Diffusion, bulk factor

Figure 2.4 Relation between diffusion impedance factor, /l, and bulk density, p, in four water-saturated rice soils. Dotted line is the theoretical relation between /l and p for a mixture of different-sized spherical particles (Kirk et al., 2003). Iloilo Epiaquult clay 21% org C 1.04% pH 3.93. Maahas Haplaquoll clay 54% org C 1.83% pH 5.89. Nueva Ecija Epiaquert clay 35 % org C 1.57 % pH 5.25. Tarlac Tropaquept clay 33 % org C 1.06% pH 6.02. Reproduced by permission of Blackwell Publishing...

The test.s of the phenomenological theory were based on diffusion coefficients deter-mined from the MD simulations) that fell in a range that corresponded to surface diffusion. This is to be expected because of the very small particle size for which the MD simulations were made. As particle size is increased, bulk diffusion proce.sses are likely to become more important. More is known about diffusion mechanisms for this process than for surface diffusion. The factors that determine bulk diffusion are reviewed in the next section. 

Further development of this model incorporating diffusion, bulk flow, transmembrane flux, and matrix shrinkage [42-44] showed that the cell membrane is the main barrier to mass transfer only for single cells or thin slices of tissue. When the thickness of the sample increases, the extracellular space may become the limiting factor [45]. 

Bulkladung (Transport) bulk container Schilttgutbehalter bulk density (BD)/ apparent density/ gross density (powder density) Rohdichte, Schiittdichte bulk diffusion Massendiffusion, Gesamtduffusion bulk factor Filllfaktor,... 

Ac this point It is important to emphasize that, by changing a and p, it is not possible to pass to the limit of viscous flow without simultaneously passing to the limit of bulk diffusion control, and vice versa, since physical estimates of the relative magnitudes of the factors and B... 

Table II.1 which depends on the pellet size, so the familiar plot of effectiveness factor versus Thiele modulus shows how t varies with pellet radius. A slightly more interesting case arises if it is desired to exhibit the variation of the effectiveness factor with pressure as the mechanism of diffusion changes from Knudsen streaming to bulk diffusion control [66,...

Several factors affect the bandshapes observed ia drifts of bulk materials, and hence the magnitude of the diffuse reflectance response. Particle size is extremely important, siace as particle size decreases, spectral bandwidths generally decrease. Therefore, it is desirable to uniformly grind the samples to particle sizes of <50 fim. Sample homogeneity is also important as is the need for dilute concentrations ia the aoaabsorbiag matrix. 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

A key factor determining the performance of ultrafiltration membranes is concentration polarization due to macromolecules retained at the membrane surface. In ultrafiltration, both solvent and macromolecules are carried to the membrane surface by the solution permeating the membrane. Because only the solvent and small solutes permeate the membrane, macromolecular solutes accumulate at the membrane surface. The rate at which the rejected macromolecules can diffuse away from the membrane surface into the bulk solution is relatively low. This means that the concentration of macromolecules at the surface can increase to the point that a gel layer of rejected macromolecules forms on the membrane surface, becoming a secondary barrier to flow through the membrane. In most ultrafiltration appHcations this secondary barrier is the principal resistance to flow through the membrane and dominates the membrane performance. 

The production of hydroxide ions creates a localized high pH at the cathode, approximately 1—2 pH units above bulk water pH. Dissolved oxygen reaches the surface by diffusion, as indicated by the wavy lines in Figure 8. The oxygen reduction reaction controls the rate of corrosion in cooling systems the rate of oxygen diffusion is usually the limiting factor. 

As noted previously, for equimolecular counterdiffusion, the film transfer coefficients, and hence the corresponding HTUs, may be expressed in terms of the physical properties of the system and the assumed film thickness or exposure time, using the two-film, the penetration, or the film-penetration theories. For conditions where bulk flow is important, however, the transfer rate of constituent A is increased by the factor Cr/Cgm and the diffusion equations can be solved only on the basis of the two-film theory. In the design of equipment it is usual to work in terms of transfer coefficients or HTUs and not to endeavour to evaluate them in terms of properties of the system. 

Despite the fact that relaxation of rotational energy in nitrogen has already been experimentally studied for nearly 30 years, a reliable value of the cross-section is still not well established. Experiments on absorption of ultrasonic sound give different values in the interval 7.7-12.2 A2 [242], As we have seen already, data obtained in supersonic jets are smaller by a factor two but should be rather carefully compared with bulk data as the velocity distribution in a jet differs from the Maxwellian one. In the contrast, the NMR estimation of a3 = 30 A2 in [81] brought the authors to the conclusion that o E = 40 A in the frame of classical /-diffusion. As the latter is purely nonadiabatic it is natural that the authors of [237] obtained a somewhat lower value by taking into account adiabaticity of collisions by non-zero parameter b in the fitting law. 

Example 10.6 A commercial process for the dehydrogenation of ethylbenzene uses 3-mm spherical catalyst particles. The rate constant is 15s , and the diffusivity of ethylbenzene in steam is 4x 10 m /s under reaction conditions. Assume that the pore diameter is large enough that this bulk diffusivity applies. Determine a likely lower bound for the isothermal effectiveness factor. 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

In early 2004, Hurlimann studied several cheese samples using D-T2 correlation experiments. The D-T2 spectrum shows predominantly two signals, one with a diffusion coefficient close to that of bulk water, and the other with a D about a factor of 100 lower. The fast diffusing component is identified as water and the other as fat globules. Two components of cheese in the D-T2 map has also been observed by Callaghan and Godefroy [65]. Recently, Hurlimann et al. have performed a systematic 2D NMR study of milk, cream, cheeses and yogurts [66], Some of the preliminary results are discussed here. 

A quantitative analysis [34], based on the adsorption isotherms and the intercrystalline porosity, yielded the remarkable result that a satisfactory fit between the experimental data and the estimates of Aong-range = Pinter Anter following Eqs. (3.1.11) and (3.1.12) only lead to coinciding results for tortuosity factors a differing under the conditions of Knudsen diffusion (low temperatures) and bulk-diffusion (high temperatures) by a factor of at least 3. Similar results have recently been obtained by dynamic Monte Carlo simulations [39—41]. 

It should be mentioned here that a different definition of the diffusion coefficient is often used in chemical engineering problems, which is more appropriate for the description of reactant or tracer transport. It takes into account the fact that the total fluid contained in a porous substance of porosity e is reduced by this factor relative to the bulk, so that an effective diffusion coefficient D of the reactants is defined such that... 

When the reaction times for Step 1 are 5 min or longer, the samples severely crack, curl, or dissolve. These results suggest that substantial reaction is occurring in the bulk of the polymer. Significant hydrophilization can occur with reaction times as short as 5 s with RTD concentrations of 0.2-0.5 M. However, 0.002-0.02 M solutions of MeTD or PhTD do not allow sufficient reaction rates for surface hydrophilization at the shorter reaction times. Thus, diffusion of MeTD and PhTD into the polymer must occur readily from the acetonitrile solutions. Acetonitrile was used because it does not swell or dissolve the polymer or RTD-polymer adduct, and the RTDs are soluble and stable in it. This solvent is quite polar (dielectric constant, 38) (25), and this is probably a major factor in the partitioning of the relatively nonpolar RTDs between the polydiene film and the solvent. As noted below, more polar RTDs show less tendency to diffuse into the polymer. 

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