Diffusion coefficients derivative

The concept of transport resistances localized in the outermost regions of NS crystals was introduced in order to explain the differences between intracrystalline self-diffusion coefficients obtained by n.m.r methods and diffusion coefficients derived from non-equilibrium experiments based on the assumption that Intracrystalline transport is rate-limiting. This concept has been discussed during the past decade, cf. the pioneering work [79-81] and the reviews [2,7,8,23,32,82]. Nowadays, one can state that surface barriers do not occur necessarily in sorption uptake by NS crystals, but they may occur if the cross-sections of the sorbing molecular species and the micropore openings become comparable. For indication of their significance, careful analysis of... 

Diffusion coefficients derived from pulsed gradient spin echo NMR-studies show lower values for the encapsulated guest identical to those of the host, see Frish, L. Matthews, S. E. Bohmer, V. Cohen, Y. J. Chem. Soc., Perkin Trans. 2 1999, 669-671. 

Kawano et al. used a microelectrode technique and the Dahms-Ruff equation to explore the diffusion mechanism of the iodide/triiodide redox couple and explain the physical diffusion accompanying an exchange reaction [22]. They showed that when high concentrations of iodide and tiiiodide are added to a RTCL as the solvent, the diffusion coefficient derived from the exchange reaction, expressed by / + —> /f + / , become significant and superior to the simple physical dif-... 

The HSDM has been successfully used to predict the adsorption of atrazine onto PAC [40]. Figure 26.4 shows the removal of atrazine as a function of time [10]. Also shown is the HSDM fit of the data, and the surface diffusion coefficient derived from the fit (DJ. This coefficient was then used to successfully predict adsorption of atrazine onto this PAC at different initial atrazine concentrations and carbon doses [10]. 

The practical application of PAC for the removal of MIB and geosmin has been aided by the application of the HSDM [63, 64, 68]. Figure 26.8 illustrates the difference in the adsorption of the two compounds, and the fit that the HSDM can give to the data. Diffusion coefficients derived from these fits can then be used to predict the adsorption of the taste and odor compounds, and consequently the PAC doses required under particular water treatment plant situations [63, 68, 69]. 

Our knowledge of eddy diffusion coefficients derives largely from observations of the spreading of natural or artificial tracers in the atmosphere and an evaluation of these data by eddy diffusion models based on Eqs. (1-12) and (1-13). The two-dimensional representation requires... 

The slower diffusion coefficient derived from the dynamics of the proton anion reactions, measured under osmotic pressure (curve B in Figure 2), implies a complex relationship between the external pressure and the ordering of water in the hydration layer. As the width shrinks, the capacity of protons to diffuse within the space by a random walk is diminished. 

Values for Da and D can also be estimated in terms of reference values for water in air and oxygen in water (Dx = Dref[MWref/MWx] ). The exponents have been estimated from experimental observations. It is now possible to estimate evaporation rates under static conditions by first estimating molecular diffusion coefficients, deriving values for the partial transfer velocities and calculating fctot-The data summarized in Table 4.6 use some laboratory observations listed in Tables 4.3-4.5 to evaluate this model. Reasonable agreement is observed between the calculated and observed values, considering possibilities for experimental error and uncertainties in the values for Henry s law constant, and so on. 

Luyben, K.C.A.M., Concentration dependent diffusion coefficients derived from experimental drying curves, in Drying 80, Vol. 2, Mujumdar, A.S. (ed.). Hemisphere Publishing, New York, 1980, pp. 233-243. 

Recently Sok et simulated the transport of He and of CH4 through a polydimethylsiloxane (PDMS) membrane. The polymer was selected because of the almost invariably amorphous condition of its samples. Thus the interpretation of experimental diffusion data is uncomplicated by the presence of crystallites. The PDMS chains were represented by oligomers consisting of 30 monomer units, and the MD simulation was based on a fine-grained representation of polymer and diffusant, i.e., all atoms were included explicitly. Like the mechanism described by Takeuchi " " here, too, transport occurred by a jump mechanism. Facilitated by fluctuations in the polymer chains, the hole surrounding the diffusant was observed to expand just prior to the jump, and a transient channel was opened, thus enabling the transition to occur. Diffusion coefficients derived for He and CH4 (respectively, 18 X 10" and 2.1 x 1(T cm s ) are consistent with at least the magnitudes of experimental measurement (10 x 1(T and 2.0 x 10" cm ... 

For the case of constant micropore diffusion coefficient, derivation of the first- and higher-order FRFs is relatively simple, for all three geometries. The first- and second-order FRFs are [56] ... 

Ion diffusivity The transport rate of dissolved ions and molecules in clays depends on their diffusivity under the influence of concentration gradients. Diffusion transport capacity is expressed by the effective diffusion coefficient. It refers to the actual effective porosity. On the other hand, the apparent diffusion coefficient derives directly from recording of the concentration profile in the clay. Thus, the importance of porosity is emphasised by the fact that cation diffusion takes place in several ways in continuous water-filled voids, along particle surfaces with electrical doublelayers, and through the interlayer space in smectites. The density of the clay plays a rather important role in ion diffusion except for clays exchanged with monovalent cations. 

FIGURE 7.1.21 Comparison of diffusion coefficients derived from the creep data and cation diffusion experiments. 

If the variation of the solid diffusion coefficient with Uthium concentration is significant, then the diffusion equation is nonlinear and the above simplification does not apply. For an electrode composed of spherical particles, a pseudo-two dimensional approach is required, in which the radial diffusion equation (Equation 17) is solved at each mesh point across the porous electrode. A set of radial nodes is then required to compute the radial solid concentration profile at each linear position in the electrode. Note that Eiquation 17 is derived using the gradient in chemical potential, and assumes only that volume changes are negligible and that aU current is carried by electrons in the solid phase. The chemical diffusion coefficient, D used in Equation 17 is related to the binary diffusion coefficient derived from the Stefan-MaxweU equations, V, (also aGled the binary interaction parameter), by the relationship presented earlier (Equation 13) for concentrated solutions ... 

Efforts to explain this case of polymer diffusion begin with a model, developed by Rouse, which represents the polymer chain as a linear series of beads connected by springs. The diffusion coefficient derived from this model is... 

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