Analytical solution of pollutant transport equation in different types of river networks considering distributed source term

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.