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

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 845 ◽  
Author(s):  
Zhongfan Zhu
Keyword(s):  
Experimental Data ◽  
Explicit Expression ◽  
Shannon Entropy ◽  
Growth Pattern ◽  
Cohesive Sediment ◽  
Tsallis Entropy ◽  
Cumulative Distribution ◽  
Power Relationship ◽  
Floc Size ◽  
Flocculation Time

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.

scholarly journals Estimating the Bed-Load Layer Thickness in Open Channels by Tsallis Entropy

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 123 ◽  
Author(s):  
Zhongfan Zhu ◽  
Jingshan Yu
Keyword(s):  
Experimental Data ◽  
Shannon Entropy ◽  
Mass Density ◽  
Particle Diameter ◽  
Tsallis Entropy ◽  
Cumulative Distribution ◽  
Open Channels ◽  
Bed Load ◽  
Deterministic Models ◽  
Prediction Ability

In the research field of river dynamics, the thickness of bed-load is an important parameter in determining sediment discharge in open channels. Some studies have estimated the bed-load thickness from theoretical and/or experimental perspectives. This study attempts to propose the mathematical formula for the bed-load thickness by using the Tsallis entropy theory. Assuming the bed-load thickness is a random variable and using the method for the maximization of the entropy function, the present study derives an explicit expression for the thickness of the bed-load layer as a function with non-dimensional shear stress, by adopting a hypothesis regarding the cumulative distribution function of the bed-load thickness. This expression is verified against six experimental datasets and are also compared with existing deterministic models and the Shannon entropy-based expression. It has been found that there is good agreement between the derived expression and the experimental data, and the derived expression has a better fitting accuracy than some existing deterministic models. It has been also found that the derived Tsallis entropy-based expression has a comparable prediction ability for experimental data to the Shannon entropy-based expression. Finally, the impacts of the mass density of the particle and particle diameter on the bed-load thickness in open channels are also discussed based on this derived expression.

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

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 605
Author(s):  
Zhongfan Zhu ◽  
Pengfei Hei ◽  
Jie Dou ◽  
Dingzhi Peng
Keyword(s):  
Experimental Data ◽  
Cross Section ◽  
Shannon Entropy ◽  
Open Channel ◽  
Lateral Distribution ◽  
Tsallis Entropy ◽  
The Other ◽  
Deterministic Models ◽  
Entire Cross Section ◽  
General Index

The velocity profile of an open channel is an important research topic in the context of open channel hydraulics; in particular, the velocity-dip position has drawn the attention of hydraulic scientists. In this study, analytical expressions for the velocity-dip position over the entire cross section and at the centerline of a rectangular open channel are derived by adopting probability methods based on the Tsallis and general index entropy theories. Two kinds of derived entropy-based expressions have the same mathematical form as a function of the lateral distance from the sidewall of the channel or of the aspect ratio of the channel. Furthermore, for the velocity-dip position over the entire cross section of the rectangular open channel, the derived expressions are compared with each other, as well as with two existing deterministic models and the existing Shannon entropy-based expression, using fifteen experimental datasets from the literature. An error analysis shows that the model of Yang et al. and the Tsallis entropy-based expression predict the lateral distribution of the velocity-dip position better than the other proposed models. For the velocity-dip position at the centerline of the rectangular open channel, six existing conventional models, the derived Tsallis and general index entropy-based expressions, and the existing Shannon entropy-based models are tested against twenty-one experimental datasets from the literature. The results show that the model of Kundu and the Shannon entropy-based expression have superior prediction accuracy with respect to experimental data compared with other models. With the exception of these models, the Tsallis entropy-based expression has the highest correlation coefficient value and the lowest root mean square error value for experimental data among the other models. This study indicates that the Tsallis entropy could be a good addition to existing deterministic models for predicting the lateral distribution of the velocity-dip position of rectangular open channel flow. This work also shows the potential of entropy-based expressions, the Shannon entropy and the Tsallis entropy in particular, to predict the velocity-dip position at the centerline of both narrow and wide rectangular open channels.

Probability Bounds Analysis Applied to the Sandia Verification and Validation Challenge Problem

2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Aniruddha Choudhary ◽  
Ian T. Voyles ◽  
Christopher J. Roy ◽  
William L. Oberkampf ◽  
Mayuresh Patil
Keyword(s):  
Experimental Data ◽  
Physical Nature ◽  
Distribution Functions ◽  
Verification And Validation ◽  
Cumulative Distribution ◽  
Probability Bounds ◽  
Model Form ◽  
Interval Representation ◽  
Model Predictions ◽  
True System

Our approach to the Sandia Verification and Validation Challenge Problem is to use probability bounds analysis (PBA) based on probabilistic representation for aleatory uncertainties and interval representation for (most) epistemic uncertainties. The nondeterministic model predictions thus take the form of p-boxes, or bounding cumulative distribution functions (CDFs) that contain all possible families of CDFs that could exist within the uncertainty bounds. The scarcity of experimental data provides little support for treatment of all uncertain inputs as purely aleatory uncertainties and also precludes significant calibration of the models. We instead seek to estimate the model form uncertainty at conditions where the experimental data are available, then extrapolate this uncertainty to conditions where no data exist. The modified area validation metric (MAVM) is employed to estimate the model form uncertainty which is important because the model involves significant simplifications (both geometric and physical nature) of the true system. The results of verification and validation processes are treated as additional interval-based uncertainties applied to the nondeterministic model predictions based on which the failure prediction is made. Based on the method employed, we estimate the probability of failure to be as large as 0.0034, concluding that the tanks are unsafe.

Modelling the activated sludge flocculation process combining laser light diffraction particle sizing and population balance modelling (PBM)

2002 ◽  
Vol 45 (6) ◽  
pp. 41-49 ◽  
Author(s):  
I. Nopens ◽  
C.A. Biggs ◽  
B. De Clercq ◽  
R. Govoreanu ◽  
B.-M. Wilén ◽  
...  
Keyword(s):  
Experimental Data ◽  
Laser Light ◽  
Population Balance ◽  
Light Diffraction ◽  
Balance Model ◽  
Floc Size ◽  
Shear Rates ◽  
Breakage Rate ◽  
On Line ◽  
Floc Size Distributions

A technique based on laser light diffraction is shown to be successful in collecting on-line experimental data. Time series of floc size distributions (FSD) under different shear rates (G) and calcium additions were collected. The steady state mass mean diameter decreased with increasing shear rate G and increased when calcium additions exceeded 8 mg/l. A so-called population balance model (PBM) was used to describe the experimental data. This kind of model describes both aggregation and breakage through birth and death terms. A discretised PBM was used since analytical solutions of the integro-partial differential equations are non-existing. Despite the complexity of the model, only 2 parameters need to be estimated: the aggregation rate and the breakage rate. The model seems, however, to lack flexibility. Also, the description of the floc size distribution (FSD) in time is not accurate.

Solving the inverse problem for aggregation in activated sludge flocculation using a population balance framework

2007 ◽  
Vol 56 (6) ◽  
pp. 95-103 ◽  
Author(s):  
I. Nopens ◽  
N. Nere ◽  
P.A. Vanrolleghem ◽  
D. Ramkrishna
Keyword(s):  
Experimental Data ◽  
Inverse Problem ◽  
Particle Size ◽  
Activated Sludge ◽  
Predictive Power ◽  
Population Balance ◽  
Model Structure ◽  
Forward Simulation ◽  
Floc Size

Many systems contain populations of individuals. Often, they are regarded as a lumped phase, which might, for some applications, lead to inadequate model predictive power. An alternative framework, Population Balance Models, has been used here to describe such a system, activated sludge flocculation in which particle size is the property one wants to model. An important problem to solve in population balance modelling is to determine the model structure that adequately describes experimentally obtained data on for instance, the time evolution of the floc size distribution. In this contribution, an alternative method based on solving the inverse problem is used to recover the model structure from the data. In this respect, the presence of similarity in the data simplifies the problem significantly. Similarity was found and the inverse problem could be solved. A forward simulation then confirmed the quality of the model structure to describe the experimental data.

scholarly journals An Expression for Velocity Lag in Sediment-Laden Open-Channel Flows Based on Tsallis Entropy Together with the Principle of Maximum Entropy

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 522 ◽  
Author(s):  
Zhongfan Zhu ◽  
Jingshan Yu ◽  
Jie Dou ◽  
Dingzhi Peng
Keyword(s):  
Maximum Entropy ◽  
Shannon Entropy ◽  
Open Channel ◽  
Tsallis Entropy ◽  
Channel Flows ◽  
Test Cases ◽  
Deterministic Models ◽  
Open Channel Flows ◽  
Principle Of Maximum Entropy ◽  
Principle Of Maximum

In the context of river dynamics, some experimental results have shown that particle velocity is different from fluid velocity along the stream-wise direction for uniform sediment-laden open-channel flows; this velocity difference has been termed velocity lag in the literature. In this study, an analytical expression for estimating the velocity lag in open-channel flows was derived based on the Tsallis entropy theory together with the principle of maximum entropy. The derived expression represents the velocity lag as a function of a non-dimensional entropy parameter depending on the average and maximum values of velocity lag from experimental measurements. The derived expression was tested against twenty-two experimental datasets collected from the literature with three deterministic models and the developed Shannon entropy-based model. The Tsallis entropy-based model agreed better with the experimental datasets than the deterministic models for eighteen out of the twenty-two total real cases, and the prediction accuracy for the eighteen experimental datasets was comparable to that of the developed Shannon entropy-based model (the Tsallis entropy-based expression agreed slightly better than the Shannon entropy-based model for twelve out of eighteen test cases, whereas for the other six test cases, the Shannon entropy-based model had a slightly higher prediction accuracy). Finally, the effects of the friction velocity of the flow, the particle diameter, and the particles’ specific gravity on the velocity lag were analyzed based on the Tsallis entropy-based model. This study shows the potential of the Tsallis entropy theory together with the principle of maximum entropy to predict the stream-wise velocity lag between a particle and the surrounding fluid in sediment-laden open-channel flows.

scholarly journals Shannon, Rényi, Tsallis Entropies and Onicescu Information Energy for Low-Lying Singly Excited States of Helium

Atoms ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 70 ◽  
Author(s):  
Jen-Hao Ou ◽  
Yew Kam Ho
Keyword(s):  
Excited States ◽  
Shannon Entropy ◽  
Electronic Structures ◽  
Tsallis Entropy ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Electron Delocalization ◽  
Information Theoretic ◽  
Molecular Systems ◽  
Information Energy

Knowledge of the electronic structures of atomic and molecular systems deepens our understanding of the desired system. In particular, several information-theoretic quantities, such as Shannon entropy, have been applied to quantify the extent of electron delocalization for the ground state of various systems. To explore excited states, we calculated Shannon entropy and two of its one-parameter generalizations, Rényi entropy of order α and Tsallis entropy of order α , and Onicescu Information Energy of order α for four low-lying singly excited states (1s2s 1 S e , 1s2s 3 S e , 1s3s 1 S e , and 1s3s 3 S e states) of helium. This paper compares the behavior of these three quantities of order 0.5 to 9 for the ground and four excited states. We found that, generally, a higher excited state had a larger Rényi entropy, larger Tsallis entropy, and smaller Onicescu information energy. However, this trend was not definite and the singlet–triplet reversal occurred for Rényi entropy, Tsallis entropy and Onicescu information energy at a certain range of order α .

Multi-qExtension of Tsallis Entropy Based Fuzzyc-Means Clustering

2014 ◽  
Vol 18 (3) ◽  
pp. 289-296 ◽  
Author(s):  
Makoto Yasuda ◽  
◽  
Yasuyuki Orito
Keyword(s):  
Membership Function ◽  
Shannon Entropy ◽  
Clustering Algorithm ◽  
Numerical Data ◽  
Tsallis Entropy ◽  
Entropy Maximization ◽  
Cluster Distribution ◽  
The Membership Function

Tsallis entropy is aq-parameter extension of Shannon entropy. Based on the Tsallis entropy, we have introduced an entropy maximization method to fuzzyc-means clustering (FCM), and developed a new clustering algorithm using a single-qvalue. In this article, we propose a multi-qextension of the conventional single-qmethod. In this method, theqs are assigned individually to each cluster. Eachqvalue is determined so that the membership function fits the corresponding cluster distribution. This is done to improve the accuracy of clustering over that of the conventional single-qmethod. Experiments are performed on randomly generated numerical data and Fisher’s iris dataset, and it is confirmed that the proposed method improves the accuracy of clustering and is superior to the conventional single-qmethod. If the parameters introduced in the proposed method can be optimized, it is expected that the clusters in data distributions that are composed of clusters of various sizes can be determined more accurately.

Tsallis Entropy-Based Fuzzy Latent Semantics Analysis

2020 ◽  
Vol 24 (1) ◽  
pp. 58-64
Author(s):  
Yuchi Kanzawa ◽  
Keyword(s):  
Free Energy ◽  
Shannon Entropy ◽  
Latent Semantic Analysis ◽  
Optimization Problem ◽  
Semantic Analysis ◽  
Tsallis Entropy ◽  
Probabilistic Latent Semantic Analysis ◽  
Numerical Examples

In this study, we present a fuzzy counterpart to the probabilistic latent semantic analysis (PLSA) approach. It is derived by solving the optimization problem of Tsallis entropy-penalizing free energy of a pseudo PLSA model by using a modified i.i.d. assumption. This derivation is similar to that of the conventional fuzzy counterpart of the PLSA, which involves solving the optimization problem of Shannon entropy-penalizing free energy. Furthermore, the proposed method is validated using numerical examples.

Confidence Interval Simulation for Systems of Random Variables

2005 ◽  
Vol 129 (3) ◽  
pp. 836-842 ◽  
Author(s):  
Thomas A. Cruse ◽  
Jeffrey M. Brown
Keyword(s):  
Experimental Data ◽  
Confidence Interval ◽  
Bayesian Network ◽  
Upper Bound ◽  
Random Variables ◽  
Network Models ◽  
Probability Of Failure ◽  
Cumulative Distribution ◽  
Probabilistic Design ◽  
Design Assessment

Bayesian network models are seen as important tools in probabilistic design assessment for complex systems. Such network models for system reliability analysis provide a single probability of failure value whether the experimental data used to model the random variables in the problem are perfectly known or derive from limited experimental data. The values of the probability of failure for each of those two cases are not the same, of course, but the point is that there is no way to derive a Bayesian type of confidence interval from such reliability network models. Bayesian confidence (or belief) intervals for a probability of failure are needed for complex system problems in order to extract information on which random variables are dominant, not just for the expected probability of failure but also for some upper bound, such as for a 95% confidence upper bound. We believe that such confidence bounds on the probability of failure will be needed for certifying turbine engine components and systems based on probabilistic design methods. This paper reports on a proposed use of a two-step Bayesian network modeling strategy that provides a full cumulative distribution function for the probability of failure, conditioned by the experimental evidence for the selected random variables. The example is based on a hypothetical high-cycle fatigue design problem for a transport aircraft engine application.

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