Estimation of diffusion coefficients from

This permits the estimation of diffusion coefficients from measurements of conductivity. 

Estimation of Diffusion Coefficients from Simulation of Polymer Microstructure... 

Solution As a rule of thumb, reactions are diffusion controlled if their rate constants are faster than 10 1/mol sec. To make this more quantitative, we can insert into Eq. 17.4-17 estimates of diffusion coefficients from Chapters 5 and 6 and sizes from solid-state radii and thus estimate the rate constants if the reactions are diffusion controlled. For reaction (a). 

This equation, called the Kubo equation, is equivalent to the Einstein equation. However, it is easier to estimate self-diffusion coefficients from the slope of the mean-square displacements. 

Zobrist J, Reichert P (2006) Bayesian estimation of export coefficients from diffuse and point sources in Swiss watersheds. J Hydrol 329 207-223... 

Although there are a lot of data in the literature regarding diffusion coefficients in liquids or then calculation from molecular properties (Appendix I, Section 1.2), it is not the case for diffusion coefficients in solids, where the phenomena appearing are more complex. In solids, the molecule may be forced to follow a longer and tortuous path due to the blocking of the cross-sectional area, and thus the diffusion is somehow impaired. Several models have been developed to take into consideration this effect in the estimation of diffusion coefficients, leading, however, to a variety of results. 

The use of isotopic models in the literature—practical limits of usage As mentioned above, simplified solutions are employed in ion exchange for the estimation of diffusion coefficients. For example, the equations of Vermeulen and Patterson, derived from isotopic exchange systems, have been successfully used, even in processes that are not isotopic. Inglezakis and Grigoropoulou (2001) conducted an extended review of the literature on the use of isotopic models for ion-exchange systems. 

Rates. Accurate estimates of diffusion coefficients in a fiber cannot be made without an accurate measurement of the fiber s diameter in the present case the water-swollen diameter of the fiber is involved. No such measurement was made. However, the rate of absorption curve for SO4 is roughly three orders of magnitude faster than a similar curve for hair from which a diffusion coefficient Z> = 4 x 10" cm /sec was calculated. The swollen diameter of the viscose monofil was estimated to be about 15% smaller than that of the hair. All in all a diffusion coefficient of order of... 

As became apparently clear from the treatment above, a reliable calculation/estimation of diffusion coefficients is a must for correct evaluation of the influence of mass transfer. 

Figure 4.2 Estimation of diffusion coefficients by autoradiography of tissue sections, (a) Digitized autoradiographic image of radiolabeled NGF released from a polymeric implant in a rat brain, reproduced from [18]. (b) The resulting concentration profile, obtained by scanning intensities from the autoradiographic image, and using calibrated standards to determine the local concentration the solid line indicates Equation 3-58.

The calculation of diffusion coefficients from equations based on some models describing the movement of matter in electrolyte solutions, in the end, a process contributing to the knowledge of their stmcture, provided we have accurate experimental data to test these equations. Thus, to understand the behavior of transport process of these aqueous systems, experimental mutual diffusion coefficients have been compared with those estimated using several equations, resulting from different models. 

The ratio of the effective diffusion coefficient in soils or mineral materials to the diffusion coefficient in free water is, however, also influenced by other effects than only the complexity of the diffusion paths in the pore spaces of a soil. For example, the viscosity of water can decrease in narrow pore spaces, with corresponding effects on the diffusion coefficients of the dissolved substances. Apparent tortuosity factors calculated from measured values can therefore be smaller than the value suggested by geometry. It is therefore justifiable to find a conservative estimate of diffusion coefficients for pollutants in soil water, for example when considering mineral landfill liners for which no measured values are available, to use this pure... 

TABLE 15.6 The Measured Oxygen in the Uncoated and Coated Emulsion Particles, the Estimated Diffusion Coefficient, and Relative Change of Diffusion Coefficient From the Uncoated Emulsion With Various Coating Thicknesses... 

Despite of these limitations, which make no possible to know a estimated values with accuracy, it is possible to have an idea of their possible range of values, considered all of these methods reasonable compromises. Consequently, for each electrolyte, either the use of a given a value from one specific method of estimation, or the use of an average value from all of the methods is legitimate (Table 4.1). In our case, we have used the average values of that parameter in the estimation of diffusion coefficients (Table 4.2), because almost all values obtained from different a values, are close each other (deviations, in general, < 2%). 

If Equation (14.12) is used for partitioned migration, some errors are expected due the violation of the prescribed conditions. For partitioned migration, Cp p longer remains zero at all time, but it increases from zero to its equilibrium value (Cp ). Therefore, the use of Equation (14.12) for partitioned migration will result in a lower CpL, a larger (8Cp/8x)lx LP> consequently a lower estimate of diffusion coefficient, compared with the corresponding actual values. 

Figure 14.9 Estimation errors of diffusion coefficients from the two models (ej and 2) and the proposed model ( 3) as a function of a. Simulated data (Mp/Mp, versus time) are generated from equation (14.27) assuming Dq = 10 " cm /s and Lp = 10 pm. (Reproduced with permission from Chung and co-workers. Food Additives and Contaminants, 2002, 19, 6, 611 [22]. 2002, Taylor Francis)...

Despite the obvious departures from the ideal behaviour in natural fibres, it has been conunon in theoretical treatments of diffusion behaviour to use the same simplifying assumptions that have been employed with synthetic fibres. While the results provide qualitative estimates of diffusion coefficients, some caution must be exercised in detailed interpretation of their meaning. Etters pointed out that a lack of appreciation of the prerequisites for mathematical apphcability of diffusion equations derived from Pick s laws probably has led to the publication of diffusion coefficients that are less than precise. However, it is likely that the values are useful for characterising the relative diffusion properties of dyes. 

The discussion of these interactions involves a somewhat different strategy than that used earlier in this book. In Chapters 1-3, we treated the diffusion coefficient as an empirical parameter, an unknown constant that kept popping up in a variety of mathematical models. In more recent chapters, we have focused on the values of these coefficients measured experimentally. In the simplest cases, these values can be estimated from kinetic theory or from solute size in more complicated cases, these values require experiments. In all these cases, the goal is to use our past experience to estimate the diffusion coefficients from which diffusion fluxes and the like can be calculated. 

Transport Properties Although the densities of SCFs can approach those of conventional liquids, transport properties are more favorable because viscosities remain lower and diffusion coefficients remain higher. Furthermore, CO2 diffuses through condensed-liquid phases (e.g., adsorbents and polymers) faster than do typical solvents which have larger molecular sizes. For example, at 35°C the estimated pyrene diffusion coefficient in polymethylmethacrylate increases by 4 orders of magnitude when the CO2 content is increased from 8 to 17 wt % with pressure [Cao, Johnston, and Webber, Macromolecules, 38(4), 1335-1340 (2005)]. 

Sheikha H., Pooladi-Darvish M., et al. Development of graphical methods for estimating the diffusivity coefficient of gases in bitumen from pressure-decay data. 2005 Energy Fuels, 19, 2041- 2049. 

One of earliest approaches of estimating the diffusion coefficient through a polymer carrier is that of Eyring (1936). In this theory, diffusion of a solute through a medium is presented as a series of jumps instead of a continuous process. Therefore, in Eq. (18) in Table I, which comes from the Eyring analysis, X is the diffusional jump of the drug in the polymer and v is the frequency of jumping. 

Starting from Eq. 12 several similar models were developed for predicting skin permeability. Probably the most popular such model was presented by Potts and Guy in 1992 [19], Instead of using Eq. 8, Potts and Guy estimated the diffusion coefficient in human skin using the relationship... 

This section describes the experimental methods and focuses on the estimation of diffusivity after the experiment. The analytical methods are not described here. Estimation of diffusivity from homogeneous reaction kinetics (e.g., Ganguly and Tazzoli, 1994) is discussed in Chapter 2 and will not be covered here. Determination of diffusion coefficients is one kind of inverse problems in diffusion. This kind of inverse problem is relatively straightforward on the basis of solutions to forward diffusion problems. The second kind of inverse problem, inferring thermal history in thermochronology and geospeedometry, is discussed in Chapter 5. 

Estimate the diffusion coefficient for 10 compounds through air at 1 atmosphere pressure from the Wilke-Lee adjustment to the Chapman-Enskog theory and compare your results with measurements. What is the percent error of the estimation (assuming that the measurements are correct) What is the primary cause of the differences between the estimated diffusivities ... 

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