مهندسی عمران شریف (Aug 2017)
حل تحلیلی معادلهی انتقال آلودگی بهازاء فعالیت چندین منبع آلایندهی نقطهیی با الگوهای زمانی دلخواه در حالات ۱ و ۲ بعدی با استفاده از روش تابع گرین
Abstract
Pollutants dispersion is one of the most important issues in surface waters. The governing equation dealing with this phenomenon is the advection-dispersion-reaction (ADRE) equation. The Application of mathematical models of pollution transport in rivers is very important. Moreover, it is necessary to utilize analytical solutions for numerical verification methods. Green's function method is a powerful method for solving nonhomogeneous partial differential equations analytically in one or multi-dimensional domains. In this research work, the analytical solution of ADRE (with constant velocity and dispersion coefficient) for different combinations of boundary conditions was derived in one and two dimensions for infinite, semi-infinite, and finite spatial domains in the integral form including the boundary and initial conditions and source term effects separately using Green's function method (GFM). First, the general solution to ADRE equation was determined. Therefore, the final explicit solution will depend on the existence of Green's function related to the original problem. In order to find the Green's function of each problem, a powerful tool, called ``Adjoint Operator'', was employed. Finally, by locating the Green's function in the general solution associated with the main boundary value problem, the final solution to ADRE equation was specified. Also, the product solution rule was used to obtain Green's functions in two-dimensional domain. Also, to accelerate the convergence of the resulting infinite series, a non-dimensional parameter was defined. As a result, the small time form of Green's function using the mathematical concept was determined. The obtained solutions were derived for multiple point sources with arbitrary time patterns. Evaluation of the derived solutions was performed using several hypothetical examples and some real data. For one- point source with simple time patterns, the evaluation was done using analytical solutions represented by other researchers. In the cases with multiple point sources in which no analytical solution exists, verification was carried out by numerical models. The Final graphs and statistical analysis show good agreement between the results of numerical models and the proposed solution. Also, finally, it can be concluded that the most comprehensive set of analytical solution to ADRE for different combinations of applicable boundary condition in rivers was presented in this research work.
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