An Analytical solution to two-dimensional unsteady pollutant transport equation with arbitrary initial condition and source term in the open channels

Document Type : Research

Authors

1 Ph.D. Student, Department of Water Structures, Tarbiat Modares University, Tehran, Iran

2 Assistant Professor, Department of Water Structures, Tarbiat Modares University, Tehran, Iran

3 Professor, Department of Water Structures, Tarbiat Modares University, Tehran, Iran

Abstract

Pollutant dispersion in environment is one of the most important challenges in the world. The governing equation of this phenomenon is the Advection-Dispersion-Reaction (ADRE) equation. It has wide applications in water and atmosphere, heat transfer and engineering sciences. This equation is a parabolic partial differential equation that is based on the first Fick’s law and conservation equation. The applications mathematical models of pollution transport in rivers is very vital. Analytical solutions are useful in understanding the contaminant distribution, transport parameter estimation and numerical model verification. One of the powerful methods in solving nonhomogeneous partial differential equations analytically in one or multi-dimensional domains is Generalized Integral Transform Technique (GITT). This method is based on eigenvalue problem and integral transform that converts the main partial differential equation to a system of Ordinary Differential Equation (ODE). In this research, an analytical solution to two-dimensional pollutant transport equation with arbitrary initial condition and source term was obtained for a finite domain in the rivers using GITT. The equation parameters such as velocity, dispersion and reaction factor were considered constant. The boundary condition was assumed homogenous. In this research, the source term is considered as point pollutant sources with arbitrary emission time pattern. To extract the analytical solution, the first step is choosing an appropriate eigenvalue problem. The eigenvalue must be selected based on Self-Adjoint operator and can be solved analytically. In the next, the eigenfunction set was extract by solving the eigenvalue problem with homogenous boundary condition using the separation of variables method. Then the forward integral transform and inverse transform were defined. By implementing the transform and using the orthogonality property, the ordinary differential equation system was obtained. The initial condition was transformed using forward transform and the ODE system was solved numerically and the transformed concentration function was obtained. Finally, the inverse transform was implemented and the main analytical solution was extracted. In order to evaluate the extracted solution, the result of the proposed solution was compared with the Green’s Function Method (GFM) solution in the form of two hypothetical examples. In this way, in the first example, the initial condition function as an impulsive one at the specific point in the domain and one point source with the exponential time pattern were considered. In the second example, the initial condition was similar to the first example and two point sources with irregular time pattern were assumed. The final results were represented in the form of the concentration contours at different times in the velocity field. The results show the conformity of the proposed solution and GFM solution and report that the performance of the proposed solution is satisfactory and accurate. The concentration gradient decreases over time and the pollution plume spreads and finally exits from the domain at the resultant velocity direction due to the advection and dispersion processes. The presented solutions have various applications; they can be used instead of numerical models for constant- parameters conditions. The analytical solution is as an exact, fast, simple and flexible tool that is conveniently stable for all conditions; using this method, difficulties associated with numerical methods, such as stability, accuracy, etc., are not involved. Also because of the high flexibility of the present analytical solutions, it is possible to implement arbitrary initial condition and multiple point sources with more complexity in emission time patterns. So it can be used as a benchmark solution for the numerical solution validation in two-dimensional mode.

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Aral, M. M. and Liao, B., 1996, Analytical solutions for two-dimensional transport equation with time-dependent dispersion coefficients., Journal of Hydrologic Engineering 1(1), 20-32.
Basha, H., 1997, Analytical model of two-dimensional dispersion in laterally nonuniform axial velocity distributions., Journal of Hydraulic Engineering 123(10), 853-862.
Cassol, M., Wortmann, S. and Rizza, U., 2009, Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace Transform techniques., Environmental Modelling Software 24(1), 144-151.
Chapra, S. C., 1997, Surface water-quality modeling, McGraw-Hill New York.
Chen, J. S., Chen, J. T., Liu, C. W., Liang, C. P. and Lin, C. W., 2011, Analytical solutions to two-dimensional advection–dispersion equation in cylindrical coordinates in finite domain subject to first-and third-type inlet boundary conditions., Journal of Hydrology 405(3-4), 522-531.
Chen, K., Zhan, H. and Zhou, R., 2016, Subsurface solute transport with one-, two-, and three-dimensional arbitrary shape sources., Journal of contaminant hydrology 190, 44-57.
Costa, C. P., Vilhena, M. T., Moreira, D. M. and Tirabassi, T., 2006, Semi-analytical solution of the steady three-dimensional advection-diffusion equation in the planetary boundary layer., 40(29), 5659-5669.
Cotta, R. M., 1993, Integral transforms in computational heat and fluid flow, CRC Press.
Cotta, R. M., Knupp, D. C. and Naveira Cotta, C. P., 2016, Analytical heat and fluid flow in microchannels and microsystems, Springer.
Cotta, R. M. and Mikhailov, M. D., 1997, Heat conduction: lumped analysis, integral transforms, symbolic computation, Wiley Chichester.
De Almeida, G. L., Pimentel, L. C. and Cotta, R. M., 2008, Integral transform solutions for atmospheric pollutant dispersion., Environmental Modeling Assessment, 13(1), 53-65.
De Barros, F. P., Mills, W. B. and Cotta, R. M., 2006, Integral transform solution of a two-dimensional model for contaminant dispersion in rivers and channels with spatially variable coefficients., Environmental Modelling Software 21(5), 699-709.
De Barros, F. P. and Cotta, R. M. J., 2007, Integral transforms for three-dimensional steady turbulent dispersion in rivers and channels., Applied Mathematical Modelling 31(12), 2719-2732.
Guerrero, J. P., Pimentel, L. C., Skaggs, TH. and Van Genuchten, M. Th., 2009, Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique., International Journal of Heat Mass Transfer 52(13-14), 3297-3304.
Mashhadgarme, N., Mazaheri, M. and Mohammad Vali Samani, J., 2017, Analytical solutions to one- and two-dimensional Advection-Dispersion-Reaction equation with arbitrary source term time pattern using Green’s function method., Sharif Journal of Civil Engineering, 33-2, 77-91.
Sanskrityayn, A., Singh, V., Bharati, V. K. and Kumar, N., 2018, Analytical solution of two-dimensional advection–dispersion equation with spatio-temporal coefficients for point sources in an infinite medium using Green’s function method. Environmental Fluid Mechanics 18(3), 739-757.
Van Genuchten, M. Th., Leij, F. J., Skaggs, T. H., Toride, N., Bradford, S. A. and Pontedeiro, E. M., 2013, Exact analytical solutions for contaminant transport in rivers 1. The equilibrium advection-dispersion equation., Journal of Hydrology Hydromechanics 61(2), 146-160.
Wortmann, S., Vilhena, M. T., Moreira, D. M. and Buske, D., 2005, A new analytical approach to simulate the pollutant dispersion in the PBL., 39(12), 2171-2178.
Xu, Zh., Travis, J. R. and Breitung, W., 2007, Green's Function Method and Its Application to Verification of Diffusion Models of GASFLOW Code, Forschungszentrum Karlsruhe.
Yadav, R. and Jaiswal, D. K. J. , 2012, Two‐dimensional solute transport for periodic flow in isotropic porous media: an analytical solution., Hydrological Processes: An International Journal, 26(22), 3425-3433.
Yadav, R. and Kumar. L. J., 2019, Solute Transport for Pulse Type Input Point Source along Temporally and Spatially Dependent Flow., Pollution, 5(1), 53-70.
Yeh, G. T., 1981, AT123D: Analytical transient one-, two-, and three-dimensional simulation of waste transport in the aquifer system, Oak Ridge National Lab., TN (USA).
Zoppou, C. and Knight, J. J., 1999, Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions., Applied Mathematical Modelling, 23(9), 667-685.