Surface Diffusion from Particle Surfaces

Surface diffusion from particle surface to neck y S iDsSsYs m RT = CsDsSsO t 4... 

Lattice diffusion from particle surface to neck 5 lODlYsVmO X = 1 RT = C,D,a t 3... 

Special reference should be made for the last section of Chapter 3 Particle analysis. Everything in connection with particle properties and basic calculations, irrespective of its specific use, is presented from particle surface area to calculations regarding its terminal velocity and diffusion coefficients. Furthermore, concerning materials used in adsorption, ion exchange, and catalysis, special paragraphs are included in Chapters 4 and 5 as well as the management of spent materials. 

Internal transfer transfer of adsorbate from particle surface to interior site by diffusion in the void space of the pores, by surface migration on the pore surface, or by volume diffusion, for example, in the holes in the chemical structure of the solid phase. 

The immobilization of enzymes may introduce a new problem which is absent in free soluble enzymes. It is the mass-transfer resistance due to the large particle size of immobilized enzyme or due to the inclusion of enzymes in polymeric matrix. If we follow the hypothetical path of a substrate from the liquid to the reaction site in an immobilized enzyme, it can be divided into several steps (Figure 3.2) (1) transfer from the bulk liquid to a relatively unmixed liquid layer surrounding the immobilized enzyme (2) diffusion through the relatively unmixed liquid layer and (3) diffusion from the surface of the particle to the active site of the enzyme in an inert support. Steps... 

One approach to describe the kinetics of such systems involves the use of various resistances to reaction. If we consider an irreversible gas-phase reaction A — B that occurs in the presence of a solid catalyst pellet, we can postulate seven different steps required to accomplish the chemical transformation. First, we have to move the reactant A from the bulk gas to the surface of the catalyst particle. Solid catalyst particles are often manufactured out of aluminas or other similar materials that have large internal surface areas where the active metal sites (gold, platinum, palladium, etc.) are located. The porosity of the catalyst typically means that the interior of a pellet contains much more surface area for reaction than what is found only on the exterior of the pellet itself. Hence, the gaseous reactant A must diffuse from the surface through the pores of the catalyst pellet. At some point, the gaseous reactant reaches an active site, where it must be adsorbed onto the surface. The chemical transformation of reactant into product occurs on this active site. The product B must desorb from the active site back to the gas phase. The product B must diffuse from inside the catalyst pore back to the surface. Finally, the product molecule must be moved from the surface to the bulk gas fluid. 

When smoke formation accompanies traces of noxious vapors, it may be called a fume—for example, a metallic oxide developing with sulfur in a melting or smelting process. The term fume is also used in a more general way to describe a particle cloud resulting from mixing and chemical reactions of vapors diffusing from the surface of a pool of liquid. 

Rates of ion exchange processes are affected by diffusional resistances of ions into and out of the solid particles as well as resistance to external surface diffusion. The particles are not really solid since their volume expands by 50% or more. For monovalent exchanges in strongly ionized resins, half times with intraparticle diffusion controlling are measured in seconds or minutes. For film diffusion, half times range from a few minutes with 0. N solutions up to several hours with 0.0017V solutions. Film diffusion rates also vary inversely with particle diameter. A rough rule is that film... 

If controlled by mass transfer in the liquid, the exchange rate is proportional to the specific surface area and thus is inversely proportional to the particle radius or diameter. If controlled by intraparticle diffusion, the rate is, in addition, inversely proportional to the distance diffusion has to cover from particle surface to center, and so is inversely proportional to the square of the particle radius or diameter. If exchange were controlled by a reaction at the exchange site, the rate would be independent of particle size. Again, the comparison is independent of specific models or equations. 

The process of growth of crystalline particles involves several steps, such as diffusion of solute molecules from the bulk of the solution to the crystal surface, adsorption on the crystal surface, diffusion over the surface, attachment to a step, diffusion along a step, and integration into a crystal kink site. The rates of these different processes depend on the type of material and on the operating conditions. The final overall molar flux of solute molecules, which has previously been indicated with /(U, , U, ), will of course be... 

The composition of exchange cations manifests its effect on the structure formation of clay sediments by changing the forces of repulsion between particles as a result of the ion-electrostatic interaction and the wedging-out action of adsorbed bound-water films. In the first case, this is caused by the varying capacity of cations to dissociate from particle surfaces and form diffuse layers of different thickness, and in the second, by the influence of cations on the specific hydrophility of clay minerals. 

Crystalline silicon, which was ground to a powder having the grain size less than 160 pm and used for the s)mthesis, had a density of 2330 kg/m. Therefore, in the molten salts it subsided to the tube bottom, where silicon could contact the refractory metal powder. However, the direct contact area of the particles is very small as the particles are separated by a layer of the salt melt. Liquid lithium and calcium, which are soluble in their own chloride melts, can diffuse from the surface to the bulk of the salt melt and form silicides. 

In charging, reaction (13.3) occurs first and hydrogen atoms are adsorbed at the active sites on the surface of the alloy particles. Subsequently, hydrogen moves from the local adsorbed state to an absorbed state on the entire surface of the alloy particles (reaction (13.4)). Finally, the absorbed hydrogen diffuses from the surface of the alloy particles into the bulk to form a hydride phase. The reverse process takes place during discharge. 

For a step change in sorbate concentration at the particle surface (r = R) at time 2ero, assuming isothermal conditions and diffusion control, the expression for the uptake curve maybe derived from the appropriate solution of this differential equation ... 

The Beckstead-Derr-Price model (Fig. 1) considers both the gas-phase and condensed-phase reactions. It assumes heat release from the condensed phase, an oxidizer flame, a primary diffusion flame between the fuel and oxidizer decomposition products, and a final diffusion flame between the fuel decomposition products and the products of the oxidizer flame. Examination of the physical phenomena reveals an irregular surface on top of the unheated bulk of the propellant that consists of the binder undergoing pyrolysis, decomposing oxidizer particles, and an agglomeration of metallic particles. The oxidizer and fuel decomposition products mix and react exothermically in the three-dimensional zone above the surface for a distance that depends on the propellant composition, its microstmcture, and the ambient pressure and gas velocity. If aluminum is present, additional heat is subsequently produced at a comparatively large distance from the surface. Only small aluminum particles ignite and bum close enough to the surface to influence the propellant bum rate. The temperature of the surface is ca 500 to 1000°C compared to ca 300°C for double-base propellants. 

In general, the foUowing steps can occur in an overall Hquid—soHd extraction process solvent transfer from the bulk of the solution to the surface of the soHd penetration or diffusion of the solvent into the pores of the soHd dissolution of the solvent into the solute solute diffusion to the surface of the particle and solute transfer to the bulk of the solution. The various fundamental mechanisms and processes involved in these steps make it impracticable or impossible to describe leaching by any rigorous theory. 

In either equation, /c 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 < R < 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 sohd diffusion and film resistance in series... 

Because of the close similarity in shape of the profiles shown in Fig. 16-27 (as well as likely variations in parameters e.g., concentration-dependent surface diffusion coefficient), a contrdling mechanism cannot be rehably determined from transition shape. If rehable correlations are not available and rate parameters cannot be measured in independent experiments, then particle diameters, velocities, and other factors should be varied ana the obsei ved impacl considered in relation to the definitions of the numbers of transfer units. 

FIG. 16-27 Constant pattern solutions for R = 0.5. Ordinant is cfor nfexcept for axial dispersion for which individual curves are labeled a, axial dispersion h, external mass transfer c, pore diffusion (spherical particles) d, surface diffusion (spherical particles) e, linear driving force approximation f, reaction kinetics. [from LeVan in Rodrigues et al. (eds.), Adsorption Science and Technology, Kluwer Academic Publishers, Dor drecht, The Nether lands, 1989 r eprinted with permission.]... 

The other mechanism appears in scrubbers. When water vapor diffuses from a gas stream to a cold surface and condenses, there is a net hydrodynamic flow of the noncondensable gas directed toward the surface. This flow, termed the Stefan flow, carries aerosol particles to the condensing surface (Goldsmith and May, in Davies, Aero.sol Science, Academic, New York, 1966) and can substantially improve the performance of a scrubber. However, there is a corresponding Stefan flow directed away from a surface at which water is evaporating, and this will tend to repel aerosol particles from the surface. 

Smaller particles, particularly those below about 0.3//m in diameter, exhibit consideroble Brownian movement and do not move uniformly along the gas streamline. These particles diffuse from the gas to the surface of the collecting body and are collected. 

---