Finite Element Method for Dissolved Oxygen and Biochemical Oxygen Demand in an Open Channel

The mathematical model for dissolved oxygen interaction with biochemical oxygen demand in an open channel flow is presented in this paper. We consider unsteady flow in one dimension. The model is in the form of partial differential equations which require to solve the dissolved oxygen coupled with the biochemical oxygen demand at the same corresponding position and time. For some specific flow cases, unsteady flows approach to certain stationary solutions which are possible to derive some corresponding exact solutions. However, for complicated flow problems, analytical solution cannot be derived. Efficient and accurate numerical method is required to solve numerically such the model. In this paper, we present the finite element method with linear basis function for solving the system. The accuracy of numerical solutions have been observed and compared with some existing exact solutions for some steady flow cases. To show the ability of our presented numerical scheme, the interaction between the dissolved oxygen and the biochemical oxygen demand is simulated for unsteady flow cases. Furthermore, the impacts of flow parameters such as flow velocity, kinetic reaction order, and diffusion coefficient have been simulated to observe the dissolved oxygen and biochemical oxygen demand dynamics.