Maxwell-Stefan relations generalized

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. 

For binary systems the generalized Maxwell-Stefan relation reduces to ... 

Equations 2.3.17 are the generalized Maxwell-Stefan (GMS) relations and the are the Maxwell-Stefan diffusion coefficients we encountered earlier. These equations are more useful when expressed in terms of the molar fluxes... 

The complexity of the Maxwell-Stefan equations and the generalized Fick s law have lead many investigators to use simpler constitutive relations that avoid the mathematical complexities (specifically, the use of matrix algebra in applications). In this chapter we examine these effective diffusivity or pseudobinary approaches. 

These relations are called the generalized Maxwell-Stefan equations and are the inverted counterparts of the Pick diffusion equations (2.281). These two descriptions contain the same information and are interrelated as proven by Curtiss and Bird [18] [19] for dilute mono-atomic gas mixtures. 

Gibbs-Duhem restriction on the chemical potential (eq. 8.5-3). Eq.(8.5-5) is the generalized Maxwell-Stefan constitutive relation. However, such form is not useful to engineers for analysis purposes. To achieve this, we need to express the chemical potential in terms of mole fractions. This is done by using eq. (8.5-2) into the constitutive flux equation (8.5-5). 

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

Another attempt to correlate transport and self-diffusivities has been based on a generalization of the Stefan-Maxwell formulation of irreversible thermodynamics [111-113]. By introducing various sets of parameters describing the facility of exchange between two molecules of the same and of different species, the resulting equations are more complex than eqs 27 and 28 They may be shown, however, to include these relations as special cases... 

The optimal Reynolds number defines the operating conditions at which the cylindrical system performs a required heat and mass transport, and generates the minimum entropy. These expressions offer a thermodynamically optimum design. Some expressions for the entropy production in a multicomponent fluid take into account the coupling effects between heat and mass transfers. The resulting diffusion fluxes obey generalized Stefan-Maxwell relations including the effects of ordinary, forced, pressure, and thermal diffusion. 

According to the general law of Stefan and Maxwell the diffusion flux of each component (xi) is related to every concentration gradient (dc/dr) ... 

The theory of diffusion in the gas phase is well developed. In general the diffusion flux of a component i (N,) depends on all of the components. According to Stefan-Maxwell theory, the diffusion flux and the concentration gradient are governed by the matrix relation... 

The general relation for the coupled fluxes of diffusing species is given by the Stefan-Maxwell equations... 

Generally, the increased complexity of the model based on Stefan-Maxwell relations does not justily improvement in model accuracy, which in most cases is redundant. Complex equations can be tolerated in numerical modelling however, in anal3dical modelling, clarity and simplicity of the resulting expressions are of the highest priority. Below we will use a simple Pick s law of diffusion to describe species transport in GDLs and in catalyst layers. 

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