Porous solids pore structure models

The severe computational burden associated with assembling and carrying out adsorption calculations on disordered model microstructures for porous solids, such as those discussed in Sections ILA and II.B, has until recently limited the development of pore volume characterization methods in this direction. While the reahsm of these models is highly appealing, their application to experimental isotherm or scattering data for interpretation of adsorbent pore structure remains cumbersome due to the structural complexity of the models and the computational resources that must be brought to bear in their utilization. Consequently, approximate pore structure models, based upon simple pore shapes such as shts or cylinders, have been retained in popular use for pore volume characterization. 

The rule of inversion may be efficiently applied to modelling. Due to it, studying of the pores structure may be substituted by studying of the solid s structure. The peirticular porous solid may be modelled in one of these systems. Corpuscular solids are more simple and exact to be modelled by the system of particles, the spongy ones — by the system of pores. 

The porous structure of either a catalyst or a solid reactant may have a considerable influence on the measured reaction rate, especially if a large proportion of the available surface area is only accessible through narrow pores. The problem of chemical reaction within porous solids was first considered quantitatively by Thiele [1] who developed mathematical models describing chemical reaction and intraparticle diffusion. Wheeler [2] later extended Thiele s work and identified model parameters which could be measured experimentally and used to predict reaction rates in... 

Abstract A simplified quintuple model for the description of freezing and thawing processes in gas and liquid saturated porous materials is investigated by using a continuum mechanical approach based on the Theory of Porous Media (TPM). The porous solid consists of two phases, namely a granular or structured porous matrix and an ice phase. The liquid phase is divided in bulk water in the macro pores and gel water in the micro pores. In contrast to the bulk water the gel water is substantially affected by the surface of the solid. This phenomenon is already apparent by the fact that this water is frozen by homogeneous nucleation. 

Taking into account the aforementioned effects of ice formation in porous materials, a macroscopic quintuple model within the framework of the Theory of Porous Media (TPM) for the numerical simulation of initial and boundary value problems of freezing and thawing processes in saturated porous materials will be investigated. The porous solid is made up of a granular or structured porous matrix (a = S) and ice (a = I), where it will be assumed that both phases have the same motion. Due to the different freezing points of water in the macro and micro pores, the liquid will be distinguished into bulk water ( a = L) in the macro pores and gel water (a = P, pore solution) in the micro pores. With exception of the gas phase (a = G), all constituents will be considered as incompressible. 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

Considering the obtained experimental data, it is possible to propose a model of the formation of a porous structure of the films of zirconia-based solid electrolytes. The model assumes the formation of pores and submicropores when vacancies, which are trapped during sputtering of the solid-electrolyte films (the sputtering temperature was Tf < 0.3Tmeit), pass to sinks and then condense [2,3,4,5], The sinks are boundaries between the crystallites forming the film structure. 

In eq 1 Dic is the effective diffusivity of species i in the reaction mixture which can be determined on the basis of various models of the diffusion process in porous solids. This aspect is discussed more fully in Section A.6.3. Difi is affected by the temperature and the pore structure of the catalyst, but it may also depend on the concentration of the reacting species (Stefan-Maxwell diffusion [9]). As Die is normally introduced on the basis of more or less empirical models, it may not be considered as a physical property, but rather as a model-dependent parameter. 

Frequently we define a porous medium as a solid material that contains voids and pores. The notion of pore requires some observations for an accurate description and characterization. If we consider the connection between two faces of a porous body we can have opened and closed or blind pores between these two faces we can have pores which are not interconnected or with simple or multiple connections with respect to other pores placed in their neighborhood. In terms of manufacturing a porous solid, certain pores can be obtained without special preparation of the raw materials whereas designed pores require special material synthesis and processing technology. We frequently characterize a porous structure by simplified models (Darcy s law model for example) where parameters such as volumetric pore fraction, mean pore size or distribution of pore radius are obtained experimentally. Some porous synthetic structures such as zeolites have an apparently random internal arrangement where we can easily identify one or more cavities the connection between these cavities gives a trajectory for the flow inside the porous body (see Fig. 4.30). 

Chargeability factor M depends on the brine/gas saturation of porous solids. Figure 3 gives the relationship between the chargeability and brine saturation for two samples. We noted that the M decreases hardly with the decrease of the brine saturation. The presence of vugs and karsts pore types (sample 9-LS8) seems to speed up the decrease of the M Chargeability factor M can be explained by a multi-linear model composed of different structures parameters such as the formation resistivity factor, water porosity, Hg-specific surface area and water permeability, e.g.. Fig. 5. 

For a few decades now cellular and porous systems have been classified in morphological terms by simulating the real systems by one or another imaginary, and always simplified, geometrical or stereometrical scheme using an artificially ordered-structure model. Such classifications have always been based on the concept that in any cellular or porous system it is possible to isolate a structural element (cell or pore). However, the diversity of pore and cell types even in small-sized real foamed systems does, in most cases, not permit a definition by only one single geometrical structural parameter, as for other types of solids (type and volume of elementary cell, interplanar or interatomic distances, etc.)... 

Among the structural models of cellular or porous materials those characterized by differently ordered packing of balls or spheres of the same diameters have most widely been used. In this approach either the spheres have been considered as real cells or the cell (pore) models have been derived from an analysis of assumed spacings between the contacting solid spheres. However, no system of packed spheres would adequately describe the properties of any real cellular system which never exhibit a regular packing. It is also impossible to describe the structure of most cellular systems via models assuming spheres of equal size. 

The industrial application of porous solids is quite widespread. Porous heterogeneous catalysts, adsorbents and membranes are used in chemical industry and in biotechnology, porous materials are common in building engineering, porous eatalysts form the basis of car mufflers, etc. The rates of processes, which take place in pore strueture of these materials, are affected or determined by the transport resistance of the pore structure. Inclusion of transport processes into the description of the whole process is essential when reliable simulations or predictions have to be made. Trends in modem chemical/biochemical reaction engineering point to utilization of more sophisticated, and therefore more reliable, models of proeesses. The basic idea is that the better the description of individual steps of the whole process the better its description and, perhaps, even extrapolation. 

Today two models are available for description of combined (diffusion and permeation) transport of multicomponent gas mixtures the Mean Transport-Pore Model (MTPM)[21,22] and the Dusty Gas Model (DGM)[23,24]. Both models enable in future to connect multicomponent process simultaneously with process as catalytic reaction, gas-solid reaction or adsorption to porous medium. These models are based on the modified Stefan-Maxwell description of multicomponent diffusion in pores and on Darcy (DGM) or Weber (MTPM) equation for permeation. For mass transport due to composition differences (i.e. pure diffusion) both models are represented by an identical set of differential equation with two parameters (transport parameters) which characterise the pore structure. Because both models drastically simplify the real pore structure the transport parameters have to be determined experimentally. 

Gas sorption porosimetry is a standard method for the characterization of the pore size distribution (PSD) of porous solids. To interpret the experimental isotherm and obtain the adsorbent PSD, one must adopt a model for the pore structure, and a theory that estimates the adsorption that will occur in pores of a particular size. If the porous solid is represented as an array of independent, noninterconnected pores of uniform geometry and identical surface chemistry, then the excess adsorption, /JP), at bulk gas pressure P is given by the adsorption integral equation... 

These are methods for the simulation of a flat amorphous surface. To simulate the atomic structure of a porous oxide adsorbent like silica gel, one may first simulate the bulk amorphous silica. Then cut out of it globules and arrange them in space to model the pore structure of silica gel. Other applications of this idea include the creation of pores such as those found in porous glass by deleting atoms from a simulated block of solid in such a way as to leave a cylindrical pore. 

Characterization of the pore structure of amorphous adsorbents and disordered porous catalysts remains an important chemical engineering research problem. Pore structure characterization requires both an effective experimental probe of the porous solid and an appropriate theoretical or numerical model to interpret the experimental measurement. Gas adsorption porosimetry [1] is the principal experimental technique used to probe the structure of the porous material, although various experimental alternatives have been proposed including immersion calorimetry [2-4], positron... 

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