  Reactive settling denotes the process of sedimentation of small solid
particles dispersed in a viscous fluid with simultaneous reactions between the
components that constitute the solid and liquid phases. This process is of
particular importance for the simulation and control of secondary settling
tanks (SSTs) in water resource recovery facilities (WRRFs), formerly known as
wastewater treatment plants. A spatially one-dimensional model of reactive
settling in an SST is formulated by combining a mechanistic model of
sedimentation with compression with a model of biokinetic reactions. In
addition, the cross-sectional area of the tank is allowed to vary as a function
of height. The final model is a system of strongly degenerate parabolic,
nonlinear partial differential equations (PDEs)that include discontinuous
coefficients to describe the feed, underflow and overflow mechanisms, as well
as singular source terms that model the feed mechanism. A finite difference
scheme for the final model is derived by first deriving a method-of-lines
formulation (discrete in space, continuous in time), and then passing to a
fully discrete scheme by a time discretization. The advantage of this
formulation is its compatibility with common practice in development of
software for WRRFs. The main mathematical result is an invariant-region
property, which implies that physically relevant numerical solutions are
produced. Simulations of denitrification in SSTs in wastewater treatment
illustrate the model and its discretization.
