## Abstract

A simple formula is developed to relate the size and settling velocity of cohesive sediment flocs in both the viscous and inertial settling ranges. This formula maintains the same basic structure as the existing formula but is amended to incorporate the fact that the flocculated sediment has an internal fractal architecture and is composed of different-sized primary particles. The input parameters needed for calculating the settling velocity include the median size and size distribution of the primary particles, the fractal dimension of the floc, the density of the sediment, and two calibrated coefficients that incorporate the effects of floc shape, permeability, and flow separation on drag. The proposed formula is compared with four data sets of settling velocity–floc size collected from the published literature, and a good agreement between the model and these data can be found.

## INTRODUCTION

Different from non-cohesive sediments such as sand and gravel, cohesive sediment is a mixture of water and fine-grained sediments (such as silt and clay) and organic matter of diverse nature (Winterwerp *et al.* 2006; Son & Hsu 2008). Fine-grained sediment has obvious cohesive characteristics and can undergo flocculation effects due to evident electrochemical and biological–chemical attractions on the surface of particles (Stone & Krishnappan 2003; Mietta *et al.* 2009; Son & Hsu 2009). When these fine-grained sediments move into rivers, reservoirs, lakes and estuarine and coastal areas, where eddy motions in a turbulent flow can induce particles to collide, they can flocculate into flocs of different sizes via the binding of primary particles (Winterwerp 1998; Shen & Maa 2015). However, the turbulent shear motion may also cause the breakup of some porous and fragile flocs, leading to some small-sized flocs and/or primary particles (Dyer 1989; Winterwerp 1998). The transport of cohesive sediment plays an important role in some geophysical processes, such as morphodynamic change and ecosystem function variation related to the water quality in rivers, reservoirs, lakes and estuarine and coastal waters (Kumar *et al.* 2010; Maggi 2013).

*et al.*2008). However, the settling of cohesive sediment plays a crucial role in determining the vertical flux of sediment transport (Dyer 1989; Camenen 2007; Song

*et al.*2008; Maggi 2013). Many formulae for predicting the settling velocity of sands, gravels and cohesive sediment flocs in a quiescent or low-turbulence water column have been proposed (e.g. Watson 1969; Hallermeier 1981; Cheng 1997; Winterwerp 1998; Ferguson & Church 2004; Khelifa & Hill 2006; Strom & Keyvani 2011; Vahedi & Gorczyca 2012; Maggi 2013). The governing terms needed in the settling velocity calculation for cohesive sediment flocs include the submerged gravity of the floc, , and the resistant drag exerted on the floc in water, , where and are the densities of the floc and the water, respectively;

*V*and

*A*are the volume and the area projected normal to the direction of motion of the floc, respectively; is the drag coefficient;

*g*is the gravitational acceleration; and is the terminal settling velocity of the floc. By considering the force balance between the submerged gravity and resistant drag terms (the equilibrium settling condition) and assuming that the floc can be treated as a sphere with modifications for floc shape and permeability, the expression for can be written as:

Here, is the floc diameter, is the effective density (excess density) of the floc (kg/m^{3}), and is a non-dimensional factor that accounts for the deviation in the relationship among the diameter, projected area, volume and settling velocity of a floc from that of a smooth solid sphere due to floc shape and floc permeability. For a smooth solid sphere, = 1.

^{2}/s)), < 1, the force balance between the creeping-flow resistant drag (, which is an analytical solution to the governing Navier–Stokes equation solved by Stokes (1850)) and submerged gravity of a floc leads to the following expression of the floc settling velocity, i.e., the well-known Stokes' law: where the drag coefficient becomes . The applicability of Stokes' law is strongly constrained to the assumption < 1, thus making it urgent to construct a more general settling velocity formula for sediment under moderate or high Reynolds number conditions, which are very common in geophysical systems (Winterwerp 1998; Khelifa & Hill 2006; Song

*et al.*2008).

*et al.*2008; Maggi 2013). Among these, the following expression for the drag coefficient given by Schiller and Naumann has commonly been presented in some works to match most empirical data (i.e., the modified Stokes' law) (Kelbaliyer 2011):

When 200, Equation (3) gives a good approximation of the real drag coefficient on spherical particles. However, it should be noted that the settling velocity formula obtained by substituting Equation (3) into Equation (1) may be not applicable for geophysical systems with higher than 200, and an iterative numerical technique is needed to solve the function of since the Reynolds number also contains the settling velocity term.

Furthermore, to the best of our knowledge, at present, all of the developed models for the settling velocity of cohesive sediment flocs have been based on an assumption that the sediment floc is composed of primary particles with a uniform size. With this assumption, sediment flocs could be further assumed to be treated as a fractal object, and fractal geometry could be conveniently adopted to characterize their physical properties, as has been widely acknowledged in the cohesive sediment research field. However, as Vahedi & Gorczyca (2011, 2012) showed, this may not be true because flocs should be a mixture of a variety of primary particle sizes in actual flocculation systems. Neglecting the effect of the multi-size property of primary particles might lead to unexpected error in predicting the setting velocity of sediment floc in a quiescent water column.

Regarding the weakness of some proposed settling velocity formulae for flocs in that there is a constraint to the Reynolds number and an iterative numerical technique is needed to solve the function of , some researchers have attempted to eliminate the need to use in models for the settling velocity calculation by non-dimensionalizing both terms of the settling velocity and particle size and constructing a direct relationship between such non-dimensional terms (Hallermeier 1981; Dietrich 1982; Cheng 1997; Ferguson & Church 2004). For example, Dietrich (1982) introduced two such non-dimensional parameters, and . By virtue of a similar non-dimensionalization method, Ferguson & Church (2004) constructed a simple and explicit relationship between the settling velocity and particle size in viscous, transitional and inertial regimes, and the proposed model had good agreement with measured experimental data of natural sands.

The work of Ferguson & Church (2004) motivates us to explore whether there exists a more general and applicable formula for estimating the settling velocity of sediment flocs. This study attempts to construct such a settling velocity formula. The structure of this paper is arranged as follows. All of the details of the model formulation are introduced in the section ‘Model formulation’. A simple comparison of the developed model and collected settling velocity–floc size data from *in situ* field measurements and laboratory settling-column data are presented in the section ‘Comparison with data’, and finally, the last section presents concluding remarks.

## MODEL FORMULATION

In the work of Ferguson & Church (2004), was chosen as the velocity scale. With this, the non-dimensional forms of the settling velocity, , and the floc diameter, , can be written as and . Introducing such expressions into Equation (1) yields the following expression: .

Under low Reynolds number conditions ( < 1), Stokes' law (Equation (2)) can be expressed in the following non-dimensional form: . However, for > 1,000, some studies have shown that is a constant and that its value depends on particle shape and roundness (Dietrich 1982; Cheng 1997; Camenen 2007). Introducing this constant as *C* can produce the following form of the settling velocity: .

In the work of Ferguson & Church (2004), Equation (4) was tested against settling column experimental data, and good agreement was presented for a range of particle shapes in the viscous, transitional, and inertial regimes.

*i*-th primary particle forming the floc, and

*k*is the number of primary particles forming the floc. Regarding the floc diameter , if a floc of diameter is assumed to be composed of

*k*primary mono-sized particles of diameter , then the floc diameter can be expressed as a function of the number of primary particles,

*k*, and the fractal dimension of the flocs, , as follows (Jiang & Logan 1991): . In the case that the floc is composed of

*k*primary multi-sized particles, we could extend this relation to the following expression, as presented in some additional studies (Khelifa & Hill 2006): . Substituting this mathematical relation into Equation (5) could yield , where , and ; the term represents the effect of the size distribution of primary particles forming natural flocs, whereas the term denotes the effect of the fractal dimension and floc size. An explanation of the internal fractal architecture of the flocs and the fractal dimension of the floc is as follows. Fractal theory describes the geometry of many natural structures that show a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole. Many studies have shown that flocs could be considered as self-similar fractal entities. According to fractal theory, the structure of a fractal entity is considered to follow a power-law behaviour. The fractal dimension is the value of the power to indicate the structure of a fractal entity and the number of primary particles in a fractal entity as follows: , as mentioned above. For the flocs, is defined to be a measure of how the primary particles fill the floc, and it can range from 1 to 3, where larger values indicate a highly compact object ( = 3 implies the floc is a highly compact sphere).

Equation (6) maintains the same basic structure as that in Ferguson & Church (2004), which should be applicable over the viscous, transitional, and inertial settling ranges but is amended to incorporate the fact that the flocculated sediment has an internal fractal architecture, characterized by , and is composed of different-sized primary particles, characterized by . On the right-hand side of Equation (6), the first term of the denominator represents the effect of flow resistance over the surface of the falling sediment floc, whereas the second term represents the effect of flow separation around the falling floc. For the case of mono-sized particles and a solid Euclidian particle for the floc, and = 3; thus, Equation (6) can be reduced to Stokes' law (Equation (2)) when .

## COMPARISON WITH DATA

To examine the ability of the proposed model (Equation (6)) to estimate the settling velocity of cohesive sediment flocs, this study attempts to compare the model with four existing data sets of the settling velocity collected from the literature: two are from *in situ* field observations, and the others are from laboratory settling column measurements, introduced as follows.

Table 1 summarizes the information for these four data sets. The second and third columns show sediment conditions and experimental conditions, respectively. The number of data points in each data set is given in the last column.

References . | Sediment condition . | Experimental condition . | Data points . |
---|---|---|---|

Medema et al. (1998) | Oocysts and cysts, fresh water | Laboratory settling column | 10 |

Smith & Friedrichs (2011) | Suspended sediment flocs, San Francisco Bay, US west coast, USA | Field observation, settling column, camera and image analysis system | 19 |

Ahn (2012) | Suspended sediments with maximum entropy classification group 1, Southern California, USA | Field observation, settling tube, LISST-ST | 12 |

Zhao et al. (2018) | Sediment, Guan Qiao Lake, Wuhan, China | Laboratory experiment, sediment settling velocity measurement system | 64 |

References . | Sediment condition . | Experimental condition . | Data points . |
---|---|---|---|

Medema et al. (1998) | Oocysts and cysts, fresh water | Laboratory settling column | 10 |

Smith & Friedrichs (2011) | Suspended sediment flocs, San Francisco Bay, US west coast, USA | Field observation, settling column, camera and image analysis system | 19 |

Ahn (2012) | Suspended sediments with maximum entropy classification group 1, Southern California, USA | Field observation, settling tube, LISST-ST | 12 |

Zhao et al. (2018) | Sediment, Guan Qiao Lake, Wuhan, China | Laboratory experiment, sediment settling velocity measurement system | 64 |

LISST: laser in situ scattering and transmissometry.

Figure 1(a)–1(d) show the comparison of these collected data sets with the proposed model (Equation (6)), using a constant fractal dimension as suggested by Strom & Keyvani (2011). In this figure, the dashed line denotes Stokes' law, and the red solid line represents the fitted model. The fitting parameters of the proposed model for each case are summarized in the fourth through seventh columns of Table 2. Here, simple values of = 20 and = 1.258 are adopted for flocs, as suggested by Camenen (2007), and = 0.7 is adopted (Khelifa & Hill 2006). The fitting correlation coefficients, *R*^{2}, of the proposed model for each case are presented in the eighth column of Table 2. It can be found that all of the examined real cases agree with the proposed model with a high correlation coefficient. It can be simply concluded that the proposed model is valid for the collected settling-velocity data sets. However, it can be seen that either the proposed model or Stokes' law does not agree well with the data when the floc size is small, especially in Figure 1(a)–1(c), possibly due to data scattering. The data scattering in both laboratory and *in situ* settling-velocity data could originate from two sources. The first source is that sediment flocs were formed in different water turbulent flow and salinity conditions, and cohesive sediment is a mixture of water, fine-grained sediments (such as silt, clay), and inorganic and organic matter of diverse nature. They greatly affect the structure of the formed flocs, as well as settling velocity. Furthermore, even for the same floc size, the flocs exhibit many different structures and different fractal dimensions due to the variety of flocculation mechanisms, floc breakup, floc restructuring and different unit masses of primary particles (Vahedi & Gorczyca 2011, 2012). Another source for data scattering is possible experimental error for measuring the settling velocity of the sediment flocs. It could be much more difficult to measure the floc size and the floc settling velocity due to the fragility of the flocs and their slow settling velocities in the water column (Kumar *et al.* 2010; Vahedi & Gorczyca 2012; Priya *et al.* 2015). It could also be difficult to control the water flow condition in the laboratory to be zero, and for *in situ* observation, a completely stagnant water environment for individual sedimentation of the floc is rare (Kumar *et al.* 2010). The camera video system and attached imaging technique used to measure the floc size properties and the floc settling velocity could have its inherent difficulties (Strom & Keyvani 2011; Vahedi & Gorczyca 2011, 2012).

Reference . | Primary particle size d_{p} (micron)
. | ρ_{s}//ρ_{w} (kg/m³)
. | c_{1}
. | c_{2}
. | Φ
. | D_{f}
. | R²
. | Experimental condition . |
---|---|---|---|---|---|---|---|---|

Medema et al. (1998) | No | 1,008//1,005 | 20 | 1.258 | 0.7 | 3 | 0.98 | Laboratory settling column |

Smith & Friedrichs (2011) | [5] | 2,650//1,000 | 20 | 1.258 | 0.7 | 2.2 | 0.99 | Field observation |

Ahn (2012) | 1.7 | 2,650//1,000 | 20 | 1.258 | 0.7 | 2.3 | 0.97 | Field observation |

Zhao et al. (2018) | 6.11 | 2,455//1,000 | 20 | 1.258 | 0.7 | 2.75 | 0.93 | Laboratory settling column |

Reference . | Primary particle size d_{p} (micron)
. | ρ_{s}//ρ_{w} (kg/m³)
. | c_{1}
. | c_{2}
. | Φ
. | D_{f}
. | R²
. | Experimental condition . |
---|---|---|---|---|---|---|---|---|

Medema et al. (1998) | No | 1,008//1,005 | 20 | 1.258 | 0.7 | 3 | 0.98 | Laboratory settling column |

Smith & Friedrichs (2011) | [5] | 2,650//1,000 | 20 | 1.258 | 0.7 | 2.2 | 0.99 | Field observation |

Ahn (2012) | 1.7 | 2,650//1,000 | 20 | 1.258 | 0.7 | 2.3 | 0.97 | Field observation |

Zhao et al. (2018) | 6.11 | 2,455//1,000 | 20 | 1.258 | 0.7 | 2.75 | 0.93 | Laboratory settling column |

The symbol ‘[ ]’ means that the value was predicted from the original reference.

## CONCLUDING REMARKS

A formula for calculating the settling velocity of cohesive sediment flocs is proposed in this study. This formula utilizes the basic structure of the formulation of Ferguson & Church (2004) proposed for natural sands but amends the previous model to incorporate the fact that the flocculated sediment has an internal fractal architecture and is composed of different-sized primary particles. The proposed formula is applicable for the settling of sediment flocs across the viscous, transitional and inertial settling regimes, and it overcomes the shortcoming of the traditional approaches in which an iterative numerical technique is needed to calculate the settling velocity.

A comparison of the proposed model with four collected groups of settling velocity data is performed in this study, and a good agreement is found.

## ACKNOWLEDGEMENTS

This work is jointly supported by the National Natural Science Foundation of China (51509004).

## REFERENCES

*Cryptosporidium*oocysts and

*Giardia*cysts in water

IX, reprinted inMathematical and Physical Papers, Vol. 3