Zhu Zhongfan
College of Water Sciences, Beijing Normal University, Xinjiekouwai Street 19, Beijing 100875, China.
Entropy (Basel). 2018 Nov 4;20(11):845. doi: 10.3390/e20110845.
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.
粘性泥沙的絮凝作用在影响海岸地区的地貌变化、航道疏浚作业、水库和湖泊中的泥沙淤积以及河口水质变化等方面起着重要作用。最近，通过理论分析、数值模拟和/或实验观测，已经开展了许多研究来建立粘性泥沙的紊流诱导絮凝模型（用絮凝时间的特征絮凝体尺寸来描述）。然而，文献中仍缺乏用于建立絮凝模型的概率研究。因此，本研究旨在基于两种常见的熵理论：香农熵和Tsallis熵，推导出粘性泥沙在紊流流体环境中絮凝的显式表达式。本研究通过假设絮凝体尺寸的累积分布函数，在约束方程的条件下最大化熵函数，推导出作为絮凝时间函数的特征絮凝体尺寸的显式表达式，该尺寸被假定为一个随机变量。结果发现，香农熵理论和Tsallis熵理论都得出了相同的表达式。此外，用文献中的实验数据对推导的表达式进行了检验，并将结果与现有确定性模型的结果进行了比较，结果表明该表达式与实验数据具有良好的一致性，并且与其他模型相比，对于数据的对数增长模式具有更好的预测精度，而对于实验数据的S形增长模式，Keyvani和Strom模型或Son和Hsu模型可能是预测絮凝体尺寸的更好选择。最后，发现该表达式中包含的一个关键参数——絮凝体尺寸增长的最大容量，与水流剪切速率呈现出经验幂律关系。
