mohammadjavad fardadi shilsar

In this study, the analytical solution of the pollution transport equation considering distributed source term and initial condition was performed by Laplace transform method for a general river network in a finite domain with constant coefficients for upstream and downstream using Dirichlet boundary conditions. Existence of the source term and initial condition increases the computational complexity to find the particular solution of the ordinary differential equation. To evaluate the existing analytical solution, two hypothetical examples were presented, that in each, modeling was performed on two branch and loop networks types considering a distributed source of pollution. Input data for modeling each of the desired river networks include values of velocity, dispersion coefficient, branch lengths, flow area, and input concentrations from boundaries and distributed sources. By calculating the diffusion and Laplace mass balance matrices (by influencing the distributed source) in the river network based on the connection and data matrix, a nonlinear equations system is created according to the Laplace s variable, which by solving it, the pollution concentration matrix and consequently the pollution concentration in each node is calculated by numerical inverse Laplace algorithm. The numerical solution used to validate the proposed analytical solution. The results showed that the statistical indices of R2, root mean square error, and mean absolute error in the best case were 99.86%, 0.0099, and 0.0067 kg/m3 for 1456 route and in the worst case were 95.20%, 0.0309 and 0.0194 kg/m3 for 23456 route of the loop network, respectively. The results showed that the two proposed solutions are well compatible together, indicating the good performance of the existing analytical solution and its replacement instead of numerical solution due to higher accuracy in the river network.

Rivers are one of the most important natural water resources in the world. Pollution transport modeling in rivers is performed by the partial advection-dispersion-reaction equation (ADRE). In the present study, using the Laplace transform, which is a powerful and useful tool in solving differential equations, the analytical solution of the ADRE equation was obtained in a finite domain with variable coefficients for the upstream and downstream Dirichlet boundary conditions and the initial zero condition in the river. To use the analytical solution in this study, three examples are presented, each of which, the river are divided into two, four, and eight parts, which, while the parameters of flow, pollution, and river geometry are variable in all three examples, for each of the examples, the accuracy of the analytical solution available when the segmentation of the intervals increases as compared to the numerical solution. By specifying the matrices of velocity, dispersion coefficient, cross-section, etc. as input to the problem, the diffusion matrix is calculated and, consequently, a complex system of equations is created that doubles the complexity of the work. The amount of pollutant concentration is calculated by solving the system of the above equations. The numerical solution is used to validate the existing analytical solution, the results showed that the greater the number of river divisions, the higher the accuracy of the solution, and the two analytical and numerical solutions will be well compatible with each other. Given the ability and performance of the existing analytical solution, it can be acknowledged that the analytical solution in this study can be used as a tool to validate and verification numerical solutions and other analytical solutions for the coefficients of the equation.