مهندسی مکانیک شریف (May 2020)
AN ANALYTICAL SOLUTION TO TWO-DIMENSIONAL UNSTEADY MASS TRANSFER EQUATION WITH ARBITRARY SOURCE TERM IN THE RIVER
Abstract
The concentration distribution in the water resources is analyzed with Mass transfer equation. Analytical solutions of this equation play an important role in understanding the mass transfer problem, mass 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. 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 Mass transfer equation with arbitrary emission time patterns of point sources was obtained at the finite domain in the open channels using Generalized Integral Transform Technique. In order to evaluate the extracted solution, the result of the proposed solution was compared with the Green's function method solution in the form of two hypothetical examples. In the first example, one point source with the compound exponential function was considered. In the second example, two point source with irregular time pattern at the distance from each other was assumed. The final results were represented in the form of the concentration contours at different times. The results show the conformity of the proposed solution and Green's function solution and report the suitable performance of the proposed solution. 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 to numerical methods, such as stability, accuracy, numerical dispersion, etc., are not involved. The extracted solution in this research can adopt multiple pollutant sources with arbitrary time patterns and can be used as a benchmark solution for the numerical solution validation in two-dimensional mode.
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