Diffusivities calculation

Z) g = mutual coefficient of diffusion AB = mutual coefficient of diffusion calculated by Fuller s method... 

Figure 9.11 Variation of c/cq with x for one-dimensional diffusion [calculated from Eq. (9.85) with D = 5 X 10 m sec ].

The value of diffusivity calculated from Eq. (12-38) must be recognized as an average value over the entire range of moisture change from W — WJ/(W — WJ = 1 to the value W — WJ/(W — WJ at which Q/d was evaluated. Further, Eq. (12-38) assumes that the theoretical curve is a straight hne for all values of time. This is not true for values W — WJ/(W — W ) less than 0.6. 

Thermo-diffusion calculations analyze the migration of hazardous material from compartment to compartment to release in containment. These calculations use physico-chemical parameters to predict the retention of hazardous materials by filtration, deposition on cold surfaces and other retention processes in the operation. Containment event trees aid in determining the amount, duration and types of hazardous material that leaves the containment. 

Fej04 . A similar correspondence between theory and practice has been found for growth of Fej04 by the solid state reactions from FeO and Fe, , between 600 and 1 200°C. The growth rate of FeO is within 10% of the theoretical rate expected from Fe lattice diffusion, calculated according to the Wagner theory . 

The diffusivity of the vapour of a volatile liquid in air can be conveniently determined by Winkelmann s method, in which liquid is contained in a narrow diameter vertical tube maintained at a constant temperature, and an air stream is passed over the top of the tube sufficiently rapidly to ensure that the partial pressure of the vapour there remains approximately zero. On the assumption that the vapour is transferred from the surface of the liquid to the air stream by molecular diffusion, calculate the diffusivity of carbon tetrachloride vapour in air at 321 K and atmospheric pressure from the following experimentally obtained data ... 

As can be seen in the graph, the Y/TUD-1 catalyst was twice as active as the commercial Y catalyst. This is primarily due to its very high calculated diffusivity of 131x10 cm /sec, which is over 10 times the diffusivity calculated for commercial zeolite Y, 11x10 cm /sec. Extrapolation of the curve to zero particle size shows that the commercial Y zeolite is in fact intrinsically more active than the Y zeolite embedded in the TUD-1. If the Y zeolite in TUD-1 had been optimized for this reaction like the commercial Y catalyst, one should expect an even greater boost in performance. 

This value is based on Cu2+ diffusivities calculated Arvia et al. (A5) from limiting-current measurements at a rotating-disk electrode by, with CuS04 concentrations below 0.1 M. In practical applications (e.g., copper refining or electrowinning) higher Cu2+ concentrations are often required, as is also the case in free-convection limiting-current measurements. 

FIGURE 5.4 MathCAD worksheet of diffusivity calculation (following Treybal). 

The average effective diffusivity calculated for the 44 compositions is 1.7 x 10"13 cm2/sec and the loss of tin 2.2%. These are in qualitative agreement with the results of laboratory testing. Cardarelli has also reported that antifouling rubber retains a considerable amount of organotin additives even after complete fouling (11). 

The object of this article is to present a concise development of the key elements of atmospheric diffusion theory leading to the formulas in common use for carrying out atmospheric diffusion calculations. The article is intended to be a unified treatment of the subject that introduces the principal ideas and shows many of the important derivations of atmospheric diffusion theory. An attempt has not been made to provide a comprehensive, evaluative literature review of current work in atmospheric diffusion theory, although we will have occasion to refer to much of the recent literature. One purpose of this article is to present usable formulas for performing atmospheric diffusion calculations. In doing so, we have not endeavored to survey in a detailed way the experiences of others gained in use of the formulas. A relatively thorough list of recent references is provided for the reader who desires to pursue the subject further. 

We have presented a relatively self-contained development of the fundamentals of atmospheric diffusion theory. The emphasis has been on elucidating the origin and applicability of the basic expressions commonly used in atmospheric diffusion calculations. This article is intended for the practicing scientist who desires a tutorial introduction to atmospheric diffusion theory or for the advanced undergraduate or graduate student who is entering atmospheric diffusion research. 

For most diffusion calculations one makes use of special cases of the above expressions. For example, for systems of constant mass density p (dilute liquid solutions at constant temperature and pressure) and for systems of constant molar density c (ideal gases at constant temperature and pressure) Eqs. (42) and (43) may be simplified to ... 

There are shown in Fig. 19 values of the eddy diffusivity calculated from the measurements by Sherwood (SI6). These data show the same trends as were found in thermal transport, indicating that the values of eddy diffusivity are determined primarily from the transport of momentum for situations where the molecular Schmidt numbers of the components do not differ markedly from each other. 

Figure 18.11 Diffusion distance, L, vs. diffusion time, t, for typical diffusivities calculated from the Einstein-Smoluchowski relation L = (2Dt)m, Eq. 18-8. The following diffusivities, D, are used (values in cm2s ) He in solid KC1 at 25°C KT10 molecular in water 1 O 5 molecular in air KT1 vertical (turbulent) in ocean 10° vertical (turbulent) in atmosphere 105 horizontal (turbulent) in ocean 106 to 108. Values adapted from Lerman (1979).

The relation between length and time scales of diffusion, calculated from the Einstein-Smoluchowski law (Eq. 18-8), are shown in Fig. 18.11 for diffusivities between 10 10 cm2s 1 (helium in solid KC1) and 108 cm2s (horizontal turbulent diffusion in the atmosphere). Note that the relevant time scales extend from less than a millisecond to more than a million years while the spatial scales vary between 1 micrometer and a hundred kilometers. The fact that all these situations can be described by the same gradient-flux law (Eq. 18-6) demonstrates the great power of this concept. 

Bell et al. (81) presented forced diffusion calculations of butene isomers in the zeolite DAF-1. DAF-1 (82) is a MeALPO comprising two different channel systems, both bounded by 12-rings. The first of these is unidimensional with periodic supercages, while the other is three-dimensional and linked by double 10-rings. The two channel systems are linked together by small 8-ring pores. It is a particularly useful catalyst for the isomerization of but-l-ene to isobutylene (S3) its activity and selectivity are greater than those of ferrierite, theta-1, or ZSM-5. 

As was the case for diffusion calculations, tremendous advances have been made recently in the simulation of the sorption locations, energetics, and conformations of adsorbates within zeolites. As far as the prediction... 

The results of experimental studies of the sorption and diffusion of light hydrocarbons and some other simple nonpolar molecules in type-A zeolites are summarized and compared with reported data for similar molecules in H-chabazite. Henry s law constants and equilibrium isotherms for both zeolites are interpreted in terms of a simple theoretical model. Zeolitic diffusivitiesy measured over small differential concentration steps, show a pronounced increase with sorbate concentration. This effect can be accounted for by the nonlinearity of the isotherms and the intrinsic mobilities are essentially independent of concentration. Activation energies for diffusion, calculated from the temperature dependence of the intrinsic mobilitieSy show a clear correlation with critical diameter. For the simpler moleculeSy transition state theory gives a quantitative prediction of the experimental diffusivity. 

Most often, only diffusive calculations in the macropore region are necessary for certain large-pore systems. Then, by applying Eqn. (3.4-83) to the macropore region we have ... 

The interpretation of the Li abundance gap using a diffusion model has been questioned because of the observed absence of abundance anomalies of heavy elements in F stars (Boesgaard and Lavery 1986 Thevenin, Vauclair and Vauclair 1986 Tomkin, Lambert and Balachandran 1985) where Be has been observed to be underabundant. Such anomalies had been predicted on account of the diffusion calculations in the absence of any mass loss (Michaud et al. 1976, Vauclair et al. 1978b). It has recently been shown that even a very small mass loss was sufficient to reduce considerably any expected overabundance in F stars. On Fig. 2c of Michaud and Charland (1986), it is shown that a mass loss rate of 10 15 Mo yr-1 is sufficient to keep the Sr overabundance, below a factor of 1.5 while Sr would be expected to be more than 100 times overabundant in the absence of mass loss (Michaud et al. 1976). The presence of even a very small mass loss rate considerably limits any overabundance when the radiative acceleration and gravity are close to each other as is the case for heavy elements in stars cooler than Teff = 7000 K. The same small mass loss rate reduces the Li overabundance in stars of Teff = 7000 K or more where Li is supported. As shown in Fig. 4 of Michaud (1986), the same mass loss rate of 10 15 Mo yr 1 eliminates the Li overabundance of a factor of 10 expected in the absence of mass loss at Teff = 7000 K. It has now been verified that the presence of mass loss cannot increase the Li underabundance that diffusion leads to beyond a total factor of 30 underabundance. 

Comparison of profundal diffusion rates with observed increases in the hypolimnion (Table III) indicated that pore-water diffusion calculated from these profiles was probably not an important transport mechanism for Hg in this seepage lake. For the June-July period, pore-water diffusion accounted for only 13% of the hypolimnetic increase. For the July-August interval, pore-water diffusion could account for only 7% of the observed increase. Therefore, we can assume that the buildup in the hypolimnion is more likely a result of redissolution of recently fallen particulate matter at the sediment surface than of direct pore-water diffusion. Our present sampling scheme (2-cm intervals) precludes evaluation of dissolution in the uppermost sediments and would require much more detail (<1 cm) in the sediment-water interfacial zone. 

The time-dependent diffusion calculated at higher density, at p = 0.7932, and at T = 0.7 is plotted in Fig. 19. The plot shows that the diffusion at higher density also does not saturate to a finite value but increases with time. In the same figure the D(t) values obtained from the simulated VACF andMSD [175] have also been plotted. The agreement between MCT and the simulations is satisfactory. 

The analysis of transfer mechanisms of drugs across the intestinal epithelial layer has passed a long way since the theory of lipid pore membrane [118] in which the total pore area of the intestinal membranes was calculated (and found to be low compared with the total surface of the mucosal aspect of the gut), through the Fickian diffusion calculations of the transport of unionized moieties of drug molecules (the Henderson-Hasselbach equation), which led to the conclusion that acidic drugs are absorbed in the stomach [119,120]. 

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