Analysis of photobioreactors in series

A photobioreactor (PBR) contains microalgae which under illumination consume carbon dioxide and substrate dissolved in water, and produce oxygen. The process is used in water recovery resource facilities with a continuous flow of wastewaster through the PBR. With several PBRs in series the reduction of substrate can be improved. This paper contains a thorough analysis of a model of PBRs in series, where each PBR is modelled with a system of three ordinary differential equations for the concentrations of dissolved substrate and biomass (algae), and the internal cell quota of substrate to biomass. Each PBR has a certain volume and irradiation. The absorption rate of substrate into the cells is modelled with Monod kinetics, whereas the biomass growth rate is modelled with Droop kinetics, in which both a minimum and a maximum internal cell quota are assumed. The main result is that the model has a unique stable steady-state solution with algae in all PBRs. Another stable steady-state solution is the wash-out solution with no algae in the system. Other steady-state solutions are combinations of these two with no algae in some of the first PBRs and algae in the rest of the PBRs in the series. Conditions on the illumination, volumetric flow and volumes of the PBRs are given for the respective solution. Numerical solutions illustrate the theoretical results and indicate further properties.