Cylindrical diffusion model

Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

Owen CS. Two dimensional diffusion theory Cylindrical diffusion model applied to fluorescence quenching. J. Chem. Phys. 1975 62 3204-3207. 

The diffusion of anions during doping and dedoping processes at PANI fibers was investigated with in situ UV-vis spectroscopy by Kanamura et al. [489]. Using a cylindrical diffusion model, it was found that the estimated diffusion coefficient increases with the radius of the PANI fiber. Data obtained with PANI prepared in nonaqueous solutions showed a similar dependence, and the overall values of the diffusion coefficient were much smaller. Whereas the kinetic measurements were made in the same solution, it was proposed, that the diffusing species were different. No suggestion of their specific identity was made. 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

The parabolic diffusion model is used to indicate that diffusion controlled phenomena are rate limiting. It was originally derived based on radial diffusion in a cylinder where the chemical compound concentration on the cylindrical surface was constant, and initially the chemical compound concentration throughout the cylinder was uniform. It was also assumed that the diffusion of the compound of interest through the upper and lower faces of the cylinder was negligible. Following Crank [119], the parabolic diffusion model can be expressed as ... 

Wheeler s treatment of the intraparticle diffusion problem invokes reaction in single pores and may be applied to relatively simple porous structures (such as a straight non-intersecting cylindrical pore model) with moderate success. An alternative approach is to assume that the porous structure is characterised by means of the effective diffusivity. (referred to in Sect. 2.1) which can be measured for a given gaseous component. In order to develop the principles relating to the effects of diffusion on reaction selectivity, selectivity in isothermal catalyst pellets will be discussed. 

Commonly [17], when the length-to-diameter ratio of a cylindrical catalyst is close to 1, the cylindrical catalyst can be simplified as a sphere, the radius of which, Rp, is calculated by 3 Kp/.S p. The one-dimensional, key-component based reaction-diffusion models of methanation system are as follows ... 

The above model has been refined based on the dusty gas model [Mason and Malinauskas, 1983] for transport through the gas phase in the pores and the surface diffusion model [Sloot, 1991] for transport due to surface flow. Instead of Equation (10-101), the following equation gives the total molar flux through the membrane pores which are assumed to be cylindrically shaped... 

The given diffusion model formulation has been used simulating the gas-liquid dynamics in cylindrical bubble column reactors [182]. 

If there are no or very few irrigated burrows present in the sediment, lateral diffusion is not significant and the r dependence of Eq. (6.12) can be ignored. In that case, the equation becomes the more traditional onedimensional transport-reaction equation used to model pore-water solute profiles where advection is relatively unimportant (Berner, 1971 1980 Lerman, 1979). Both the cylindrical microenvironment model and the onedimensional Cartesian coordinate model will be used here to quantify the Mn distributions at NWC and DEEP. 

Pore-water profiles were used together with solid-phase dissolution rates in diagenetic models to determine first-order anoxic precipitation rate constants for both Mn and Fe. A two-dimensional cylindrical coordinate model was employed to account for the effects of biogenic irrigation of burrows on pore-water Mn " distributions. Two-dimensional diffusion can result in a decrease in Mn " with depth that would be interpreted as evidence for precipitation and cause overestimation of precipitation rates in a one-dimensional model. 

For the sake of comparison hydrogenation experiments with large cylindrical catalyst particles were carried out. The increase of the particle size diminished the velocity of catalytic hydrogenation. These experimental results provide a path for the process scale-up, i.e. a prediction of the hydrogenation rate on large catalyst particles starting from crushed particles. The values of the kinetic constants obtained for crushed particles were utilized and the ratio of porosity to tortuosity from the reaction-diffusion model was adjusted (0.167) to fit successfully the experimental data (Figure 10.40). 

A regular pore structure is found in crystalline zeolites or molecular sieves but when these materials are used as catalysts, tiny zeolite crystals (1-2 fj,m) are combined with a binder to make practical-size pellets (1-5 mm). Spaces between the cemented crystals are macropores of irregular shape and size, and diffusion in these macropores has to be considered as well as diffusion in the micropores of the zeolite crystals. The cylindrical capillary model is used to describe diffusion in zeolite catalyst and other catalysts and porous solids because of its simplicity and because most of the literature values for average pore size are based on this model. However, the... 

There are mainly two types of Slurry seepage paths named cylindrical and surface-shaped, during the slurry diffusion process, cylindrical diffusion refers to slurry along the approximate space of the cylinder movement. The planar diffusion model of the slurry, is refers to the slurry along the approximate plane of the spatial movement... 

WTien the hydrogen pressure is 1 atm and the temperature is 77 K, the experimentally observed (apparent) rate constant is 0.159 cm /(s- g catalyst). Determine the mean pore radius, the combined diffusivity for hydrogen in the pores of this catalyst, and the catalyst effectiveness factor based on the straight cylindrical pore model. 

Mathematical Model Considering the Conditions of Spherical and Cylindrical Diffusion... 

Model of the Cylindrical Diffusion Around the Top Edge of a Surface Protrusion-Deposition to the Line... 

Precursors of cadmium dendrites [47] obtained by the processes of electrochemical deposition from 0.1 M CdS04 in 0.50 M H2SO4 onto cadmium wire electrodes at different overpotentials are shown in Fig. 2.22. It is obvious that further growth of the dendrite precursors shown in Fig. 2.22 leads to the formation of 2D dendrites (Fig. 2.23). Around the tips of dendrite precursors, as well as around the tips of dendrites, spherical or cylindrical diffusion control can occur, which is in good agreement with the requirements of the mathematical model. 

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