Servicios Personalizados
Revista
Articulo
Indicadores
- Citado por SciELO
- Accesos
Links relacionados
- Similares en SciELO
Compartir
Revista mexicana de ingeniería química
versión impresa ISSN 1665-2738
Rev. Mex. Ing. Quím vol.8 no.3 Ciudad de México dic. 2009
Derivation and application of the StefanMaxwell equations
Desarrollo y aplicación de las ecuaciones de StefanMaxwell
Stephen Whitaker*
Department of Chemical Engineering & Materials Science University of California at Davis. * Corresponding author. Email: whitaker@mcn.org
Received 5 of June 2009
Accepted 9 of November 2009
Abstract
The StefanMaxwell equations represent a special form of the species momentum equations that are used to determine species velocities. These species velocities appear in the species continuity equations that are used to predict species concentrations. These concentrations are required, in conjunction with concepts from thermodynamics and chemical kinetics, to calculate rates of adsorption/desorption, rates of interfacial mass transfer, and rates of chemical reaction. These processes are central issues in the discipline of chemical engineering.
In this paper we first outline a derivation of the species momentum equations and indicate how they simplify to the StefanMaxwell equations. We then examine three important forms of the species continuity equation in terms of three different diffusive fluxes that are obtained from the StefanMaxwell equations. Next we examine the structure of the species continuity equations for binary systems and then we examine some special forms associated with Ncomponent systems. Finally the general Ncomponent system is analyzed using the mixedmode diffusive flux and matrix methods.
Keywords: continuum mechanics, kinetic theory, multicomponent diffusion.
Resumen
Las ecuaciones de StefanMaxwell representan una forma especial de las ecuaciones de cantidad de movimiento de especies que son usadas para determinar las velocidades de especies. Estas velocidades de especies aparecen en las ecuaciones de continuidad de especies que son usadas para predecir las concentraciones de especies. Estas concentraciones son requeridas, en conjunción con los conceptos de termodinámica y cinética química, para calcular las velocidades de adsorción/desorción, las velocidades de transferencia de masa interfacial, y las velocidades de reacción química. Estos procesos son elementos centrales en la disciplina de la ingeniería química.
En este artículo presentamos primeramente un desarrollo de las ecuaciones de cantidad de movimiento de especies e indicamos como se simplifican a las ecuaciones de StefanMaxwell. Posteriormente examinamos tres formas importantes de la ecuación de continuidad de especies en términos de tres diferentes fluxes difusivos que se obtienen de las ecuaciones de StefanMaxwell. Más adelante examinamos la estructura de las ecuaciones de continuidad de especies para sistema binarios y examinamos algunas formas especiales asociados con sistemas de Ncomponentes. Finalmente se analiza el sistema general de Ncomponentes usando métodos matriciales y de flux difusivo de modo mixto.
Palabras clave: mecánica del continuo, teoría cinética, difusión multicomponente.
DESCARGAR ARTÍCULO EN FORMATO PDF
Acknowledgment
This paper grew out of a presentation at the Second International Seminar on Trends in Chemical Engineering, the XXI Century, Mexico City, January 28 29, 2008. The encouragement of students from Puebla to prepare a more complete discussion of the StefanMaxwell equations is greatly appreciated. In addition, the thoughtful comments of Francois MathieuPotvin helped to clarify some of the issues treated in this work. Finally, the comments of Professor R.B. Bird have clarified my understanding of the complex process of multicomponent mass transfer.
References
Aris, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, PrenticeHall, Englewood Cliffs, New Jersey. [ Links ]
Bearman, R.J. and Kirkwood, J.G. (1958). Statistical mechanics of transport Processes. XI Equations of transport in Multicomponent systems. Journal of Chemical Physics 28, 136145. [ Links ]
Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (1960). Transport Phenomena, First Edition. John Wiley and Sons, Inc., New York. [ Links ]
Bird, R.B. (1995) personal communication. [ Links ]
Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena, Second Edition. John Wiley and Sons, Inc., New York. [ Links ]
Bird, R.B. (2009). Notes for the 2nd edition of Transport Phenomena, http://www.engr.wisc.edu/che/faculty/bird_byron.html. [ Links ]
Birkhoff, G. (1960). Hydrodynamics, A Study in Logic, Fact, and Similitude, Princeton University Press, Princeton, New Jersey. [ Links ]
Chapman, S. and Cowling, T.G. (1939). The Mathematical Theory of Nonuniform Gases, First Edition, Cambridge University Press. [ Links ]
Chapman, S. and Cowling, T.G. (1970). The Mathematical Theory of Nonuniform Gases, Third Edition, Cambridge University Press. [ Links ]
Curtiss, C.F. and Bird, R.B. (1996). Multicomponent diffusion in polymeric liquids. Proceedings of the National Academy of Sciences USA 93, 74407445. [ Links ]
Curtiss, C.F. and Bird, R.B. (1999). Multicomponent diffusion. Industrial and Engineering Chemistry Research 38, 25152522. [ Links ]
Deen, W. M. (1998). Analysis of Transport Phenomena. Oxford University Press, New York. [ Links ]
Gibbs, J.W. (1928). The Collected Works of J. Willard Gibbs, Volume I: Thermodynamics, Longmans, Green and Co., New York. [ Links ]
Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B. (1954). Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., New York. [ Links ]
Quintard, M., Bletzacker, L., Chenu, D. and Whitaker, S. (2006). Nonlinear, multicomponent mass transfer in porous media, Chemical Engineering Science 61, 26432669. [ Links ]
Rutten, Ph.W.M. (1992). Diffusion in Liquids (PhD thesis), Delft University Press, The Netherlands. [ Links ]
Serrin, J. (1959). Mathematical Principles of Classical Fluid Mechanics, in Handbuch der Physik, Vol. VIII, Part 1, edited by S. Flugge and C. Truesdell, Springer Verlag, New York. [ Links ]
Slattery, J.C. (1999). Advanced Transport Phenomena, Cambridge University Press, Cambridge. [ Links ]
Stein, S.K. and Barcellos, A. (1992). Calculus and Analytic Geometry, McGrawHill, Inc., New York. [ Links ]
Truesdell, C. and Toupin, R. (1960). The Classical Field Theories, in Handbuch der Physik, Vol. III, Part 1, edited by S. Flugge, Springer Verlag, New York. [ Links ]
Truesdell, C. (1962). Mechanical basis of diffusion, Journal of Chemical Physics 37, 23362344 [ Links ]
Truesdell, C. (1968). Essays in the History of Mechanics, SpringerVerlag, New York. [ Links ]
Truesdell, C. (1969). Rational Thermodynamics, McGrawHill Book Company, New York. [ Links ]
Truesdell, C. (1971). The Tragicomedy of Classical Thermodynamics, SpringerVerlag, New York. [ Links ]
Whitaker, S. and Pigford, R.L. (1958). Thermal diffusion in liquids. Measurements and a molecular model. Industrial and Engineering Chemistry 50, 10261032. [ Links ]
Whitaker, S. (1981). Introduction to Fluid Mechanics, R.E. Krieger Pub. Co., Malabar, Florida. [ Links ]
Whitaker, S. (1986). Transport processes with heterogeneous reaction, pages 1 to 94 in Concepts and Design of Chemical Reactors, edited by S. Whitaker and A.E. Cassano, Gordon and Breach Publishers, New York. [ Links ]
Whitaker, S. (1988). Levels of simplification: The use of assumptions, restrictions, and constraints in engineering analysis. Chemical Engineering Education 22, 104108. [ Links ]
Whitaker, S. (1989). Heat transfer in catalytic packed bed reactors, in Handbook of Heat and Mass Transfer, Vol. 3, Chapter 10, Catalysis, Kinetics & Reactor Engineering, edited by N.P. Cheremisinoff, Gulf Publishers, Matawan, New Jersey. [ Links ]
Whitaker, S. (1999). The Method of Volume Averaging, Kluwer Academic Publishers, Dordrecht. [ Links ]
Whitaker, S. (2009a). Newton's laws, Euler's laws, and the speed of light. Chemical Engineering Education, Spring. [ Links ]
Whitaker, S. (2009b). Chemical engineering education: Making connections at interfaces, Revista Mexicana de Ingeniería Química 8, 132. [ Links ]