Solids, diffusion rate

However, this difference would not seem to be large enough to account for the very large differences in solvent losses that have been reported. Another factor may be that high solids coatings may reach a stage where solvent loss is controUed by diffusion rate much eadier than is the case in low solids coatings (8,43). 

The diffusion coefficient in these phases D,j is usually considerably smaller than that in fluid-filled pores however, the adsorbate concentration is often much larger. Thus, the diffusion rate can be smaller or larger than can be expected for pore diffusion, depending on the magnitude of the flmd/solid partition coefficient. 

Combined Intraparticle Resistances When solid diffusion and pore diffusion operate in parallel, the effec tive rate is the sum of these two rates. When solid diffusion predominates, mass transfer can be represented approximately in terms of the LDF approximation, replacing/c in column 2 of Table 16-12 with... 

The practical importance of vacancies is that they are mobile and, at elevated temperatures, can move relatively easily through the crystal lattice. As illustrated in Fig. 20.21b, this is accompanied by movement of an atom in the opposite direction indeed, the existence of vacancies was originally postulated to explain solid-state diffusion in metals. In order to jump into a vacancy an adjacent atom must overcome an energy barrier. The energy required for this is supplied by thermal vibrations. Thus the diffusion rate in metals increases exponentially with temperature, not only because the vacancy concentration increases with temperature, but also because there is more thermal energy available to overcome the activation energy required for each jump in the diffusion process. 

Tenet (v). Experimental studies of the interaction of a solid with a gas, liquid or solute must ensure that there is uniform availability of the homogeneous participant at all surfaces within an assemblage of reactant crystallites if meaningful kinetic measurements relating to the chemical step are to be obtained. If this is not achieved, then diffusion rates will control the overall rate of product formation. Such effects may be particularly significant in studies concerned with finely divided solids. 

The retarding influence of the product barrier in many solid—solid interactions is a rate-controlling factor that is not usually apparent in the decompositions of single solids. However, even where diffusion control operates, this is often in addition to, and in conjunction with, geometric factors (i.e. changes in reaction interfacial area with a) and kinetic equations based on contributions from both sources are discussed in Chap. 3, Sect. 3.3. As in the decompositions of single solids, reaction rate coefficients (and the shapes of a—time curves) for solid + solid reactions are sensitive to sizes, shapes and, here, also on the relative dispositions of the components of the reactant mixture. Inevitably as the number of different crystalline components present initially is increased, the number of variables requiring specification to define the reactant completely rises the parameters concerned are mentioned in Table 17. 

In any catalyst selection procedure the first step will be the search for an active phase, be it a. solid or complexes in a. solution. For heterogeneous catalysis the. second step is also deeisive for the success of process development the choice of the optimal particle morphology. The choice of catalyst morphology (size, shape, porous texture, activity distribution, etc.) depends on intrinsic reaction kinetics as well as on diffusion rates of reactants and products. The catalyst cannot be cho.sen independently of the reactor type, because different reactor types place different demands on the catalyst. For instance, fixed-bed reactors require relatively large particles to minimize the pressure drop, while in fluidized-bed reactors relatively small particles must be used. However, an optimal choice is possible within the limits set by the reactor type. 

One possibility for increasing the minimum porosity needed to generate disequilibria involves control of element extraction by solid-state diffusion (diffusion control models). If solid diffusion slows the rate that an incompatible element is transported to the melt-mineral interface, then the element will behave as if it has a higher partition coefficient than its equilibrium partition coefficient. This in turn would allow higher melt porosities to achieve the same amount of disequilibria as in pure equilibrium models. Iwamori (1992, 1993) presented a model of this process applicable to all elements that suggested that diffusion control would be important for all elements having diffusivities less than... 

The reactant solid B is porous and the reaction occurs in a diffuse zone. If the rate of the chemical reaction is much slower compared to the rate of diffusion in the pores, the concentration of the fluid reactant would be uniform throughout the pellet and the reaction would occur at a uniform rate. On the other hand, if the chemical reaction rate is much faster than the pore diffusion rate, the reaction occurs in a thin layer between the unreacted and the completely reacted regions. The thickness of the completely reacted layer would increase with the progress of the reaction and this layer would grow towards the interior of the pellet). 

Many heterogeneous catalytic organic reactions are run in the liquid-phase, and liquid phase reactions present special mass transfer problems. Diffusion barriers exist between the gas and the liquid and between the liquid and the solid, so there are gas-liquid-solid diffusion barriers. When these barriers are too large, the true chemical rate at the surface is not observed. 

In either equation, k° is given by Eq. (16-84) for parallel pore and surface diffusion or by Eq. (16-85) for a bidispersed particle. For nearly linear isotherms (0.7 < ft < 1.5), the same linear addition of resistance can be used as a good approximation to predict the adsorption behavior of packed beds, since solutions for all mechanisms are nearly identical. With a highly favorable isotherm (R —> 0), however, the rate at each point is controlled by the resistance that is locally greater, and the principle of additivity of resistances breaks down. For approximate calculations with intermediate values of R, an overall transport parameter for use with the LDF approximation can be calculated from the following relationship for solid diffusion and film resistance in series ... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Npe, for axial dispersion. For pore and solid diffusion, = r/rp is a dimensionless radial coordinate, where rp is the radius of the particle. If a particle is bidis-perse, then rp can be replaced by rs, the radius of a subparticle. For preliminary calculations, Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

The performance of a reactor for a gas-solid reaction (A(g) + bB(s) -> products) is to be analyzed based on the following model solids in BMF, uniform gas composition, and no overhead loss of solid as a result of entrainment. Calculate the fractional conversion of B (fB) based on the following information and assumptions T = 800 K, pA = 2 bar the particles are cylindrical with a radius of 0.5 mm from a batch-reactor study, the time for 100% conversion of 2-mm particles is 40 min at 600 K and pA = 1 bar. Compare results for /b assuming (a) gas-film (mass-transfer) control (b) surface-reaction control and (c) ash-layer diffusion control. The solid flow rate is 1000 kg min-1, and the solid holdup (WB) in the reactor is 20,000 kg. Assume also that the SCM is valid, and the surface reaction is first-order with respect to A. 

The preceding data, though limited in nature, represent one of the first attempts to measure solid state diffusion rates of alkali elements into the near-surface region of feldspars and natural glasses at low temperature. As such, interesting comparisons can be made with diffusion coefficients and activation energies calculated from numerous high temperature isotope and tracer diffusion studies f 11-181. 

Fig. 14.13 Graphical representation of the effect of MW on T2 (dashed-dotted), on the translational diffusion rate D (solid), on the steady state NOE (dashed) and on the build-up of the NOE (dotted). All values are normalized to a 300 Da molecular weight molecule. For the calculation of the parameters involving dipolar relaxation we used a formula that can be found in the literature...

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