Modelagem matemática e computacional do modelo de monod fracionário no estudo de biocombustíveis para possíveis aplicação em foguetes

Abstract

This dissertation addresses the mathematical and computational modeling of bioprocesses oriented toward biofuel production, with potential applications to the study of rocket propellants. The work demonstrates the limitations of the classic Monod kinetic model and proposes an innovative approach using fractional calculus to describe the kinetics of reactions in biological systems. The introduction of fractional-order derivatives, within the Riemann-Liouville and Caputo formulations, allows the incorporation of memory effects and non-locality, which are inherent characteristics of complex bioprocesses. The general objective consists of the creation and computational application of a fractional Monod model for the analysis and optimization of biofuel production. The adopted methodology includes implementation in the Python language using the Modified Iterative Regularized Finite Difference Method (MIRDF), designed for the numerical resolution of non-linear fractional differential equations. Simulations were carried out in a batch reactor configuration, with parameters μmax = 0.009 h−1 and Ks = 4 × 10−8 mg/L. The results demonstrate that the fractional model, with order α < 1, describes microbial growth phases with greater fidelity compared to the classic model (α = 1.0), producing smoother transition curves between the lag and exponential phases and incorporating the historical dependence of the system. The numerical validation of the MIRDF, grounded on linear test problems for rigorous convergence analysis before application to the coupled nonlinear Monod system, yielded absolute errors on the order of 10−7, confirming the precision and robustness of the method under the conditions of Banach’s Fixed-Point Theorem. Comparative analysis with similar studies shows that the fractional parameter α acts as a calibration factor, expanding the predictive capacity of the model. It is concluded that fractional calculus constitutes an effective engineering tool for optimizing bioprocesses aimed at producing highperformance renewable propellants for aerospace applications, contributing to the development of clean energy sources in the space sector