# The Influence of Temperature on the Bulk Settling of Cohesive Sediment in Still Water with the Lattice Boltzmann Method

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{−6}m

^{2}s

^{−1}at 20 °C, 1.31 × 10

^{−6}m

^{2}s

^{−1}at 10 °C and 0.80 × 10

^{−6}m

^{2}s

^{−1}at 30 °C) and accelerates the sediment settling velocity according to Stokes’s Law [9,15]. Most properties of the flocculation process can usually be deduced from the flocs’ size and their effective density. With the increasing development of computer technology, mathematical models have emerged as one of the most important methods to study the settlement and flocculation of cohesive sediment, and some researchers have relied on the theory of fractal dimensions to explain the relationship between floc size and floc effective density [16,17,18,19,20] and conducted other relevant works [21,22,23,24,25] by computer. However, most of the models did not take into account the effect of temperature. The mathematical model can eliminate the interference of other factors, and, hence, the influential mechanism can be deeply studied. Qiao et al. [26] studied the mechanism of temperature effects on the flocculation process of two cohesive sediment particles via the Lattice Boltzmann Method (LBM).

## 2. Methods and Model

#### 2.1. Lattice Boltzmann Method

_{i}(x,t), which describes the particle status at a lattice location x at time t with the discrete velocity

**e**

_{i}. The Boltzmann equation is used to solve the collision-induced evolution of the fluid particle, and the equation can be written as:

_{i}(f

_{i}) is the collision operator; in Nguyen and Ladd’s model [30], Ω

_{i}(f

_{i}) can be written as follows:

_{j}

^{eq}is the local equilibrium function, and f

_{j}

^{neq}is the non-equilibrium function, i.e., ${f}_{j}^{neq}={f}_{j}^{}-{f}_{j}^{eq}$. The hydrodynamic parameters, such as the mass density ρ and momentum ρ

**u**, are the function of the distribution function f and discrete velocity

**e**

_{i}, which can be written as follows:

**|e**

_{i}

**|**equals 0 (i = 0), 1(i = 1, 2, …, 6) and $\sqrt{2}$(i = 7, 8, …, 18). The coefficients of the three speeds ${a}^{{e}_{i}}$ are 0 (i = 0), 1/18 (i = 1, 2, …, 6) and 1/36 (i = 7, 8, …, 18).

_{ij}are the matrix elements of the linearized collision operator, which must satisfy the following eigenvalue equations:

_{v}are related to the shear viscosity η and bulk viscosity η

_{v}, respectively, which are within the range of −2 < λ < 0, in which $\eta =-\rho {c}_{s}^{2}\left(1/\lambda +1/2\right)$ and ${\eta}_{\nu}=-\rho {c}_{s}^{2}\left[2/(3{\lambda}_{\nu})+1/3\right]$.

_{i}

^{∗}can be written in the same form as f

_{i}

^{eq}:

#### 2.2. The Extended Derjaguin–Landau–Verwey–Overbeek Theory

^{LW}

_{i-j}, the electrostatic double-layer repulsive force F

^{EL}

_{i-j}, and the Lewis acid‒base force F

^{AB}

_{i-j}. Each of them is the negative derivative of the corresponding potential to the distance. Among them, temperature only affects the electrostatic double-layer repulsive force by changing the Debye length κ

^{−}

^{1}. For more details, please refer to [20,26,31,32,33,34,35].

_{1}and R

_{2}can be written as follows [20]:

_{ij}= r

_{ij}− (R

_{1}+ R

_{2}) is the net distance between spheres, and r

_{ij}is the distance between sphere centers; ε

_{0}is the dielectric permittivity in a vacuum with a value of 8.854 × 10

^{−12}C

^{2}/(J∙m); ε

_{r}is the relative dielectric constant, and for water, it is 78.5; ψ

_{0}is the surface potential of particles, which is very sensitive to salinity and pH but not sensitive to temperature when the temperature is below approximately 150 °C [36]; κ

^{−}

^{1}is the Debye length, related to temperature, cation valence, and ion concentration, which can be written as follows:

^{−19}C; ${N}_{\mathrm{A}}$ is Avogadro’s number 6.022 × 10

^{23}/mol; k is the Boltzmann constant 1.38 × 10

^{−23}J/K; T is the absolute temperature with a unit of K; c is the cation concentration with a unit of mol/L; and z represents the cation valence and is dimensionless.

#### 2.3. Criterion Distance of Flocculation

^{−}

^{1}, with ψ

_{0}on the particle surface and ζ at the distance δ. Both ψ

_{0}and ζ can be measured in experiments, thus the slipping layer thickness δ can be calculated as follows:

## 3. Computational Conditions

^{3}(volume concentration of 0.049%). Eight hundred sediment particles with diameters from 5 μm to 10 μm were scatted randomly in the whole calculation area. The calculation time step was set as 10

^{−6}s, and the total time was 20 s. In all cases, the cation was selected as Na

^{+}, with a valence of +1 and a concentration of 0.085 mol/L or salinity of 5 ppt. Illite was considered to be the clay mineral with a surface potential of −27.22 mV [38] as illite is widely distributed and its chemical and physical properties that needed in this study are easy to obtain as they are studied more extensively. The sediment density was ρ

_{s}= 2650 kg/m

^{3}. The water density was ρ = 1000 kg/m

^{3}, without considering the influence of salinity and temperature as the error of the density at the values adopted will be less than 4% according to the sea water state equation proposed by UNESCO [39]. The other parameters are listed in Table 1.

## 4. Results

#### 4.1. Floc Size and Floc Volume

#### 4.2. Settling and Flocculation Process

#### 4.3. Suspended Sediment Concentration

_{c}is the attenuation coefficient, in which a large value indicates a faster incline.

_{0}) of water depths shallower than 4.5 mm, in which layer the SSC is somewhat unchanged as shown in Figure 6, are draw in Figure 7a. As shown in Figure 7a, the fitting results of Equation (11) (the dash lines) were less superior in these cases, therefore, some trials of indexes of n for C in the right side had been done. As a result, 0.3 was the best fitted index (the solid lines in Figure 7a), thus yielding the following formula:

_{1/2}. From Equation (12), the half-life period t

_{1/2}can be written as follows:

#### 4.4. Sediment Settling Velocity

_{1/2}and the settling distance [6]. The bulk velocity can be expressed simply as follows:

_{1/2}) bulk settling velocity at water depth H, in which depth the SSC changes slightly during settling, and that is why 4.5 mm (4.0–5.0 mm) is chosen in Figure 7a. The derivation of Equation (14) can refer to You’s work [41]. The water depth H and time t

_{1/2}in Equation (14) can be obtained easily through macroscopic observation.

_{1/2}in the 4.5 mm layer (4.0–5.0 mm) were listed in Table 2, as velocity 2. The results of the two methods were almost identical, with a small relative error of approximately 2%. Equation (14) can be taken as the connection between the microscopic statistical methods and physical test methods.

^{3}and 1600 kg/m

^{3}, but closer to the line of 1600 kg/m

^{3}, probably due to the small size of the flocs, so they have a larger effective density [16,42]. The simulation results overlap with the in situ observation data of Xia et al. [12] for July 1999 and January 2000. It should be noted that the simulation velocities are each floc’s speed, while that of other studies are bulk velocity, or statistical macro data of suspended samples.

## 5. Discussion

_{glo}, the ratio of collisions that created new flocs or enlarged original flocs to the total collisions is the capture frequency η

_{cap}, and the product of the two ratios is the aggregation frequency E

_{agg}= η

_{glo}× η

_{cap}. The three parameters were analyzed and are illustrated in Figure 9.

_{glo}continued to grow up, while η

_{cap}began to decline slightly after its peak at about 20 °C after an initial increase. Sterling et al. [46] stated that the collision frequency was due to three factors: Brownian motion, turbulence shear, and differential settling. In this simulation, the sediment particles were much larger than the molecule and thus the Brownian motion could be ignored; the turbulence effect was omissible in still water; therefore, the main factor was the differential settling. Figure 9b showed a positive correlation between η

_{glo}and settling velocity, which was consistent with the viewpoint of Sterling et al. [46]. η

_{glo}in the simulations ranges from 2.1% to 4.2%, very close to the results of Kim and Stolzenbach [45] (from 2.96% to 3.20%). The potential reasons for the difference between the two results might lie in Kim and Stolzenbach’s [45] presumption that their model neglected the repulsive colloidal interaction, which is very sensitive to temperature (in Section 2.2) but was not mentioned in their study, so they might not take the temperature into account in their work.

_{cap}remained unchanged, while in the 20 °C case this collision made the η

_{cap}larger as it enlarged the floc. From the aspect of force, when a third particle was involved, the difference between the two double-layer forces acting on particle #681, F

_{EL681–689}–F

_{EL681–721}, was the same in magnitude as the particle gravity (Figure 10b), and the macroscopic force, including the gravity and hydrodynamic force, was highlighted. The higher the temperature, the more obvious the macro forces were.

_{cap}. However, the reason is different for the low-temperature case, where the large repulsion force makes it more difficult for the particles to get close enough to form flocs, resulting in a small η

_{cap}.

## 6. Conclusions

- (1)
- The mean floc size and floc volume increased with increasing temperature. The maximum floc size initially increased and then decreased slightly with its peak at 10 °C and trough at 5 °C. The floc was not easily formed at low temperature but was unstable and cracked easily at high temperature. The aggregation process, aggregation frequency and forces between particles can be explained by the above. At low temperatures, the collision frequency η
_{glo}and capture frequency η_{cap}were low, which meant the floc was not easily formed; at high temperatures, the large flocs were easily broken as the weighting of the macro force increased to have the same magnitude as the short-distance force. - (2)
- During settling, the SSC time series curves fit well with the equation $\mathrm{d}C/\mathrm{d}t=-{k}_{c}{C}^{0.3}$, from which the settlement half-life period and bulk setting velocity were deduced. Increasing the temperature had a negative effect on the settlement half-life, indicating a faster SSC incline at high temperatures than at low temperatures.
- (3)
- The macroscopic bulk velocity derived from the SSC change agreed well with the microscopic statistical settling velocity of each particle and floc. Both velocities agreed well with the existing physical test results, on-site observation data, and formulas, indicating that the LBM is a reasonable choice for simulating cohesive sediment bulk settling.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The possible velocity directions in the D3Q19 topology. The speed of discrete velocity |

**e**

_{i}| equals 0 when particle keeps its original position after a time step, equals 1 when it moves to the faces of the cubic and $\sqrt{2}$ to the edges of the cubic.

**Figure 2.**The XDLVO potentials of different net distances h

_{ij}between particles. The Lifshitz‒van der Waals potential is always attractive; the electrostatic double-layer potential is repulsive and the Lewis acid‒base potential’s sign depends on the properties of colloids. XDLVO potential is the summation of the aforementioned three potentials, resulting in potential wells and potential barriers. Particles whose interactive forces conquer the potential barrier can form a stable floc.

**Figure 3.**Illustration of slipping layer thickness, equaling the thickness of the electrostatic double layer in Yang et al.’s study [37]. This zone is comprised of an absorbed layer and a diffuse layer. The slipping layer thickness can be calculated by the potentials ψ

_{0}on the particle surface and ζ at the distance δ as the potential decays exponentially. When the particles are close enough and wrapped by a common slipping layer, they are considered as forming a new or enlarging an old floc.

**Figure 4.**Floc properties of each case. (

**a**) Time histories of maximum floc size (solid symbols) and mean floc size (hollow symbols) for the 5 °C, 10 °C, 20 °C, and 30 °C cases; (

**b**) The maximum floc diameter, mean floc diameter (the left axis, with unit of μm) and floc volume (the right axis, with unit of 10

^{−15}m

^{3}) at the end of the simulation. In (

**b**), the thin solid line and dashed line represent the line fit of maximum and mean floc diameters, respectively, and the thick solid line is the exponential fit of floc volume.

**Figure 5.**Time series of particle velocities and particles’ net distance, i.e., h

_{ij}in Equation (8) and r

_{ij}− R

_{i}− R

_{j}in Figure 3. Solid symbols are net distances, and hollow symbols mean velocities. d1 represents the distance between particles #681 and #689; d2 represents that between particles #689 and #721; d3 represents that between particles #681 and #721; u1, u2, and u3 represent the settling velocities of particles #681, #689, and #721, respectively. The left axes are for the distance, and the right ones are for the velocities. (

**a**) T = 5 °C; (

**b**) T = 10 °C; (

**c**) T = 20 °C; and (

**d**) T = 30 °C.

**Figure 6.**SSC of each water depth in (

**a**) 5 °C, (

**b**) 10 °C, (

**c**) 20 °C, and (

**d**) 30 °C. The symbols between two labeled depths represent the SSC between those two depths. The SSC can be statistically obtained by the sediment weight in this zone. It was generally stable at depth of 4.5 mm, but there were some small differences between the cases.

**Figure 7.**The SSC time series and its relative parameters. (

**a**) The time series curves of relative SSC (ratio of c(t) at time t to initial c

_{0}) of a water system shallower than 4.5 mm; (

**b**) The k

_{c}in Equation (12) and t

_{1/2}in Equation (13). In (

**a**), the symbols are the experiment results, the solid lines and dashed lines are the fitting curves for n = 0.3 and n = 1.0, respectively. In (

**b**), the solid square and hollow triangles symbols are t

_{1/2}and k

_{c}, both of which are deduced from the solid curves in (

**a**). The solid line and dashed line in (

**b**) are the linear fittings for t

_{1/2}(t

_{1/2}= −0.380T + 21.366) and k

_{c}(k

_{c}= 0.120T + 2.661).

**Figure 8.**The settling velocities of flocs of different diameter and temperature in many studies. (

**a**) Floc diameter and settling velocities from the observational data of Xia et al. [12], Khelifa and Hill [43], Guo and He [13], Dyer and Manning [42], and Manning et al. [43]; theory results of Manning et al. [43], Winterwerp [44], and Khelifa and Hill [43]; and the simulation results of LBM under temperature conditions ranging from 5 °C to 30 °C; (

**b**) floc velocities and temperature from the observational data of Xia et al. [12], Wan et al. [6], Guo and He [13], and this simulation’s results.

**Figure 9.**The change of Collision frequency η

_{glo}, capture frequency η

_{cap}and aggregation frequency E

_{agg}with (

**a**) temperature and (

**b**) settling velocity.

**Figure 10.**Comparisons of F

_{EL}and the gravity of the primary particle with diameters of 9.36 μm and efficient densities ρ

_{s}-ρ of 1650 kg/m

^{3}. (

**a**) The electrostatic double layer force between particles #681 and #689, F

^{EL}

_{681–689}of different particle net distances at temperatures of 5 °C, 10 °C, 20 °C, and 30 °C; (

**b**) the difference in electrostatic double-layer forces acting on particle #681, F

^{EL}

_{681–689}–F

^{EL}

_{681–721}with different particle net distances in the cases of 5 °C, 10 °C, 20 °C, and 30 °C.

Case ID | #1 | #2 | #3 | #4 |
---|---|---|---|---|

Temperature/(°C) | 5 | 10 | 20 | 30 |

2δ/(nm) | 18.7 | 19.1 | 20.0 | 20.7 |

Water viscosity ν/(10^{−6} m^{2} s^{−1}) | 1.52 | 1.31 | 1.00 | 0.80 |

**Table 2.**Bulk settling velocity of different methods (velocity 1 v

_{1}is derived from H and t

_{1/2}; velocity 2 v

_{2}is the statistical result of all particles and flocs at water depths from 4.0 mm to 5.0 mm; ABS(1-2) |v

_{1}−v

_{2}| means the absolute difference between velocity 1 and velocity 2; |v

_{1}−v

_{2}|/(|v

_{1}+v

_{2}|/2) represents the relative difference between velocity 1 and velocity 2).

Temperature/(°C) | 5 | 10 | 20 | 30 |
---|---|---|---|---|

(1) velocity 1 v_{1}/(mm/s) | 0.099 | 0.117 | 0.155 | 0.189 |

(2) velocity 2 v_{2}/(mm/s) | 0.101 | 0.117 | 0.152 | 0.185 |

(3) ABS(1-2) |v_{1}−v_{2}|/(mm/s) | 0.002 | 0.000 | 0.003 | 0.004 |

(4) |v_{1}−v_{2}|/(|v_{1}+v_{2}|/2)/(%) | 2.0% | 0.0% | 2.0% | 2.1% |

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**MDPI and ACS Style**

Qiao, G.-q.; Zhang, J.-f.; Zhang, Q.-h.; Feng, X.; Lu, Y.-c.; Feng, W.-b.
The Influence of Temperature on the Bulk Settling of Cohesive Sediment in Still Water with the Lattice Boltzmann Method. *Water* **2019**, *11*, 945.
https://doi.org/10.3390/w11050945

**AMA Style**

Qiao G-q, Zhang J-f, Zhang Q-h, Feng X, Lu Y-c, Feng W-b.
The Influence of Temperature on the Bulk Settling of Cohesive Sediment in Still Water with the Lattice Boltzmann Method. *Water*. 2019; 11(5):945.
https://doi.org/10.3390/w11050945

**Chicago/Turabian Style**

Qiao, Guang-quan, Jin-feng Zhang, Qing-he Zhang, Xi Feng, Yong-chang Lu, and Wei-bing Feng.
2019. "The Influence of Temperature on the Bulk Settling of Cohesive Sediment in Still Water with the Lattice Boltzmann Method" *Water* 11, no. 5: 945.
https://doi.org/10.3390/w11050945