Modeling bacteria flocculation as density‐dependent growth

Biological reactors are commonly used to remove pollutants from wastewater. One standard technology is the twostep activated sludge (AS) process. Both in the reaction and the settling tank, bacteria naturally aggregate and form flocs. It is well known—but poorly understood—that both floc formation and settling capacity strongly depend on the loading rate. To optimize this bioprocess it is, therefore, necessary to better understand the flocculation phenomenon. Mathematical modeling has proven to be a valuable tool in the study of wastewater treatment plants. The activated sludge models describe the different biological processes (for example, chemical oxygen demand removal, (de) nitrification and phosphorus removal) involved in the AS process. Its core consists of the mass-balance equations, including the reaction kinetics as a function of the limiting substrates, which read in their simplest form where x is the biomass concentration, s the substrate concentration, h(s) the specific growth rate, D the dilution rate, and sin the substrate concentration in the inflow. The description of floc formation and settling remains the weakest part of AS models. The problem has been studied by a variety of approaches (see for reviews). Population balance models (PBM) describe floc aggregation and breakage and allow to compute the floc-size distribution as a function of time. Computational-fluid dynamics (CFD) simulators describe the hydrodynamics in the clarification tank and try to predict the settling properties of the flocs. Individualbased models (IBM) take both physicochemical and biological processes into account at the level of a single floc. These modeling approaches have in common a high-dimensional parameter space. Although these parameters can be identified from experiments, the resulting model is often too complex to provide insight in the governing mechanisms. Moreover, to compute the settling properties of the ensemble of interacting flocs in the clarifier, one has to combine a CFD with a PBM approach, which leads to even more intricate models. Instead of using advanced simulators, we propose to take the simple model (Eq. 1) as a starting point. In particular, we investigate how these equations are modified when the biomass is organized in flocs. We propose a PBM-like model where both the floc interactions (as in standard PBM), and the bacteria growth are included. This qualitative model is dx dt 1⁄4 hðsÞx Dx