Analysis of a Conservation PDE With Discontinuous Flux: A Model of Settler

Abstract

A dynamic model of the settling process in the secondary settler of a wastewater treatment plant is given by a nonlinear scalar conservation law $c_t + \psi ( x,c )_x = 0$ for the sludge concentration $c( t,x )$, where the flux function$\psi ( x,c )$ presents discontinuities. The authors analyze this partial differential equation (PDE) with emphasis both on the existence of stationary solutions and on the evolution of the shock corresponding to the rising of a sludge blanket. Theoretical and numerical simulations are compared with real data. A model with two classes of particles in interaction is introduced to take into account the thickening process, which appears to improve the fit with the data. Furthermore, regulation strategies of the rising of a sludge blanket in case of important water admission to the plant are proposed.