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Comparison between Shannon and Tsallis entropies for prediction of shear stress distribution in open channels

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Abstract

The concept of Tsallis entropy was applied to model the probability distribution functions for the shear stress magnitudes in circular channels (with filling ratios of 0.506, 0.666, 0.826), circular with flat bed (filling ratios of 0.333, 0.666), rectangular channel (1.34, 2, 3.94, 7.37 aspect ratios) and compound channel (with relative depths of 0.324, 0.46). The equation for the shear stress distribution was obtained according to the entropy maximization principle, and is able to estimate the shear stress distribution as much on the walls as the channel bed. The approach is also compared with the predictions obtained based on the Shannon entropy concept. By comparing the two prediction models, this study highlights the application of Tsallis entropy to estimate the shear stress distribution of open channels. Although the results of the two models are similar in the circular cross-section, the differences between them are more significant in circular with flat bed and rectangular channels. For a wide range of filling ratio values, experimental data are used to illustrate the accuracy and reliability of the proposed model.

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Acknowledgments

The authors would like to express their appreciation to the anonymous reviewers for their helpful comments and to Ellen Vuosalo Tavakoli for the painstaking editing of the English text.

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Correspondence to Hossein Bonakdari.

Appendices

Appendix 1

This appendix presents the way Eq. (15) is derived. The first constraint applied concerns the general probability of the variable τ, i.e.,

$$ \int_{0}^{{\tau_{\rm{max }} }} {f(\tau )} d\tau = 1, $$
(21)

by substitution of the probability density function [Eq. (12)] into Eq. (21), one obtains:

$$ \int_{0}^{{\tau_{\rm{max }} }} {\left( {\frac{q - 1}{q}\left[ {\lambda^{'} + \lambda_{2} \tau } \right]} \right)^{{\frac{1}{q - 1}}} d\tau = 1} , $$
(22)

integrating between 0 and τ max gives:

$$ \left. {\left( \frac{1}{k} \right)^{k} \frac{1}{{\lambda_{2} }}\left[ {\lambda^{'} + \lambda_{2} \tau } \right]^{k} } \right]_{0}^{{\tau_{\rm{max }} }} = 1, $$
(23)

the definite integral over that interval 0 and τ max is given by:

$$ \left( \frac{1}{k} \right)^{k} \frac{1}{{\lambda_{2} }}\left[ {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right]^{k} - \left( \frac{1}{k} \right)^{k} \frac{1}{{\lambda_{2} }}[\lambda^{'} ]^{k} = 1, $$
(24)

thus:

$$ \left[ {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right]^{k} - [\lambda^{'} ]^{k} = \lambda_{2} k^{k} . $$
(25)

Appendix 2

This appendix presents the way Eq. (16) is derived. The second constraint concerns the continuity of the variable τ, i.e.:

$$ \int_{0}^{{\tau_{\rm{max }} }} {\tau \left( {\frac{q - 1}{q}\left[ {\lambda^{'} + \lambda_{2} \tau } \right]} \right)}^{{\frac{1}{q - 1}}} d\tau = \tau_{mean} , $$
(26)

this can be written in the form:

$$ \left( {\frac{q - 1}{q}} \right)^{{\frac{1}{q - 1}}} \int_{0}^{{\tau_{\rm{max }} }} {\tau \left( {\left[ {\lambda^{'} + \lambda_{2} \tau } \right]} \right)}^{{\frac{1}{q - 1}}} d\tau = \tau_{mean} , $$
(27)

by applying integration by parts one obtains:

$$ \left( {\frac{q - 1}{q}} \right)^{{\frac{1}{q - 1}}} \left[ {\tau_{\rm{max }} \left( {\frac{q - 1}{q}} \right)\frac{1}{{\lambda_{2} }}\left( {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right)^{{\frac{q}{q - 1}}} - \underbrace {{\int_{0}^{{\tau_{\rm{max }} }} {\frac{q - 1}{q}\frac{1}{{\lambda_{2} }}\left( {\left[ {\lambda^{'} + \lambda_{2} \tau } \right]} \right)}^{{\frac{q}{q - 1}}} d\tau }}_{A}} \right] = \tau_{mean} , $$
(28)

to solve Eq. (28), part A can be expressed as:

$$ A = \int_{0}^{{\tau_{\rm{max }} }} {\frac{q - 1}{q}\frac{1}{{\lambda_{2} }}\left( {\left[ {\lambda^{'} + \lambda_{2} \tau } \right]} \right)}^{{\frac{q}{q - 1}}} d\tau , $$
(29)

after integrating between 0 and τ max we will have:

$$ A = \frac{q - 1}{q}\frac{1}{{\lambda_{2}^{2} }}\frac{q - 1}{2q - 1}\left. {\left( {\lambda^{'} + \lambda_{2} \tau } \right)^{{\frac{2q - 1}{q - 1}}} } \right|_{0}^{{\tau_{\rm{max }} }} , $$
(30)

the definite integral over that interval 0 and τ max is given by:

$$ A = \frac{q - 1}{q}\frac{1}{{\lambda_{2}^{2} }}\frac{q - 1}{2q - 1}\left( {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right)^{{\frac{2q - 1}{q - 1}}} - \frac{q - 1}{q}\frac{1}{{\lambda_{2}^{2} }}\frac{q - 1}{2q - 1}(\lambda^{'} )^{{\frac{2q - 1}{q - 1}}} , $$
(31)

by inserting Eq. (31) into Eq. (28) one obtains:

$$ \begin{gathered} \left( {\frac{q - 1}{q}} \right)^{{\frac{1}{q - 1}}} \left[ {\tau_{\rm{max }} \left( {\frac{q - 1}{q}} \right)\frac{1}{{\lambda_{2} }}\left( {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right)^{{\frac{q}{q - 1}}} - \frac{q - 1}{q}\frac{1}{{\lambda_{2}^{2} }}\frac{q - 1}{2q - 1}\left( {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right)^{{\frac{2q - 1}{q - 1}}} } \right. \hfill \\ \left. { + \frac{q - 1}{q}\frac{1}{{\lambda_{2}^{2} }}\frac{q - 1}{2q - 1}(\lambda^{'} )^{{\frac{2q - 1}{q - 1}}} } \right] = \tau_{mean} , \hfill \\ \end{gathered} $$
(32)

this can be written in the form:

$$ \left( \frac{1}{k} \right)^{k} \frac{{\tau_{\rm{max }} }}{{\lambda_{2} }}\left[ {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right]^{k} - \left( \frac{1}{k} \right)^{k} \frac{1}{{\lambda_{2}^{2} }}\frac{1}{k + 1}\left[ {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right]^{k + 1} + \left( \frac{1}{k} \right)^{k} \frac{1}{{\lambda_{2}^{2} }}\frac{1}{k + 1}[\lambda^{'} ]^{k + 1} = \tau_{mean} , $$
(33)

thus:

$$ \frac{{\tau_{\rm{max }} }}{{\lambda_{2} }}\left[ {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right]^{k} - \frac{1}{{\lambda_{2}^{2} }}\frac{1}{k + 1}\left[ {\lambda^{'} + \lambda_{2} \tau_{\rm{max }} } \right]^{k + 1} + \frac{1}{{\lambda_{2}^{2} }}\frac{1}{k + 1}[\lambda^{'} ]^{k + 1} = k^{k} \tau_{mean} . $$
(34)

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Bonakdari, H., Sheikh, Z. & Tooshmalani, M. Comparison between Shannon and Tsallis entropies for prediction of shear stress distribution in open channels. Stoch Environ Res Risk Assess 29, 1–11 (2015). https://doi.org/10.1007/s00477-014-0959-3

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