Capturing stirring–diffusion and product inhibition in fed-batch fermentation process: A distributed parameter modeling framework

Abstract

Agitation intensity exerts a decisive influence on product yield in microbial fermentation, yet most published work treats mixing effects qualitatively and lacks a quantitative link between stirring conditions and product synthesis. This study develops a distributed parameter system (DPS) model that couples partial differential equations (PDEs) with ordinary differential equations (ODEs) to capture how agitation shapes axial substrate gradients and product formation. Building on the Birol kinetics, the model introduces an axial diffusion coefficient $D_Z$ (m² s^$-1$) to translate stirring speed into mixing efficiency, and a product-inhibition constant to represent feedback from local product accumulation on microbial metabolism. Theoretical analysis establishes the well-posedness of the model, and numerical simulations illustrate that insufficient agitation leads to strong substrate gradients and reduced productivity, whereas stronger agitation promotes uniform mixing and higher yields. In addition, a laboratory visualization experiment confirmed the agitation-dependent attenuation of gradients, providing further support for the proposed model. Together, these results highlight the DPS model’s potential to guide the design of effective agitation strategies and improve the efficiency of large-scale fermentation processes.