# Analysis of Existing Equations for Calculating the Settling Velocity

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## Abstract

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## 1. Introduction

_{d}) on a sediment particle and the buoyancy is equal to the downward force of gravity (F

_{G}), and the particle stops accelerating and continues falling at a constant velocity that is known as terminal velocity. The particle acceleration ceases by the combined action of fluid drag and submerged weight of the particle. In other words, the fall velocity of a particle is measured by equating the gravity and drag forces. To know the insight of sediment transport processes in combined sewers and rivers, fall velocity is an essential factor in accurate prediction. The sediment fall velocity depends on the fluid’s density and viscosity and the particle’s density, size, and shape [1,2,3,4,5,6,7,8]. Researchers have established many empirical and semi-empirical equations to answer problems such as the fall velocity of particle and sediment transport processes, started by Stokes in 1851. Initially, the settling velocity of sediment equations was developed by assuming the particles to be spheres [1,9,10,11,12]. Practically, sediment particles’ shapes depart from the sphere, which causes a reduction in settling velocity compared with the sphere [13,14,15,16]. As a result of these practical implications, many equations have been developed to compute the settling velocity of natural sediments with an assumption of sphere with the nominal diameter (i.e., the diameter of a sphere with the same volume as a particle of natural sediment) [2,3,4,5,6,7,8]. Previous studies on concentrated suspensions of cohesive sediment flocs [5,16] and non-cohesive sediment particles [5,8] revealed that the three unique mechanisms associated to the existence of the cohesive floc/non-cohesive particle concentration cause the impeded settling characteristics, viz.: (i) generating a return flow and forming a wake; (ii) an increase in the viscosity of the mixture; and (iii) influence of buoyancy. In this regard, the settling properties of monodisperse non-cohesive or flocculated suspensions have been widely studied and the approach proposed by Rijn [5], or some related variation, are used in the majority of impeded settling models in cohesive streams. When it comes to cohesive sediments, precise estimation of the settling fall velocity characteristics of cohesive flocs is a key factor and it is important to determine their interactions, transport, and fate within the cohesive stream beds. Characterises of cohesive flocs are critical for maintaining and managing navigation channels, ports, and harbours, as well as determining the consequences of increased turbidity on water quality and aquatic habitats in cohesive bed streams. The size and density of the flocs in cohesive sediments are both affected by the flow field and each other. There are several processes that can influence the settling velocity of individual particles in a concentrated suspension. In this paper, we categorized the equations based on the particle size ranges with and without particle shape factors. Most of the previous studies on settling velocity of natural sediment particles that have been carried out explicitly did not take the particle shape factor and roundness into account, which was assumed as 0.7 and 3.5, respectively [7,9,10,11,12,13,14]. More recently, Jimenez and Madsen [8], Wu and Wang [15], and Camenen [16] developed some empirical equations, which include the particle size, shape, and roundness directly. These equations may perform differently because of the different methods and data sets employed in their calibration.

## 2. Existing Equations for Settling Velocity of Sediments

_{f}; and three with shape factor, S

_{f}) previously developed empirical equations [2,3,5,6,7,9,10,12,15,16] are selected for verifying their accuracy and these equations are listed below. A shape factor is defined as an irregularity in the shape of a particle from the sphere. Here, csf (Corey shape factor) is used to measure irregularity, which is formulated as csf = c/${\left(ab\right)}^{0.5}$, where a, b, and c are the lengths of the longest axis, the intermediate axis, and the shortest axis, respectively.

_{gr}is the dimensionless particle size calculated as ${D}_{gr}={\left[\frac{g\left(s-1\right)}{{\vartheta}^{2}}\right]}^{\frac{1}{3}}d$.

_{gr}is a non-dimensional particle parameter calculated as:

_{f}is the safe factor.

_{f}), and roundness (P):

## 3. Data Description

## 4. Results and Discussions

#### 4.1. Performances of Existing Equations

_{f}; three with S

_{f}) equations proposed by earlier investigators [2,3,5,6,7,8,9,10,12,15,16] were checked with 226 field and laboratory experimental data sets [4,7,19,20,24]. The accuracy and reliability of these equations were analysed both graphically and statistically. In Figure 1a–k and Figure 2a–c, values on X-axis and Y-axis represent the observed and predicted data sets of the settling velocity of particles, respectively. The data set has been divided into two groups: one is without the shape factor (assumed as a natural particle, S

_{f}= 0.7); the second is with the shape factor (shape factor considered explicitly). The first group consists of data sets of Engelund and Hansen [19], Hallermeier [4], and Cheng [7]. The second group consists of the data sets of Briggs [24] and Raudkivi [20].

_{f}assumed as 0.7), and a slight under prediction is observed through scatter plot 1(g). Zhang’s [9] expression was over predicted for lower settling velocity between 2–9 cm/s and further observed good agreement with measured settling velocity data, shown in Figure 1d. The agreement between observed and predicted data of Zanke [3], Soulsby [12], and Julien [10] expressions are shown in Figure 1b,f,h, respectively. Due to similarities between equations, these three expressions show analogous trends in their scatter plots, and Julien’s [10] equation showed lower accuracy graphically and statistically. Rijn [5] gave over prediction with moderate performance as shown in Figure 1c. Rijn [5] divided datasets into three sediment diameter ranges (d < 0.01 cm, d > 0.1 cm, and d = 0.01–0.1 cm) and most of the collected datasets belonged to d > 0.01 cm, which may be the reason for over prediction with moderate performance. Figure 1j and Figure 2b show the agreement between observed and predicted settling velocity by the equation of Wu and Wang [15], with and without the shape factor, respectively. Wu and Wang [15] included the importance of the shape factor and excluded the particle roundness factor. This was done so that it could be used only for data that excludes the roundness factor; if not, an error in the particle’s settling velocity calculations could occur.

#### 4.2. Statistical Performance Analysis of Equations

^{2}) explains the fraction of total variance in observed data sets, and it ranges from 0 to 1:

^{2}, NSE, KGE, PBIAS, and MAE for different equations are listed in Table 2. The R

^{2}, NSE, and KGE of Equation (14), i.e., proposed by Wu and Wang [15], are highest, and PBIAS and MAE are lowest among all other equations. Statistical performances indicate that the expression of Wu and Wang [15] predicts the better settling velocity of sediment particles than all other equations. This was observed graphically and statistically. The expressions proposed by Camenen [16] and Jimenez and Madsen [8] without the shape factor provided the second highest agreements between observed and predicted data, as can be seen in Table 2 and Figure 1.

## 5. Conclusions

_{f}; three with S

_{f}) were used for verifying the accuracy and performance of particle settling velocity. The authors graphically observed that the relationship proposed by Wu and Wang [15], with and without the shape factor, provides superior agreements, as is shown in Figure 1j and Figure 2b. Statistically, the relationships proposed by Wu and Wang [15], with and without the shape factor, and Camenen [16], Jimenez and Madsen [8], and Cheng [7], without the shape factor, provide approximately similar values among all. However, Wu and Wang [15] show slightly higher values than the other equations, as shown in Table 2. Graphically and statistically, it was observed that the expressions of Zanke [3], Soulsby [12], and Julien [10] show the same pattern with medium performance. The low performance of Jimenez and Madsen’s [8] and Camenen’s [16] equations may be strictly due to particular datasets. There may be a scope for checking the accuracy of the above equations with broad datasets, including particle shape factor and roundness factor.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Parameters | No. of Data | d (mm) | S (-) | CSF (-) | $\mathit{\vartheta}$$\left(\frac{\mathit{c}{\mathit{m}}^{2}}{\mathit{s}}\right)$ | ${\mathit{\omega}}_{\mathit{s}}$$\left(\frac{\mathit{c}\mathit{m}}{\mathit{s}}\right)$ | |
---|---|---|---|---|---|---|---|

Authors | |||||||

Briggs [24] | 110 | 0.09–0.5 | 3.97–5.07 | 0.049–0.881 | 0.01 | 0.9–9.5 | |

Engelund and Hansen [19] | 21 | 0.01–2.0 | 2.65 | 0.7 | 0.01–0.0131 | 0.5–17.0 | |

Hallermeier [4] | 21 | 0.09–1.3 | 2.65 | 0.7 | 0.0084–0.0114 | 0.54–14 | |

Raudkivi [20] | 36 | 0.2–2.0 | 2.65 | 0.5–0.9 | 0.009–0.0131 | 1.68–24.0 | |

Cheng [7] | 38 | 0.06–4.5 | 2.65 | 0.7 | 0.0114–0.0141 | 0.235–28.1 |

Researchers | NSE | KGE | PBIAS | MAE | R^{2} |
---|---|---|---|---|---|

Wu and Wang [15] with S_{f} | 0.9937 | 0.976 | −1.06 | 0.2691 | 0.9942 |

Wu and Wang [15] without S_{f} | 0.9931 | 0.9616 | −1.463 | 0.4107 | 0.9948 |

Camenen [16] without S_{f} | 0.9936 | 0.9733 | −2.3895 | 0.4376 | 0.9944 |

Jimenez and Madsen [8] without S_{f} | 0.9929 | 0.9512 | 2.6308 | 0.4164 | 0.9952 |

Cheng [7] without S_{f} | 0.9924 | 0.96 | 3.8275 | 0.4342 | 0.9941 |

Zhang [9] without S_{f} | 0.9925 | 0.964 | −1.7206 | 0.4835 | 0.9937 |

Zanke [3] without S_{f} | 0.9847 | 0.9143 | −2.4174 | 0.7276 | 0.9914 |

Soulsby [12] without S_{f} | 0.9861 | 0.9241 | −4.5574 | 0.7057 | 0.9915 |

Rijn [5] without S_{f} | 0.9824 | 0.8998 | −9.1874 | 0.7705 | 0.9933 |

Camenen [16] with S_{f} | 0.9581 | 0.8958 | 3.4168 | 0.5018 | 0.9661 |

Julien [10] without S_{f} | 0.9714 | 0.8582 | 1.0309 | 0.9229 | 0.9901 |

Jimenez and Madsen [8] with S_{f} | 0.9234 | 0.8648 | −6.1132 | 0.8406 | 0.9354 |

Ruby [2] without S_{f} | 0.9069 | 0.7104 | 12.4768 | 1.3657 | 0.989 |

Zhu and Cheng [6] without S_{f} | 0.8707 | 0.7057 | −28.4017 | 2.1954 | 0.9599 |

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**MDPI and ACS Style**

Shankar, M.S.; Pandey, M.; Shukla, A.K.
Analysis of Existing Equations for Calculating the Settling Velocity. *Water* **2021**, *13*, 1987.
https://doi.org/10.3390/w13141987

**AMA Style**

Shankar MS, Pandey M, Shukla AK.
Analysis of Existing Equations for Calculating the Settling Velocity. *Water*. 2021; 13(14):1987.
https://doi.org/10.3390/w13141987

**Chicago/Turabian Style**

Shankar, M. Shiva, Manish Pandey, and Anoop Kumar Shukla.
2021. "Analysis of Existing Equations for Calculating the Settling Velocity" *Water* 13, no. 14: 1987.
https://doi.org/10.3390/w13141987