A Simple Explicit Expression for the Flocculation Dynamics Modeling of Cohesive Sediment Based on Entropy Considerations

Abstract

The flocculation of cohesive sediment plays an important role in affecting morphological changes to coastal areas, to dredging operations in navigational canals, to sediment siltation in reservoirs and lakes, and to the variation of water quality in estuarine waters. Many studies have been conducted recently to formulate a turbulence-induced flocculation model (described by a characteristic floc size with respect to flocculation time) of cohesive sediment by virtue of theoretical analysis, numerical modeling, and/or experimental observation. However, a probability study to formulate the flocculation model is still lacking in the literature. The present study, therefore, aims to derive an explicit expression for the flocculation of cohesive sediment in a turbulent fluid environment based on two common entropy theories: Shannon entropy and Tsallis entropy. This study derives an explicit expression for the characteristic floc size, assumed to be a random variable, as a function of flocculation time by maximizing the entropy function subject to the constraint equation using a hypothesis regarding the cumulative distribution function of floc size. It was found that both the Shannon entropy and the Tsallis entropy theories lead to the same expression. Furthermore, the derived expression was tested with experimental data from the literature and the results were compared with those of existing deterministic models, showing that it has good agreement with the experimental data and that it has a better prediction accuracy for the logarithmic growth pattern of data in comparison to the other models, whereas, for the sigmoid growth pattern of experimental data, the model of Keyvani and Strom or Son and Hsu model could be the better choice for floc size prediction. Finally, the maximum capacity of floc size growth, a key parameter incorporated into this expression, was found to exhibit an empirical power relationship with the flow shear rate.

An Entropic Approach to Estimating the Instability Criterion of People in Floodwaters

“…Figure 2 a–g shows a comparison of the derived entropy-based model for each experimental dataset, and the calculated error parameters: , RBIAS, and RMSE , are presented in Table 2 . The proposed entropy-based model is consistent with the experimental results of Foster and Cox [ 14 ], Takahashi et al [ 15 ], Karvonen et al [ 12 ], Yee [ 30 ], and especially Xia et al [ 6 ]. A deviation for some experimental datasets, including that of Abt et al [ 11 ] ( Figure 2 b) and Gomariz et al [ 29 ] ( Figure 2 g), is also noticeable, which is discussed as follows.…”

“…Error estimation is carried out to quantitatively evaluate the accuracy of the derived entropy-based model against collected experimental datasets by calculating the correlation coefficient between the estimated and the observed datasets, the relative bias (RBIAS) between the estimated and the observed datasets, which is defined as RBIAS = , and the root-mean-square error (RMSE), which is defined as RMSE = , where and are the estimated and observed data points, respectively, and is the total number of observed data points, as also adopted by Zhu [ 30 ]. A larger value and smaller RBIAS and RMSE values indicate a model with a better goodness-of-fit.…”

Recent Achievements in Polymer Bio-Based Flocculants for Water Treatment

“…A detailed description of flocculation mechanisms, supported by theoretical considerations, has recently been published in several articles [ 28 , 33 , 34 , 35 , 36 ]. Modeling of the process allowed determining the time of adsorption (τ ads ) and aggregation (τ agg ), which is different in Brownian diffusion and shear-induced flocculation.…”

Evaluating Different Methods for Determining the Velocity-Dip Position over the Entire Cross Section and at the Centerline of a Rectangular Open Channel

“…These works motivate us to explore the possibility of using two more general entropy theories, Tsallis entropy and general index entropy, to predict the velocity-dip position over the entire cross section and at the centerline of open channel flow. In fact, in recent years, Tsallis entropy has been widely applied to tackling some hydraulic engineering problems, such as the velocity profile prediction of open channel flow (e.g., [ 24 , 25 ]), the prediction of shear stress distribution in open channels (e.g., [ 26 , 27 ]), the estimation of suspended sediment concentration (e.g., [ 25 , 28 ]), the calculation of water distribution networks (e.g., [ 29 ]) and the flocculation dynamics modeling of cohesive sediment [ 30 ]. General index entropy has also been adopted to estimate the two-dimensional velocity distribution profile in open channel flow (e.g., [ 31 ]).…”