Driving Forces and Fluxes for Diffusion

In general, the properties of crystals and other types of materials, such as composites, vary with direction (i.e., macroscopic materials properties such as mass diffusivity and electrical conductivity will generally be anisotropic). It is possible to generalize the isotropic relations between driving forces and fluxes to account for... 

Clays are generally considered to be effective barriers for flow of water and solutes due to their low permeability and high ion adsorption capacity. However, as environmental criteria for the emission of contaminants and water from clay barriers become increasingly stringent, it is crucial to be aware of all relevant driving forces and fluxes and to take them into account in model assessments. In this respect the processes of chemical and electro-osmosis may not be neglected in clayey materials of hydraulic conductivity < 10-9 m/s [7], At these low conductivities the surface charge of the clay particles and the counter-ion accumulation in diffuse double layers enable explanation and quantification of osmotic processes and semi-permeability in clays [1],... 

The basic forms of transport are sununarised in figure V - 2. In die case of multi-component mixtures, fluxes often caimot be described by simple phenonenological equations because the driving forces and fluxes are coupled. In practice, this means that the individual components do not permeate independently from each other. For example a pressure difference across the membrane not only results in a solvent flux but also leads to a mass flux and the development of a solute concentration gradient. On the other hand, a concencrauon gradient not only results in diffusive mass transfer but also leads to a buildup of hydrostatic pressure. 

The diffusion model and the hydraulic permeation model differ decisively in their predictions of water content profiles and critical current densities. The origin of this discrepancy is the difference in the functions D (T) and /Cp (T). This point was illustrated in Eikerling et al., where both flux terms occurring in Equation (6.46) were converted into flux terms with gradients in water content (i.e., VA) as the driving force and effective transport coefficients for diffusion, A), and hydraulic permeation,... 

The approach pursued in this and the next chapter is focused on the common mathematical characteristics of boundary processes. Most of the necessary mathematics has been developed in Chapter 18. Yet, from a physical point of view, many different driving forces are responsible for the transfer of mass. For instance, air-water exchange (Chapter 20), described as either bottleneck or diffusive boundary, is controlled by the turbulent energy flux produced by wind and water currents. The nature of these and other phenomena will be discussed once the mathematical structure of the models has been developed. 

The setting up of the constitutive relation for a binary system is a relatively easy task because, as pointed out earlier, there is only one independent diffusion flux, only one independent composition gradient (driving force) and, therefore, only one independent constant of proportionality (diffusion coefficient). The situation gets quite a bit more complicated when we turn our attention to systems containing more than two components. The simplest multicomponent mixture is one containing three components, a ternary mixture. In a three component mixture the molecules of species 1 collide, not only with the molecules of species 2, but also with the molecules of species 3. The result is that species 1 transfers momentum to species 2 in 1-2 collisions and to species 3 in 1-3 collisions as well. We already know how much momentum is transferred in the 1-2 collisions and all we have to do to complete the force-momentum balance is to add on a term for the transfer of momentum in the 1-3 collisions. Thus,... 

Equation (2.3-55) is in the form of a rate being governed by two resistances in series—diffusion and chemical reaction. If I k SIOAB (fast surface reaction), die rale is governed by diffusion, while if Ilk 6/Dar (slow reaction rate), the rate is governed by cheraical kinetics. This additivity of resistances is only obtained when linear expressions relate rates and driving forces and wonld not be obtained, for example, if Ihe surface reaction kinetics were second order. More complex kinatic situations can be analyzed in a similar fashion where reaction stoichiometry at the surface provides information on (be flux ratio of various species. 

When there are only two components in the mixture, diffusional fluxes are written in terms of a driving force and a binary molecular diffusion coefficient via Pick s first law. For example, if the diffusional mass flux with respect to a reference frame that translates at the mass-averaged velocity of the mixture is based on a mass fraction driving force, then Pick s first law for component A is... 

The membrane permeability for toluene was determined from independent measurements of the pure toluene flux at different applied pressures. Docosane and TOABr membrane permeabilities were determined from the nanofiltration data assuming a concentration driving force and a solute flux experimentally determined at a low applied pressure of 4 bar, to avoid the influence of the exponential term in the solution-diffusion model and, the effect of concentration polarization. The model parameter values are summarized in Tab. 4.3. 

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