Diffusion heterogeneous materials

While catalytic HDM results in a desirable, nearly metal-free product, the catalyst in the reactor is laden with metal sulfide deposits that eventually result in deactivation. Loss of catalyst activity is attributed to both the physical obstruction of the catalyst pellets pores by deposits and to the chemical contamination of the active catalytic sites by deposits. The radial metal deposit distribution in catalyst pellets is easily observed and understood in terms of the classic theory of diffusion and reaction in porous media. Application of the theory for the design and development of HDM and HDS catalysts has proved useful. Novel concepts and approaches to upgrading metal-laden heavy residua will require more information. However, detailed examination of the chemical and physical structure of the metal deposits is not possible because of current analytical limitations for microscopically complex and heterogeneous materials. Similarly, experimental methods that reveal the complexities of the fine structure of porous materials and theoretical methods to describe them are not yet... 

Once the multi-step reaction sequence is properly chosen, the bifunctional catalytic system has to be defined and prepared. The most widely diffused heterogeneous bifunctional catalysts are obtained by associating redox sites with acid-base sites. However, in some cases, a unique site may catalyse both redox and acid successive reaction steps. It is worth noting that the number of examples of bifunctional catalysis carried out on microporous or mesoporous molecular sieves is not so large in the open and patent literature. Indeed, whenever it is possible and mainly in industrial patents, amorphous porous inorganic oxides (e.g. j -AEOi, SiC>2 gels or mixed oxides) are preferred to zeolite or zeotype materials because of their better commercial availability, their lower cost (especially with respect to ordered mesoporous materials) and their better accessibility to bulky reactant fine chemicals (especially when zeolitic materials are used). Nevertheless, in some cases, as it will be shown, the use of ordered and well-structured molecular sieves leads to unique performances. 

The diffusion of small molecules into the skin from the external world is limited by the stratum comeum (SC). There is now considerable evidence that a major pathway for such diffusion through the stratum comeum itself consists of the intercellular spaces and, in particular, the lipid component of the intercellular spaces. Such lipids are arranged in lamellae that may well be bilayers and that in other respects also resemble biological membranes. In addition, there are other components within the intercellular spaces (e.g., proteins and an aqueous phase) that, although not well understood, mean that the intercellular diffusion path is a heterogeneous material. Within such material, lipids appear to play a very important role (Potts and Guy, 1992), and... 

The quantity kldc is the thermal diffusivity. For heterogeneous materials, the effective thermal conductivity is used in conjunction with Fourier s law. 

Two general theoretical approaches have been applied in the analysis of heterogeneous materials. The macroscopic approach, in terms of classical electrodynamics, and the statistical mechanics approach, in terms of charge-density calculations. The first is based on the application of the Laplace equation to calculate the electric potential inside and outside a dispersed spherical particle (11, 12). The same result can be obtained by considering the relationship between the electric displacement D and the macroscopic electric field Em a disperse system (12,13). The second approach takes into account the coordinate-dependent concentration of counterions in the diffuse double layer, regarding the self-consistent electrostatic poton tial of counterions via Poisson s equation (5, 16, 17). Let us consider these approaches briefly. 

Clough AS and Jenneson P (1998) Ion beam analysis of diffusion in heterogeneous materials. Nuclear Instruments and Methods in Physics Research B 139 51-57. 

Despite the benefits of Nernst-Planck equations, this modeling approach provides only a macroscopic view of the permeation process. The different models derived from this formalism cannot afford the description of heterogeneous materials, including impurities and occluded bubbles, as is the case of most real perovskite layers. To this aim, the development of meso- or microscale models with a proper description of diffusion effects and vacancy generation would be desirable. 

In the initial constant rate drying period, a knowledge of the heat diffusivity and heat conductivity coefficients of the wet material is necessary because they are the controlling factors for heat transport within the material. In this section of drying it is presumed that the pores of the wet material are saturated by moisture. Consequently, the above-mentioned characteristics of a heterogeneous material consisting of a solid skeleton and, within this, a capillary system (filled... 

D. Bernin and D. Topgaard, NMR Diffusion and Relaxation Correlation Methods. New Insights in Heterogeneous Materials, Curr. Opin. Colloid Interface Set, 2013,18,166. 

Problems involving transport through porous media occur in many disciplines. Although the most frequently studied individual problem is the movement of fluid through porous soils or rocks, examples of transport in porous media occur in many distinct types of systems. For example, tissues In the body are composed of cells and extracellular regions, two phases which often differ dramatically in resistance to diffusion of solutes. In recent years -as researchers have learned more about the structure of heterogeneous materials like soils, porous polymers, and animal tissues--the application of techniques developed to understand transport in porous media increase. 

MD and BD are widely nsed numerical tools to calculate diffusion coefficients in bulk materials. Given the fact that the former is computationally expensive, MD cannot consider dispersed phase in a continuum. On the condary, BD shows its advantages in calculating the effective diffusion coefficient of heterogeneous material. In this section, both methods are introduced in sequence in each subsection. 

In this chapter, analytical expressions that are used to calculate the effective diffusion coefQcient of two-phase materials are firstly presented, and are followed by numerical methods. MD can be used to calculate the self-diffusion coefficient, while BD can be applied to obtain chemical diffusion coefficient for both homogenous materials and heterogeneous materials. A BD-based model is also presented in detail, which is used to calculate the diffusion coefficient of small polymer chains in solute. Subsequently,... 

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