Estudo de modelos fenomenológicos anômalos nos processos de transferência de calor e de massa

Abstract

Traditionally, Fourier and Fick Laws are used for the characterization of heat and mass transfer models, respectively. Despite producing good results in many cases, there are some situations in which these laws do not have a good precision. This is observed when the scales of spatial and time variables are very low and the gradient of temperature and concentration are very high. In these cases, the anomalous phenomena is characterized. One of strategies that have been used to generalize both laws corresponds to use of models with non-integer order derivatives. In this work, one model that presents first and second order temporal derivatives, one time delay factor and spatial derivatives, being one with first order and other with fractional order, is considered. To solve the model, the Legendre Pseudo-Spectral Method and the Fractional Finite Difference Method were proposed. Each one considers a definition for fractional derivative. Both methods were applied in purely mathematic problems, for validation purposes, and heat and mass transfer problems. To estimate the parameters of the model, one inverse heat transfer problem in the skin using synthetic experimental data and three inverse mass transfer problems using real experimental data were formulated and solved by considering the Differential Evolution Algorithm. To guarantee the dimensional consistency of fractional diffusive terms, a correction factor was proposed. Both strategies used to solve direct problems demonstrated a good accuracy when approximate and exact solutions were compared. From the physical point of view, it was possible to verify the influence of model parameters on obtained temperature and concentration profiles. The obtained results considering all inverse problems are qualitatively coherent in relation with those presented in specialized literature and indicate that the proposed spatial fractional hyperbolic model configure a good alternative to describe the anomalous phenomena in heat and mass transfer.