A theoretical and computational approach for the behaviour of heterogeneous reaction-diffusion process in a batch reactor with an immobilised enzyme (Penicillin G Acylase)

This study presents a comprehensive theoretical and computational approach for analysing heterogeneous reaction-diffusion processes in a batch reactor with immobilised Penicillin G Acylase (PGA). We develop a mathematical model to describe the complex interplay between reaction kinetics and mass transport phenomena within porous catalyst particles. The governing partial differential equations incorporate multiple inhibition mechanisms such as uncompetitive inhibition by Penicillin G (Pen-G), competitive inhibition by Phenyl Acetic Acid (PAA), and non-competitive inhibition by 6-Aminopenicillanic Acid (6-APA). Using the method Laplace transform, we derive analytical solutions for the dimensionless concentration profiles of substrate and products under both steady and unsteady state conditions. A convergence analysis confirms numerical stability of our solution system, with optimal convergence achieved. The mean integrated effectiveness factor (MIEF) is calculated to evaluate catalytic efficiency, revealing that diffusion limitations significantly impact reaction performance. Our sensitivity analysis demonstrates that the effective diffusion coefficient is the predominant parameter affecting catalyst efficiency, with a normalised sensitivity value of 100%, while kinetic parameters show minimal influence (approximately 3.6%). The model successfully acquires the spatial and temporal evolution of concentration gradients within the catalyst, showing how varying Thiele moduli influence substrate consumption and product formation. The analysis further establishes explicit design guidelines for minimising diffusion limitations, demonstrating that catalyst transport properties dominate performance under diffusion-limited conditions. This theoretical framework provides valuable insights for optimising immobilised enzyme systems in pharmaceutical applications, particularly for β-lactam antibiotic production.