Gas-filled pores

Diffusivity in Pores and Fick s Laws Diffusion in Gas-Filled Pores Knudsen Effect Diffusion in Liquid-Filled Pores Renkin Effect Diffusion in the Unsaturated Zone of Soils... 

Due to the small dimensions of the channels in porous media, viscous forces usually suppress turbulence. Hence, diffusion through the pore space occurs by molecular motions. If the size of the pores is small, molecular motions are reduced. In gas-filled pores, this is the case if the pore size is similar to or smaller than the... 

If the typical pore diameter, dp, is of the same order of magnitude as the mean free path kj (Eq. 18-42) of the molecules migrating through gas-filled pores, then the molecules will frequently collide with the pore wall and the effective mean free path will be reduced. This effect is accounted for by the nondimensional Knudsen number, Kn ... 

First note that benzene is primarily moving through the gas-filled pores. Diffusion through the water-filled pores is too slow to account for much of the total flux. To calculate the steady-state diffusive flux through the 3-meter-thick gas-filled pores, use Eq. 18-56 and replace Dipm by diffusivity in the unsaturated zone, Djuz, and ( > by 0g. The latter is the gas-filled void which amounts to 75% of the 40% total porosity. That is, 0 = 0.30. 

The flux per gas-filled pore cross-sectional area is ... 

The resistances to the mass transport that a species encounters when is transferred from the gas to the hquid phase are reported in Figure 38.3. Gas and liquid phases contribute to the overall resistance because of the formation of boundary layers close to the membrane surface. This imphes that the concentration of a generic species i in the bulk of the two phases is different from its concentration at the membrane surfaces. The resistance offered by the membrane with gas-filled pores will be different (generally lower) than that with liquid-filled pores, due to the different effective diffusion coefficients. The overall mass-transport coefficient is given by... 

At the same time, there is no crude physical controversy in this approach. The largest pores are those which are obtained at given Pc by capillary condensation. Gas filled pores, that is, pores with larger radii than those corresponding to Eq. (3) do not exist, and water-filled pores do not conduct gas. Equilibrium with the gas outside the membrane is established. 

The effective diffusion coefficient depends on the particle porosity, the pore diameter, the tortuosity, and the nature of the diffusing species. For gas-filled pores, the above factors can be allowed for to make a reasonable estimate of the effective diffusivity in the gas phase. However, diffusion of adsorbed molecules along the pore walls, called surface diffusion, often contributes much more to the total flux than diffusion in the gas phase. This is particularly evident in the adsorption of water vapor on silica gel and the adsorption of hydrocarbon vapors on carbon, where the measured values of correspond to internal and external coefficients of comparable magnitude or even to external film control, For adsorption of solutes from aqueous solutions, surface migration is much less important, and the internal diffusion resistance generally dominates the transfer process. 

The concentration gradients in an asymmetric membrane are complex because the driving force for diffusion in the skin layer is the concentration gradient of gas dissolved in the dense polymer, and the driving force in the porous support layer is a concentration or pressure gradient in the gas-filled pore. When the porous layer is thick, diffusion does not contribute very much to the flux, and gas flows by laminar flow in the tortuous pores. For high-flux membranes, there may also be significant mass-transfer resistances in the fluid boundary layers on both sides. 

Another driving force for methane transport is the generation of critical pressures in the gas phase (Flemings et al. 2003 Trehu et al. 2004). Interconnection of gas-filled pores below the GHSZ transmits hydrostatic pressures from greater depths because of the low density of the gas phase. The excess (non-hydrostatic) pressure at the top of the gas layer may be sufficient to... 

The multilayered films of Cu/V, Cu/Nb, and Fe/W with the thickness of some layers varying in the range of 2.5-200 nm were exposed to irradiation with helium ions and the formation of gas-filled pores has been analyzed [81-84]. Whatever the combination of metals is, helium bubbles accumulate along the boundaries between layers. Moreover, as the thickness of layers decreases the pore size reduces (Figure 24.20a). At a thickness of 2.5 nm, the resolution of electron microscope failed to show any helium formations. In addition, it was noticed that the hardening of multilayered structures degrades as the thickness of individual films decreases (Figure 24.20b). 

Membrane gas absorption (MGA) is a gas-liquid contacting operation [1,2,24,25]. The key element in the process is a microporous hydrophobic HFM. The process is illustrated in Figure 4.3 for removal of component X from a gas stream. The gas stream is fed along one side of the membrane where an absorption liquid is flowing at the other side of the membrane. The hydrophobic membrane wall keeps gas phase and absorption liquid separated from each other. The absorption liquid is chosen in such a way that it has a high affinity for component X. Component X will now diffuse through the gas-filled pores of the membrane to the other side of the membrane where it is absorbed in a liquid phase. Absorption in the liquid phase takes place either by physical absorption or by a chemical reaction. This determines the selectivity of the process. The membrane used... 

The fraction of dry catalyst was found contribute significantly or even dominate the overall rate of reaction. The effect was attributed to the higher rates of mass transfer and lower rates of heat transfer in gas-filled pores compared with liquid-filled pores. Similarly, Wood and Gladden [15] have assumed that gas-phase reaction dominates the overall rate in their simulations of hydrodesulfurization of diethyl sulfide, by neglecting liquid-phase reaction. Ostrovskii et al. [7] have... 

The effects of capillary condensation were included in the network model, by calculating the critical radius below which capillary condensation occurs based on the vapor composition in each pore using the multicomponent Kelvin Equation (23.2). Then the pore radius was compared with the calculated critical radius to determine whether the pore is liquid- or vapor-filled. As a significant fraction of pores become filled with capillary condensate, regions of vapor-filled pores may become locked off from the vapor at the network surface by condensate clusters. A Hoshen and Kopelman [30] algorithm is used to identify vapor-filled pores connected to the network surface, in which diffusion and reaction continue to take place after other parts of the network filled with liquid. It was assumed that, due to the low hydrogen solubility in the liquid, most of the reaction takes place in the gas-filled pores. The diffusion/reaction simulation is repeated, including only vapor-filled pores connected to the network surface by a pathway of other vapor-filled pores. 

Capillary condensation was also shown to influence catalyst deactivation, although sometimes deactivation was slower in the liquid-filled pores compared with the gas-filled pores. Coking of hydro-treatment catalysts was found to lead to significant losses in pore volume and increases in tortuosity. 

As with liquid flow, the intrinsic permeability Kg with respect to gas flow is a function of only the porous medium. However, the ability of a gas to migrate through a porous medium will depend on the liquid content in the porous medium. When the porous medium is completely diy, Kg represents the continuity of flow through the pores attributable only to the porous solids (i.e., Kg = K). However, as the pores are filled with liquid (e.g., water) such that the percentage of gas-filled pores decreases, the gas may become occluded in some pores such that some of the pathways for gas-phase migration are cut off. When the porous medium is sufficiently saturated with liquid such... 

Catalyst particles in three-phase fixed-bed reactors are usually completely filled with liquid. Then intraparticle temperature gradients are negligible due to the low effective diffusivities in the liquid phase, as pointed out by Satterfield [13] and Baldi [92]. However, if the limiting reactant and the solvent are volatile, vapor-phase reaction may occur in the gas-filled pores, causing significant intraparticle temperature gradients [109, 110]. In these conditions, intraparticle heat transfer resistance is necessary to describe the heat transfer. 

Fiber sorption properties mainly determine the evaporation process and therefore the heat and mass transfer by evaporation of water, diffusion of water vapor, and condensation. Water evaporates from hot regions and moves across the gas-filled pores by diffusion and condenses on the cold region, thus releasing its latent heat of vaporization [19—21]. 

The first applications employed symmetric microporous hollow fibers in which the fiber materials were hydrophobic or hydrophilic. The membrane liquid may or may not wet the pores spontaneously. To separate a gas mixture in a permeator employing symmetric hydrophobic microporous fibers with gas-filled pores and nonwetting aqueous liquid membranes (Figure 2a), the liquid membrane pressure (PjJ is maintained higher than those of the feed gas (Ppo) and the sweep gas (Psg) to prevent their dispersion in the membrane liquid (4), However, the excess pressure of the liquid membrane over that of either gas stream must be less than the breakthrough pressure for the membrane liquid (which is a function of the membrane material, pore size, and the interfacial tension between the pore fluid (here, gas) and the membrane liquid). For other fibers and configurations. Figures 2b and 2c provide appropriate details. 

The presence of a liquid phase and a liquid-solid interface in multiphase reactors results in added transport resistances. For instance, the effective diffusivity in liquid-filled pores (of the order of 10 to 10 cmVsec) is much smaller than that in gas-filled pores (of the order of 10 cmVsec). The solubility of the gaseous reactant is an important factor since the gaseous reactant has to be dissolved into the liquid reactant for the reaction to take place on the catalyst surface. As emphasized in Chapter 4, the Biot number for heat is much larger than the Biot number for mass for liquid-solid systems the opposite is true for gas-solid systems. Therefore, the major external resistance lies in the mass transport, and the pellet is not necessarily isothermal. In many cases, however, the equilibrium gas concentration in the liquid is quite low and, thus, the heat evolved is small in spite of high heats of reaction. The pellet can be considered isothermal in such a case,... 

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