Conservation Laws, Numerical Schemes and Control Strategies for Sedimentation and Wastewater Treatment

Sammanfattning: A consistent theme throughout this thesis is quasilinear partial differential equations (PDEs) appearing in one-dimensional models of sedimentation for solid-liquid separation of suspensions. We mainly focus on continuous sedimentation in so-called clarifier-thickeners (CTs), which are found in mineral processing and wastewater treatment. In addition to complications with shocks and non-unique solutions that are standard for quasilinear first-order PDEs, the considered CT models offer challenges such as spatially discontinuous fluxes and strongly-degenerate diffusion terms. We are primarily concerned with numerical methods for two CT models: one previously published scalar PDE model relying on the assumption that all suspended solids have the same settling properties, and one system of strongly coupled PDEs that accounts for solids with non-homogeneous settleability. A method of Godunov type is proposed for the scalar model. This method has been well received by the wastewater community and it is implemented both on benchmark simulation platforms and in commercial software. During the development of numerical schemes for the second model, we encounter a hyperbolic system with a delicate feature: a nonlinear contact field. Generalising the problem setting to a family of systems, also containing the celebrated Aw-Rascle-Zhang traffic flow model, we construct a novel random sampling scheme to capture discontinuities with jumps along curves in such fields. Convergence to weak solutions of the hyperbolic system is proved for that scheme. The research presented also covers steady-state analyses and design of control strategies for stand-alone CTs and for a wastewater treatment plant with a biological reactor preceding the sedimentation process. The plant-wide models of interest are given as systems of ordinary differential equations for the reactor coupled to some PDEs for the CT.