Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics

Barik Swarup; Dalal D. C.

doi:10.1007/s10409-021-09037-y

Volume 38 Issue 3

Feb 2022

Turn off MathJax

Article Contents

JOURNAL OF MECHANICAL ENGINEERING > 2022 > 38(3): 321506.

S. Barik, and D. C. Dalal, Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics, Acta Mech. Sin. 38, 321506 (2022), http://www.w3.org/1999/xlink' xlink:href='https://doi.org/10.1007/s10409-021-09037-y'>https://doi.org/10.1007/s10409-021-09037-y

Citation:

S. Barik, and D. C. Dalal, Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics, Acta Mech. Sin. 38, 321506 (2022), http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.1007/s10409-021-09037-y">https://doi.org/10.1007/s10409-021-09037-y

Citation:

S. Barik, and D. C. Dalal, Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics, Acta Mech. Sin. 38, 321506 (2022), http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.1007/s10409-021-09037-y">https://doi.org/10.1007/s10409-021-09037-y

PDF( 0 KB)

Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics

doi: 10.1007/s10409-021-09037-y

Barik Swarup^{1, 2,*
,
,},
Dalal D. C.¹

1.
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India;
2.
Department of Mathematics, SRM Institute of Science and Technology, Chennai, 603203, India;

More Information

Corresponding author: Barik Swarup, E-mail address: swarupb@srmist.edu.in (Swarup Barik)
Accepted Date: 08 Nov 2021

Available Online: 01 Aug 2022

Publish Date: 28 Jan 2022
Issue Publish Date: 01 Mar 2022

Abstract

Abstract

This study is about an analytical attempt that explores the two-dimensional concentration distribution of a solute in an open channel flow. The solute undergoes reversible sorption at the channel bed. The method of multiple scales is used to find the two-dimensional concentration distribution, which is important for modern day application in industry, environmental risk assessment, etc. Study deduces an analytic expression of two-dimensional concentration distribution for an open channel flow with sorptive channel bed. Effects of retention parameter, Damkohler number on the solute dispersion are also discussed in this paper. Results reveal that slow or strong kinetics (small value of Damkohler number) increases solute dispersion. It is also observed that for slow phase exchange kinetics between bulk flow and small retentive channel bed, solute concentration distribution will uniform faster than their inert counterpart.
- Multi-scale method,
- Dispersion coefficient,
- Two-dimensional concentration distribution,
- Phase exchange kinetics

FullText(HTML)

References(56)

References

[[1]]	M. R. Davidson, and R. C. Schroter, A theoretical model of absorption of gases by the bronchial wall, J. Fluid Mech. 129, 313 (1983).
[[2]]	J. B. Grotberg, B. V. Sheth, and L. F. Mockros, An analysis of pollutant gas transport and absorption in pulmonary airways, J. Biomech. Eng. 112, 168 (1990).
[[3]]	A. Sarkar, and G. Jayaraman, The effect of wall absorption on dispersion in annular flows, Acta Mech. 158, 105 (2002).
[[4]]	H. B. Fischer, Longitudinal dispersion and turbulent mixing in open-channel flow, Annu. Rev. Fluid Mech. 5, 59 (1973).
[[5]]	H. B. Fischer, Mixing and dispersion in estuaries, Annu. Rev. Fluid Mech. 8, 107 (1976).
[[6]]	H. B. Fischer, E. J. List, R. C. Y. Koh, J. Imberger, and N. H. Brooks, Mixing in Inland and Coastal Waters (Academic Press, New York, 1979)
[[7]]	Z. Wu, Z. Li, and G. Q. Chen, Multi-scale analysis for environmental dispersion in wetland flow, Commun. Nonlinear Sci. Numer. Simul. 16, 3168 (2011).
[[8]]	Z. Wu, L. Zeng, G. Q. Chen, Z. Li, L. Shao, P. Wang, and Z. Jiang, Environmental dispersion in a tidal flow through a depth-dominated wetland, Commun. Nonlinear Sci. Numer. Simul. 17, 5007 (2012).
[[9]]	Z. Wu, L. Zeng, and G. Q. Chen, in Analytical modeling for environmental dispersion in wetland: Developments in Environmental Modelling, edited by S. E. Jø rgensen, N. B. Chang, and F. L. Xu (Elsevier, 2014), pp. 251-274
[[10]]	P. Wang, and G. Q. Chen, Solute dispersion in open channel flow with bed absorption, J. Hydrol. 543, 208 (2016).
[[11]]	A. Shankar, and A. M. Lenhoff, Dispersion and partitioning in short coated tubes, Ind. Eng. Chem. Res. 30, 828 (1991).
[[12]]	F. P. Carvalho, J. P. Villeneuve, C. Cattini, J. Rendón, and J. M. de Oliveira, Ecological risk assessment of PCBs and other organic contaminant residues in Laguna de Terminos, Mexico, Ecotoxicology 18, 403 (2009).
[[13]]	G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. Lond. A 219, 186 (1953).
[[14]]	R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc R Soc Lond A 235, 67–77 (1956).
[[15]]	T. Takahashi, T. Korenaga, and F. Shen, A numerical solution for the dispersion in laminar flow through a circular tube, Can. J. Chem. Eng. 68, 191 (1990).
[[16]]	A. N. Stokes, and N. G. Barton, The concentration distribution produced by shear dispersion of solute in Poiseuille flow, J. Fluid Mech. 210, 201 (1990).
[[17]]	K. Ekambara, and J. B. Joshi, Axial mixing in laminar pipe flows, Chem. Eng. Sci. 59, 3929 (2004).
[[18]]	M. W. Lau, and C. O. Ng, On the early development of dispersion in flow through a tube with wall reactions, Int. J. Mech. Aerosp. Ind. Mech. Manuf. Eng. 1, 485 (2007).
[[19]]	K. K. Mondal, S. Dhar, and B. S. Mazumder, On dispersion of solute in steady flow through a channel with absorption boundary: an application to sewage dispersion, Theor. Comput. Fluid Dyn. 34, 643 (2020).
[[20]]	S. Dhar, N. Poddar, R. R. Kairi, B. S. Mazumder, and K. K. Mondal, Numerical study on dispersion of fine settling particles in a depth dominated wetland flow, Commun. Nonlinear Sci. Numer. Simul. 96, 105707 (2021).
[[21]]	R. Aris, On the dispersion of a solute in pulsating flow through a tube, Proc. R. Soc. Lond. A 259, 370 (1960).
[[22]]	W. N. Gill, A note on the solution of transient dispersion problems, Proc. R. Soc. Lond. A 298, 335 (1967).
[[23]]	W. N. Gill, Exact analysis of unsteady convective diffusion, Proc. R. Soc. Lond. A 316, 341 (1970).
[[24]]	W. N. Gill, Dispersion of a non-uniform slug in time-dependent flow, Proc. R. Soc. Lond. A 322, 101 (1971).
[[25]]	S. Paul, and C. O. Ng, On the time development of dispersion in electroosmotic flow through a rectangular channel, Acta Mech. Sin. 28, 631 (2012).
[[26]]	J. Rana, and P. V. S. N. Murthy, Unsteady solute dispersion in non-Newtonian fluid flow in a tube with wall absorption, Proc. R. Soc. A. 472, 20160294 (2016).
[[27]]	S. Paul, and B. S. Mazumder, Effects of nonlinear chemical reactions on the transport coefficients associated with steady and oscillatory flows through a tube, Int. J. Heat Mass Transfer 54, 75 (2011).
[[28]]	S. Barik, and D. C. Dalal, On transport coefficients in an oscillatory Couette flow with nonlinear chemical decay reactions, Acta Mech. 228, 2391 (2017).
[[29]]	H. Wang, Z. Zhu, S. Li, and W. Huai, Solute dispersion in wetland flows with bed absorption, J. Hydrol. 579, 124149 (2019).
[[30]]	A. K. Roy, A. K. Saha, and S. Debnath, Effect of multiple reactions on the transport coefficients in pulsatile flow through an annulus, Int. Commun. Heat Mass Transfer 110, 104369 (2020).
[[31]]	N. Poddar, S. Dhar, B. S. Mazumder, and K. K. Mondal, An exact analysis of scalar transport in hydromagnetic flow between two parallel plates: a multi-scale approach, Proc. R. Soc. A. 477, rspa.2020.0830(2021).
[[32]]	R. Smith, A delay-diffusion description for contaminant dispersion, J. Fluid Mech. 105, 469 (1981).
[[33]]	J. W. Westhaver, Theory of open-tube distillation columns, Ind. Eng. Chem. 34, 126 (1942).
[[34]]	R. Aris, On the dispersion of a solute by diffusion, convection and exchange between phases, Proc. R. Soc. Lond. A 252, 538 (1959).
[[35]]	C. G. Phillips, and S. R. Kaye, Approximate solutions for developing shear dispersion with exchange between phases, J. Fluid Mech. 374, 195 (1998).
[[36]]	C. O. Ng, and T. L. Yip, Effects of kinetic sorptive exchange on solute transport in open-channel flow, J. Fluid Mech. 446, 321 (2001).
[[37]]	C. O. Ng, and Y. C. Bai, Dispersion in oscillatory Couette flow with sorptive boundaries, Acta Mech. 178, 65 (2005).
[[38]]	C. O. Ng, Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions, Proc. R. Soc. A. 462, 481 (2006).
[[39]]	C. O. Ng, Dispersion in open-channel flow subject to the processes of sorptive exchange on the bottom and air-water exchange on the free surface, Fluid Dyn. Res. 38, 359 (2006).
[[40]]	C. O. Ng, Chemical transport associated with discharge of contaminated fine particles to a steady open-channel flow, Phys. Fluids 12, 136 (2000).
[[41]]	P. C. Chatwin, The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe, J. Fluid Mech. 43, 321 (1970).
[[42]]	W. N. Gill, and V. Ananthakrishnan, Laminar dispersion in capillaries: Part IV. The slug stimulus, AIChE J. 13, 801 (1967).
[[43]]	L. Zeng, G. Q. Chen, H. S. Tang, and Z. Wu, Environmental dispersion in wetland flow, Commun. Nonlinear Sci. Numer. Simul. 16, 206 (2011).
[[44]]	B. Chen, Contaminant transport in a two-zone wetland: dispersion and ecological degradation, J. Hydrol. 488, 118 (2013).
[[45]]	L. Zeng, Y. J. Zhao, B. Chen, P. Ji, Y. H. Wu, and L. Feng, Longitudinal spread of bicomponent contaminant in wetland flow dominated by bank-wall effect, J. Hydrol. 509, 179 (2014).
[[46]]	Z. Wu, and G. Q. Chen, Axial diffusion effect on concentration dispersion, Int. J. Heat Mass Transfer 84, 571 (2015).
[[47]]	Z. Wu, and G. Q. Chen, Approach to transverse uniformity of concentration distribution of a solute in a solvent flowing along a straight pipe, J. Fluid Mech. 740, 196 (2014).
[[48]]	Z. Wu, and G. Q. Chen, Analytical solution for scalar transport in open channel flow: slow-decaying transient effect, J. Hydrol. 519, 1974 (2014).
[[49]]	C. C. Mei, J. L. Auriault, and C. O. Ng, Some applications of the homogenization theory, Adv. Appl. Mech. 32, 277 (1996).
[[50]]	C. C. Mei, and B. Vernescu, Homogenization Methods for Multiscale Mechanics, (World Scientific, Singapore, 2010)
[[51]]	H. C. W. Chu, S. Garoff, T. M. Przybycien, R. D. Tilton, and A. S. Khair, Dispersion in steady and time-oscillatory two-dimensional flows through a parallel-plate channel, Phys. Fluids 31, 022007 (2019).
[[52]]	H. C. W. Chu, S. Garoff, R. D. Tilton, and A. S. Khair, Dispersion in steady and time-oscillatory flows through an eccentric annulus, AIChE J 66, (2020).
[[53]]	S. Barik, and D. C. Dalal, Transverse concentration distribution in an open channel flow with bed absorption: a multi-scale approach, Commun. Nonlinear Sci. Numer. Simul. 65, 1 (2018).
[[54]]	C. O. Ng, and N. Rudraiah, Convective diffusion in steady flow through a tube with a retentive and absorptive wall, Phys. Fluids 20, 073604 (2008).
[[55]]	G. Ramon, Y. Agnon, and C. Dosoretz, Solute dispersion in oscillating electro-osmotic flow with boundary mass exchange, Microfluid Nanofluid 10, 97 (2011).
[[56]]	S. Barik, and D. C. Dalal, Multi-scale analysis for concentration distribution in an oscillatory Couette flow, Proc. R. Soc. A. 475, 20180483 (2019).

Relative Articles

Supplements(0)

Cited By

Proportional views

Proportional views

Figures(8) / Tables(3)

Get Citation

PDF

XML

Article Metrics

Article views(45) PDF downloads(0)

Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics

doi: 10.1007/s10409-021-09037-y

Abstract

References

Proportional views

Catalog

Article Metrics

Proportional views

Related

Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics

doi: 10.1007/s10409-021-09037-y

Abstract

References

Proportional views

Catalog

Article Metrics

Proportional views

Related

Export File

Citation

Format

Content