Analytical and Numerical Solutions of Pollution Concentration with Uniformly and Exponentially Increasing Forms of Sources

Abstract

The study of pollution movement is an important basis for solving water quality problems, which is of vital importance in almost every country. This research proposes the motion of flowing pollution by using a mathematical model in one-dimensional advection-dispersion equation which includes terms of decay and enlargement process. We are assuming an added pollutant sources along the river in two cases: uniformly and exponentially increasing terms. The unsteady state analytical solutions are obtained by using the Laplace transformation, and the finite difference technique is utilized for numerical solutions. Solutions are compared by relative error values. The result appears acceptable between the analytical and numerical solutions. Varying the value of the rate of pollutant addition along the river (q) and the arbitrary constant of exponential pollution source term (λ) is displayed to explain the behavior of the incremental concentration. It is shown that the concentration increases as q and λ increase, and the exponentially increasing pollution source is a suitable model for the behavior of incremental pollution along the river. The results are presented and discussed graphically. This work can be applied to other physical situations described by advection-dispersion phenomena which are affected by the increase of those source concentrations.

Advection-Dispersion Equation for Concentrations of Pollutant and Dissolved Oxygen

“…So the concentrations C(x, t) and X(x, t) satisfy advection-dispersion equations. The coupled equations [8,12] are expressed in one dimension as…”

“…Manitcharoen and Pimpunchat [8] suggested unsteady state solutions for the advection-dispersion equations governing pollutant concentration. Both analytical and numerical solutions have been achieved using the Laplace transformation technique.…”

Study of solute dispersion under linear sorption in a semi-infinite porous formation

“…Mathematically these modeled are formulated in the form second order partial differential equation of elliptic type known as Advection-Dispersion Equation (ADE). Analytical and numerical approaches have been used by many researchers to solve ADE with uniform/temporal dependent solute concentration [1][2][3][4][5][6]. The sorption of solute in the porous medium is also taken into consideration by many researchers along with its transport [7][8][9][10].…”

Analytical Solution for Advection-Dispersion Equation of the Pollutant Concentration using Laplace Transformation

“…The solutions were obtained by using Laplace transformation technique. According to the study of Manitcharoen and Pimpunchat [9], the unsteady state solutions of pollutant concentration by considering advection-dispersion equations in one dimension were proposed by using the Laplace transform technique and the explicit finite difference technique, for analytical and numerical solutions, respectively. In recent years many techniques have been developed to find the solution of partial differential equations [4,5,14,15].…”