Spherical diffusion model

Fig. 4.2.1 Spherical diffusion model for the growth of a tabular grain. (From Ref. 6.)...

Spherical Diffusion Model. This simple model assumes that diffusion occurs within a spherical particle. The model, however, cannot yield the diffusion coefficient directly since it contains a dimensional length parameter whose numerical value depends on the assumed diffusion mechanism. If intradiffusion predominates, the characteristic length parameter is assumed to be the size of a single crystal of the adsorbent. Consequently, the resulting diffusion coefficients are very small. If interdiffusion predominates, the characteristic length parameter is assumed to be the adsorbent particle diameter. The diffusion coefficient values in this case are much higher than the former ones. 

As shown in Figure 10 the spherical diffusion model does not fit the experimental results except at the last stage of the adsorption process where the adsorption rate is rather low. 

To proceed with the analysis, we need to be able to compare rates of interdiffusion among the various experiments. We calculate diffusion coefficients by fitting the data to a spherical diffusion model which satisfies Pick s laws of diffusion, eq (4),... 

PasquiU Atmo.spheric Diffusion, Van Nostrand, 1962) recast Eq, (26-60) in terms of the dispersion coefficients and developed a number of useful solutions based on either continuous (plume) or instantaneous (puff) releases, Gifford Nuclear Safety, vol, 2, no, 4, 1961, p, 47) developed a set of correlations for the dispersion coefficients based on available data (see Table 26-29 and Figs, 26-54 to 26-57), The resulting model has become known as the Pasquill-Gifford model. 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

As a last example in this section, let us consider a sphere situated in a solution extending to infinity in all directions. If the concentration at the surface of the sphere is maintained constant (for example c — 0) while the initial concentration of the solution is different (for example c = c°), then this represents a model of spherical diffusion. It is preferable to express the Laplace operator in the diffusion equation (2.5.1) in spherical coordinates for the centro-symmetrical case.t The resulting partial differential equation... 

Figure 1. Outline of the uptake model showing the spherical diffusion of species M through the medium towards two different sites where adsorption is followed by internalisation. The radius of the organism is taken as ro...

The Diffusion Model. The uptake of a solute by a sorbent can be analyzed by a diffusion model, which has been used successfully to model adsorption rates onto activated carbon (74, 75), ion exchangers (72), heterogeneous catalysts (76), and soil columns (77). For the purpose of illustration, we can consider the diffusion of a compound into a spherical sorbent grain under conditions of linear sorption and no exterior mass transfer limitations (73), which is described by... 

X-ray and diffuse neutron scattering and diffraction studies of PMN have been interpreted in terms of the spherical layer model of Vakhrushev et al. [25,26]. The Pb atom is not situated at the (000) position as it should be for an ideal perovskite lattice, but is distributed over a sphere of radius R around this position. 

We consider a spherical particle aggregate with radius r0 surrounded by a concentric boundary layer of thickness 8 (Fig. 19.16). Transport into the aggregate is described by the linear approximation of the radial diffusion model. Thus, the total flux from the particle to the fluid is given by Eq. 19-85 ... 

The second term represents a correction for spherical diffusion. This result is approximate and assumes that there is a steady state in the reaction layer. Numerical solution using the expanding-plane model leads to the approximate equation... 

Diffusion models of geminate pair combination connect the time-dependent pair survival probability, P t), with the macroscopic properties of the host solvent. Radicals are treated as spherical particles immersed in a uniformly viscous medium. The pair is assumed to undergo random Brownian movements that ultimately lead to either recombination or escape. The expression of P i) depends on the degree of sophistication of the theory chosen for analyzing the process. In the simplest theory,... 

It has been recognized for a long time that the orientation dependence of a vector fixed to a polymer chain could not be represented by a simple isotropic rotational diffusion model. In such a model the orientation is assumed to follow a vector joining the center of a sphere to a point performing a random brownien diffusion on the surface of that sphere. According to this model which describes well the orientation of spherical objects or infinitely thin rigid rods, the OACF is an exponential function... 

One possibility that would lead to larger inferred porosities for the U-series was introduced by Qin (1992, 1993), who proposed that the retained melt was only in complete equilibrium with the surface of minerals and that solid-state diffusion limited the re-equilibration of the retained melt with the solid. In other respects, this model is identical to the ACM model of Williams and Gill (1989). Qin introduced a specific microscopic melting/diffusion model for spherical grains and coupled it to the larger-scale dynamic melting models. The net affect of this... 

The simplest of the diffusive models proposed, the Solid core model, is based on a spherical catalyst particle with a spherical shell of polymer growing around it. 

In this, diffusion model the rate of energy transfer can be derived by considering the flux of excitons crossing the spherical surface of a single trap... 

The result is that the present authors model and Brainina s earlier assumption become increasingly identical and will lead to the same results (i) at high microscopic coverage (ii) with large metal (microparticle) center radii and/or (iii) at slow scan rates. In other words, the role of spherical diffusion becomes more important if the metal (microparticle) center radii and the microscopic coverage are reduced. Spherical diffusion leads to non-uniform local coverage (Glc), while on the metal center... 

Ideally, first the measurement modeling should be carried out. The number and the nature of the circuit elements should be identified and then the process modeling should be carried out. Such a procedure is relatively elementary for a circuit containing simple elements R, C, and L. It may also be carried out for circuits containing distributed elements that can be described by a closed-form equation CPE, semi-infinite, finite length, or spherical diffusion, etc. However, many different conditions arise from the numerical calculations (e.g., for correct solution for porous electrodes, for... 

Joslin and Goldman [105] in 1992 studied this problem by using the Diffusive Quantum Monte Carlo Methods. By resorting to the hard spherical box model, they performed calculations, not only on the ground state of helium atom, but also for H- and Li+. In this method the Schrodinger equation is... 

The profile side-pore diffusion model is the most complex and, perhaps, the most realistic of the four models. Without the Freundlich sorption mechanism, the model is the same as that developed by Kipp ( ). The case of radial diffusion with linear sorption was considered by van (ienuchten et al. ( ) whereas, spherical diffusion that had linear sorption was described by... 

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