Diffusion in pores

Segment diffusion in pores suggests itself as a typical model scenario representing the premisses of the tube/reptation model. NMR diffusometry is suitable to probe the time or length scales of the Doi/Edwards limits (II)de (III)de and beyond. Since the wall adsorption effect in the PHEMA system is expected to be negligible, one can therefore expect that the reptation features of the anomalous segment diffusion regime, especially with respect to limit (III)de are faithfully rendered by the experiments. 

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For the effective diffusivity in pores, De = (0/t)D, the void fraction 0 can be measured by a static method to be between 0.2 and 0.7 (Satterfield 1970). The tortuosity factor is more difficult to measure and its value is usually between 3 and 8. Although a preliminary estimate for pore diffusion limitations is always worthwhile, the final check must be made experimentally. Major results of the mathematical treatment involved in pore diffusion limitations with reaction is briefly reviewed next. 

Fig. 7.80 A schematic thermodynamic phase stability diagram for the A-C-O system, showing three reaction paths. Paths 2 and 3 are only possible if gaseous diffusion in pores in the oxide product results in a carbon activity increase through the scale, as shown in Fig. 7.81 (after...

Thus, considering diffusion in pores leads to very similar results to those we obtained when describing diffusion in catalyst particles. 

Because of the similarity of transport in biotilms and in stagnant sediments, information on the parameters that control the conductivity of the biofilm can be obtained from diagenetic models for contaminant diffusion in pore waters. Assuming that molecular diffusion is the dominant transport mechanism, and that instantaneous sorption equilibrium exists between dissolved and particle-bound solutes, the vertical flux ( ) through a stagnant sediment is given by (Berner, 1980)... 

Table 18.2 The Role of Diffusion in Pores for First-Order Reactions in Series...

To summarize the goal of this section, we must start with the microscopic description of the catalytic reaction, then consider diffusion in pores, and then examine the reactant composition around and within the pellet, in order finally to describe the reactor maSS-balance equations in terms of z alone. The student should understand the logic of this procedure as we go from micrscopic to macroscopic, or the following sections will be unintelligible (or even more unintelligible than usual). 

Figure 4.51 Diffusion in pores (a) Knudsen diffusion, (b) Fickian diffusion, and (c) transitional diffusion.

Using resistance-in-series model (Bosanquet formula), the diffusivity in pore... 

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... 

Typical transfer velocity across liquid layer 10"3 cm s 1 (range 10 5 to 10 2 cm s see Section 20.2 and Illustrative Examples 19.3, 19.4). Km is the equilibrium partition coefficient with typical values between 1 and 104 (see Table 19.1). DA is the aqueous molecular diffusivity in pore space (typical size 10"5 cmV) divided... 

Other results also confirm the important role of internal diffusion. Experimental activation energies (67—75 kJ mol"1) of the sucrose inversion catalysed by ion exchangers [506—509] were considerably lower than those of a homogeneously catalysed reaction (105—121 kJ mol"1) [505, 506,508] and were close to the arithmetic average of the activation energy for the chemical reaction and for the diffusion in pores. The dependence of the rate coefficient on the concentration in the resin of functional groups in the H+-form was found to be of an order lower than unity. A theoretical analysis based on the Wheeler—Thiele model for a reaction coupled with intraparticle diffusion in a spherical bead revealed [510,511] that the dependence of the experimental rate coefficient on acid group concentration should be close to those found experimentally (orders, 0.65 and 0.53 for neutralisation with Na+ and K+ ions respectively [511] or 0.5 with Na+ ions [510]). 

The discussion of diffusion in pores of a catalyst requires taking into account the specific features of diffusion in narrow capillaries. 

At present two models are available for description of pore-transport of multicomponent gas mixtures the Mean Transport-Pore Model (MTPM)[4,5] and the Dusty Gas Model (DGM)[6,7]. Both models permit combination of multicomponent transport steps with other rate processes, which proceed simultaneously (catalytic reaction, gas-solid reaction, adsorption, etc). These models are based on the modified Maxwell-Stefan constitutive equation for multicomponent diffusion in pores. One of the experimentally performed transport processes, which can be used for evaluation of transport parameters, is diffusion of simple gases through porous particles packed in a chromatographic column. 

The mouth of the micropores may be narrowed by adsorbed CO2 molecules, which block N2 molecules from entering the pores. Furthermore, CO2 diffuses in pores of the zeolite membrane at a faster rate than Nj. This selection mechanism is plausible for micropores with a width of up to six molecules [7]. When CO2 molecules are strongly adsorbed on the pore wall, the COj permeation rate will be low even if CO2 is concentrated in the pore. If the pore size is close to the size of molecules, CO2 molecules cannot pass N2 molecules. Thus, balances between pore size and molecule size and between adsorptivity and mobility as well as difference in polarity of competitive species are important to attain both high permeance and selectivity. 

Transport of dissolved Mn(II) accumulated by reductive dissolution is governed by molecular diffusion in pore waters and by eddy diffusion in the stratified hypolimnia of lakes. Transport follows the concentration gradients,... 

Molecular diffusion is significant in large pores and imder high pressures. In these cases, the effects of molecule-molecule collisions dominate over those of molecule-wall collisions. However, compared to molecular diffusion in the free bulk phase, molecular diffusion in pores is hindered by the pore walls and may be slower or even much slower. 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

The actual rate determining step cannot be infered from the course of the kinetic curves, but some assessment is possible by means of specially arranged experiment. Interruption test makes it possible to assess the role intraparticle diffusion (diffusion in pores of the adsorbent) as potential rate determining step. The isotope exchange experiment is started in usual way vide infra), and k kinetic data points are collected (equilibration times t, . .., ft), but at the time ijj the solid is separated from the liquid, then both phases are stored separately for a time much... 

Cadmium sorption in soils is known to be a fast process, with 95% of the Cd sorption taking place in the first 10 minutes and reaching equilibrium in 1 hour (Christensen, 1984). Although the initial sorption of trace elements is rapid, further sorption is usually quite slow, which was ascribed to inter or intraparticle diffusion in pores, sites of low reactivity, and surface precipitation (Waychunas et al., 1993 Sparks, 1999). An important factor affecting the degree of slow sorption of trace elements is the resident time of the sorbate with the sorbent. 

DIFFUSION IN PORES OF CATALYSTS AND CATALYTIC REACTION RATES... 

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