently insufficient 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 10 min) 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 measurements. It appears then, that the lower mixing intensity was more favorable for producing less compact aggregate structures (lower D2 ), though alum dose and particle concentration also affected D2 : Similarly, an examination of data in Table 1 shows that higher alum
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Fig. 5. Images from the mixing jar for multiple aggregates after (a) 10 min, (b) 20 min, and (c) 30 min.
Relative Frequency (%)
35
30 min
30 25
20 min 10 min
20 15 10 5 0 0.5
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35 Relative Frequency (%)
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Fig. 6. Particle size distributions at three sampling times for experiments (a) 1, and (b) 2.
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.
R.K. Chakraborti et al. / Water Research 37 (2003) 873–883
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 (reactionlimited, 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 dimension 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. Chemical 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
Fig. 7. Temporal changes in aggregate size and fractal dimension; data points represent ranges of values and averages obtained from the present experiments.
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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 arguments, 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 associated 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 experiments 1 and 2 are 1.6270.02 and 1.7570.03, respectively). 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
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Fig. 8. Change in fractal dimension as a response to two levels of shear rate.
calculated according to fractal theory could be orders of magnitude greater than standard values based on a spherical particle assumption. This suggests that significant 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 conditions 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 measurements. 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 procedure 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 assumption that interacting particles are solid spheres. Fractal dimension is difficult to incorporate directly in the traditional Smoluchowski equation, so alternative approaches 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.
R.K. Chakraborti et al. / Water Research 37 (2003) 873–883 [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.
883
[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. Nonintrusive characterization of fractal aggregates formed by alum. AWWA Annual Meeting, Universities’ Forum, Denver, June, 2000.