%0 Journal Article
%T Analytical Solution of Non-Isothermal Diffusion-Reaction Processes and Effectiveness Factors
%A Vembu Ananthaswamy
%A Lakshmanan Rajendran
%J ISRN Physical Chemistry
%D 2013
%R 10.1155/2013/487240
%X The mathematical modeling of nonlinear boundary value problems in catalytically chemical reactor is discussed. In this paper, we obtain the approximate analytical solution and the effectiveness factors for the evolution of single-step transformations under non-isothermal conditions using homotopy perturbation method. We have applied it to many reaction models and obtained very simple analytical expressions for the shape of the corresponding transformation rate peaks. These analytical solutions represent a significant simplification of the systemĄ¯s description allowing easy curve fitting to experiment. The accuracy achieved with our method is checked against several reaction models and numerical results. A satisfactory agreement is noted. 1. Introduction Non-isothermal systems, where reaction and diffusion take place, are typical in the chemical process industry [1] and also in biological systems [2¨C4]. The chemical reaction is always central in these systems, because the rate of the reaction often will determine how fast chemicals can be produced. A high rate can be realized when the reaction is far from equilibrium. But an operation far from equilibrium is also an operation in which the energy dissipation is large. With the present interest to save valuable resources, chemical reactors should be studied also from the perspective of obtaining a more energy-efficient operation, in addition to maintaining the production of chemicals. In biological systems, one may expect that energy efficiency is an issue of survival, especially under harsh conditions [5]. In such cases and probably many others, a thermodynamic description will be important to understand the transport phenomena involved [4, 6]. Studies of minimum energy dissipation start with an expression for the entropy production [7¨C9]. Chemical reactions are inherently non-linear processes, and are most successfully described in the field of reaction kinetics by the law of mass action [10, 11]. The reaction rate is not commonly expressed as a function of the reaction Gibbs energy. This is not surprising, because classical non-equilibrium thermodynamics [12, 13] assumes a linear relation between these two variables, and experimental evidence indicates that this is only correct very close to chemical equilibrium. The first to address this problem successfully was Kramers [14] who described the reaction as a diffusion process along a reaction coordinate. The extension in the context of non-equilibrium thermodynamics was first proposed by Prigogine and Mazur [15¨C17]. By integrating over these variables to
%U http://www.hindawi.com/journals/isrn.physical.chemistry/2013/487240/