Diffusion resistance factor

If substance B is a porous solid the molecular diffusion coefficient D has to be replaced by the effective diffusion coefficient DeB. This is smaller than the molecular diffusion coefficient because the movement of the molecules is impeded by the pores. It is common practice to define a diffusion resistance factor... 

FIGURE 6.1 A Poppe plot for the required plate number in conventional HPLC. The parameters are taken from Poppe s original paper (Poppe, 1997). The parameters are maximum pressure AP = 4x 107 Pa, viscosity / = 0.001 Pa/s, flow resistance factor

diffusion coefficient D= lx 1CT9 m2/s, and reduced plate height parameters using Knox s plate height model are A — 1, B— 1.5, C = 0.05. 

Chen et al. [54] have reported a model for the assessment of the combined effects of the intrinsic reaction kinetics and dye diffusion into phosphorylated polyvinyl alcohol (PVA) gel beads. The analysis of the experimental data in terms of biofilm effectiveness factor highlighted the relevance of intraparticle diffusion to the effective azo-dye conversion rate. On the basis of these results, they have identified the optimal conditions for the gel bead diameter and PVA composition to limit diffusion resistance. 

The presence (or absence) of pore-diffusion resistance in catalyst particles can be readily determined by evaluation of the Thiele modulus and subsequently the effectiveness factor, if the intrinsic kinetics of the surface reaction are known. When the intrinsic rate law is not known completely, so that the Thiele modulus cannot be calculated, there are two methods available. One method is based upon measurement of the rate for differing particle sizes and does not require any knowledge of the kinetics. The other method requires only a single measurement of rate for a particle size of interest, but requires knowledge of the order of reaction. We describe these in turn. 

The activity calculated from (7) comprises both film and pore diffusion resistance, but also the positive effect of increased temperature of the catalyst particle due to the exothermic reaction. From the observed reaction rates and mass- and heat transfer coefficients, it is found that the effect of external transport restrictions on the reaction rate is less than 5% in both laboratory and industrial plants. Thus, Table 2 shows that smaller catalyst particles are more active due to less diffusion restriction in the porous particle. For the dilute S02 gas, this effect can be analyzed by an approximate model assuming 1st order reversible and isothermal reaction. In this case, the surface effectiveness factor is calculated from... 

Pharmaceutical scientists assess and express drug permeation across membrane barriers in terms of flux. Flux measures the molar unit of a drug that permeates a resistant barrier (e.g., skin or gastrointestinal epithelial cells) per unit time and surface area (Box 13.1). Permeation enhancers, such as alcohols and surfactants, increase flux by modulating resistance factors that counteract drug diffusion across barriers at the site of administration. 

The employment of small particles results in effectiveness factors near unity. In other words, the intraparticle diffusion resistance is low. 

It is useful to introduce the external effectiveness factor i]c . as the ratio of the observed overall rate r to the chemical reaction rate r without diffusion resistance (C = C,) ... 

The following calculation as made for the Saline Water Project (6) shows the relation between pressure applied and production rate. The dominant factors are (1) the salt solution whose osmotic pressure must be overcome, (2) the pressure, as an energy source, (3) the diffusion of heat and (4) vapor as resistance factors, and (5) viscous losses within the cellophane capillaries. 

Powders vary dramatically in particle size on the basis of their origin. It is common for catalyst manufacturers to classify powders in order to assure users of consistency from batch to batch since suspension, settling rates, filtration, and performance in slurry-phase reactions are all dependent on particle size. The effect on suspension, settling rates, and filtration is obvious. However, factors that favor these are unfavorable for kinetics. For reactions controlled by transport rates from the bulk fluid to the surface of the catalyst, the overall reaction rate is a strong function of geometric surface area and thus is favored by small particles. Pore diffusion resistance is also minimized by smaller particles since reaction paths to active sites are smaller. The only mode of reaction control not influenced by particle size is for those reactions in which rate is controlled by reaction at active sites. Therefore, a compromise for optimum filtration and maximum reaction rates must be made. 

The support has an internal pore structure (i.e., pore volume and pore size distribution) that facilitates transport of reactants (products) into (out of) the particle. Low pore volume and small pores limit the accessibility of the internal surface because of increased diffusion resistance. Diffusion of products outward also is decreased, and this may cause product degradation or catalyst fouling within the catalyst particle. As discussed in Sec. 7, the effectiveness factor Tj is the ratio of the actual reaction rate to the rate in the absence of any diffusion limitations. When the rate of reaction greatly exceeds the rate of diffusion, the effectiveness factor is low and the internal volume of the catalyst pellet is not utilized for catalysis. In such cases, expensive catalytic metals are best placed as a shell around the pellet. The rate of diffusion may be increased by optimizing the pore structure to provide larger pores (or macropores) that transport the reactants (products) into (out of) the pellet and smaller pores (micropores) that provide the internal surface area needed for effective catalyst dispersion. Micropores typically have volume-averaged diameters of 50 to... 

A gas-solid reaction usually involves heat and mass transfer processes and chemical kinetics. One important factor which complicates the analysis of these processes is the variations in the pore structure of the solid during the reaction. Increase or decrease of porosity during the reaction and variations in pore sizes would effect the diffusion resistance and also change the active surface area. These facts indicate that the real mechanism of gas-solid noncatalytic reactions can be understood better by following the variations in pore structure during the reaction. 

It plays the same role as the effectiveness factor in heterogeneous catalysis and is a measure of the film thickness uniformity. It represents the ratio of the total reaction rate on each pair of wafers to that we would obtain if the concentration in the cell formed by the two wafers were equal to the bulk concentration everywhere. Thus, if the surface reaction is the rate controlling step, n = 1, whereas if the diffusion between the wafers controls, n < 1. In the limit of strong diffusion resistance the deposition is confined to a narrow outer band of the wafers and a strongly nonuniform film results. 

A simplification is often employed for effectiveness factor calculations in the asymptotic limit of strong intraparticle diffusion resistance (12,13). In this situation, an alternative form of the key component mass balance can be written as follows ... 

The effectiveness factor 17 is a measure of how much the reaction rate is lowered because of the diffusion resistance, and Eq. (9.24) is defined by... 

Figure 9.3 is a plot of Eq. (9.28), which shows that if r/> -> 0 then rj -> 1, which means there is no considerable diffusion resistance. As diffusion resistance increases, we have (j> - oo and hence rj -> 0. The latter can occur not for small diffusivity, for large pellet size L, or for very fast reaction rate, or for all three factors. This regime where the diffusion strongly affects the rate of reaction is called strong pore resistance. For a first-order reaction, a general criterion of... 

Subsequent tests with velvetleaf, Kodkia, Jerusalem artichoke, and cocklebur showed that their allelopathic action altered water balance (55,94,95). Growth reductions in sorghum and soybean seedlings in nutrient solution amended with extracts from these weeds correlated with high diffusive resistances and low leaf water potentials. Stomatal closure occurred in plants treated with the more concentrated extracts. Depressions in water potential were due to a reduction in both turgor pressure and osmotic potential. A lower relative water content was also found in velvetleaf-treated plants. These impacts on water balance were not from osmotic factors. Allelochemicals from these weeds have not been thoroughly ascertained, but the present evidence shows that some contain phenolic inhibitors. Lodhi (96) reported that Kodkia contains ferulic acid, chlorogenic acid, caffeic acid, myricetin, and quercetin. As noted earlier, an effect on plant-water relationships is one mechanism associated with the action of ferulic acid. 

The first three types (pellets, extrudates and granules) are primarily used in packed bed operations. Usually two factors (the diffusion resistance within the porous structure and the pressure drop over the bed) determine the size and shape of the particles. In packed bed reactors, cooled or heated through the tube wall, radial heat transfer and heat transfer from the wall to the bed becomes important too. For rapid, highly exothermic and endothermic reactions (oxidation and hydrogenation reactions, such as the ox-... 

Because internal diffusion resistance is also significant, not all of the interior surface of the pellet is accessible to the concentration at the external surface of the pellet, C j. We have already learned that the effectiveness factor is a measure of this surface accessibility [see Equation (12-38)] ... 

The rate theory examines the kinetics of exchange that takes place in a chromatographic system and identifies the factors that control band dispersion. The first explicit height equivalent to a theoretical plate (HETP) equation was developed by Van Deemter et al. in 1956 [1] for a packed gas chromatography (GC) column. Van Deemter et al. considered that four spreading processes were responsible for peak dispersion, namely multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase, and resistance to mass transfer in the stationary phase. 

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