Mathematical Modeling and Solution of the Moving-Boundary Problem Related to Substrate Diffusion and Reaction in Enzymatic Catalytic Particles

This study presents a transient mathematical model and its numerical solution for the moving-boundary problem related to substrate diffusion and reaction in enzymatic catalytic particles. The main focus is on bioreactor startup, where the initial substrate concentration inside the particles is zero, forming a dead core that shrinks over time and makes the catalytic effectiveness factor time-dependent. The substrate mass balance leads to a partial differential equation with a moving boundary, solved using the method of lines coupled with Newton’s method (MLN), implemented in Wolfram Mathematica (WM). The proposed approach was validated for zero- and first-order kinetics at steady state, whose analytical solutions are available. Compared to the method of orthogonal collocation on finite elements, the MLN offers advantages such as not requiring an initial concentration profile and simple implementation in WM. The results demonstrate that the proposed method provides accurate and physically consistent solutions, contributing to a better understanding of dead-core dynamics and supporting the design of heterogeneous bioreactors with immobilized enzymes.