Maxwell-flux law

Extended Stefan-Maxwell constitutive laws for diffusion Eq. 4 resolve a number of fundamental problems presented by the Nemst-Planck transport formulation Eq. 1. A thermodynamically proper pair of fluxes and driving forces is used, guaranteeing that all the entropy generated by transport is taken into account. The symmetric formulation of Eq. 4 makes it unnecessary to identify a particular species as a solvent - every species in a solution is a solute on equal footing. Use of velocity differences reflects the physical criterion that the forces driving diffusion of species i relative to species j be invariant with respect to the convective velocity. Finally, all possible binary solute/solute interactions are quantified by distinct transport coefficients each species i in the solution has a diffusivity or mobility relative to every other species j, Djj or up, respectively. 

Figures 7(a)-(c) show a comparison between the numerically computed absorption flux and the absorption flux obtained from expression (31), using eqs (24), (30) and (34)-(37). From these figures it can be concluded that for both equal and different binary mass transfer coefficients absorption without reaction can be described well with eq. (24), whereas absorption with instantaneous reaction can be described well with eq. (30). If the Maxwell-Stefan theory is used to describe the mass transfer process, the enhancement factor obeys the same expression as the one obtained on the basis of Fick s law [eq. (35)]. Finally, from Figs 7(b) and 7(c) it appears that the use of an effective mass transfer coefficient m the Hatta number again produces satisfactory results.

For the diffusion flux (N] various approaches are possible, ranging from the complete Stefan-Maxwell set of equations to the simple law of Pick (7). The symbols of eqs. (l)-(2) are defined in Notation. 

The mass diffusive flux m, of Equation (3.2) generally depends on the operating conditions, such as reactant concentration, temperature and pressure and on the microstructure of material (porosity, tortuosity and pore size). Well established ways of describing the diffusion phenomenon in the SOFC electrodes are through either Fick s first law [21, 34. 48, 50, 51], or the Maxwell-Stefan equation [52-55], Some authors use more complex models, like for example the dusty-gas model [56] or other models derived from this [57, 58], A comparison between the three approaches is reported by Suwanwarangkul et al. [59], who concluded that the choice of the most appropriate model is very case-sensitive, and should be selected, according to the specific case under study. 

The droplet current / calculated by nucleation models represents a limit of initial new phase production. The initiation of condensed phase takes place rapidly once a critical supersaturation is achieved in a vapor. The phase change occurs in seconds or less, normally limited only by vapor diffusion to the surface. In many circumstances, we are concerned with the evolution of the particle size distribution well after the formation of new particles or the addition of new condensate to nuclei. When the growth or evaporation of particles is limited by vapor diffusion or molecular transport, the growth law is expressed in terms of vapor flux equation, given by Maxwell s theory, or... 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

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. 

The j° term denotes the ordinary concentration diffusion (i.e., multi-component mass diffusion). In general, the concentration diffusion contribution to the mass flux depends on the concentration gradients of all the substances present. However, in most reactor systems, containing a solvent and one or only a few solutes having relatively low concentrations, the binary form of Pick s law is considered a sufficient approximation of the diffusive fluxes. Nevertheless, for many reactive systems of interest there are situations where a multi-component closure (e.g., a Stefan-Maxwell equation formulated in terms... 

Wilke [103] proposed a simpler model for calculating the effective diffusion coefficients for diffusion of a species s into a multicomponent mixture of stagnant gases. For dilute gases the Maxwell-Stefan diffusion equation is reduced to a multicomponent diffusion flux model on the binary Pick s law form in which the binary diffusivity is substituted by an effective multicomponent diffusivity. The Wilke model derivation is examined in the sequel. 

An alternative to the complete Maxwell-Stefan model is the Wilke approximate formulation [103]. In this model the diffusion of species s in a multicomponent mixture is written in the form of Tick s law with an effective diffusion coefficient instead of the conventional binary molecular diffusion coefficient. Following the ideas of Wilke [103] we postulate that an equation for the combined mass flux of species s in a multicomponent mixture can be written as ... 

However, if convective transport of heat and species mass in porous catalyst pellets have to be taken into account simulating catal3dic reactor processes, either the Maxwell-Stefan mass flux equations (2.394) or dusty gas model for the mass fluxes (2.427) have to be used with a variable pressure driving force expressed in terms of mass fractions (2.426). The reason for this demand is that any viscous flow in the catalyst pores is driven by a pressure gradient induced by the potential non-uniform spatial species composition and temperature evolution created by the chemical reactions. The pressure gradient in porous media is usually related to the consistent viscous gas velocity through a correlation inspired by the Darcy s law [21] (see e.g., [5] [49] [89], p 197) ... 

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

Additionally, the higher viscosity of the organic phase makes it more likely that the diffusion resistance is predominant in the organic phase in respect to the aqueous one. A correct description is then to model diffusion inside the droplet with the Maxwell-Stefan diffusion law [39], where the chemical potential and not concentration differences represent the driving force. However, a problem up to now is to get information in respect to an exact value of the cross-diffusion coefficients. They represent the frictional coupling of the fluxes of the species involved and can only be neglected in very dilute systems. 

The effect of convection will be significant if we consider a higher temperature situation. Take the case where the temperature is 60 °C and the total pressure is 1 atm. The vapor pressure of benzene at this temperature is 400 Torr. Using the Stefan-Maxwell result (eq. 8.2-75), we calculate the evaporation flux as 1.06 x 10 moles/cm /hr, compared to 7.47 xlO " moles/cm /hr calculated by the Pick law equation. An error of nearly 30% underpredicted by the Pick s law shows the importance of the convection term in the Stefan-Maxwell equation. 

By using the Grahams law equation (8.6-22) into the Stefan-Maxwell equation (8.6-20a), we obtain the following equation expressing the flux in terms of concentration gradient for the component 1 ... 

In 1868, 12 years after Tick s definitive publication of his theory, James Clerk Maxwell published a paper on a different approach to studying the diffusivity of gases, hi 1871 Josef Stefan extended Maxwell s theory and anticipated multiconponent effects (Cussler. 2009). Although the Maxwell-Stefan theory has had many strong adherents in the more than 140 years since its development, it always seems to be playing catch-up to the earlier Fickian theory. Three perceived difficulties have prevented wider acceptance of the Maxwell-Stefan theory. First, the Fickian model is well-entrenched in textbooks and diffusivity data collections, and it works well for many binary systems. Second, the Maxwell-Stefan theory gives one fewer flux N than is needed to conpletely solve the problem. However, this is really no different than choosing a reference velocity for Tick s law, and, as will be shown later, for most... 

Maxwell s rule n. A law stating that every part of an electric circuit is acted upon by a force tending to move it in such a direction as to enclose the maximum amount of magnetic flux. 

In catalytic cracking the gas oil feed reacts to much lighter compounds, which causes a high convective flux from the catalyst surface to the bulk of the fluid. Therefore Fick s diffusion law is not applicable (assumes equimolar counter-diffusion) in the mass transfer calculations and as a result rigorous Maxwell-Stefan equations must be used. Due to the... 

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