On dispersion of solute in steady flow through a channel with absorption boundary: an application to sewage dispersion
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O R I G I NA L A RT I C L E
Kajal Kumar Mondal · Subham Dhar Mazumder
· Bijoy Singha
On dispersion of solute in steady flow through a channel with absorption boundary: an application to sewage dispersion
Received: 21 December 2019 / Accepted: 18 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The paper describes the longitudinal dispersion of passive tracer materials released into an incompressible viscous fluid, flowing through a channel with walls having first-order reaction. Its model is based on a steady advection–diffusion equation with Dirichlet’s and mixed boundary conditions, and whose solution represents the concentration of the tracers in different downstream stations. For imposing the boundary conditions properly, artanh transformation is used to convert the infinite solution space to a finite one. A finite difference implicit scheme is used to solve the advection–diffusion equation in the computational region, and an inverse transformation is employed for the solution in the physical region. It is shown how the mixing of the tracer molecule influenced by the shear flow and due to the action of the absorption parameter at both the walls of the channel. For convection-dominated flow, uniform mesh is failed to capture the layer phenomena along the different downstream stations and a piecewise uniform mesh; namely, Shishkin mesh is used. The results are compared with existing experimental and numerical data available in the literature, and we have achieved an excellent agreement with them. The study plays a significant role to understand the basic mechanisms of sewage dispersion. Keywords Dispersion · Finite difference scheme · Convection · Layer adapted mesh · Absorption
1 Introduction Mixing of pollutants from chemical industries or plants, altering a river and water channel is a common occurrence; this is creating ecological risks and disturbing conditions for the entire ecosystem. From a practical point of view, the processes governing the dispersion of dissolved contaminants in natural flows are numerous and very complex. The consequences of such contamination can be very dangerous to the environment and are irreversible. The longitudinal dispersion of pollutants left in a circular pipe has been extensively studied by many researchers following the classical works of Taylor [1]. Aris [2] extended Taylor’s theory to include longitudinal Communicated by Harindra Joseph Fernando. K. K. Mondal · S. Dhar (B) Department of Mathematics, Cooch Behar Panchanan Barma University, Cooch Behar 736101, India E-mail: [email protected] K. K. Mondal E-mail: [email protected] B. S. Mazumder Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India E-mail: [email protected]
K. K. Mondal et al.
dispersion within an infinite tube and developed a “method of moments” to analyze the convection process in steady flow using some first integral moments. Aris’s mechanism was modified by Barton [3] to obtain shorter and mean time moments after the rel
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