Diffusion derivation

Kedem-Katchalsky equations Phenomenological equations for combined convection and diffusion, derived from nonequilibrium thermodynamics. See Eqs. (19) and (20). 

Equation (9) is Fick s second law of diffusion, derived on the assumption that D is constant. Fick s second law essentially states that the rate of change in concentration in a volume within the diffusional field is proportional to the rate of change in the spatial concentration gradient at that point in the field, the proportionality constant being the diffusion coefficient. 

Hikosaka presented a chain sliding diffusion theory and formulated the topological nature in nucleation theory [14,15]. We will define chain sliding diffusion as self-diffusion of a polymer chain molecule along its chain axis in some anisotropic potential field as seen within a nucleus, a crystal or the interface between the crystalline and the isotropic phases . The terminology of diffusion derives from the effect of chain sliding diffusion, which could be successfully formulated as a diffusion coefficient in our kinetic theory. 

Lamb, R. G., and Duran, D. R. (1977). Eddy diffusivities derived from a numerical model of the convective boundary layer. Nuov. Cimento [1] 1C, 1-17. 

In this review, we focus on the information at an atomic/molecular level that is obtainable via the different techniques. The precise methods and techniques used are not extensively discussed instead we summarize the relevant details and direct the reader toward key references. Nor do we review the potentials that are used in the classical simulations of sorption and diffusion. Derivation and evaluation of these parameters require extensive comparison with detailed spectroscopic data and are beyond the scope of this work. Similarly, the volume of experimental results that may be used in comparison to the calculations is vast. We use representative data taken largely from reviews or books. 

Further, mixing/turbulent diffusivity derived by considering that the mixing process in a liquid is caused by the random movement of inner substances based on the turbulent flow can also be used as an index for the evaluation of the local mixing rate. The mass balance in the flow field is written as... 

The governing phenomenological equation for ionic conduction, as in electronic conduction, is Ohm s law (Eq. 6.21). Concentration gradient induced processes, on the other hand, follow Pick s laws of diffusion, derived by Adolf Eugen Pick (1829-1901) in 1855 (Pick, 1855). 

In what follows, the equation of diffusion derived in Chapter 2 is generalized to take into account the effect of flow. For point particles (dp = 0), rates of convective diffusion can often be predicted from theory or from experiment with aqueous solutions because the Schmidt numbers are of the same order of magnitude. There is an extensive literature on this subject to which the reader is directed. For particle diffusion, there is a difference from the usual theory of convective diffusion because of the special boundary condition The concentration vanishes at a distance of one particle radius from the surface. This has a very large effect on particle deposition rates and causes considerable difficulty in the mathematical theory. As discussed in this chapter, the theory can be simplified by incorporating the particle radius in the diffusion boundary condition. 

Abstract In this chapter the main macroscopic experimental methods for measuring diffusion in microporous solids are reviewed and the advantages and disadvantages of the various techniques are discussed. For several systems experimental measurements have been made by more than one technique, and in Part 3 the results of such comparative studies are reviewed. While in some cases the results show satisfactory consistency, there are also many systems for which the apparent intracrystaUine diffusivities derived from macroscopic measurements are substantially smaUer than the values from microscopic measurements such as PFG NMR. Recent measurements of the transient intracrystalline concentration profiles show that sirnface resistance and intracrystalline barriers are both... 

Despite extensive work in the last decade, large discrepancies still persist between the various experimental techniques which measure diffusion in zeohtes. One of the difficulties is that one has to compare self-diffusivities, obtained by PFG NMR or QENS methods, with transport diffusivities derived from macroscopic experiments. The transport diffusivity is defined as the proportionahty factor between the flux and a concentration gradient (Fick s first law)... 

Fig. 12 Diffusivities of linear alkanes in different zeolites obtained at 475 K as a function of the carbon number a self-diffusivities derived from PPG NMR in Na-X zeolite, b self-diffusivities measured by QENS in ZSM-5, c transport diffusivities obtained by NSE in 5A [48]...

Despite the large difference in the apparent diffusivities derived from uptake rate measurements, measurements of the intracrystalline self-diffusivity (for C2H6-5A) by the NMR (PFG) method show little difference between the large laboratory synthesized crystals and the small Linde crystals, thus favoring the surface barrier hypothesis. The increase in activation energy, which is observed for this system on severe hydrothermal pretreatment of the smaller crystals, is also consistent with a change from intracrystalline diffusion to surface barrier control. Detailed sorption rate studies with the system n-butane-5A, however, support the opposite conclusion that the differences in... 

TABLE 6.5. Diffusivities Derived from Single-Particle Uptake Experiments... 

If we are simulating a reaction where Cq varies with time for all involved species then we would like to compute these accurately also. For simplicity, take again the case of chronopotentiometry, only one species being involved. The diffusion equation results, by the process described in Sect. 5.4.1, in system 5.129. To get Cq (which is needed to solve that system), we cannot simply add its diffusion-derived equation we need an anchor for it. This we get from the constant current or gradient G at U = 0. Again Eq. 5.124 is the key, expressed as Eq. 5.132. This yields Cq as a function of all other C and can then be substituted for in system 5.129. Then, Cq will adjust itself along with the whole profile. Without any special effort, Cq is found implicitly in the solution. This was developed by Pons (1981A). 

There is a fairly large body of experimental data to support selection of realistic elemental diffusivities (derived from both sorption and diffusion measurements) for the bentonite backfill. Repeating the calculations with more conservative sorption distribution coefficients (generally a factor of ten smaller) has little effect on calculated releases from the geosphere, even though the near-field release profiles change somewhat. A major effect of such increased diffusivity is to allow release from the bentonite of some isotopes which otherwise decay within the near-field ( Ni, Sn, Pu, " Pu). However, these radionuclides are not dominant contributors to dose and, in any case, will decay within the far-field. 

FIG. 4.23. Hydrogen-water vapor effective diffusivity derived from data for the reduction of hematite to iron compared with that obtained by direct measurements and calculated from pore structure. From the work of Turkdogan et al. [62]. V, previous work A, present work (both direct measurements) , calculated from pore structure 0, derived from reduction data. 

Example 2.2-1 Membrane diffusion Derive the concentration profile and the flux for a single solute diffusing across a thin membrane. As in the preceding case of a film, the membrane separates two well-stirred solutions. Unlike the film, the membrane is chemically different from these solutions. 

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