# Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model

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## Abstract

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## 1. Introduction

_{s}), launch radius (representing initial particulate concentration, R

_{e}), and finite motion step (representing the motion energy of the particles, M

_{s})—on the morphology, fractal dimensions (D

_{f}), and porosities (V

_{f}) of the two-dimensional (2D) as well as three-dimensional (3D) DLA aggregates, in order to get a deeper understanding of the fractal features of the DLA aggregates, which would assist in describing flocculation processes more exactly.

## 2. Model

_{0}) in the center of a square divided into an L × L grid (or a sphere divided into an L × L × L grid) and draw a large circle (or a big sphere) centering around the “seed” particle; the radius of the circle (or the sphere) is the so-called launch radius (R

_{e}). Then, release a free particle (not the “seed” particle) from any site on the perimeter of the circle (or the sphere). The launched particle moves randomly, following a Brownian path, which means that it walks in a random direction at each step, with no correlation to the previous direction, in which the new position of the free particle is given by

_{k}) is a limiting radius that is used to identify a disposal event when the walking particle moves beyond the maximum circle (or sphere) region. Another particle is then launched and halted when adjacent to the two occupied sites, and so forth. As this process repeats a large number of times, a fractal-like DLA aggregate can be formed.

_{0}is the prefactor, R

_{g}is the gyration radius denoting the maximum radius of the aggregate, R

_{0}is the radius of the primary particle, and D

_{f}is the Hausdorff dimension (fractal dimension). Equation (4) in logarithmic form can be expressed as follows:

_{g}/R

_{0}), a straight line can be obtained, and the slope of the fitting curve is equal to the fractal dimension D

_{f}.

_{p}is the radius of the primary particle and V

_{f}is the porosity of the DLA aggregate.

## 3. Results and Discussion

#### 3.1. Effect of Particle Number

_{s}= 2), a launch radius of 2 (R

_{e}= 2), and different particle numbers; the effect of particle number on the fractal features and structures of the DLA aggregates is shown in Figure 2 and Figure 3. Generally, the particle number relates to the flocculation time: the higher the number of particles, the longer the flocculation time. For the 2D DLA simulation, when the particle number is fairly small (N < 1000), the aggregates are fairly small and only show local fractal features, and their scale-invariances as well as self-similarities are insignificant. The aggregate gradually grows with rising particle numbers, and it presents typical fractal morphology with properties of global scale-invariances when the particle number reaches a certain value (N ≥ 1000). The fractal dimension increases rapidly as the particle number rises from 50 to 1000, and then changes little as the particle number increases further (Figure 4). The fractal dimension slightly fluctuates in the range 1.66–1.74, which coincides with the fractal dimension value range of typical 2D DLA aggregates [12]. The porosity is smaller (around 0.8) when the particle number is less than 500, as it is easier to form a compact DLA aggregate when the particle numbers are lower, and it becomes increasingly compact and grows to a two-dimensional branched aggregate with rising particle numbers, leading to the increase in the fractal dimension and porosity. Therefore, the optimal particle number for 2D DLA simulation is 2000. For 3D DLA simulation, when the particle number is very small (N < 400), the aggregates are also small with lower fractal dimensions (Figure 3 and Figure 4). The aggregate grows to a highly assembled three-dimensional aggregation structure with the rising particle numbers; the fractal dimension rises sharply as the particle number increases from 50 to 400, and then it varies little as the particle number continues to increase. When the particle number is above 1000, the fractal dimension generally stays at around 2.20, which agrees with the fractal dimension values of representative 3D DLA aggregates [30]. However, the porosity only slightly fluctuates with varying particle numbers because the 3D DLA aggregate has a much bigger packing volume compared with the 2D DLA aggregate. Thus, the optimal particle number for 3D DLA simulation is 1000. For 2D and 3D DLA aggregates, the fractal dimensions generally increase with flocculation time elapsed (rising particle numbers) when N

_{s}< 1000, and then fluctuate little.

#### 3.2. Effect of Motion Step Length

_{e}= 2), and motion step lengths varying from 0.5 to 10; the effect of motion step length on the fractal features and structures of the DLA aggregates is illustrated in Figure 5 and Figure 6. It is well known that water temperature has a big influence on flocculation processes: High water temperature can accelerate the movement of colloidal particles and hydrolytic processes of flocculants, which contributes to the enhancement of flocculation efficiency. Thus, the motion step length is adopted to denote the water temperature of the practical flocculation processes: the bigger the motion step length, the higher the water temperature. When the motion step length is small (N

_{s}< 4), the 2D DLA aggregates present two-dimensional slender dendritic planar structures of typical fractal features with scale-invariant properties, and the 3D DLA aggregates show fairly loose three-dimensional spatial structures. The DLA aggregate gradually grows to an increasingly compact structure as the motion step length rises from 4 to 10: The 2D DLA aggregate generates a dense cake-like structure of inconspicuous fractal features with higher fractal dimension (D

_{f}= 1.92), and the 3D DLA aggregate forms a highly compact ball-like structure with higher fractal dimension of 2.40 when the motion step length approaches 10. With increasing motion step length, the fractal dimensions gradually increase while the porosities decrease correspondingly (Figure 7). The reason for this is that the smaller the motion step length is, the smaller the particle motion region is, and the released particles can be more easily captured by the external branches of the aggregates, leading to the rapider growth of the aggregate branches with highly slender structures, which contributes to the decrease in fractal dimensions and increase in porosities. Conversely, when the motion step length is bigger, the motion region of the particle becomes larger, and it is more difficult for the aggregate external branches to capture the launched particles that can more easily go into the interior of the aggregate to collide and coagulate with the “seed” particle or clusters. This can eliminate the shielding effect and make the aggregate grow more slowly with less branches to form a highly compact structure, leading to the increase in fractal dimension and decrease in porosity. Thus, the DLA aggerates will grow into increasingly compact objects with inconspicuous fractal features as the water temperature (the motion step length) increases. It should be noted that when the motion step length is equal to 2 (N

_{s}= 2), the DLA aggregate presents prominent fractal structure characteristics.

#### 3.3. Effect of Launch Radius

_{s}= 2), and a versatile launch radius ranging from 1 to 50. The effect of launch radius on the fractal features and structures of the DLA aggregates is displayed in Figure 8 and Figure 9. The launch radius can represent the initial particulate concentration of the practical flocculation processes: A bigger launch radius means that the initial particulates in the wastewater are further away from each other; that is, the initial particulate concentration is lower. As we can see, the aggregates gradually grow outward with the increasing launch radius. When the launch radius rises from 1 to 50, the fractal dimensions of the 2D DLA aggregates generally decrease from 1.69 to 1.53, and the porosities slightly increase from 0.82 to 0.89; the fractal dimensions of the 3D DLA aggregates generally drop from 2.24 to 2.01, and the porosities slowly rise from 0.94 to 0.98 (Figure 10). When the launch radius exceeds 10, the fractal dimensions of the 2D DLA aggregates fluctuate between 1.53 and 1.71, and the porosities stay at around 0.82–0.90; the fractal dimensions of the 3D DLA aggregates fluctuate between 2.01 and 2.16, and the porosities stay at 0.96–0.98. The bigger the launch radius, the larger the motion radius of the particle and, thus, it is easier for the released particles to go outward, resulting in the outward growth of the aggregate as well as the decrease in fractal dimension and increase in porosity. Thus, the DLA aggregates will grow increasingly outward with rising initial particulate concentration (launch radius) accompanying the reduction of fractal dimensions and rise of porosities. It can be concluded that the DLA aggregates display conspicuous fractal features when the launch radius is between 1 and 10.

#### 3.4. Effect of Finite Motion Step

_{s}= 2), and launch radius of 2 (R

_{e}= 2) with various finite motion steps. The effect of the finite motion step on the fractal features and structures of the DLA aggregates is presented in Figure 11 and Figure 12.

_{f}< 1.60) and larger porosity (V

_{f}> 0.87). They grow increasingly discrete and the fractal dimension reduces rapidly with the gradual decrease of the finite motion step numbers. When M

_{s}exceeds 800 (but is less than 3000), the aggregates grow to increasingly compact branched structures with typical scale-invariant features, and fractal dimension between 1.62 and 1.69 (Figure 13). However, the fractal dimension slightly increases to 1.64–1.74 when the finite motion step numbers rise from 3000 to 5000. For the 3D simulation, when the finite motion step numbers are less than 200, the aggregates exhibit a highly scattered spatial structure composed of a few isolated small clusters with lower fractal dimensions (D

_{f}< 1.95) and larger porosity (V

_{f}> 0.88). They also grow increasingly scattered and the fractal dimension drops sharply with decreasing finite motion step numbers. When M

_{s}exceeds 200, the aggregates produce increasingly compact spatial structures, and the fractal dimension stays at 2.00–2.30.

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Sketch of particle aggregation of DLA model (

**a**) and fractal dimension calculation by using gyration method (

**b**).

**Figure 4.**Effect of particle number on the fractal dimension and porosity of 2D and 3D DLA aggregates.

**Figure 7.**Effect of motion step length on the fractal dimension and porosity of 2D and 3D DLA aggregates.

**Figure 10.**Effect of launch radius on the fractal dimension and porosity of 2D and 3D DLA aggregates.

**Figure 13.**Effect of finite motion step numbers on the fractal dimension and porosity of 2D and 3D DLA aggregates.

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**MDPI and ACS Style**

Liu, D.; Zhou, W.; Song, X.; Qiu, Z.
Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model. *Fractal Fract.* **2017**, *1*, 12.
https://doi.org/10.3390/fractalfract1010012

**AMA Style**

Liu D, Zhou W, Song X, Qiu Z.
Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model. *Fractal and Fractional*. 2017; 1(1):12.
https://doi.org/10.3390/fractalfract1010012

**Chicago/Turabian Style**

Liu, Dongjing, Weiguo Zhou, Xu Song, and Zumin Qiu.
2017. "Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model" *Fractal and Fractional* 1, no. 1: 12.
https://doi.org/10.3390/fractalfract1010012