Fenómenos de transporte

 

Solution approach with green's functions for predicting the concentration of the slurry within a stirred tank reactor with nonlinear kinetics

 

Método de solución con funciones de Green para predecir la concentración de lodos dentro de un reactor tanque agitado con cinética no lineal

 

J. A. Ochoa-Tapia* and F.J. Valdés -Parada

 

Departamento de Ingeniería de Procesos e Hidráulica. División de Ciencias Básicas e Ingeniería. Universidad Autónoma Metropolitana-Iztapalapa. Av. San Rafael Atlixco 186 col. Vicentina, CP. 09340, México D.F., México. * Corresponding author. E-mail: jaot@xanum.uam.mx Tel. 58-04-46-00.

 

Received July 7 , 2014.
Accepted September 24, 2014.

 

Abstract

The purpose of this work is to set the basis for a numerical scheme to solve the model that describes the diffusion and reaction, with nonlinear kinetics, in the dispersed catalytic pellets part of the slurry contained in a reactor tank. The method presented is based on the use of Green's functions for the solution of the linear problem. However, different analytical solution approaches can lead to identical expressions for the solution; some of these results are compared and discussed. The numerical solution for the nonlinear case relies on the use of an iterative procedure. At this point, it is evident that the main drawback of the method proposed for the solution of the nonlinear transient problem is the infinite Fourier series that represent the Green's function. For such reason, the presented method is also used to obtain fluid and pellet concentration profiles for the quasi-steady state and steady-state cases. The resulting expressions for such two simpler cases are used to predict the concentration profiles that are also compared with those resulting from the numerical solution of the problem using finite differences. The good agreement of the predictions indicates that more compact expressions for the Green's function will improve the efficiency of the new numerical scheme.

Keywords: Green's function, analytical solution, iterative scheme, nonlinear kinetics, stirred tank reactor.

 

Resumen

El propósito de este trabajo es sentar las bases de un esquema numérico para resolver el modelo que describe la difusión y reacción, con cinética no lineal, en la parte del lodo que contiene pellets catalíticos dispersos en un reactor tanque agitado. El método presentado se basa en el uso de funciones de Green para la solución del problema lineal. Sin embargo, diferentes métodos de solución analítica pueden llevar a expresiones idénticas de la solución; algunos de estos métodos son comparados y discutidos. La solución numérica del caso no lineal se basa en el uso de un procedimiento iterativo. En este punto, es evidente que la principal desventaja del método propuesto para la solución del problema no lineal transitorio son las series de Fourier infinitas que representan a la función de Green. Por estas razones, el método presentado es usado también para obtener los perfiles de concentración en el fluido y las partículas para los casos de estados cuasi-estacionario y estacionario. Las expresiones resultantes para estos casos más simples se usan para predecir los perfiles de concentración que son comparados con los resultantes de la solución numérica usando diferencias finitas. La buena concordancia de las predicciones indica que expresiones más compactas para las funciones de Green mejorarán la eficiencia del esquema numérico.

Palabras clave: funciones de Green, solución analítica, método iterativo, cinética no lineal, reactor tanque agitado.

 

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Acknowledgments

FVP expresses his gratitude to Fondo Sectorial de Investigación para la educación from CONACyT (Project number: 12511908; Arrangement number: 112087) for the financial aid provided.

 

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