Diffusion theory background

In classical diffusion theory, the diffusing mass flux is treated as the sum of molecular diffusion, pressure diffusion, thermal diffusion by the Soret effect, and so on, whereas the physical background of these effects are not completely discussed. 

Basic Equations AU of the processes described in this sec tion depend to some extent on the following background theory. Substances move through membranes by several meoianisms. For porous membranes, such as are used in microfiltration, viscous flow dominates the process. For electrodialytic membranes, the mass transfer is caused by an elec trical potential resulting in ionic conduction. For aU membranes, Ficldan diffusion is of some importance, and it is of dom-... 

Figure 16. Schematic illustration of envelopes of gas species i, in this case Mg, surrounding a volatilizing molten chondrule in space. The size of the gas envelope is a function of ambient background pressure P by virtue of the effect that pressure has on the gas molecule diffusivity D,. The diffusion coefficient can be calculated from the kinetic theory of gases, as shown. The level of isotopic fractionation associated with volatilization of the molten chondrule depends upon the balance between the evaporative flux J vap and the condensation flux Tom When the fluxes are equal (i.e., when = 0), there is no mass-dependent isotope fractionation associated with volatilization. This will be the case when the local partial pressure P, approaches the saturation partial pressure P,...

The increased understanding of turbulence and the extension of the analysis of potential flow have made possible the consideration of many thermal and material transfer problems which formerly were not susceptible to analysis. However, at present the application of such methods is hampered by the absence of adequate information concerning the thermal conductivities and diffusion coefficients of the components of petroleum. The diffusion coefficient in particular is markedly influenced by the state of the phase. For this reason much experimental effort will be required to obtain the requisite experimental background to permit the quantitative application of the recent advances in fluid mechanics and potential theory to dynamic transfer problems of practical interest. 

After presenting some background information on theories of turbulent combustion in the following section, we shall discuss theoretical descriptions of turbulent diffusion flames (Section 10,2) and of turbulent premixed flames (Section 10,3), Although there are situations in which turbulence can tend to wash out the distinction between premixed and non-premixed systems, the best approaches to analyses of these two limits remain distinct today, and thus it is convenient to treat them separately. 

The residual difference after a successful DDM refinement or/and decomposition can be considered as a scattering component of the powder pattern free of Bragg diffraction. The separation of this component would facilitate the analysis of the amorphous fraction of the sample, the radial distribution function of the non-crystalline scatterers, the thermal diffuse scattering properties and other non-Bragg features of powder patterns. The background-independent profile treatment can be especially desirable in quantitative phase analysis when amorphous admixtures must be accounted for. Further extensions of DDM may involve Bayesian probability theory, which has been utilized efficiently in background estimation procedures and Rietveld refinement in the presence of impurities.DDM will also be useful at the initial steps of powder diffraction structure determination when the structure model is absent and the background line cannot be determined correctly. The direct space search methods of structure solution, in particular, may efficiently utilize DDM. 

Diffusion and mass transfer in multicomponent systems are described by systems of differential equations. These equations are more easily manipulated using matrix notation and concepts from linear algebra. We have chosen to include three appendices that provide the necessary background in matrix theory in order to provide the reader a convenient source of reference material. Appendix A covers linear algebra and matrix computations. Appendix B describes methods for solving systems of differential equations and Appendix C briefly reviews numerical methods for solving systems of linear and nonlinear equations. Other books cover these fields in far more depth than what follows. We have found the book by Amundson (1966) to be particularly useful as it is written with chemical engineering applications in mind. Other books we have consulted are cited at various points in the text. 

Presently, practitioners of EER either have a background in namral sciences or in social sciences, or in both disciplines. As a consequence, EER uses methods both from the natural sciences and from the social sciences. Researchers will have to motivate their choice between quantitative and qualitative, as well as mixed methods research designs. The choice of methods depends on the worldview and of course on the research questions. In general, quantitative research implies that there are some well-developed theories that need further confirmation. Qualitative and mixed methods research aims at building theoretical nnderstanding from a more diffuse starting point (Creswell 2009). 

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