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Fractional entropy-based modeling of suspended concentration distribution of type I and type II and sediment discharge in pipe and open-channel turbulent flows

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Abstract

Fractional entropy based on probability concept is applied to study the type I and type II suspended particle concentration distribution along the vertical direction in pipe and open-channel turbulent flows. Taking dimensionless concentration as a random variable and maximizing the fractional entropy, proposed models for suspension distribution of both the types are derived by considering a realistic nonzero particle concentration at the free surface. At first, the model of type I is developed by considering the whole flow depth as one region; and then, for the type II profile, the flow zone is divided into two regions. Finally, using the concept of the asymptotic matching method, the model for the whole flow depth of the type II profile is obtained. To get the model, at first the density function and cumulative distribution functions are derived in the concentration domain, and then the cumulative distribution function of concentration is described in the spatial domain. This study introduces for the first time a type II model using the fractional entropy-based probability density function for suspension distribution. Apart from it, a new cumulative distribution function for suspension distribution based on the Mittag–Leffler function is suggested for further generalization and application of the study. The derived sediment concentration distribution models for both type I and II are validated with experimental and field data, and good agreement is obtained for both the models. Further, the proposed models are also compared with existing Renyi entropy-based concentration model of type I profile and with a deterministic model of type II. Comparison results and error analysis show that for both types of concentration distribution, fractional entropy-based models have better prediction accuracy than the existing entropy-based model and the deterministic concentration model. Finally, the model is applied to compute sediment discharge. The computed values using the model and observed values from experimental data of sediment discharge show a good agreement, and the present study improves the result of previous predictions using Renyi entropy.

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Author notes

  1. Nizamuddin Ahamed

    Present address: Triveni Devi Bhalotia College, Raniganj, West Bengal, 713347, India

Authors and Affiliations

  1. Department of Mathematics, NIT Jamshedpur, Jamshedpur, Jharkhand, 831014, India

    Nizamuddin Ahamed & Snehasis Kundu

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  1. Nizamuddin Ahamed
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  2. Snehasis Kundu
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Contributions

Dr. Kundu originally formulated the idea of the work. Mr. Ahamed initiated and contributed to the preliminary concept of the study under the supervision of Dr. Kundu. Afterward, Mr. Ahamed prepared the initial mathematical models and Dr. Kundu modified and corrected them. Mr. Ahamed completes the data analysis, modeling, programming and fitting parts and prepared the initial draft of the paper. Dr. Kundu edited, revised and completed the final draft. Finally, both the authors approved the final version.

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Correspondence to Snehasis Kundu.

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Ahamed, N., Kundu, S. Fractional entropy-based modeling of suspended concentration distribution of type I and type II and sediment discharge in pipe and open-channel turbulent flows. Z. Angew. Math. Phys. 74, 101 (2023). https://doi.org/10.1007/s00033-023-01988-w

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