The Maxwell-Stefan Relations

Das Studium der Maxwell schen Abhandlung ist nicht leicht. 

Diffusion is the intermingling of the atoms or molecules of more than one species it is the inevitable result of the random motions of the individual molecules that are distributed throughout space. The development of a rigorous kinetic theory to describe this intermingling in gas mixtures is one of the major scientific achievements of the nineteenth century. A simplified kinetic theory of diffusion, adapted from Present (1958), is the main theme of Section 2.1. More rigorous (and complicated) developments are to be found in the books by Hirschfelder et al. (1964), Chapman and Cowling (1970), and Cunningham and Williams (1980). An extension to cover diffusion in nonideal fluids is developed thereafter. 

Your objectives in studying this section are to be able to  

Write down the Maxwell-Stefan equations for a binary system and for multicomponent systems. 

Define the concepts Maxwell-Stefan (MS) diffusivity and thermodynamic factor. 

Express the driving force for mass transfer in terms of mole fraction gradients for ideal and nonideal systems. 

Consider a binary mixture of ideal gases 1 and 2 at constant temperature and pressure. From a momentum balance describing collisions between molecules of species 1 and molecules of species 2, we obtain (Taylor and Krishna, 1993) 

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With all of the above assumptions the Maxwell-Stefan relations (Eq. 2.1.16) reduce to a system of first-order linear differential equations... 

The method of Taylor and Smith (1982) is a generalization of the method of Burghardt and Krupiczka for Stefan diffusion. We use the determinacy condition (Eq. 7.2.10) to eliminate the nth flux from the Maxwell-Stefan relations (Eq. 2.1.16) and combine the first n-1 equations in matrix form as... 

Chapter 1 serves to remind readers of the basic continuity relations for mass, momentum, and energy. Mass transfer fluxes and reference velocity frames are discussed here. Chapter 2 introduces the Maxwell-Stefan relations and, in many ways, is the cornerstone of the theoretical developments in this book. Chapter 2 includes (in Section 2.4) an introductory treatment of diffusion in electrolyte systems. The reader is referred to a dedicated text (e.g., Newman, 1991) for further reading. Chapter 3 introduces the familiar Fick s law for binary mixtures and generalizes it for multicomponent systems. The short section on transformations between fluxes in Section 1.2.1 is needed only to accompany the material in Section 3.2.2. Chapter 2 (The Maxwell-Stefan relations) and Chapter 3 (Fick s laws) can be presented in reverse order if this suits the tastes of the instructor. The material on irreversible thermodynamics in Section 2.3 could be omitted from a short introductory course or postponed until it is required for the treatment of diffusion in electrolyte systems (Section 2.4) and for the development of constitutive relations for simultaneous heat and mass transfer (Section 11.2). The section on irreversible thermodynamics in Chapter 3 should be studied in conjunction with the application of multicomponent diffusion theory in Section 5.6. 

We also feel that portions of the material in this book ought to be taught at the undergraduate level. We are thinking, in particular, of the materials in Section 2.1 (the Maxwell-Stefan relations for ideal gases). Section 2.2 (the Maxwell-Stefan equations for nonideal systems). Section 3.2 (the generalized Fick s law). Section 4.2 (estimation of multicomponent diffusion coefficients). Section 5.2 (multicomponent interaction effects), and Section 7.1 (definition of mass transfer coefficients) in addition to the theory of mass transfer in binary mixtures that is normally included in undergraduate courses. 

The corresponding combined mass flux form of the Maxwell-Stefan relation can be expressed like ... 

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