Geometry pores

Apparently a negative AP with Q < 90° can be found for particular pore geometries [53]. A different type of water repellency is desired to prevent the deterioration of blacktop roads consisting of crushed rock coated with bituminous materials. Here the problem is that water tends to spread into the stone-oil interface, detaching the aggregate from its binder [54]. No entirely satisfactory solution has been found, although various detergent-type additives have been found to help. Much more study of the problem is needed. 

It is only for smooth field models, in this sense, that partial differential equations relating species concentrations to position in space can be written down. However, a pore geometry which is consistent with the smooth... 

Another important distinction relating to pore geometry is that between "through" pores, with two open ends, and "dead-end" pores with only one. 

Che pore size distribution and Che pore geometry. Condition (iil). For isobaric diffusion in a binary mixture Che flux vectors of Che two species must satisfy Graham s relation... 

In the higher pressure sub-region, which may be extended to relative pressure up to 01 to 0-2, the enhancement of the interaction energy and of the enthalpy of adsorption is relatively small, and the increased adsorption is now the result of a cooperative effect. The nature of this secondary process may be appreciated from the simplified model of a slit in Fig. 4.33. Once a monolayer has been formed on the walls, then if molecules (1) and (2) happen to condense opposite one another, the probability that (3) will condense is increased. The increased residence time of (1), (2) and (3) will promote the condensation of (4) and of still further molecules. Because of the cooperative nature of the mechanism, the separate stages occur in such rapid succession that in effect they constitute a single process. The model is necessarily very crude and the details for any particular pore will depend on the pore geometry. 

The limits of pore size corresponding to each process will, of course, depend both on the pore geometry and the size of the adsorbate molecule. For slit-shaped pores the primary process will be expected to be limited to widths below la, and the secondary to widths between 2a and 5ff. For more complicated shapes such as interstices between small spheres, the equivalent diameter will be somewhat higher, because of the more effective overlap of adsorption fields from neighbouring parts of the pore walls. The tertiary process—the reversible capillary condensation—will not be able to occur at all in slits if the walls are exactly parallel in other pores, this condensation will take place in the region between 5hysteresis loop and in a pore system containing a variety of pore shapes, reversible capillary condensation occurs in such pores as have a suitable shape alongside the irreversible condensation in the main body of pores. 

A surprisiagly large number of important iadustrial-scale separations can be accompHshed with the relatively small number of zeoHtes that are commercially available. The discovery, characterization, and commercial availabiHty of new zeoHtes and molecular sieves are likely to multiply the number of potential solutions to separation problems. A wider variety of pore diameters, pore geometries, and hydrophobicity ia new zeoHtes and molecular sieves as weU as more precise control of composition and crystallinity ia existing zeoHtes will help to broaden the appHcations for adsorptive separations and likely lead to improvements ia separations that are currently ia commercial practice. 

Glandt, ED, Noncircular Pores in Model Membranes A Calculation of the Effect of Pore Geometry on the Partition of a Solute, Journal of Membrane Science 8, 331, 1981. 

In addition, mercury intrusion porosimetry results are shown together with the pore size distribution in Figure 3.7.3(B). The overlay of the two sets of data provides a direct comparison of the two aspects of the pore geometry that are vital to fluid flow in porous media. In short, conventional mercury porosimetry measures the distribution of pore throat sizes. On the other hand, DDIF measures both the pore body and pore throat. The overlay of the two data sets immediately identify which part of the pore space is the pore body and which is the throat, thus obtaining a model of the pore space. In the case of Berea sandstone, it is clear from Figure 3.7.3(B) that the pore space consists of a large cavity of about 85 pm and they are connected via 15-pm channels or throats. 

Pore shape is a characteristic of pore geometry, which is important for fluid flow and especially multi-phase flow. It can be studied by analyzing three-dimensional images of the pore space [2, 3]. Also, long time diffusion coefficient measurements on rocks have been used to argue that the shapes of pores in many rocks are sheetlike and tube-like [16]. It has been shown in a recent study [57] that a combination of DDIF, mercury intrusion porosimetry and a simple analysis of two-dimensional thin-section images provides a characterization of pore shape (described below) from just the geometric properties. 

Detailed analysis of the DDIF spectra showed a constant microporosity for almost all samples. This lends strong support to the hypothesis that the original rocks after burial, compaction and the initial diagenesis had a common pore geometry. Such a base rock appears to be dominated by the micropores with approximately 2/3 micropores, 1 /3 mesopores and very few macropores. The effect... 

The pore geometry described in the above section plays a dominant role in the fluid transport through the media. For example, Katz and Thompson [64] reported a strong correlation between permeability and the size of the pore throat determined from Hg intrusion experiments. This is often understood in terms of a capillary model for porous media in which the main contribution to the single phase flow is the smallest restriction in the pore network, i.e., the pore throat. On the other hand, understanding multiphase flow in porous media requires a more complete picture of the pore network, including pore body and pore throat. For example, in a capillary model, complete displacement of both phases can be achieved. However, in real porous media, one finds that displacement of one or both phases can be hindered, giving rise to the concept of residue saturation. In the production of crude oil, this often dictates the fraction of oil that will not flow. 

DDIF has been applied to understand two-phase flow (air and water) in a Berea sandstone sample and the relationship to the pore geometry [65], Several different states of saturation were studied full saturation and partial saturation by three methods, i.e., centrifugation, co-current imbibition and counter-current imbibition. Imbibition is a process in which a porous sample absorbs the wetting fluid through capillary force. In the case of co-current imbibition, the bottom of the rock sample was kept in contact with water, so the water is imbibed into the rock and the water and air flowed in the same direction. For counter-current imbibition, the whole sample was immersed and the water was drawn into the center of the rock as, the air was forced out in this case, the water and air flowed in opposite directions. 

Velocity variations resulting from variations in pore geometry and the fact that water velocity is higher in the center of a pore space than that for water moving near the pore wall... 

The first term on the right side of Equation 5 corresponds to the rate of stretching/squeezing and is sensitive to pore geometry. The sharper the constriction (i.e., high R /R ), the... 

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. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

The bi-functional conversion of 2,2,4-trimethylpentane over Pt/DAY has been recently reported by Jacobs et al. (104). It was compared to the corresponding conversion over Pt/H-ZSM-5 and Pt/H-ZSM-11. All three zeolites had the same chemical composition. The authors found that 2,2,4-trime-thylpentane underwent 3-scission over Pt/DAY, while the formation of feed isomers was favored over the other two catalysts. The differences in reaction products were related to differences in the pore geometry of the zeolites. A similar study was carried out with n-decane. 

Thus, either type I or type IV isotherms are obtained in sorption experiments on microporous or mesoporous materials. Of course, a material may contain both types of pores. In this case, a convolution of a type I and type IV isotherm is observed. From the amount of gas that is adsorbed in the micropores of a material, the micropore volume is directly accessible (e.g., from t plot of as plot [1]). The low-pressure part of the isotherm also contains information on the pore size distribution of a given material. Several methods have been proposed for this purpose (e.g., Horvath-Kawazoe method) but most of them give only rough estimates of the real pore sizes. Recently, nonlocal density functional theory (NLDFT) was employed to calculate model isotherms for specific materials with defined pore geometries. From such model isotherms, the calculation of more realistic pore size distributions seems to be feasible provided that appropriate model isotherms are available. The mesopore volume of a mesoporous material is also rather easy accessible. Barrett, Joyner, and Halenda (BJH) developed a method based on the Kelvin equation which allows the calculation of the mesopore size distribution and respective pore volume. Unfortunately, the BJH algorithm underestimates pore diameters, especially at... 

Schmidt, V., McDonald, D. A., and Platt, R. L., 1977, Pore Geometry and Reservoir Aspects of Secondary Porosity in Sandstones Bulletin of the Canadian Petroleum Geology, Vol. 25, pp. 271-290. 

RET provides a spectroscopic method of probing local pore geometries. Levitz et al. (1988) critically evaluated the application of RET in probing the morphology of porous solids (e.g. silica gels). 

Another consequence of pore geometry is that for crystalline electrodes, other crystal planes are exposed to the electrolyte at the pore tip than at the pore walls. The dependence of pore growth on crystal orientation of the silicon electrode is discussed in Chapters 8 and 9. 

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