Analytical Solutions for Spatially Variable Transport-Dispersion of Non-Conservative Pollutants

Document Type : Full Length Research Article

Author

Professor, Department of Mathematics

Abstract

Analytical solutions have been obtained for both conservative and non-conservative forms of one-dimensional transport and transport-dispersion equations applicable for pollution as a result of a non-conservative pollutant-disposal in an open channel with linear spatially varying transport velocity and nonlinear spatially varying dispersion coefficient on account of a steady unpolluted lateral inflow in accordance to the channel. A logarithmic transformation in the space variable has been applied in order to derive a general solution of the transport equation for spatially variable initial pollutant distribution and upstream time-dependent pollutant concentration. The logarithmic transformation reduces both conservative and non-conservative forms of transport-dispersion equation to a form with constant coefficients that is solvable by analytical methods. An analysis of these solutions indicates that only the solution of a conservative form of the governing equation yields appropriate results that are conceptually acceptable in a real physical situation.
The solution lends to analyze the damping effect of such transport on the pollutant with an initial Gaussian profile, in contrast with that of the initial quasi-Gaussian profile available in the literature. It is noteworthy to mention that the solution of conservative form of the transport equation implies that mass of the non-conservative pollutant in the channel decreases with an increase in time, and finally reaches to a constant value that is a ratio of product of the transport velocity coefficient and upstream concentration to the coefficient of decay of the pollutant.

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References

[1] M. Th. van Genucheten and W. J. Alves, “Analytical solution of one-dimensional convective-dispersion solute transport equation.” Technical Bulletin No. 1661, U.S. Dept. of Agriculture, Washington, DC( 1982).
[2] C. V. Chrysikopoulos and Y. Sim, “Onedimensional Virus Transport Homogeneous Porous Media with Time-dependent Distribution Coefficient,” J. Hydrol., 185(1–4), 199–219 (1996).
[3] M. K. Singh, V. P.Singh, P. Singh and D. Shukla, “Analytical solution for conservative solute transport in one-dimensional homogeneous porous formations with time-dependent velocity,” J. Eng. Mech., 135(9), 1015–1021 (2009).
[4] M. K. Singh, S. Ahamad, and V. P. Singh, “Analytical Solution for One-Dimensional Solute Dispersion with Time-Dependent Source Concentration along Uniform Groundwater Flow in a Homogeneous Porous Formation,” J. Eng. Mech., 138, 1045-1056 (2012).
[5] A. Kumar, D. K. Jaiswal and N. Kumar, “Analytical solution to one-dimensional advection-diffusion equation with variable coefficient in semi-infinite media,” J. Hydrol., 380 (3–4), 330–337 (2010).
[6] M. Th. van Genuchten, F. J. Leij, T. H. Skaggs, N. Toride, , A. B. Scott, M. P. Elizabeth, “Exact analytical solutions for contaminant transport in rivers 1. The equilibrium advectiondispersion equation,” J. Hydrol. Hydromech. 61(2), 146–160 (2013). [7] W.H. Streeter and E.B. Phelps, “A study of the pollution and natural purification of the Ohio River, III. Factors concerned in the phenomena of oxidation and reaeration,” February Public Health Bull. 146, U.S. Public Health Service, Government Printing Office, Washington, D.C. (1925). 132 V. P. Shukla / JHMTR 6 (2019) 125-132
[8] H.A. Basha, and F.S. El-Habel, "Analytical Solution of the One-dimensional Time Dependent Transfer Equation." Water Resour. Res. 29(9), 2209-3214 (1993).
[9] J.R. Philip, "Some exact solutions of convection-diffusion and diffusion equations," Water Resour. Res. 30(12), 3541- 3551 (1994).
[10] C. Zoppou and J. H. Knight, "Comment on ‘A space time accurate method for solving solute transport problems,' by S.G.Li, E. Ruan and D.McLaughlin" Water Resour. Res., 30(11), 3233-3235 (1994).
[11] C. Zoppou and J.H. Knight, "Analytical solutions for advection and advectiondiffusion equations with spatially variable coefficients,” J. Hyd. Engng. 123(2), 144-148 (1997).
[12] J.S. Pérez Guerrero, T. H. Skaggs, “Analytical solution for one-dimensional advectiondispersion transport equation with distancedependent coefficients,” Journal of Hydrology, 390(1), 57-65 (2010).
[13]V. P. Shukla, “Analytical Solutions for Unsteady Transport Dispersion of Nonconservative Pollutant with Time-Dependent Periodic Waste Discharge Concentration,” Journal of Hydraulic Engineering, 128 (9), 866–869 (2002).
[14]G. de Marsily, Quantitative Hydrogeology: Groundwater Hydrology for engineers. Academic Press, Inc., San Diego, California, (1986). [15]I.N. Sneddon, The use of Integral Transforms. McGraw Hill, New York, (1972).

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History

  • Receive Date: 09 August 2018
  • Revise Date: 23 July 2019
  • Accept Date: 23 July 2019

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