Molecular diffusion, in porous

Steady-State Binary Molecular Diffusion in Porous Solids... 

Example 1.20 Steady-State Molecular Diffusion in Porous Solid... 

A brief review of the limited number of in situ measurements is then presented. This is followed by a review of the available correlations proposed for estimating these kinds of MTCs in Section 12.4. Section 12.5 is concerned with the subject of classical molecular diffusion in porous media at steady state. The presentation includes a brief description of the upper sediment layers, measurement techniques, laboratory measurement data of effective diffusion coefficients, and models for prediction and extrapolation. A guide appears in Section 12.6 to steer users to suggested procedures for estimating these two types of MTCs. The chapter ends with some example problems and their solutions in Section 12.7. 

In accordance with Pick s Law, diffusive flow always occurs in the direction of decreasing concentration and at a rate, which is proportional to the magnitude of the concentration gradient. Under true conditions of molecular diffusion, the constant of proportionality is equal to the molecular diffusivity of the component i in the system, D, (m /s). For other cases, such as diffusion in porous matrices and for turbulent diffusion applications, an effective diffusivity value is used, which must be determined experimentally. 

The main focus of the following considerations is on catalysis using inorganic materials. Similar considerations come into play for catalysis with molecular compounds as catalytic components of course, issues related to diffusion in porous systems are not applicable there as molecular catalysts, unless bound or attached to a solid material or contained in a polymeric entity, lack a porous system which could restrict mass transport to the active center. It is evident that the basic considerations for mass transport-related phenomena are also valid for liquid and liquid-gas-phase catalysis with inorganic materials. 

A variety of spin probe methods have also been used to study the morphological features of the nano-channels present within MCM 41, as well as dynamical aspects connected to molecular diffusion in the inner pores,186-188 EPR has been used to investigate the adsorption and interactions of nitroxide-labelled de-ndrimers within porous silica.181 This method allows one to investigate the effective porosity of a solid surface (as a host) which is determined by the accessibility of the host surface to an adsorbed guest molecule. Information on the adsorption and interaction of dendrimers with the porous surface arises from computer-aided analysis of the EPR spectra based on of the well-established procedure proposed by Schneider and Freed.189... 

The thermal movement of molecules often serves as a prototype of random motion. In fact, molecular diffusion is the result of the random walk of atoms and molecules through gaseous, liquid, solid, or mixed media. This section deals with molecular diffusion of organic substances in gases (particularly air) and in aqueous solutions. Diffusion in porous media (i.e., mixes of gases or liquids with solids) and in other media will be discussed in the following section. 

Diw is the molecular diffusion coefficient of the chemical in water, x is tortuosity, and aL is the (longitudinal) dispersivity (dimension L). The first term describes molecular diffusion in a porous medium (Eq. 18-57), the second the effect of dispersion (Eq. 22-52). Typical values of the dispersivity aL for field systems with flow distances of up to about 100 m lie between 1 and 100 m. Since aL depends strongly on the scale... 

The mass flux inside the green body due to molecular diffusion in a porous network is... 

The process of analyte retention in high-performance liquid chromatography (HPLC) involves many different aspects of molecular behavior and interactions in condensed media in a dynamic interfacial system. Molecular diffusion in the eluent flow with complex flow dynamics in a bimodal porous space is only one of many complex processes responsible for broadening of the chromatographic zone. Dynamic transfer of the analyte molecules between mobile phase and adsorbent surface in the presence of secondary equilibria effects is also only part of the processes responsible for the analyte retention on the column. These processes just outline a complex picture that chromatographic theory should be able to describe. 

Molecular transport and tracer diffusion in porous solids are conveniently correlated in terms of diffusivities defined in accordance with Pick s first equation ... 

It has been demonstrated that the combined application of various NMR techniques for observing molecular rotations and migrations on different time scales can contribute to a deeper understanding of the elementary steps of molecular diffusion in zeolite catalysts. The NMR results (self-diffusion coefficients, anisotropic diffiisivities, jump lengths, and residence times) can be correlated with corresponding neutron scattering data and sorption kinetics as well as molecular dynamics calculations, thus giving a comprehensive picture of molecular motions in porous solids. 

Figure 4.18 Tortuosity as a function of porosity for randomly oriented porous media. For diffusion in porous materials, the length of the diffusional path is increased. If the pores are randomly oriented, and large enough to permit random molecular trajectories, the tortuosity is a function of total porosity. Equation 4-46 (dashed line). The tortuosity predicted for diffusion around a lattice of sparsely populated spheres is obtained from Equation 4-43 assuming completely impermeable spheres (heavy dashed line). Tortuosities for ensembles of cuboidal cells are also included (triangles). The solid lines without symbols indicate tortuosity for a Bethe lattice of coordination number 4 or 7.

The interphase mass transfer coefficient of reactant A (i.e., a,mtc), in the gas-phase boundary layer external to porous solid pellets, scales as Sc for flow adjacent to high-shear no-slip interfaces, where the Schmidt number (i.e., Sc) is based on ordinary molecular diffusion. In the creeping flow regime, / a,mtc is calculated from the following Sherwood number correlation for interphase mass transfer around solid spheres (see equation 11-121 and Table 12-1) ... 

Abstract As a non-invasive technique, NMR spectroscopy allows the observation of molecular transport in porous media without any disturbance of their intrinsic molecular dynamics. The space scale of the diffusion phenomena accessible by NMR ranges from the elementary steps (as studied, e.g., by line-shape analysis or relaxometry) up to macroscopic dimensions. Being able to follow molecular diffusion paths from ca. 100 nm up to ca. 100 xm, PPG NMR has proven to be a particularly versatile tool for diffusion studies in heterogeneous systems. With respect to zeolites, PFG NMR is able to provide direct information about the rate of molecular migration in the intracrystalline space and through assemblages of zeolite crystallites as well as about possible transport resistances on the outer surface of the crystallites (surface barriers). 

Diffusion in porous media Ion and molecular transport in mesopores Nanofluidics in Continuous model Distributed parameter model mesopores... 

Gas diffusion in porous solid. In this type a gas phase is present on both sides of the membrane, which is a microporous solid. The rates of molecular diffusion of the various gas molecules depend on the pore sizes and the molecular weights. This type of diffusion in the molecular, transition, and Knudsen regions was discussed in detail in Section 7.6. 

The mechanisms of mass transport can be divided into convective and molecular flow processes. Convective flow is either forced flow, for example, in pipes and packed beds, or natural convection induced by temperature differences in a fluid. For diffusive flow we have to distinguish whether we have molecular diffusion in a free fluid phase or a more complicated effective diffusion in porous solids. Like heat transport, diffusion may be steady-state or transient. 

A. dispersion caused by mokcular diffusion Dispersion due to molecular diffusion in the interparticle void spaces is described by the void fraction of the bed, e, and the tortuosity of the diffusion path in the void space. Unlike the diffusion in porous particles reviewed in Chapter 4, the latter is considered close to unity for diffusion in packed beds. Then,... 

Liquid and gaseous molecules have been known to exhibit characteristic transport behaviors in each type of porous material. For example, mass transport can be obtained via viscous flow and molecular diffusion in a macroporous material, through surface diffusion and capillary flow in a mesoporous material and by activated diffusion in a microporous material. 

Figure 3. Tortuosity as a function of porosity for randomly oriented porous media. For diffusion in porous materials, the length of the diffusional path Is increased. If the pores are randomly oriented, and large enough to permit random molecular trajectories, the tortuosity is a function of total porosity, as shown.

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