Mathematical modeling by fractional calculus applied to separation processes

Abstract

Process modeling is a way of representing real problems through equations and while the traditional models of mass and heat transfer processes consider the transient term as a first-order derivative, fractional order modeling is applied to verify that the order of the derivative of the transient term would not be represented by a non-integer number. In this context, the objective of this work was to apply fractional calculus to the kinetics study of adsorption and extraction processes and compare the results obtained with traditional models. The Caputo derivative was used to solve the proposed equations, and the analyzed processes were the adsorption of glycerol for biodiesel at different temperatures and the solid–liquid extraction of antioxidant compounds from corn for different proportions of solvent. For the adsorption, the fractional order model was compared with the pseudo-first, pseudo-second, and first-order models, and the fractional order model represented adsorption satisfactorily, with method efficiencies of 0.9603 and 0.9433 for the temperatures of 45 °C and 60 °C, respectively, while the performance of traditional models depends on the mechanisms that control the process. In addition, the α values obtained with the fractional order model were less than 1, indicating that glycerol adsorption does not follow an integer order derivative and implies that the adsorption exhibits a level of complexity that is not fully characterized by traditional models. The extraction of antioxidant compounds was analyzed and compared with the first-order and So and MacDonald’s models. It was verified that the derivative order values for the fractional order model were close to 1 and, therefore, the behavior is similar to the first order model and with indications that diffusion occurs as described by Fick’s law of diffusion. Thus, the results indicated that fractional order modeling can be used efficiently to describe chemical engineering processes such as adsorption and extraction.