Diffusion model considerations

Membrane performance characteristics in the hydraulic and diffusion limits are compared to each other in Fig. 9. Figure 9(a) illustrates that in the diffusion model considerable deviations from the purely ohmic performance of the saturated membrane arise already at small jv/Jj, well below the critical current density. This is in line with the comparison of the water-content profiles calculated in the diffusion model, Fig. 9(b), with those from the hydraulic permeation model, in Fig. 7. Indeed, membrane dehydration is much stronger in the diffusion model, affecting larger membrane domains at given values of jp/./j. Moreover, the profiles exhibit different curvature from those in Fig. 7. 

Equations (12.13) and (12.14) are themselves approximations, since in Chapter 3 we eliminated thermal transpiration and diffusion from consideration before constructing the dusty gas model flux relations. See Appendix 1. 

The proposed specification of the kernel for m- and J-diffusion models is mathematically closed, physically clear and of quite general character. In particular, it takes into consideration that any collisions may be of arbitrary strength. The conventional m-diffusion model considers only strong collisions (0(a) = 1 /(27c)), while J-diffusion considers either strong (y = 0) or weak (y = 1) collisions. Of course, the particular type of kernel used in (1.6) restricts the problem somewhat, but it does allow us to consider kernels with arbitrary y < 1. 

From a mathematical perspective either of the two cases (correlated or non-correlated) considerably simplifies the situation [26]. Thus, it is not surprising that all non-adiabatic theories of rotational and orientational relaxation in gases are subdivided into two classes according to the type of collisions. Sack s model A [26], referred to as Langevin model in subsequent papers, falls into the first class (correlated or weak collisions process) [29, 30, 12]. The second class includes Gordon s extended diffusion model [8], [22] and Sack s model B [26], later considered as a non-correlated or strong collision process [29, 31, 32],... 

Discovery of the hydrated electron and pulse-radiolytic measurement of specific rates (giving generally different values for different reactions) necessitated consideration of multiradical diffusion models, for which the pioneering efforts were made by Kuppermann (1967) and by Schwarz (1969). In Kuppermann s model, there are seven reactive species. The four primary radicals are eh, H, H30+, and OH. Two secondary species, OH- and H202, are products of primary reactions while these themselves undergo various secondary reactions. The seventh species, the O atom was included for material balance as suggested by Allen (1964). However, since its initial yield is taken to be only 4% of the ionization yield, its involvement is not evident in the calculation. 

Kinetic models proposed for sorption/desorption mechanisms including first-order, multiple first-order, Langmuir-type second-order, and various diffusion rate laws are shown in Sects. 3.2 and 3.4. All except the diffusion models conceptualize specific sites to or from which molecules may sorb or desorb in a first-order fashion. The following points should be taken into consideration [ 181,198] ... 

We have presented a general reaction-diffusion model for porous catalyst particles in stirred semibatch reactors applied to three-phase processes. The model was solved numerically for small and large catalyst particles to elucidate the role of internal and external mass transfer limitations. The case studies (citral and sugar hydrogenation) revealed that both internal and external resistances can considerably affect the rate and selectivity of the process. In order to obtain the best possible performance of industrial reactors, it is necessary to use this kind of simulation approach, which helps to optimize the process parameters, such as temperature, hydrogen pressure, catalyst particle size and the stirring conditions. 

The calculated and experimental curves deviate considerably in shape, and it seems that the simple diffusion model is not adequate to describe the kinetics of the uptake process in this case. Also the value of D (9.25 X 10"9 cm.2/sec.) required to make the two curves coincide at early times seems to be much smaller than expected. The only data available on the diffusion of molybdenum oxide in liquid silicates is from a report by Norman et al. (11), who measured the diffusion constant of... 

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. 

It is particularly interesting and instructive to note that application of Henry + Langmuir dual-mode sorption and diffusion models is not confined to glassy polymer-gas systems. Sorption and transport of high affinity ionic species, exemplified by anionic dyes, in charged polymers, exemplified by polyamides at low pH, has been treated in the same way. These systems are of considerable importance both from the bio-mimetic and from the textile processing point of view, but have received limited atten-... 

In one dimensional diffusion experiments (e.g., starting with a thin film source of A on a B crystal surface) one often finds an exponential decrease in the A concentration at the far tail of the concentration profile. This behavior has been attributed to pipe diffusion along dislocation lines running perpendicular to the surface. Models have been introduced which assume a (constant) pipe radius, rp, inside which Dl = p-D, b and p denoting here bulk and dislocation respectively. P values of 103 have been obtained in this way. It is difficult to assess the validity of these observations. The model considerably simplifies the real situation. During diffusion annealing, the structure of the dislocation networks is likely to change because of self-stress (see Chapter 14) and chemical interactions. 

The temporal evolution of the concentration profiles of the adspecies with allowance for their interaction seems to have been studied for the first time by Bowker and King [158]. Initially, the distribution of the adspecies density has been given up in the form of a step (this technique is often applied to surface diffusion studies). Consideration has been given to the concentration profiles in the case of attraction and repulsion of the adspecies to conclude that they can be used to estimate the lateral interactions. The applicability of the model to the description of diffusion in the O/W (110) system [159] is discussed. 

This is because of the effect of the total pressure gradient which develops which must be accounted for in the diffusion model. Clearly, although this may not play an important role in carefully designed experiments for determining diffusion coefficients using this technique, it has a considerable bearing on the use of such information where non-equimolal fluxes can arise, as with chemical reaction (6). 

The first section of this chapter is devoted to the presentation of some diffusion models constructed on the basis of phenomenological considerations. These so-called heuristic models are often cited in the literature and used for the interpretation of experimental results. A special emphasis is to discuss how the mathematical formulae of these models can correlate with experimetal data and moreover to predict diffusion coefficients beyond the ranges experimentally investigated. This latter aspect is of great interest not only from a fundamental point of view but also in many practical fields where the possibility to predict a diffusion process might be a more economic alternative to its experimental investigation. 

The use of these diffusion models to progress the evaluation process of a food packaging plastic will be discussed shortly. In those cases where assessment by mass balance considerations under equilibrium conditions, including partitioning effects, does not provide a clear picture of the plastics conformity status, then the different diffusivities of polymer types and the influence of the migrant molecule size or its molecular weight on its mobility within a plastic can be taken into account to achieve more distinguished views on QM/SML ratios. 

Lickly, T. D., Rainey M. L. Burgert L. C., Breder C. V., Borodinsky L. 1997, Using a simple diffusion model to predict residual monomer migration - considerations and limitations. Food Additives and Cntaminants, 14. 65-74. 

The sedimentation diffusion model, when applied to the iron oxide system, gave solid settling velocities in agreement with theory. Solid dispersion coefficients were in the range predicted hy the Kato correlation, hut showed considerable experimental scatter. 

It therefore appears that the cluster approximation with AB pairs would improve the model considerably. This was first done by Dickman [60]. His description has been adapted [53] and extended later by several groups, in order to include diffusion [62], unreactive desorption [63,64] and Eley-Rideal steps [64]. (Note that in these papers the cluster approximation is also called a mean-field approximation. They are distinguished by the terminology site-approximation, pair-approximation etc.)... 

Values of measured for reversible reactions under conditions permitting absorption equilibration between gas and solid products are often comparable with the reaction enthalpy [1,47]. Under such conditions the identification of the rate expression, g(nr) = kt, on the single criterion that this equation gives the most acceptable correlation coefficient is not a sufficient foimdation to characterize a geometric reaction model. The use of additional information, for example microscopy, can provide confirmatory evidence concerning interface development. Similarly, the value of studies which conclude that kinetic results are satisfactorily described by equations based on diffusion models is increased considerably if the identity of the migrating species is established [53]. 

An important effort in this investigation was the thermal decomposition study of the shales. Considerable effort has been made to find a simple kinetic model which will accurately describe the weight loss curves for non-isothermal pyrolysis at various heating rates. In the past, many researchers have proposed and tested theoretical kinetic models for this reaction Q-4), however, most attempts at finding a suitable model have been focused on finding a very accurate fit to experimental data. Successive studies have increasingly emphasized microscopic details (i.e., diffusion models, exact chemical composition, etc.) in an attempt to find a precise model to fit the weight loss curves. In this... 

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