## Preprint 2023-11

## Raimund Bürger, Julio Careaga, Stefan Diehl, Romel Pineda:

### Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors

### Abstract:

Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.

This preprint gave rise to the following definitive publication(s):

**Raimund BüRGER, Julio CAREAGA, Stefan DIEHL, Romel PINEDA: ***Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors*. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 57, no. 5, pp. 2931–2976, (2023).