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On the solution of a mathematical model for the dynamical analysis of the chemical reaction

Abstract:

In this paper, we present a qualitative study of the solutions of a mathematical model that is formulated to analyze the dynamical behavior of the exothermic and reversible chemical reaction, in a catalytic fixed bed reactor, adiabatically operated, and in presence of vanadium pentoxide . The model is a Cauchy problem for two coupled non-linear ordinary differential equations. These equations are coupled through the sulfur dioxide conversion, the temperature of the system (chemical reaction and chemical reactor) and its characteristic physicochemical parameters. We prove that the Cauchy problem has a unique solution (system orbits) for every initial condition that belongs to the domain for the directional field of the problem. We also show that the system orbits tend asymptotically to some stationary state located on an attracting manifold, embedded on the phase plane, when the time is large enough. These theoretical results allow us to describe the dynamic of a case given in the literature, in which it is reported the value of the physicochemical parameters, temperature ranges, and the reachable conversion levels in the industry. The dynamical behavior is as expected on the phase plane, and numerical results show that temperature changes of the system cause significantly changes in the conversion from to sulfur trioxide, when time evolves.