q-formalismo na dinâmica adaptativa por Extremum Seeking e na Termodinâmica de eletrólitos no contexto da digestão anaeróbia

Abstract

Inspired by the mathematical works developed in Nonadditive Statistical Mechanics, with their functions, algebra and trigonometry, extensively applied to complex systems, are realized two proposals in this thesis which imply in consequences in the context of the non- linear dynamic system and in the thermodynamic of electrolytes, that orbit surround the anaerobic digestion in reactors. The first contribution talks about the proposition of an algorithm extremum seeking modified by a function from 𝑞-trigonometry of Borges with consequence in the increase of the speed in the search of the extreme and the minimum of the external disturbances. This result is analytically demonstrated by the presentation of the asymptotic stability according to Lyapunov. The second proposal, is in the context of the long-range interaction context typically presented by the behavior in electrolyte solutions and modelled by the potential of Debye-Hückel, which this potential is generalized by the 𝑞-exponential function, which allows, in a simple way, to represent the superficial tension properties and mean ionic activity coefficient, important to many applications in chemical, biological and environmental systems. In the investigation of thermodynamic hypothesis simplifying adopted in the ADM1 to the balance of the biomethane, it is verified that they are enough to represent the system of the study case, different from the behavior of the carbon dioxide, which is dependable from the chemical and electrolyte equilibrium reactions. In addition to that, it is confirmed the increase of velocity, the breadth of the domain and the reduce of the residual disturbance of the application of the generalized extremum seeking algorithm. Lastly, it is demonstrated how the potential of generalized Debye-Hückel is capable of estimate properties such as the superficial tension and the average of mean ionic activity coefficient to the wide ranges of ionic concentrations.