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

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.