Coefficient, -diflusion

The equations used to describe dopant incorporation are identical to those used to describe the deposition of the semiconductor. Thus equations 12-14 are applicable to a diflusion-limited model, with the number of components, n, increased by the number of dopants added. The equilibrium distribution coefficient, ki9 is defined as... 

The analytical solutions to Fick s continuity equation represent special cases for which the diflusion coefficient, D, is constant. In practice, this condition is met only when the concentration of diffusing dopants is below a certain level ( 1 x 1019 atoms/cm3). Above this doping density, D may depend on local dopant concentration levels through electric field effects, Fermi-level effects, strain, or the presence of other dopants. For these cases, equation 1 must be integrated with a computer. The form of equation 1 is essentially the same for a wide range of nonlinear diffusion effects. Thus, the research emphasis has been on understanding the complex behavior of the diffusion coefficient, D, which can be accomplished by studying diffusion at the atomic level. 

A completely different result comes out if one pursues the experimental course of Fig. lib. The variation of the apparent diffusion coefficient as a function of quench depth AT = T — Ts, where Ts is the spinodal point, is shown in Fig. 13. Initially, the apparent diflusity increases with increasing quench depth AT which is in accord with the mean-field theory since (% — Xs)/Xs AT. However, when the quench depth is raised further the apparent diffusion coefficient starts to decrease and finally, apparently levels off. 

Useful for measurement of physicochemical properties of the catalyst, e.g.. diflusivity in the catalyst pellets, adsorplion coefficient, etc., under reaction conditions. Relative importance of adsorption and surface rate processes can be determined from this type of a reactor. 

Here J = R, good for particles smaller than 5-10 pm [130, 131]. Both and are related Ihrou the solubility product. For the case of a 1 1 [a 6] stoichiometry wilh similar diflusion coefficients,... 

In this equation, is the total interaction energy between the two colliding particles defined in the previous section. The stability ratio, W, for the system gives the ratio of rapid coagulation, Jp, to slow coagulation, J[= J W], DQi) is the position-dependent diflusion equation. This diffusion coefficient ratio is a factor that decreases the collision rate because of the difficulty in draining the liquid between the two solid surfaces. This diffiision coefficient ratio is given by [60,61]... 

Higher fluxes and lower effective pore diflusion coefficients wiU give a higher partial pressure gradient and result in a larger stress due to capillary pressure. 

The above equations are written for component A. Similar equations apply for component B. The concentration dependence of pore diflusivity followed Darken s equation. However, the concentration dependence of the barrier coefficient was much stronger than that expected from the analog of Darken s equation, which following other studies in this... 

Ledwell J. R. (1984) The variation of the gas transfer coefficient with molecular diflusivity. In Gas Transfer at Water Surfaces (eds. W. Brutsaert and G. H. Jirka). Reidel, Dordrecht, pp. 293—302. 

DIMENSIONAL ANALYSIS. From the mechanism of mass transfer, it can be expected that the coefficient k would depend on the diflusivity D and on the variables that control the character of the fluid flow, namely, the velocity w, the viscosity fi, the density p, and some linear dimension D. Since the shape of the interface can be expected to influence the process, a different relation should appear for each shape. For any given shape of transfer surface... 

The actual coefficient is greater than because frequent acceleration and deceleration of the particle raises the average slip velocity and because small eddies in the turbulent liquid penetrate close to the particle surface and increase the local rate of mass transfer. However, if the particles are fully suspended, the ratio kJk T falls within the relatively narrow range of 1.5 to 5 for a wide range of particle sizes and agitation conditions.The effects of particle size, diflusivity, and viscosity follow the trends predicted for k p, but the density difference has almost no effect until it exceeds 0.3 g/cm . For suspended particle, varies with only the... 

The contribution of the subsurface-surface transfer has also been considered by Ravera et al. [69], however this will not be discussed here further in detail. The complete adsorption kinetics problem consists now of the transport by diffusion and the boundary condition (4.30). In order to estimate the influence of the three main processes going on simultaneously, a comparison of the characteristic times is helpful. The characteristic time of the diffusion process is given by the diflusion relaxation time as defined above in Eq. (4.26), which depends on the diffusion coefficient D and the surface properties of the surfactant expressed by the ratio Fo/co- The characteristic time of the orientation process is found by assuming the other processes to be at equilibrium [69]... 

Ca, is the fluid reactant concentration in the pore, Rp the pore radius. D,p in this model may be a harmonic mean of the bulk and Knudsen diflusion coefficient with real geometries it would be a true effective difTusivity including the tortuosity factor and an internal void fraction. D p is an effective diffiisivity for the mass transfer inside the solid and is a correction factor accounting for the restricted availability of reactant surface in the region where the partially reacted zones interfere. For Jt(y) < LJ2 (shown in Fig. 4.5-2) or j>2 < J f e factor ( = 1 for L/ > R y) > L/2 or >i < y < yj the factor = 1 — (40/x) where tgB = (2/L) Ji (y) - (L/2) for y < yi the factor C 0, where R(y) is the radial position of the reaction front. It is clear from Eq. 4.S-1 that no radial concentration gradient of A is considered within the pore. 

These trends reflect the competition between nucleation, growth, coalescence and Ostwald ripening, particularly through the influence of the diflusion coefficient, which is inversely proportional to the viscosity of the solution [24]. 

The solvent viscosity was measured by an Automated Microviscometer AMVn (Anton Paar GmbH, Austria, Graz) that is based on the rolling-ball principle. The diameter of the capillary having an inclination of 30° or 40° and ball were 1.8 mm and 1.5 mm, respectively. The filled solvent in the capillary was kept at 60 °C for 15 min before measurements. All measured values were calibrated by the value of EGi (4.95 mPas) at 60 °C. The accuracy was 1%, which was taken into account when the diflusion coefficients were converted to the hydrodynamic radii. 

The models of van Baten and Krishna (2004) and Vandu et al. (2005), for gas-Uquid bubble flows, showed little or no agreement with the experimental results. Van Baten and Krishna (2004) developed their model (Eq. 7.1.1) over a wide range of parametric values (ID = 1.5-3 mm, Luc = 0.015-0.05 m). Their model underestimated the current mass transfer coefficients for all the channels. It is worth noting that in this work the length of the unit cells (Luc) and the velocity of the dispersed phase (Up) were one order of magnitude lower than those used by Van Baten and Krishna (2004). In the model by Vandu et al. (2005) (Eq. 7.1.2), which was evaluated for channel sizes from 1 to 3 mm ID and unit cell lengths from 5 to 60 mm, the only contribution on the mass transfer coefficient is by the film. The kuu obtained for 0.5 and 1 mm ID channel seem to fall within the predictions of their model (for C = 8.5), whilst mass transfer is underestimated in all cases for the 2 mm ID channel with a relative error from 40 to 60 %. The discrepancies between the experimental results and the gas-liquid models may be attributed to the more complex hydrodynamics in the liquid-liquid systems. In addition, there is less resistance to mass transfer by diflusion within a gas plug compared to a liquid one. 

Hydrodynamic theories for prediction of liquid-phase diffusion coefficients at infinite dilution are represented Iqr the Stokes-Einstein equation which views the diflusion process as the motion of a spherical solute molecule A through a continuum made up of solvent molecules B. 

Mass transfer during drop formation can be quite significant. After formation the drop falls (or rises) through the continuous phase at its terminal velocity. Small drops (<2 mm), those in the presence of surfactants, or those for which the continuous-phase viscosity is much less than the drop viscosity behave as rigid spheres with little internal circulation. For this situation the continuous-phase coefficient can be obtained from correlations such as Eq. (2.4-38) indeed, much of the data for this correlation were obtained from evaporation rates of pure liquid drops in a gas. If no circulation is occurring within the drop, the mass transfer mechanism within the drop is that of transient molecular diflusion into a sphere for which solutions are readily available (see Section 2.3). 

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