Porous, diffusive transport

Saltzman, W. M., Pasternak, S. H., and Langer, R., Micro-structural models for diffusive transport in porous polymers, in Controlled-Release Technology, ACS Symposium Series 348... 

Kim, JH Ochoa, JA Whitaker, S, Diffusion in Anisotropic Porous Media, Transport in Porous Media 2, 327, 1987. 

In order to simplify the situation, we assume that our porous sample under investigation covers the bottom of an open straight-walled can and fills it to a height d (Figure 1). Such a sample will exhibit the same areal exhalation rate as a free semi-infinite sample of thickness 2d, as long as the walls and the bottom of the can are impermeable and non-absorbant for radon. A one-dimensional analysis of the diffusion of radon from the sample is perfectly adequate under these conditions. To idealize the conditions a bit further we assume that diffusion is the only transport mechanism of radon out from the sample, and that this diffusive transport is governed by Fick s first law. Fick s law applied to a porous medium says that the areal exhalation rate is proportional to the (radon) concentration gradient in the pores at the sample-air interface... 

This method is commonly used to obtain the diffusion coefficient through porous membranes. The schematic diagram illustrating the best technique for application of the time-lag method for determination of diffusion transport is shown in Fig. 4. As in the test setup shown in Fig. 4 a, the soil is contained between the source and collection reservoirs. Using this technique for diffusion coefficient determination of pollutants requires that the following conditions are satisfied ... 

In dense membranes, no pore space is available for diffusion. Transport in these membranes is achieved by the solution diffusion mechanism. Gases are to a certain extent soluble in the membrane matrix and dissolve. Due to a concentration gradient the dissolved species diffuses through the matrix. Due to differences in solubility and diffusivity of gases in the membrane, separation occurs. The selectivities of these separations can be very high, but the permeability is typically quite low, in comparison to that in porous membranes, primarily due to the low values of diffusion coefficients in the solid membrane phase. 

The species diffusivity, varies in different subregions of a PEFC depending on the specific physical phase of component k. In flow channels and porous electrodes, species k exists in the gaseous phase and thus the diffusion coefficient corresponds with that in gas, whereas species k is dissolved in the membrane phase within the catalyst layers and the membrane and thus assumes the value corresponding to dissolved species, usually a few orders of magnitude lower than that in gas. The diffusive transport in gas can be described by molecular diffusion and Knudsen diffusion. The latter mechanism occurs when the pore size becomes comparable to the mean free path of gas, so that molecule-to-wall collision takes place instead of molecule-to-molecule collision in ordinary diffusion. The Knudsen diffusion coefficient can be computed according to the kinetic theory of gases as follows... 

At catalytically active centers in the center of carrier particles, external mass transfer (film diffusion) and/or internal mass transfer (pore diffusion) can alter or even dominate the observed reaction rate. External mass transfer limitations occur if the rate of diffusive transport of relevant solutes through the stagnating layer at a macroscopic surface becomes rate-limiting. Internal mass transfer limitations in porous carriers indicate that transport of solutes from the surface of the particle towards the active site in the interior is the slowest step. 

Microstructural Models for Diffusive Transport in Porous Polymers... 

The results from this analysis can now be used to construct geometrically accurate models of the diffusive transport in porous polymers. Previous models of diffusion in these polymers have used an empirically determined tortuosity factor as a lumped parameter to account for the retardation of release by all mechanisms (7-8). 

The boundary condition of zero accumulation on the interface between macropores and solid phase is imposed. The effective diffusivity of the porous sample G1 with bimodal pore size distribution is summarized in Fig. 16, where the sample macro-porosity macro is varied on the horizontal axis. This effective diffusivity is compared with a situation where the diffusion transport in nanopores is omitted. The contribution of the transport through the nano-porous solid phase to the total diffusion flux is significant. The calculated effective... 

Hindered diffusion, the primary transport mechanism in porous solids, can be qualitatively described as a series of hops by the analyte, via gas-phase diffusion, from one surface site to the next. Thus, hindered diffusion is composed of two main components a pure diffusion-related term, often Fickian in nature, associated with movement of the analyte in the gas phase and a term describing the noninstantaneous equilibration between gas-phase analyte and the solid surface at each point where the analyte touches down (adsorbs). In extended porous solids (e.g., a chromatographic column tightly packed with porous beads), transport is often more complex, requiring the consideration of such factors as eddy diffusion and Knudsen effusion. This is important if there is a significant pressure drop along the path of the analyte [109]. Finally, the presence of any external fields (thermal, electric, etc.) must be considered as well. 

The usual interpretation of the parameter P, referred to here as the deposition modulus, is that it is the square of the ratio of the characteristic time for diffusion to the characteristic time for surface deposition. In this view it is equivalent to the square of the Thiele modulus commonly appearing in analyses of porous-bed catalysis. Another useful interpretation of this parameter is that it is the ratio of two rates - the rate of deposition on the preform fiber surfaces, Ss a, to the maximum rate of diffusive transport, pDDf/a. Thus when P is small, the actual rate of diffusive transport will be less than this maximum, and the mean gradient of the reactant fraction will be smaller than the maximum value off/a. Under any of these interpretations, small values of P are associated with high uniformity of both the reactant fraction and coating thickness. 

The water transport mechanism changes from the flow mechanism in porous membrane to the diffusive transport in nonporous homogeneous membrane due to the deposition of a homogeneous LCVD layer that fills the pore, i.e., water transport changes from bulk flow to diffusive flow when pores are covered by LCVD film. 

Here is the molecular diffusion coefficient of the pair C-T and K. 3 (2/3)V(8RgT/7tMT) is the Knudsen constant for the tracer T, Rg is the gas constant, T temperature, and Mt the tracer molecular weight. v t and vi/ are parameters characterizing the porous medium (transport parameters). stands for the integral mean radius of pores through which the... 

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