# Stochastic Flocculation Model for Cohesive Sediment Suspended in Water

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Previous FGM of Son Hsu [1]

_{c}is a characteristic fractal dimension and ${D}_{fc}$ is a characteristic size of floc. The typical vales of F

_{c}and ${D}_{fc}$ are suggested to be F

_{c}= 2.0 and ${D}_{fc}$ = 2000 $\text{\mu}$m by Khelifa and Hill [20].

#### 2.2. Application of Stochastic Approach to FGM

_{y}), the minimum stress for floc breakup (see [1,8] for more details). However, turbulent shear exceeding F

_{y}does not necessarily cause the breakup of flocs (e.g., [20]). Therefore, ${k}_{B}^{\prime}$ is assumed to be a random number in this study. As the size of flocs increases, the density of flocs decreases, resulting in higher probability of breakup. Thus, the distribution of floc size should be skewed positively (skewed to the right; skewness ≥ 0). Based on this idea and previous studies mentioned in the first paragraph of this section, ${k}_{B}^{\prime}$ is empirically assumed to be a random number having a log-normal distribution (see Equation (3)).

_{l}and σ

_{l}are the mean and the standard deviation of log-normal distribution, respectively. In Equation (3), X is a pseudorandom number. Incorporating Equation (3) into Equation (1), the stochastic FGM of this study is developed. μ

_{l}is equal to ${k}_{B}^{\prime}$ of the deterministic FGM. σ

_{l}is calibrated with the distribution shape of experimental results. Compared to the previous deterministic Son and Hsu FGM [1], the new stochastic FGM has one more parameter (σ

_{l}).However, the new FGM has a capability to calculate the size distribution of flocs ensuring the mass conservation. μ

_{l}and σ

_{l}are calibrated with three experimental results in Section 3.

## 3. Results and Discussion

^{3}, respectively. The average value of G produced by the impeller is 50 s

^{−1}. The size and density of polystyrene particles (c and ${\text{\rho}}_{s}$) of the experiment are 0.87 μm and 1050 kg/m

^{3}, respectively. In Biggs and Lant [9], the size distribution of activated sludge is reported under the experimental condition of G = 19.4 s

^{−1}and $\text{\varphi}$ = 0.05. Under the assumption that ${\text{\rho}}_{s}$ is 2650 kg/m

^{3}and the density of sludge is 1300 kg/m

^{3}, the mass concentration is calculated to be 24.19 kg/m

^{3}. The activated sludge is mixed in a baffled batch vessel using a flat six-blade impeller. Burban et al. [24] also report the results of laboratory experiments with Detroit River sediment. c and G of the experiment are 0.8 kg/m

^{3}and 200 s

^{−1}, respectively. d and ${\text{\rho}}_{s}$ of Detroit River sediment are assumed to be 4 μm and 2650 kg/m

^{3}due to absence of further information. The detailed conditions of experiments are summarized in Table 1. The calibrated values of empirical coefficients are shown in Table 2.

**Figure 1.**Example of numerical simulation and analysis data. Dots represent the experimental result of Spicer et al. [23]. Solid and dotted lines are the stochastic FGM (SFGM) of this study and the deterministic FGM (DFGM), respectively.

Experiment | $\mathbf{\text{\varphi}}$ | c (kg/m^{3}) | ${\rho}_{\mathit{s}}\text{}$
(kg/m^{3}) | d (μm) | G (s^{−1}) |
---|---|---|---|---|---|

Spicer et al., (1998) | $1.4\times {10}^{-5}$ | 0.0147 | 1050 | 0.87 | 50.0 |

Biggs and Lant (2000) | 0.05 | 24.19 (Assumed) | 2650 (Assumed) | 4.00 (Assumed) | 19.4 |

Burban et al., (1989) | N.A. | 0.05 | 2650 (Assumed) | 4.00 (Assumed) | 200.0 |

Experiment | ${\mathit{k}}_{\mathit{A}}^{\mathit{\prime}}$ | ${\mu}_{\mathit{l}}$ | ${\sigma}_{\mathit{l}}^{\mathbf{2}}$ | B_{1} |
---|---|---|---|---|

Spicer et al., (1998) | 6.74 | $4.39\times {10}^{-6}$ | $4.47\times {10}^{-6}$ | $2.63\times {10}^{-14}$ |

Biggs and Lant (2000) | 0.02 | $2.61\times {10}^{-5}$ | $5.62\times {10}^{-5}$ | $4.20\times {10}^{-13}$ |

Burban et al., (1989) | 0.30 | $3.95\times {10}^{-5}$ | $3.16\times {10}^{-3}$ | $4.10\times {10}^{-12}$ |

^{3}). The intensity of turbulent shear is not low (G = 50 s

^{−1}). However, the mean size of flocs is about 200 μm. Therefore, it is clear that the polystyrene particles are easily aggregated. Under this assumption, the coefficient determining the aggregation efficiency has a relatively high value (${k}_{A}^{\prime}=6.74$). In terms of the mean size of flocs, the simulation results are in satisfactory agreement with experimental results. The mean size is determined by calibrating ${k}_{A}^{\prime}$ and ${\text{\mu}}_{l}$. That is, the model has the capability to calculate mean size of flocs reasonably well. It is shown in Figure 2 that the calculated size distribution is positively skewed slightly more than the measured result. The log-normal distribution function has only 2 parameters, the mean and variance. Therefore, the skewness and kurtosis of log-normal distribution is indirectly determined by the values of mean and variance. Three- or four-parameter distribution functions such as generalized extreme value distribution and Wakeby distribution have the capability to determine the skewness or kurtosis by calibrating additional parameters. However, more empirical and site-specific parameters need to be calibrated when a 3- or 4-parameter distribution function is incorporated into FGM.

**Figure 2.**Simulation results of stochastic FGM and experimental results of Spicer et al. [23]. The bars and the solid lines represent the experimental and simulation results, respectively. (

**a**) semi-log axis; (

**b**) normal (Cartesian) axis.

^{3}). The intensity of turbulence is relatively low (G = 19.4 s

^{−1}). Under these conditions, the mean size of flocs is not large (about 120 μm). Therefore, it is deduced that the aggregation process of activated sludge used for the experiment has a low efficiency resulting in a low value for the aggregation efficiency parameter (${k}_{A}^{\prime}=0.02$). The simulation result is in a good agreement with laboratory measurements. Both the mean size and the distribution are reasonably replicated by the proposed stochastic FGM. The sudden increase of volume fraction around floc size 700 μm is due to the random property of stochastic simulation. Figure 4a shows the mean, maximum, and minimum floc sizes of 30 simulation cases. The dotted line is the equilibrium floc size calculated by the deterministic Son and Hsu FGM [1]. Figure 4b shows the temporal evolution of mean floc size measured by Biggs and Lant [9], a simulation result of deterministic FGM ([1]), and one case of simulation by stochastic FGM. As mentioned in the first paragraph of this section, the last 1000 values of floc size for each stochastic simulation are analyzed in this study. Therefore, the size distribution of this study is determined using 30,000 values for floc size. Figure 4 shows that a stochastic FGM calculates the floc size very randomly after the temporal evolution of floc size approaches the equilibrium state (around 15 min in Figure 4b). The maximum floc size among the values in the last 1000 time steps has a wide range (200 μm to 700 μm). Furthermore, the minimum floc size is also in the range of 10 μm to 50 μm whereas the mean values are between 70 and 120 μm in most cases.

**Figure 3.**Simulation results of stochastic FGM and experimental results of Biggs and Lant [9]. The bars and the solid lines represent the experimental and simulation results, respectively. (

**a**) semi-log axis; (

**b**) normal (Cartesian) axis.

^{−1}) and a moderate mass concentration condition (c = 0.8 kg/m

^{3}). Due to the high turbulence intensity, a high breakup efficiency is expected. Thus, the mean of ${k}_{B}^{\prime}$ is calibrated to be relatively high (μ

_{l}= 3.95 × 10

^{−5}) compared to the two previous cases ([9,23]). Turbulent shear plays two conflicting and simultaneous roles in the flocculation process. The first one is to increase the chance of particle collisions resulting in floc aggregation. Turbulence is the most important mechanism of aggregation ([6,33,50,51]). The second role is disaggregation of flocs. Based on laboratory experiments, Tsai and Hwang [38] insist that a floc usually disaggregates into two roughly equal-sized daughter flocs. From this finding, it is deduced that the main mechanism of breakup is turbulent shear rather than particle collision. Figure 5 shows the effects of concentration and turbulent shear on mean floc size. Figure 5 shows that the high turbulent shear enhances the breakup process compared to aggregation process causing reduction of floc size ([2]). Therefore, the mean floc size of Burban et al. [24] is relatively small (about 20 μm) resulting in a high value for ${\text{\mu}}_{l}$. Figure 6 shows the results of numerical simulation by stochastic FGM and measurements from Burban et al. [24]. The volume fraction near the mean value is slightly overestimated by the stochastic FGM. However, the overall shape of the distribution is in good agreement with measurement results. The volume fraction slightly increases around floc size 80 μm due to the random nature of the stochastic model. A stochastic model is usually based on a random number generation technique (pseudorandom number generator in this study). Therefore, perturbations due to randomness are inevitable.

**Figure 4.**The random calculation of stochastic FGM. The dotted line of (

**a**) shows the simulation results of deterministic FGM (DFGM). The circle, square, and triangle symbols represent the maximum, mean, and minimum values of each simulation, respectively; the solid and dotted lines of (

**b**) are the results of deterministic FGM (DFGM) and stochastic FGM (SFGM), respectively; (a) mean, maximum, and minimum floc size (b) temporal evolution of floc size.

**Figure 6.**Simulation results of stochastic FGM and experimental results of Burban et al. [24]. The bars and the solid lines represent the experimental and simulation results, respectively. (

**a**) semi-log axis; (

**b**) normal (Cartesian) axis.

^{−5.0}and 5.623 × 10

^{−5.0}, respectively. Figure 8 shows the result of stochastic FGM using a Pearson type 3 distribution (Gamma distribution). A Pearson type 3 distribution needs 2 parameters: shape parameter (k

_{p}) and scale parameter (θ). The mean and variance of Pearson type 3 distributions are indirectly determined by k

_{p}θ and k

_{p}θ

^{2}, respectively. Therefore, it is not simple to calibrate a stochastic FGM. The determined values of k

_{p}and θ for the simulation are 4.780 × 10

^{−1.0}and, 5.248 × 10

^{−5.0}, respectively. Compared to results of the log-normal distribution (Figure 3), FGMs using a normal distribution and a Pearson type 3 distribution overestimate the mean floc size, whereas the overall shape of size distribution is underestimated. Figure 9 represents the results calculated by stochastic FGM using a generalized extreme value (GEV) distribution. A GEV distribution has 3 parameters: mean, variance, and shape parameter ($\text{\xi}$). In this study, the mean, variance, and $\text{\xi}$ of the GEV distribution are set to be 1.147 × 10

^{−5.0}, 1.0 × 10

^{−5.0}and 0.5, respectively. In the case of the GEV distribution, the floc size larger than the mean value is overestimated. Although a GEV distribution has one more degree of freedom ($\text{\xi}$) compared to log-normal distribution, no significant advantage is found.

## 4. Conclusions

**Figure 7.**Simulation result of stochastic FGM using a normal distribution (compared to the experimental results of Biggs and Lant [9]). The bars and the solid lines represent the experimental and simulation results, respectively. (

**a**) semi-log axis; (

**b**) normal axis.

**Figure 8.**Simulation result of stochastic FGM using a Pearson type 3 distribution (compared to the experimental results of Biggs and Lant [9]). The bars and the solid lines represent the experimental and simulation results, respectively. (

**a**) semi-log axis; (

**b**) normal (Cartesian) axis.

**Figure 9.**Simulation result of stochastic FGM using a generalized extreme value distribution (compared to the experimental results of Biggs and Lant [9]). The bars and the solid lines represent the experimental and simulation results, respectively. (

**a**) semi-log axis; (

**b**) normal (Cartesian) axis.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Shin, H.J.; Son, M.; Lee, G.-h.
Stochastic Flocculation Model for Cohesive Sediment Suspended in Water. *Water* **2015**, *7*, 2527-2541.
https://doi.org/10.3390/w7052527

**AMA Style**

Shin HJ, Son M, Lee G-h.
Stochastic Flocculation Model for Cohesive Sediment Suspended in Water. *Water*. 2015; 7(5):2527-2541.
https://doi.org/10.3390/w7052527

**Chicago/Turabian Style**

Shin, Hyun Jung, Minwoo Son, and Guan-hong Lee.
2015. "Stochastic Flocculation Model for Cohesive Sediment Suspended in Water" *Water* 7, no. 5: 2527-2541.
https://doi.org/10.3390/w7052527