Last modified: 2014-03-28

#### Abstract

Biofilms are layered bacterial communities, attached to a surface in an aqueous environment, constructing a three-dimensional structure that is significantly different from the surrounding that free floating bacteria are found in. Bacteria within biofilms act together as a macrostructure, exhibiting typical characteristics such as a diffusion-limited substrate consumption and a strong antimicrobial resistance. The latter is often the reason why biofilms are so difficult to eradicate, which is a critical issue in cases when biofilms are harmful to their surroundings, for example in dental plaque. In wastewater treatment, however, biofilms are considered beneficial as they are used in biological treatment processes for degradation and collection of organic matter as well as nitrogen and phosphorus. The bacteria grow by consumption of a substrate, which is thereby removed from the wastewater, and produce a compound that is either harmless to the environment or that proceeds through further treatment, before the treated water is released into a receiving water body.

Mathematical models of wastewater treatment systems are useful tools for process understanding, design, control and optimization and can prevent lengthy empirical studies. Contrary to the reality of a biofilm reactor, in which a certain amount of suspended biomass always remains present due to erosion from the biofilm, most biofilm reactor models do not include the suspended biomass, assuming its contribution to the process performance is negligible.

In this work, we focus on a biofilm reactor with concurrent suspended growth and investigate mathematically the optimal substrate removal in the reactor with respect to the amount of removed substrate and with respect to treatment process duration. For this purpose we assume a reactor setup where the wastewater is fed from a storage reactor into a biological treatment reactor. The resulting two-objective optimal control problem is constructed with the flow rate between the reactors as the selected control and the treatment reactor is modeled by a system of three ordinary differential equations, which indirectly contain a two-point boundary value problem.

Due to the singularity of the optimal control problem, it is impractical to determine its solution in the class of measurable functions and unfeasible to implement in reality. By instead choosing a class of off-on functions, motivated by the underlying biological process, we solve a simpler problem of reactor performance optimization. The off-on control functions initially have a no-flow period before switching to a constant flow rate that empties the storage reactor. For this optimal control problem we approximate the Pareto Front numerically and study the system behavior and its dependence on reactor and initial data. In general, we find that the limited potential to improve reactor performance through different control strategies is mainly due to an initial transient period during which the bacteria adapt to the environmental conditions in the reactor. The determination of the length of the transient period depends strongly on the initial state of the dynamic system, which is, thus, often unknown in real applications, wherefore the efficiency of reactor optimization, compared to the uncontrolled system with constant flow rate, is limited.