Spherical diffusion model particles

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. 

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 ... 

Because particles of different sizes are distributed throughout the bulk randomly, developing an exact model that couples diffusion to particle size evolution is daunting. However, a mean-field approximation is reasonable because diffusion near a spherical sink (see Section 13.4.2) has a short transient and a steady state characterized by steep concentration gradients near the surface. The particles act as independent sinks in contact with a mean-field as in Fig. 15.2. 

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,... 

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. 

The mathematical model for the mass transfer of an adsorbate in the LC column packed with the silicalite crystal particles is based on the assumptions of (1) axial—dispersed plug—flow for the mobile phase with a constant interstitial flow velocity (2) Fickian diffusion in the silicalite crystal pore with an intracrystalline diffus— ivity independent of concentration and pressure and (3) spherical silicalite crystal particles with a uniform particle size distribution. A detailed discussion of these assumptions can be found in (13). The differential mass balances over an element of the LC column and silicalite crystal result in the following two partial differential equations ... 

According to this simple model, the Agl actually forms at the surface and not in the core at all. The Agl phase that emerges onto the external silica surface grows by spherical diffusion in the bulk solution, where the I2 concentration is much higher, and diffusion is unimpeded. This may explain the hemispherical particle shape adopted by many of the Agl particles. It is apparent that such complex morphology changes could not have been predicted from the spectroscopic data shown in Figure 51.17 alone. 

For most practical solids (at least in gas phase application), the Henry constant K is usually of the order of 10 to 1000, that is the adsorbed phase is approximately 10 to 1000 times more dense than the gas phase hence the apparent diffusivity is approximately equal to the surface diffusivity. This extreme is called the solid or surface diffusion model, commonly used to describe diffusion in solids such as ion-exchange resins and zeolite crystals. The half time for the case of surface diffusion model in a spherical particle is (Section 9.2.1.4) ... 

The cathode of a modem Ni-Cd battery consists of controlled particle size spherical NiO(OH)2 particles, mixed with a conductive additive (Zn or acetylene black) and binder and pressed onto a Ni-foam current collector. Nickel hydroxide cathode kinetics is determined by a sohd state proton insertion reaction (Huggins et al. [1994]). Its impedance can therefore be treated as that of intercalation material, e.g. considering H+ diffusion toward the center of sohd-state particles and specific conductivity of the porous material itself. The porous nature of the electrode can be accounted for by using the transmission line model (D.D. Macdonald et al. [1990]). The equivalent circuit considering both diffusion within particles and layer porosity is given in Figure 4.5.9. Using the diffusion equations derived for spherical boundary conditions, as in Eq. (30), appears most appropriate. 

Figure 10.15 shows a model of thermally induced phase transformation in a spherical HAp powder particle, considering a paraboUc temperature gradient according to Fourier s law. During the short residence time of the particle in the plasma jet, the innermost core wiU remain at a temperature well below 1550°C owing to the low thermal diffusivity, showing HAp and OAp as stable phases (see Steps 1 and 2 in Scheme 10.1). The second shell has been heated to a temperature above the incongruent melting point of hydroxyapatite (1570 °C), and consists of...

One way to treat the obstruction effect for spherical particles has been considered by Jonstromer et al, (16). These authors express the reduction in self-diffusion at low concentrations according to a cell-diffusion model, as shown in the following equation ... 

The fluctuations in the pulse height in the first part of the curve are related to particle orientation and ruggedness and we see that the curve becomes smoother as the particle size decreases (indicating that the particle has become more uniform in shape). When the last portion of the curve in Figure 7 is transformed into the equivalent radii squared (assuming spherical particles) it can be fitted quite well to a straight line in accordance with the diffusion model presented above, without surface tension. 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

Bartle et al. [286] described a simple model for diffusion-limited extractions from spherical particles (the so-called hot-ball model). The model was extended to cover polymer films and a nonuniform distribution of the extractant [287]. Also the effect of solubility on extraction was incorporated [288] and the effects of pressure and flow-rate on extraction have been rationalised [289]. In this idealised scheme the matrix is supposed to contain small quantities of extractable materials, such that the extraction is not solubility limited. The model is that of diffusion out of a homogeneous spherical particle into a medium in which the extracted species is infinitely dilute. The ratio of mass remaining (m ) in the particle of radius r at time t to the initial amount (mo) is given by ... 

Gas diffusion in the nano-porous hydrophobic material under partial pressure gradient and at constant total pressure is theoretically and experimentally investigated. The dusty-gas model is used in which the porous media is presented as a system of hard spherical particles, uniformly distributed in the space. These particles are accepted as gas molecules with infinitely big mass. In the case of gas transport of two-component gas mixture (i = 1,2) the effective diffusion coefficient (Dj)eff of each of the... 

The biofilm thickness (Lf) and density (X = 50 g/L) were assumed uniform and the biofilm treated as a continuum. A substrate diffusion-reaction model assuming spherical particle was used. Diffusion coefficient of phenol and oxygen in the biofilm were assessed according to Fan et al. [64] ... 

This study was carried out to simulate the 3D temperature field in and around the large steam reforming catalyst particles at the wall of a reformer tube, under various conditions (Dixon et al., 2003). We wanted to use this study with spherical catalyst particles to find an approach to incorporate thermal effects into the pellets, within reasonable constraints of computational effort and realism. This was our first look at the problem of bringing together CFD and heterogeneously catalyzed reactions. To have included species transport in the particles would have required a 3D diffusion-reaction model for each particle to be included in the flow simulation. The computational burden of this approach would have been very large. For the purposes of this first study, therefore, species transport was not incorporated in the model, and diffusion and mass transfer limitations were not directly represented. 

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