Pore geometries, complex examples

The ratio of volume to area within a pore depends upon the pore geometry. For example, the volume to area ratios for cylinders, parallel plates and spheres are, respectively, r/2, r/2 and r/3, where r is the cylinder and sphere radii or the distance of separation between parallel plates. If the pore shapes are highly irregular or consist of a mixture of regular geometries, the volume to area ratio can be too complex to express mathematically. In these cases, or in the absence of specific knowledge of the pore geometry, the assumption of cylindrical pores is usually made, and equation (8.6) becomes... 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

This expression is more widely applicable than Eq. 2.43 because few pores are true cylinders and dc, but not 5, loses meaning for noncylindrical pore geometries. Equation 2.44 can consequently be used as an approximation for other pore shapes and even for more complex pore space. For example, Eq. 2.44 proves to be exactly applicable to long pores of square cross section [27] Eq. 2.43 cannot be applied without arbitrarily defining an apparent pore diameter to replace dc. For any given pore geometry, s l is proportional to mean pore size. 

Calculation of meniscus curvature in pores bordered by spheres is still too difficult for a full mathematical analysis. However, there is one class of pore geometry that is complex but can still be analysed by a simple theory. It is the geometry of a uniform non-axi-symmetric tube (or tubes). For example the capillary behaviour of a tube of triangular cross-section can be analysed quite simply. Even tubes assembled from parallel rods do not cause much difficulty. 

Many processes and structures that are difficult to describe by means of traditional Euclidean geometry can thus be precisely characterized using fiactal geometry, for example the complex and disordered microstructures of advanced materials, adsorbents, polymers and minerals. Recent studies have shown that using fiactal dimensions enables the real sizes of pore radii to be determined and pore-size distribution functions to be calculated from the data of programmed thermodesorption of liquids [35],... 

Analytical expressions for effective diffusion coefficients in complex media can be obtained only when the geometry is simple. Consider, for example, the diffusion of solute through a periodic array of spherical obstructions, in which the solute diffuses through the continuous interstitial space with a diffusion coefficient i A.pore and through the spheres with a diffusion coefficient Z>a,s-The effective diffusion coefficient for such a composite material can be determined exactly [94] ... 

The ratio of the effective diffusion coefficient in soils or mineral materials to the diffusion coefficient in free water is, however, also influenced by other effects than only the complexity of the diffusion paths in the pore spaces of a soil. For example, the viscosity of water can decrease in narrow pore spaces, with corresponding effects on the diffusion coefficients of the dissolved substances. Apparent tortuosity factors calculated from measured values can therefore be smaller than the value suggested by geometry. It is therefore justifiable to find a conservative estimate of diffusion coefficients for pollutants in soil water, for example when considering mineral landfill liners for which no measured values are available, to use this pure... 

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