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Revista mexicana de ingeniería química

Print version ISSN 1665-2738

Rev. Mex. Ing. Quím vol.6 n.3 México Dec. 2007

 

Fenómenos de transporte

 

Análisis de problemas de transporte de masa y reacción mediante funciones de green

 

Analysis of mass transport and reaction problems using green's functions

 

F. J. Valdés–Parada*, J. Alvarez–Ramírez y J. A. Ochoa–Tapia

 

Departamento de Ingeniería de Procesos e Hidráulica, Universidad Autónoma Metropolitana–Iztapalapa, Apartado Postal 55–534, 09340 México D.F., México. * Autor para la correspondencia: E–mail: iqfv@xanum.uam.mx Tel. 58 04 46 48 ext 219, Fax 58 04 49 00

 

Recibido 10 de Septiembre 2007
Aceptado 31 de Octubre 2007

 

Resumen

Se presenta una metodología para resolver problemas típicos de transferencia de masa y reacción en ingeniería de las reacciones químicas en términos de funciones de Green. La idea fundamental consiste en invertir analíticamente un operador diferencial a partir de la solución de un problema de valor a la frontera asociado al problema original de transporte y reacción. La variable dependiente queda expresada en función de la solución de dicho problema asociado que es la función de Green. Entre las ventajas que presenta esta metodología son el suavizado de los errores de redondeo así como la incorporación en forma exacta de las condiciones de frontera. Un requisito indispensable para la aplicación de la metodología es que el operador diferencial sea autoadjunto. Para ilustrar la habilidad del método, se estudian problemas de difusión y reacción en una partícula catalítica involucrando cinéticas tanto lineales como no lineales; además se analiza el efecto de las resistencias externas a la transferencia de masa y se discute la aplicación a problemas de transporte no isotérmico, convectivo, en estrado transitorio y multicomponentes. Las predicciones se comparan con las que resultan de la solución numérica mediante diferencias finitas. El análisis se lleva a cabo en términos de parámetros tanto numéricos (tiempo de cómputo, tamaño de malla) como de transporte (módulo de Thiele, número de Biot).

Palabras clave: transporte de masa y reacción, función de Green, diferencias finitas, métodos numéricos.

 

Abstract

A methodology to solve typical problems of mass transfer and chemical reaction engineering in terms of Green's functions is presented. The fundamental idea consists on analytically inverting a differential operator by means of the solution of a boundary value problem associated to the original transport and reaction problem. The dependent variable is expressed then as function of the solution of such associated problem, which is the Green's function. Among the advantages that this methodology presents are the smoothing of round–off errors as well as the exact incorporation of boundary conditions. A mandatory requirement for the application of this methodology is that the differential operator must me self–adjoint. To illustrate the potential of the method, diffusion and reaction problems are studied in a catalytic particle involving both linear and non–linear reaction kinetics; in addition, the effect of the external mass transfer resistances is analyzed and the application to non–isothermal, convective, transient and multicomponent problems is discussed. The predictions are compared with those resulting from the numeric solution using finite differences. The analysis is carried out in terms of both numeric (computer time, mesh size) and transport (Thiele modulus, Biot number) parameters.

Keywords: mass transport and reaction, Green's function, finite differences, numeric methods.

 

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Agradecimientos

FJVP desea agradecer al Consejo Nacional de Ciencia y Tecnología (CONACyT) por la beca de posdoctorado otorgada a través del convenio 49705–Y.

 

Referencias

Alvarez–Ramirez, J., Valdés–Parada, F.J., Alvarez, J., Ochoa–Tapia, J.A. (2007). A Green's function formulation for finite difference schemes. Chemical Engineering Science 62, 3083–3091.         [ Links ]

Amundson, N.R., Schilson, R.E. (1961). Intraparticle reaction and conduction in porous catalysts–I. Single reactions. Chemical Engineering Science 13, 226–236.         [ Links ]

Aris, R. (1975). The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Vol. I. Clarendon Press, Oxford.         [ Links ]

Axelsson, O., Gololobov S.V. (2003). A combined method of local Green's functions and central difference method for singularly perturbed convection–diffusion problems. Journal of Computational and Applied Mathematics 161, 245–257.         [ Links ]

Denn, M.M., Aris, R. (1965a). Green's Functions and Optimal Systems. Necessary Conditions and Iterative Technique. Industrial and Engineering Fundamentals 4, 7–16.         [ Links ]

Denn, M.M., Aris, R. (1965b). Green's functions and optimal systems. The gradient direction in decision space. Industrial and Engineering Fundamentals 4, 213–222.         [ Links ]

Denn, M.M., Aris, R. (1965c). Green's Functions and Optimal Systems. Complex Interconnected Structures. Industrial and Engineering Fundamentals 4, 248–257.         [ Links ]

Dixit, R.S., Tavlarides L.L. (1982). Integral analysis of Fischer–Tropsch synthesis reactions in a catalyst pellet. Chemical Engineering Science 37, 539–544.         [ Links ]

Greenberg, M.D. (1971). Application of Green's Functions in Science and Engineering. Prentice–Hall, New York.         [ Links ]

Kesten, A.S. (1969). An integral equation method for evaluating the effects of film and pore diffusion of heat and mass on reaction rates in porous catalyst particles. AIChE Journal 15, 128–131.         [ Links ]

Haberman, R. (2004). Applied Partial Differential Equations, 4th edition, Prentice Hall, USA.         [ Links ]

Mishra, M., Peiperl L., Reuven Y., Rabitz H., Yetter R.A., Smooke M.D. (1991). Use of Green's functions for the analysis of dynamic couplings: Some examples from chemical kinetics and quantum dynamics. Journal of Physical Chemistry 95, 3109–3118.         [ Links ]

Mishra M., Yetter R., Reuven Y., Rabitz H., Smooke M.D. (1994). On the role of transport in the combustion kinetics of a steady–state premixed laminar CO + H2+ O2 flame. International Journal of Chemical Kinetics 26, 437–453.         [ Links ]

Mukkavilli S., Tavlarides, L.T. & Wittmann, Ch.V. (1987a). Integral method of analysis for chemical reaction in a nonisothermal finite cylindrical catalyst pellet–I. Dirichlet problem. Chemical Engineering Science 42, 27–33.         [ Links ]

Mukkavilli S., Tavlarides, L.T. & Wittmann, Ch.V. (1987b) Integral method of analysis for chemical reaction in a nonisothermal finite cylindrical catalyst pellet–II. Robin problem. Chemical Engineering Science 42, 35–40.         [ Links ]

Valdés–Parada, F.J., Sales–Cruz, M., Ochoa–Tapia, J.A., Alvarez–Ramirez, J. (2007). On Green's function methods to solve nonlinear reaction–diffusion systems. Computers and Chemical Engineering, en prensa.         [ Links ]

 

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