Skip to main content
SpringerLink
Log in

Analytical solutions for one-dimensional advection–dispersion equation of the pollutant concentration

  • Published:
Journal of Earth System Science Aims and scope Submit manuscript

Abstract

We present simple analytical solutions for the unsteady advection–dispersion equations describing the pollutant concentration C(x, t) in one dimension. The solutions are obtained by using Laplace transformation technique. In this study we divided the river into two regions x ≤ 0 and x≥0 and the origin at x = 0. The variation of C(x, t) with the time t from t = 0 up to t → ∞ (the steady state case) is taken into account in our study. The special case for which the dispersion coefficient D = 0 is studied in detail. The parameters controlling the pollutant concentration along the river are determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5

Similar content being viewed by others

References

  • Angelakis A N and Rolston D E 1985 Transient movement and transformation of carbon species in soil during wastewater application; Water Resour. Res. 21 (9) 1141–1148.

    Article  Google Scholar 

  • Carslaw H S and Jaeger J C 1959 Conduction of Heat in solids; Clarendon Press, Oxford.

    Google Scholar 

  • Djordjevich A and Savovic S 2013 Solute transport with longitudinal and transverse diffusion in temporally and spatially dependent flow from a pulse type source; Int. J. Heat Mass Transfer 65 321–326.

    Article  Google Scholar 

  • Dobbins W E 1964 BOD and oxygen relationships in streams; J. Sanitary Engineering Division, Proc. ASCE 90 53–78.

    Google Scholar 

  • Doetsch G 1970 Introduction to the Theory and Application of the Laplace Transform; Springer-Verlag.

  • Ge S and Lu N 1996 A semi-analytical solution of one-dimensional advective–dispersive solute transport under an arbitrary concentration boundary condition; Ground Water 34 (3).

  • Kumar A, Jaiswal D K and Kumar N 2009 Analytical solutions of one-dimensional advection–diffusion equation with variable coefficients in a finite domain; J. Earth Syst. Sci. 118 539–549.

    Article  Google Scholar 

  • Kumar A, Jaiswal D K and Yadav R R 2011 One-dimensional solute transport for uniform and varying pulse type input point source with temporally dependent coefficients in longitudinal semi-infinite homogeneous porous domain; Int. J. Math. Sci. Comput. 1 (2).

  • Leij F J and Van Genuchten M T. 1995 Approximate analytical solutions for solute transport in two-layer Porous media; Kluwer Academic Publishers; Transport in Porous Media 18 65–85.

    Article  Google Scholar 

  • Li G 2006 Stream temperature and dissolved oxygen modeling in the Lower Flint River Basin (Athens, Georgia), PhD thesis, 45p.

  • Massabo M, Cianci R and Paladino O 2006 Some analytical solution for two-dimensional convection–dispersion equation in cylindrical geometry; Environmental and Software 21 681–688.

    Article  Google Scholar 

  • Mourad F D, Ali S W and Fayez N I 2013 The effect of added pollutant along a river on the pollutant concentration described by one-dimensional advection diffusion equation; Int. J. Engg. Sci. Tech. 5 1662–1671.

    Google Scholar 

  • Mikhailov M D and Ozisik M N 1984 Unified analysis and solutions of heat and mass diffusion (John Wiley, New York).

  • Pimpunchat B, Sweatman W L, Triampo W, Wake G C and Parshotam A 2007 Modeling river pollution and removalby aeration; In: Modsim 2007, International Congress on Modelling and Simulation (eds) Oxley L and Kulasiri D, Land, Water & Environmental Management: Integrated Systems for Sustainability, Modelling and Simulation Society of Australia and New Zealand, pp. 2431–2437, ISBN: 978-0-9758400-4-7.

  • Pimpunchat B, Sweatman W L, Triampo W, Wake G C and Parshotam A 2009 A mathematical model for pollution in a river and its remediation by aeration; Appl. Math. Lett. 22 304–308.

    Article  Google Scholar 

  • Roberts G E and Kaufman H 1969 Table of Laplace Transforms (W.E. Saunders Co., Philadelphia and London).

  • Savovic S and Caldwell J 2003 Finite difference solution of the one-dimensional Stefan problem with periodic boundary conditions; Int. J. Heat Mass Transfer. 46 2911–2916.

    Article  Google Scholar 

  • Savovic S and Caldwell J 2009 Numerical solution of Stefan problem with time dependent boundary conditions by variable space grid method; Therm. Sci. 13 165–174.

    Article  Google Scholar 

  • Savovic S and Djordjevich A 2012 Finite difference solution of the one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media; Int. J. Heat Mass Transfer. 55 4291–4294.

    Article  Google Scholar 

  • Savovic S and Djordjevich A 2013 Numerical solution for temporally and spatially dependent solute dispersion of pulse type input concentration in semi-infinite media; Int. J. Heat Mass Transfer. 60 291–295.

    Article  Google Scholar 

  • Shukla J B, Misra A K and Chandra P 2008 Mathematical modeling and analysis of the depletion of dissolved oxygen in eutrophied water bodies affected by organic pollutants; Nonlinear Analysis. Real Applications 9 1851–1865.

    Article  Google Scholar 

  • Sudicky E A and Neville C J 2008 Solution of partial differential equations (PDE’s) using integral transform method; The Laplace transform, pp. 6–12.

  • Van Genuchten M T. and Alves W J 1982 Analytical solutions of the one-dimensional convective–dispersive solute transport equation; U.S.D.A. Tech. Bull. 1661, 149p.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, AinShams University, Cairo, 0020, Egypt.

    Ali S Wadi, Mourad F Dimian & Fayez N Ibrahim

Authors
  1. Ali S Wadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mourad F Dimian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fayez N Ibrahim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali S Wadi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wadi, A.S., Dimian, M.F. & Ibrahim, F.N. Analytical solutions for one-dimensional advection–dispersion equation of the pollutant concentration. J Earth Syst Sci 123, 1317–1324 (2014). https://doi.org/10.1007/s12040-014-0468-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12040-014-0468-2

Keywords

Search

Navigation