Complex pores

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

There are two major types of filtration "cake" and "filter-medium" filtration. In the former, solid particulates generate a cake on the surface of the filter medium. In filter-medium filtration (also referred to as clarification), solid particulates become entrapped within the complex pore structure of the filter medium. The filter medium for the latter case consists of cartridges or granular media. Among the most common examples of granular materials are sand or anthracite coal. 

Models of regular structures, such as zeolites, have been extensively considered in the catalysis literature. Recently, Garces [124] has developed a simple model where the complex pore structure is represented by a single void with a shell formed by n-connected sites forming a net. This model was found to work well for zeolites. Since polymer gels consist of networks of polymers, other approaches, discussed later, have been developed to consider the nature of the structure of the gel. 

Mapping of transport parameters in complex pore spaces is of interest for many respects. Apart from classical porous materials such as rock, brick, paper and tissue, one can think of objects used in microsystem technology. Recent developments such as lab-on-a-chip devices require detailed knowledge of transport properties. More detailed information can be found in new journals such as Lab on a Chip [1] and Microfluidics and Nanofluidics [2], for example, devoted especially to this subject. Electrokinetic effects in microscopic pore spaces are discussed in Ref. [3]. 

In the light of the wide variety of complex pore spaces where microscopic transport features are of interest, we decided to concentrate on a special type of microporous systems suitable for testing NMR mapping methods and to examine... 

M. T. Myers 1991, Pore Combination Modeling A Technique for Modeling the Permeability and Resistivity Properties of Complex Pore Systems, paper SPE 22662 presented at the 1991 SPE ATC E, Dallas, TX, 6-9 October, 1991. 

The technique of DDIF provides a quantitative characterization of the complex pore space of the rocks to supplement conventional mineralogy, chemistry and petrology analyses. A combination of DDIF, Hg intrusion, NMR T2 and image analysis has become the new paradigm to characterize porous rocks for petroleum applications [62, 61]. 

Fukasawa, T., Ando, M., Ohji, T. Fabrication of porous ceramics with complex pore structure by freeze drying process. Ceram. Trans. 112, Innovative Process-ing/Synthesis Ceramics, Glasses and Composites IV, 217-226, 2001... 

Another method applied to produce porous polymers is based on the addition of an inorganic matrix of well-known porosity, for example, silica gel or aluminum oxide, to the reacting mixture [210-212], Subsequent to the polymerization process, the inorganic pattern is eliminated by dissolution without destruction of the produced polymer [211], These materials develop a complex pore system [211-213],... 

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. 

We have already dealt with stationary phase processes and have noted that they can be treated with some success by either macroscopic (bulk transport) or microscopic (molecular-statistical) models. For the mobile phase, the molecular-statistical model has little competition from bulk transport theory. This is because of the difficulty in formulating mass transport in complex pore space with erratic flow. (One treatment based on bulk transport has been developed but not yet worked out in detail for realistic models of packed beds [11,12].) Recent progress in this area has been summarized by Weber and Carr [13]. 

Restricted transport or, by a different name, configurational diffusion [13] occurs when the diffusing molecules are comparable in size to the pores within which they diffuse. This happens, for example, in hydrodemetallation over alumina-supported Co-Ni catalysts [14]. The observation that the effective diffusivity depends on the fourth power of the molecule-to-pore size ratio is important, but it is not yet evident how to correlate complex pore size distributions with effective diffusivities in the configurational... 

Although many adsorbents possess exceedingly complex pore structures, the mesopore size analysis carried out by several groups (Dollimore and Heal, 1973 Havard and Wilson, 1976) appears to indicate that the calculated pore size distribution is rather insensitive to the model. However, as was pointed out by Haynes (1975) such results may be due in part to the over-simplification of the computations. 

Since highly active adsorbents generally contain complex pore structures, it is not suiprising that many physisorption isotherms are of a composite nature. Thus, most of die Type IV isotherms reported for oxide gels are exceedingly complex with features which are difficult to identify. 

The available transport models are not reliable enough for porous material with a complex pore structure and broad pore size distribution. As a result the values of the model par ameters may depend on the operating conditions. Many authors believe that the value of the effective diffusivity D, as determined in a Wicke-Kallenbach steady-state experiment, need not be equal to the value which characterizes the diffusive flux under reaction conditions. It is generally assumed that transient experiments provide more relevant data. One of the arguments is that dead-end pores, which do not influence steady state transport but which contribute under reaction conditions, are accounted for in dynamic experiments. Experimental data confirming or rejecting this opinion are scarce and contradictory [2]. Nevertheless, transient experiments provide important supplementary information and they are definitely required for bidisperse porous material where diffusion in micro- and macropores is described separately with different effective diffusivities. 

In the case of zeolites, studies have also been performed to distinguish sites located at the external surface from those present in the cavities, and also to distinguish the position in different cavities for zeolites with a complex pore structure. This can be performed using molecules having different molecular sizes but similar chemical behavior. Some of the useful molecules are shown in Scheme 3.6. 

The complex pore structure of MCM-22 is also reflected in the unique three step uptake profile of bulky 2,2-dimethylbutane (DMB) observed in the dynamic sorption experiment [12], shown in Figure 4. Each step is attributed to adsorption into different sections of MCM-22, but specific assignment is ambiguous. 

For determination of the overall porosity, pycnometric methods are recommended that use imbibition of the material in a light inert gas and mercury. Mercury porosimetry is the method of choice for assessing macro- and meso-pores. The use of intrusion and extrusion measurements is necessary to understand more complex pore structures. 

For porous media with unknown or complex pore structure, permeability must be determined experimentally. It can be measured by injecting fluid through... 

Models that are used to predict transport of chemicals in soil can be grouped into two main categories those based on an assumed or empirical distribution of pore water velocities, and those derived from a particular geometric representation of the pore space. Velocity-based models are currently the most widely used predictive tools. However, they are unsatisfactory because their parameters generally cannot be measured independently and often depend upon the scale at which the transport experiment is conducted. The focus of this chapter is on pore geometry models for chemical transport. These models are not widely used today. However, recent advances in the characterization of complex pore structures means that they could provide an alternative to velocity based-models in the future. They are particularly attractive because their input parameters can be estimated from independent measurements of pore characteristics. They may also provide a method of inversely estimating pore characteristics from solute transport experiments. 

The literature on solute transport in porous media is voluminous. For a general introduction to this subject the reader is referred to Leij and Dane (1989), El-rick and Clothier (1990), and Jury and Fliihler (1992). Sahimi (1993) has reviewed some of the advances made in modeling solute transport within complex pore structures. Of the numerous older review articles, those by Bear (1969) and Fried and Combamous (1971) are especially thorough, and are still relevant today. Other important contributions that discuss aspects of pore geometry as related to solute transport include those by Greenkom and Kessler (1969), Rose (1977), Brusseau and Rao (1990), and Celia et al. (1995). In addition, several books are relevant to this topic, including those by Bear (1972), Dullien (1992), Adler (1992), and Sahimi (1995). 

We apply simple effective medium models in an attempt to understand the diffusion process in the complex pore network of a porous SiC sample. There is an analogy between the quantities involved in the electrostatics problem and the steady state diffusion problem for a uniform external diffusion flux impinging on a coated sphere. Kalnin etal. [17] provide the details of such a calculation for the Maxwell Garnett (MG) model [18]. The quantity involved in the averaging is the product of the diffusion constant and the porosity for each component of the composite medium. The effective medium approach does not take into account possible effects due to charges on the molecules and/or pore surfaces, details in the size and shape of the protein molecules, fouling (shown to be negligible in porous SiC), and potentially important features of the microstructure such as bottlenecks. 

The parameter is the acmal porosity and q is a measure of the tortuosity of the pores for complex pore structures. For the case of straight cylindrical pores, q is simply equal o unity. 

Characterization of porous media based on the pore (microscopic) level is carried out for the purpose of understanding, modeling, and sometimes controling the macroscopic behavior and properties of the medium. The macroscopic (bulk) properties needed to relate to the pore description are porosity, permeability, tortuosity, and connectivity. When one examines a sample of a porous medium, for example, sandstone, it is obvious that the number of pore sizes, shapes, orientations, and interconnections is enormous. Furthermore, even the identification of a pore is not unique. Because of this complexity, pore structure is often characterized based on an idealized model. A true description is not realistic for a natural porous medium. 

T. Fukasawa and M, Ando, Synthesis of Porous Ceramics with Complex Pore Structure by Freeze-dry Processing, J. Am. Ceram. Soc., 84 [1] 230-232(2001). 

Interpretation of fundamental transport parameters in terms of rate constants for the simple pore and for the complex pore... 

Parameter Simple-pore interpretation Complex-pore interpretation... 

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