On the asymptotic stability of advection-diffusion equations of mass transport in a bubble column bioreactor

This study presents an asymptotic stability analysis of a model of a bioreactor converting carbon monoxide (CO) gas into ethanol through a C. autoethanogenum biocatalyst. The configuration is a bubble column reactor with co-current gas-liquid flows where gas feed is introduced by a gas distributor placed at the bottom of the column. A pure culture of C. autoethanogenum is subsequently injected at the bottom of the column; therein, cells are dispersed in the liquid and consume the dissolved gas and release by-products such as ethanol and acetic acid. Cellular growth and byproduct secretion are affected by spatially varying dissolved gas concentrations due to advection-diffusion mass transports which are induced by the effect of the injection pressure and gravitational force. The model accounts for four species representing the biomass, the CO substrate in the liquid phase, and two by-products - ethanol and acetic acid. Substrate dynamics is described by an advection-diffusion equation. We investigate the asymptotic stability of the biomass dynamics that is a requirement for the system’s controllability, i.e. for the possibility to steer a dynamical system from an arbitrary initial state to an arbitrary final state using a set of controls. The concept of stability of the controls is extremely relevant to controllability since almost every workable control system is designed to be stable. If a control system is not stable, it is usually of no use in practice in industrial processes. In the case of a bioreactor, the control is the biomass and controllability is the possibility of modulating through this control the ethanol production. We present a test for asymptotic stability, based on the analysis of the properties of the dynamic function defining its role as storage function.