Single-component diffusion model

The diffusion coefficients for Rb, Cs and Sr in obsidian can be calculated from the aqueous rate data in Table 1 as well as from the XPS depth profiles. A simple single-component diffusion model (9j characterizes onedimensional transport into a semi-infinite solid where the diffusion coefficient (cm2-s 1) is defined by ... 

Ammonia synthesis reaction is a complex multi-component and reversible reaction, and the kinetic equation can be expressed by power function. For simplification, the multi-component diffusion model is usually simply treated in engineering as a single-component diffusion model of key component, and the utilization ratio of internal surface is obtained by an approximate method from a simplified first-order reaction model. 

Fig. 14 Separation of C2H6/CH4 mixtures by permeation through a silicalite membrane, a Flux b selectivity. Continuous lines show the predictions of the Maxwell-Stefan model (Eq. 44) based on single-component diffusivities (Dqa> F>ob) with Dab from the Vignes correlation (Eq. 46). Dotted lines show predictions from the simplified Habgood model in which mutual diffusion effects are ignored (Eq. 45). From van de Graaf et al. [53] with permission...

The current PEP setup allows two types of experiments to measure diffusion in microporous materials, hi the first type, labeled molecules are injected as a small pulse into a steady-state feed stream of either an inert carrier gas or of unlabeled molecules of the same kind. The propagation of the pulse through the reactor is followed using the PEP detector. Information about the diffusive processes can be obtained from the delay and broadening of the pulse, and quantitative information can be obtained by analysis of the measurements using an appropriate model, as will be discussed in more detail in the next section. This type of experiments is especially suited for diffusion measurements under zero loading conditions. A drawback of this method is that it is limited to the determination of single-component diffusion coeffi-... 

The study of the behavior of reactions involving a single species has attracted theoretical interest. In fact, the models are quite simple and often exhibit IPT. In contrast to standard reversible transitions, IPTs are also observed in one-dimensional systems. The study of models in ID is very attractive because, in some cases, one can obtain exact analytical results [100-104]. There are many single-component nonequilibrium stochastic lattice reaction processes of interacting particle systems [100,101]. The common feature of these stochastic models is that particles are created autocatalytically and annihilated spontaneously (eventually particle diffusion is also considered). Furthermore, since there is no spontaneous creation of particles, the zero-particle... 

Surface Polarization in TFF The simplified model of polarization shown in Fig. 20-47 is used as a basis for analyzing more complex systems. Consider a single component with no reaction in a thin, two-dimensional boundary layer near the membrane surface. Axial diffusion is negligible along the membrane surface compared to convection. 

A possible modification of this expression is presented elsewhere (82). The value of t, can be related to a diffusion coefficient (e.g., tj = l2/6D, where / is the jump distance), thereby making the Ar expressions qualitatively similar for continuous and jump diffusion. A point of major contrast, however, is the inclusion of anisotropic effects in the jump diffusion model (85). That is, jumps perpendicular to the y-ray direction do not broaden the y-ray resonance. This diffusive anisotropy will be reflected in the Mossbauer effect in a manner analogous to that for the anisotropic recoil-free fraction, i.e., for single-crystal systems and for randomly oriented samples through the angular dependence of the nuclear transition probabilities (78). In this case, the various components of the Mossbauer spectrum are broadened to different extents, while for an anisotropic recoil-free fraction the relative intensities of these peaks were affected. 

Non-isothermal and non-adiabatic conditions. A useful approach to the preliminary design of a non-isothermal fixed bed reactor is to assume that all the resistance to heat transfer is in a thin layer near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the approximate design of reactors. Neglecting diffusion and conduction in the direction of flow, the mass and energy balances for a single component of the reacting mixture are ... 

Whereas the dual sorption and transport model described above unifies independent dilatometric, sorption and transport experiments characterizing the glassy state, an alternate model offered recently by Raucher and Sefcik provides an empirical and fundamentally contradictory fit of sorption, diffusion and single component permeation data in terms of parameters with ambiguous physical meanings (28), The detailed exposition of the dual mode model and the demonstration of the physical significance and consistency of the various equilibrium and transport parameters in the model in the present paper provide a back drop for several brief comments presented in the Appendix regarding the model of Raucher and Sefcik,... 

Figure 2.16 Illustration of isotopic fractionation effects in diffusion. The model is that 132Xe and 134Xe are initially uniformly distributed throughout spheres in the ratio 134Xe/132Xe = 0.382 and then allowed to escape by diffusion with the boundary condition that the concentration vanishes on the surface. The figure shows the instantaneous composition of the released gas at various stages, assuming that the diffusion coefficients varies as m 112. The single-component locus is for all spheres having the same radius the mixed-component locus is for distribution of sizes. Reproduced from Funk, Podosek, and Rowe (1967).

The evaluation of D by fitting an error function, which is based upon an undisturbed diffusion model in a single component system, will not lead to a proper description of the fluorine uptake in a natural multi-component system. 

Diffusivities are often measured under conditions which are far from those of catalytic reactions. Moreover, corresponding to their different nature, the various measuring techniques are limited to special ranges of application. The possibility of a mutual transformation of the various diffusivities would therefore be of substantial practical relevance. Since each of the coefficients of self-diffusion and transport diffusion in single-component and multicomponent systems refers to a particular physical situation, one cannot expect that the multitude of information contained in this set of parameters can in general be adequately reflected by a smaller set of parameters. Any correlation which might be used in order to reduce the number of free parameters must be based on certain model assumptions. 

Wu et al. (1993] have developed a mathematical model based on Knudsen diffusion and intermolecular momentum transfer. Their model applies the permeability values of single components (i.e., pure gases) to determine two parameters related to the morphology of the microporous membranes and the reflection behavior of the gas molecules. The parameters are then used in the model to predict the separation performance. The model predicts that the permeability of carbon monoxide deviates substantially from that based on Knudsen diffusion alone. Their model calculations are able to explain the low gas separation efficiency. Under the transport regimes considered in their study, the feed side pressure and pressure ratio (permeate to feed pressures) are found to exert stronger influences on the separation factor than other factors. A low feed side pressure and a tow pressure ratio provide a maximum separation efficiency. 

From this modelling approach it seems that the combined surface difiusion and activated gas translational difiusion can describe the observed single component permeation behaviour. The interpretation for the latter type of diffusion is not well-crystallized at present. It might have to do with an increasing deformation of the silicalite-1 structure with increasing temperature that causes this apparently activated process. This aspect has not been considered up to now, the silicalite-1 has been considered as a rigid structure. 

Furthermore, the theoretical analysis of the single-component problem in the ideal model provides some of the fimdamental concepts in nonlinear chromatography, such as the notions of the velocity associated with a concentration, of concentration shocks, and of diffuse bormdaries [1,2]. It also provides an understanding of the relationship between the thermod5mamics of phase equilibria, the shape of the isotherm (i.e., convex upward, linear, convex downward, or S-shaped) and the band profiles. Finally, it provides an explanation of the relative importance of the influences of the thermodynamics and the kinetics on the band profile. These concepts will provide a most useful framework for imderstanding the phenomena that occur in preparative chromatography. 

Numerical Solution of the Lumped Pore Diffusion Model (or FOR Model) for Single Component Systems... 

In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measru ments improve and more interest is paid to complex... 

Figures 16.23a to d compare experimental profiles of mixtures of the enantiomers of 1-indanol on cellulose tribenzoate with those calculated with the GMS-GRM model of these authors [57]. For the numerical calculations, they assmned that surface diffusion plays the dominant role in mass transfer across the particles and neglected the contribution of pore diffusion to the fluxes. Unfortunately, it was impossible independently to measure or even estimate the surface diffusion parameters. So, the numerical values of the surface diffusion coefficients needed for the calculation were estimated by minimizing the discrepancies between the measured and the calculated band profiles i.e., by parameter adjustment). Yet, it is impressive that, using a unique set of diffusion coefficients, it was possible to calculate band profiles of single components of binary mixtures in the whole range of relative composition, for loading factors between 0 and 10%.

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