A Mathematical Analysis of a Membrane Bioreactor Containing a Sludge Disintegration System

The activated sludge process is widely used to treat domestic and industrial wastewater. A significant drawback of this process is the production of sludge, the disposal of which can comprise a significant proportion of the total operating costs of a wastewater treatment plant. We analyze the steady-state operation of a membrane bioreactor system (MBR) incorporating a sludge disintegration unit (SDU) to reduce sludge production. We provide a qualitative understanding of the model by finding analytically the steady-state solutions of the model and determining its stability as a function of the residence time. In practice a target value of the mixed liquor suspended solids (MLSS) content within the membrane reactor is specified. Applying the mathematical technique of singularity theory we show that if the sludge disintegration factor is sufficiently high then the MLSS content is guaranteed to be below the target value. This model prediction, of key interest from a practical perspective, was not identified in the original investigation of this model, which relied upon numerical integration of the governing equations.