Particle mass diffusion

Modulated Dyiuimic LS [53,54] Scattering intensity fluctuations of a single particle Particle mass, diffusion, velocity and shape study Flow... 

A derivation for particle-phase diffusion accompanied by fluid-side mass transfer has been carried out by Rosen []. Chem. Phy.s., 18,1587 (1950) ibid., 20, 387 (1952) Jnd. Eng. Chem., 46,1590 (1954)] with a limiting form a.t N > 50 of... 

Principles and Characteristics Supercritical fluid extraction uses the principles of traditional LSE. Recently SFE has become a much studied means of analytical sample preparation, particularly for the removal of analytes of interest from solid matrices prior to chromatography. SFE has also been evaluated for its potential for extraction of in-polymer additives. In SFE three interrelated factors, solubility, diffusion and matrix, influence recovery. For successful extraction, the solute must be sufficiently soluble in the SCF. The timescale for diffusion/transport depends on the shape and dimensions of the matrix particles. Mass transfer from the polymer surface to the SCF extractant is very fast because of the high diffusivity in SCFs and the layer of stagnant SCF around the solid particles is very thin. Therefore, the rate-limiting step in SFE is either... 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

These results take into account three possible processes in series mass transfer of fluid reactant A from bulk fluid to particle surface, diffusion of A through a reacted product layer to the unreacted (outer) core surface, and reaction with B at the core surface any one or two of these three processes may be rate-controlling. The SPM applies to particles of diminishing size, and is summarized similarly in equation 9.1-40 for a spherical particle. Because of the disappearance of the product into the fluid phase, the diffusion process present in the SCM does not occur in the SPM. 

Since the absolute thickness of the effective hydrodynamic boundary layer is very small, below a particular size range minimum, no hydrodynamic effects are perceived experimentally with varying agitation. This, however, does not mean, that there are no such influences Further, the mechanisms of mass transfer and dissolution may change for very small particles depending on a number of factors, such as the fluid viscosity, the Sherwood number (the ratio of mass diffusivity to molecular diffusivity), and the power input per unit mass of fluid. 

During the studies carried out on this process some unusual behavior has been observed. Such results have led some authors to the conclusion that SSP is a diffusion-controlled reaction. Despite this fact, the kinetics of SSP also depend on catalyst, temperature and time. In the later stages of polymerization, and particularly in the case of large particle sizes, diffusion becomes dominant, with the result that the removal of reaction products such as EG, water and acetaldehyde is controlled by the physics of mass transport in the solid state. This transport process is itself dependent on particle size, density, crystal structure, surface conditions and desorption of the reaction products. 

The theory of particle diffusion was first advanced in 1934 by Lewis and von Elbe [5] in dealing with the ozone reaction. Tanford and Pease [6] carried this concept further by postulating that it is the diffusion of radicals that is all important, not the temperature gradient as required by the thermal theories. They proposed a diffusion theory that was quite different in physical concept from the thermal theory. However, one should recall that the equations that govern mass diffusion are the same as those that govern thermal diffusion. 

When mass diffuses into or out of a fluid particle, the concentration within the particle changes with time. Therefore the time derivatives must be retained in the diffusion equations for both internal and external fluids. The internal and external concentration fields are related at the interface. If there is no... 

Diffusion of the gas to the surface (rg). As described by Fuchs and Sutugin (1970, 1971) in their comprehensive treatment of highly dispersed aerosols, the rate of transfer of mass to the surface of a spherical particle by diffusion of a gas is described by... 

It should be noted here that while in catalytic systems the rate is based on the moles disappearing from the fluid phase - dddt, and the rate has the form ( —ru) = f(k, C), in adsorption and ion exchange the rate is normally based on the moles accumulated in the solid phase and the rate is expressed per unit mass of the sohd phase dqldt where q is in moles per unit mass of the solid phase (solid loading). Then, the rate is expressed in the form of a partial differential diffusion equation. For spherical particles, mass transport can be described by a diffusion equation, written in spherical coordinates r ... 

Equations (33) and (34) show that diffusion studies combined with sedimentation studies, either under the force of gravity or in a centrifuge, yield information about particle masses with no assumptions about the shape of the particle. 

Human hemoglobin, for example, has a sedimentation coefficient of 4.48 S and a diffusion coefficient of 6.9 10 m2 s l in aqueous solution at 20°C. The density of this material is 1.34 g cm -3. Substituting these values into Equation (34) shows the particle mass to be... 

EXAMPLE 2.4 Solvation and Ellipticity of Human Hemoglobin from Sedimentation Data. The diffusion coefficient of the human hemoglobin molecule at 20°C is 6.9 10 11 m2 s "1. Use this value to determine f for this molecule. Evaluate f0 for hemoglobin using the particle mass calculated in Equation (35). Indicate the possible states of solvation and ellipticity that are compatible with the experimental flfQ ratio. 

Sedimentation and diffusion data allow for the unambiguous determination of particle mass and also allow the suspended particles to be placed on a contour in a plot such as that of Figure 2.9. This is as far as these experiments can take us toward the characterization of the particles. Of course, additional data from other sources, such as the x-ray diffraction results just cited, may lead to still further specification of the system. One such source of information is intrinsic viscosity data for the same dispersion. In Chapter 4 we discuss the complementarity between viscosity data and sedimentation-diffusion results (see Section 4.7b). 

The choice of an appropriate model is heavily dependent on the intended application. In particular, the science of the model must match the pollutant(s) of concern. If the pollutant of concern is fine PM, the model chemistry must be able to handle reactions of nitrogen oxides (NOx), sulphur dioxide (SO2), volatile organic compounds (VOC), ammonia, etc. Reactions in both the gas and aqueous phases must be included, and preferably also heterogeneous reactions taking place on the surfaces of particles. Apart from correct treatment of transport and diffusion, the formation and growth of particles must be included, and the model must be able to track the evolution of particle mass as a function of size. The ability to treat deposition of pollutants to the surface of the earth by both wet and dry processes is also required. 

Internal diffusional limitations are possible any time that a porous immobilized enzymatic preparation is used. Bernard et al. (1992) studied internal diffusional limitations in the esterification of myristic acid with ethanol, catalyzed by immobilized lipase from Mucor miehei (Lipozyme). No internal mass diffusion would exist if there was no change in the initial velocity of the reaction while the enzyme particle size was changed. Bernard found this was not the case, however, and the initial velocity decreased with increasing particle size. This corresponds to an efficiency of reaction decrease from 0.6 to 0.36 for a particle size increase from 180 pm to 480 pm. Using the Thiele modulus, they also determined that for a reaction efficiency of 90% a particle size of 30 pm would be necessary. While Bernard et al. found that their system was limited by internal diffusion, Steytler et al. (1991) found that when they investigated the effect of different sizes of glass bead, 1 mm and 3 mm, no change in reaction rate was observed. 

The form of equation (21) is interesting. It shows that the uptake curve for a system controlled by heat transfer within the adsorbent mass has an equivalent mathematical form to that of the isothermal uptake by the Fickian diffusion model for mass transfer [26]. The isothermal model hag mass diffusivity (D/R ) instead of thermal diffusivity (a/R ) in the exponential terms of equation (21). According to equation (21), uptake will be proportional to at the early stages of the process which is usually accepted as evidence of intraparticle diffusion [27]. This study shows that such behavior may also be caused by heat transfer resistance inside the adsorbent mass. Equation (22) shows that the surface temperature of the adsorbent particle will remain at T at all t and the maximum temperature rise of the adsorbent is T at the center of the particle at t = 0. The magnitude of T depends on (n -n ), q, c and (3, and can be very small in a differential test. 

For a system containing spherical particles, D = RT/6rrr]aNA - i.e. D oc 1/m1, where m is the particle mass. For systems containing asymmetric particles, D is correspondingly smaller (see Table 2.3). Since D = k77/, the ratio D/D0 (where D is the experimental diffusion coefficient and D0 is the diffusion coefficient of a system containing the equivalent unsolvated spheres) is equal to the... 

It is evident from the above expressions that the appropriate diffusion coefficient must also be measured in order that molecular or particle masses may be determined from sedimentation velocity data. In this respect, a separate experiment is required, since the diffusion coefficient cannot be determined accurately in situ, because there is a certain self-sharpening of the peak due to the sedimentation coefficient increasing with decreasing concentration. 

When a state of sedimentation-diffusion equilibrium has been reached, the molecular or particle mass can, therefore, be evaluated without a knowledge of the diffusion coefficient (and, hence, independently of shape and solvation) by determining relative concentrations at various distances from the axis of rotation. Molecules as small as sugars have been studied by this technique. 

The first detailed study on ion exchange rates, and particularly mechanisms, appeared in the very definitive and elegant studies of Boyd et al. (1947) with zeolites. Working in conjunction with the Manhattan Project, these researchers clearly showed that ion exchange is diffusion-controlled, and that the reaction rate is limited by mass-transfer phenomena that are either film (FD) or particle (PD) diffusion-controlled. Boyd et al. (1947) were also the first to derive rate laws for FD, PD, and CR. Additionally, they demonstrated that particle size had no effect on reaction control, that in FD the rate was inversely proportional to particle size, and that the PD rate was inversely proportional to the square of the particle size. 

Film diffusion With a fast surface reaction on a nonporous particle, mass transfer limitations can arise in the fluid phase. 

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