Diffusion restrictions

Fig. 11a and b. Decay of the alignment echo height as a function of the mixing time x2 for different motional mechanisms, a Tetrahedral jumps as a model for conformational changes b Diffusive motion, the solid lines correspond to unrestricted rotational diffusion, the dashed lines to diffusion restricted to an angular region of 8°. Note the strong dependence of the decay curves on the evolution time t, in case of diffusive motion... 

Fig. 12a and b. Calculated 2H spin alignment spectra for diffusive motion, a unrestricted rotational diffusion for different mixing times x2 b diffusion restricted to angular regions as indicated for long mixing times t2... 

The endgroup diffusion restricts the number of endgroups which takes part actively in the reaction. The rate of reaction depends on the active endgroup concentration which does not only change with progress of reaction but also by the diffusion of endgroups. 

Food products can generally be considered as a mixture of many components. For example, milk, cream and cheeses are primarily a mixture of water, fat globules and macromolecules. The concentrations of the components are important parameters in the food industry for the control of production processes, quality assurance and the development of new products. NMR has been used extensively to quantify the amount of each component, and also their states [59, 60]. For example, lipid crystallization has been studied in model systems and in actual food systems [61, 62]. Callaghan et al. [63] have shown that the fat in Cheddar cheese was diffusion-restricted and was most probably associated with small droplets. Many pioneering applications of NMR and MRI in food science and processing have been reviewed in Refs. [19, 20, 59]. 

A comparison of the data for the unsupported and the supported catalysts reveals that the activity per Co atom present as Co-Mo-S is much lower for the former catalyst system. This is probably related to the fact that in these catalysts many of the Co atoms are either inaccessible to the reactants or are subjected to diffusion restrictions. 

The silica carrier of a sulphuric acid catalyst, which has a relatively low surface area, serves as an inert support for the melt. It must be chemically resistant to the very corrosive pyrosulphate melt and the pore structure of the carrier should be designed for optimum melt distribution and minimum pore diffusion restriction. Diatomaceous earth or synthetic silica may be used as the silica raw material for carrier production. The diatomaceous earth, which is also referred to as diatomite or kieselguhr, is a siliceous, sedimentary rock consisting principally of the fossilised skeletal remains of the diatom, which is a unicellular aquatic plant related to the algae. The supports made from diatomaceous earth, which may be pretreated by calcination or flux-calcination, exhibit bimodal pore size distributions due to the microstructure of the skeletons, cf. Fig. 5. 

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... 

Based on the previous analysis of the different transport phenomena, which determine the overall mass transport rate, the structure of the solid phase matrix is of extreme importance. In the case of any chromatographic process, the different diffusion restrictions increase the time required for separation, since any increase of the flow rate of the mobile phase leads to an increase of the peak broadening [12]. Thus, the improvement of the existing chromatographic separation media (column packing of porous particles) and hence the speed of the separation should enable the following tasks ... 

It has been supposed that the surface diffusion is a form of molecular diffusion, restricted by adsorption interaction. Consequently ... 

Is the diffusion restriction (by the membrane as shown) of any consequence Note that the enzymes must be ordered in this system. 

Pyrolysis of Sodium-Tetramethylammonium Zeolite Omega. Preliminary calcinations of the Q zeolites showed that intracrystalline diffusion restrictions interfered greatly with transport both of oxygen and of calcination products. Under mild conditions, coking was observed, and even under favorable conditions (550° C, thin beds, good venting) the reaction was slow. Some samples of zeolite were pyrolyzed under vacuum, and the products were identified by low resolution mass spectrometry. 

Maines A, Ashworth D, Vadgama P. Diffusion restricting outer membranes for greatly extended hnearity measurements with glucose oxidase enzyme electrodes. Analytica Chimica Acta 1996, 333, 223-231. 

For both samples a small decrease of the effective diffusion coefficient for short A is found. This could indicate some diffusion restriction. However, the samples consisted of small pieces of material pressed together in the NMR tube. Unfortunately, at the moment the possibility that the initial decrease of Deff is caused by a motion (approximately 1 pm would be sufficient ) of some of the particles caused by the pulsed field gradients cannot be excluded. For short diffusion times such a motion has a large effect, for longer A such motions can be neglected. 

Thus, at subcritical, gas-like conditions, there is no restricted diffusion. As the density increases beyond the critical point, the extent of pore-diffusion restriction increases. 

The functional and morphological heterogeneity of a lamellar system of chloroplasts indicates that pH values in different compartments (in granal and intergranal thylakoids) differ. This type of structure makes it difficult to measure local pH values at different sites. Therefore, mathematical models taking into account the spatial structure of chloroplasts provide a tool for studying the effect of diffusion restrictions on pH distributions over the thy lakoid on the rates of electron transport, proton transport, and ATP synthesis. The rate of ATP synthesis depends on the osmotic properties of a chloroplast-incubation medium and, therefore, on topological factors. 

The basis for comparing the ratios of the free diffusion coefficients and permeability coefficients was the assumption that hindrance considerations could be ignored. In the instance that this assumption is valid (i.e., the case of large pore dimensions relative to solute radii), the free diffusion coefficients are a reasonable approximation to the diffusion coefficients of the solutes in the membrane. In the instance that hindrance considerations are not negligible, due to pore dimensions that lead to diffusion-restricting hydrodynamic interactions between the solute and the membrane, the diffusion coefficient of the solute in the membrane is a function of both the solute parameters and the properties of the membrane. In this case, the effective diffusion coefficient can be approximated by the product of the free diffusion coefficient and a diffusional hindrance factor, HQC) (Deen, 1987) ... 

Catalyst selectivity differences have been found for sulfur and metals removal in residuum hydroprocessing (19). Mass transfer limitations are believed to be important (20). The data reported here show that the metal-containing molecules are larger consequently, they should be more subject to diffusion restrictions than the sulfur-containing molecules. Therefore, it will be more difficult for a small pore catalyst to demetallate a residuum than to desulfurize it. 

Experimental Measurements of Reaction Kinetics. The reaction expressions discussed in the following model the intrinsic reaction on the catalyst surface, free of mass-transfer restrictions. Experimental measurements, usually made with very fine particles, are described by theoretically deduced formulas, the validity of which is tested experimentally by their possibility for extrapolation to other reaction conditions. Commonly the isothermal integral reactor is used with catalyst crushed to a size of 0.5-1.5 mm to avoid pore diffusion restriction and heat-transfer resistance in the catalyst particles. To exclude maldistribution effects and back mixing, a high ratio of... 

Catalyst particle size (Section 3.6.1.2) smaller catalyst particles give higher conversion because of lower diffusion restrictions (higher pore efficiency)... 

Figure IB shows how veil the data fits the relationship with a zero intercept. Most runs, however produced non-zero intercepts the value of which was difficult to reproduce. This was probably due to erratic coking during the first 2 hours for inknown reasons. Despite this initial uncertainty the slope of the line was surprisingly repeatable. It is esclmaced that over 90% of runs gave slopes with an accuracy of 10%, Little change in coking rates was observed on varying linear space velocities indicating the absence of mass diffusion restrictions.

The change in sample size has been shown to influence product ratios, fractional conversions, and rates of pyrolysis for several polymers [14]. However, the dependence of pyrolysis results on sample size does not have only a negative impact on data reproducibility. This effect may reveal mechanistic features of the pyrolysis process, such as secondary reactions that occur during the residence time of primary products in the melt and diffusion-restriction of the interaction of long-chain radicals, when the thickness of the degrading film is comparable with the diameter of the hydrodynamic volume of the polymer molecules. 

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