Porous materials models

In a steady-state situation when gas flows through a porous material at a low velocity (laminar flow), the following empirical formula, Darcy s model, is valid ... 

Table 6.2 summarizes the low pressure intercept of observed shock-velocity versus particle-velocity relations for a number of powder samples as a function of initial relative density. The characteristic response of an unusually low wavespeed is universally observed, and is in agreement with considerations of Herrmann s P-a model [69H02] for compression of porous solids. Fits to data of porous iron are shown in Fig. 6.4. The first order features of wave-speed are controlled by density, not material. This material-independent, density-dependent behavior is an extremely important feature of highly porous materials. 

With porous materials, a slow diffusion in the pores can sometimes control the rate of desorption. This may give rise to complications because diffusion in the pores may be complex and difficult to treat mathematically. Cvetanovi6 and Amenomiya (48) gave a model treatment for their modification of the thermal desorption technique. 

Figure 13 Schematic model for moisture uptakes in porous materials.

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. 

Affected by multiple scattering are, in particular, porous materials with high electron density (e.g., graphite, carbon fibers). The multiple scattering of isotropic two-phase materials is treated by Luzatti [81] based on the Fourier transform theory. Perret and Ruland [31,82] generalize his theory and describe how to quantify the effect. For the simple structural model of Debye and Bueche [17], Ruland and Tompa [83] compute the effect of the inevitable multiple scattering on determined structural parameters of the studied material. 

PALS is based on the injection of positrons into investigated sample and measurement of their lifetimes before annihilation with the electrons in the sample. After entering the sample, positron thermalizes in very short time, approx. 10"12 s, and in process of diffusion it can either directly annihilate with an electron in the sample or form positronium (para-positronium, p-Ps or orto-positronium, o-Ps, with vacuum lifetimes of 125 ps and 142 ns, respectively) if available space permits. In the porous materials, such as zeolites or their gel precursors, ort/zo-positronium can be localized in the pore and have interactions with the electrons on the pore surface leading to annihilation in two gamma rays in pick-off process, with the lifetime which depends on the pore size. In the simple quantum mechanical model of spherical holes, developed by Tao and Eldrup [18,19], these pick-off lifetimes, up to approx. 10 ns, can be connected with the hole size by the relation ... 

In turn, porous space of many real and model porous materials can be considered as a lattice of expansions cavities (or sites), connected with narrower windows or necks (bonds). With such a definition of sites and bonds it is acceptable to have the whole volume of pores concentrated only in cavities of different sizes and forms. In this case, windows are considered as volumeless figures that correspond to the flat sections in places of the smallest narrowings between the neighbors (as well as bond in a lattice of particles) [3,61], This approach seems to be the most... 

The given discussion shows that rather universal and simple classification of porous materials equivalent to classification of crystals is absent. However, one can consider a system of interrelating classifications that take into account order, morphology and sizes at different hierarchical levels, degrees of integrity, structure, heterogeneity of a various type, etc. Such a systematic approach can be used as well for adequate modeling of various hierarchical levels of a porous material structure. 

While modeling the structure and properties of porous materials one usually is interested in structural properties of a desirable hierarchical level. For example, for chemical properties the molecular structure is major, and the specific adsorption and catalytic properties are guided by the structure and composition of particle surface. Diffusion permeability is determined by the supramolecular... 

The energy equation entails a detailed account of heat generation due to irreversible heat of the electrochemical reaction, reversible (or entropic) heat, and Joule heating. The heat generation term in a CFCD model must be unambiguous and location specific. More discussion is deferred to section 3.3. In addition, the heat accumulation in a porous material consisting of the matrix and fluid is given... 

The numerical simulations of the stress distributions are carried out on porous materials submitted to uniaxial loading. In order to check the validity of the numerical simulations, macroporous epoxies are prepared via the CIPS technique. Cyclohexane is selected as the solvent, thus resulting in the formation of a closed porosity, and the statistical distribution of the voids coincides with the random distribution of the model system. The structural characteristics of these materials prepared by curing at T=80 °C are summarized in Table 4. 

In Equation 24, t is the tortuosity, a term well established for gas diffusion into porous materials (10). It is unfortunate, but necessary to introduce t into our model. The value of t cannot be obtained a-priori cuid must be obtained experimentally since it is an almost impossible task to describe the complicated pore geometry in a gel. Given idealised perfect pore geometry it has been possible to estimate t for gas diffusion processes. In our work, of necessity x becomes an adjustable pareuneter to help achieve better agreement between the model predictions eind experimental results. Since x is unmeasurable our only concern has been to use reasonable values in our simulations. 

The most fundamental model of a porous structure is provided by the dusty gas model as critically reviewed and described by Jackson [19]. The basis of the theory, as suggested by Maxwell as early as 1860, is to suppose that the action of the porous material is similar to that of a number of particles fixed in space and obstructing the motion of molecules. Thus, if a number of gaseous species are diffusing through three-dimensional space, the imposition of one further species of much... 

Turner (T14) has proposed two detailed models of packed beds which try to closely approximate the true physical picture. The first model considers channels of equal diameter and length but with stagnant pockets of various lengths opening into the channels. There is no flow into or out of these pockets, and all mass transfer occurs only by molecular diffusion. The second model considers a collection of channels of various lengths and diameters. We will briefly discuss each of these models, which are probably more representative of consolidated porous materials than packed unconsolidated beds. 

Karadimitra, K., Lorentzou, S., Agrafiotis C., and Konstandopoulos A. G. Modeling of Catalytic Particle Synthesis via Spray Pyrolysis In-Situ Deposition on Porous Materials . PARTEC 2004, International Conference for Particle Technology, Nuremberg, Germany, 2004, March 16-18. 

The adsorption of gases and vapors on mesoporous materials is generally characterized by multilayer adsorption followed by a distinct vertical step (capillary condensation) in the isotherm accompanied by a hysteresis loop. Studies of adsorption on MCM-41 have also demonstrated the absence of hysteresis for materials having pore size below a critical value. While this has been reported for silica gel and chromium oxide containing some mesopores, no consistent explanation has been offered [1], However, conventional porous materials, having interconnected pores with a broader size distribution, are generally known to display a hysteresis loop with a point of closure which is characteristic of the adsorptive. These materials have an independent method of estimating the pore size from XRD and TEM, that allows comparison with theoretical results. Consequently, we have chosen these materials to test the proposed model. 

Adsorption is often studied using powders or porous materials because the total surface area is large even for small amounts of adsorbent. In a typical experiment the volume (V) or the mass (to = V/p) adsorbed per gram of adsorbent, is measured. Theoretical models always describe an adsorption per surface area. In order to compare theoretical isotherms to experimentally determined adsorption results, the specific surface area needs to be known. The specific surface area (in m2/kg) is the surface area per kg of adsorbent. Once the specific surface area is known, the area can be calculated by A = madT, where mad is the mass of the adsorbent. 

Detailed modeling of a porous material under compression is a challenging task of applied structural mechanics. The reduced compression model employed in the current study is based on the unidirectional morphological displacement of solid voxels in the GDL structure under load and with the assumption of negligible transverse strain. The reduced compression model is detailed in our recent work.33 However, with the reduced compression model, it is difficult to find a relation between the compression ratio and the external load. The compression ratio is defined as the ratio of the thickness of compressed sample to that of the uncompressed sample. Nevertheless, this approach leads to reliable 3-D morphology of the non-woven GDL structures under compression. Figure 17a shows compressed, reconstructed non-woven GDL... 

EMCPMT models wifi be described that can simulate transient electro-mechano-chemo diffusion, convection, and osmosis in one-dimensional FEMs composed of one and/or multiple layers of porous material with prescribed no(Xi) and FCD, < [ (X,) in the solid. The left (L) and right (R) interfaces are water baths containing prescribed concentrations of up to three charged species (p, m, b). Mechanical force (stress) or displacement fluid pressure, and electrical potential will also be prescribed on these interfaces. The first example is... 

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. 

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