Diffusion theory large particles

The velocity, viscosity, density, and channel-height values are all similar to UF, but the diffusivity of large particles (MF) is orders-of-magnitude lower than the diffusivity of macromolecules (UF). It is thus quite surprising to find the fluxes of cross-flow MF processes to be similar to, and often higher than, UF fluxes. Two primary theories for the enhanced diffusion of particles in a shear field, the inertial-lift theory and the shear-induced theory, are explained by Davis [in Ho and Sirkar (eds.), op. cit., pp. 480-505], and Belfort, Davis, and Zydney [/. Membrane. Sci., 96, 1-58 (1994)]. While not clear-cut, shear-induced diffusion is quite large compared to Brownian diffusion except for those cases with very small particles or very low cross-flow velocity. The enhancement of mass transfer in turbulent-flow microfiltration, a major effect, remains completely empirical. 

The two fundamental theories for calculating the attachment coefficient, 3, are the diffusion theory for large particles and the kinetic theory for small particles. The diffusion theory predicts an attachment coefficient proportional to the diameter of the aerosol particle whereas the kinetic theory predicts an attachment coefficient proportional to the aerosol surface area. The theory... 

When the radius of an aerosol particle, r, is of the order of the mean free path, i, of gas molecules, neither the diffusion nor the kinetic theory can be considered to be strictly valid. Arendt and Kallman (1926), Lassen and Rau (1960) and Fuchs (1964) have derived attachment theories for the transition region, r, which, for very small particles, reduce to the gas kinetic theory, and, for large particles, reduce to the classical diffusion theory. The underlying assumptions of the hybrid theories are summarized by Van Pelt (1971) as follows 1. the diffusion theory applies to the transport of unattached radon progeny across an imaginary sphere of radius r + i centred on the aerosol particle and 2. kinetic theory predicts the attachment of radon progeny to the particle based on a uniform concentration of radon atoms corresponding to the concentration at a radius of r + L... 

For small particle sizes the kinetic theory is applicable, whereas for large particle sizes the diffusion theory applies. A useful approximation is therefore to use the kinetic theory in the small particle range and the diffusion theory in the large size region. 

Figures 3 and 4 show the variation of the average attachment coefficient with CMD. It can be seen that for particles of CMD less than 0.06 ym and Og = 2 the kinetic theory predicts an attachment coefficient similar to the hybrid theory, whereas for CMD greater than about 1 ym the diffusion theory and the hybrid theory give approximately the same results. For a more polydisperse aerosol (Og = 3) the kinetic theory deviates from the hybrid theory even at a CMD = 0.01 ym. The diffusion theory is accurate for a CMD greater than about 0.6 ym. These results are easily explained since as the aerosol becomes more polydisperse, there are more large diameter particles (CMD >0.3 ym) which attach according to the diffusion theory. In contrast, the kinetic theory becomes more inaccurate as the aerosol becomes more polydisperse.

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

For noninteracting particles D b is + D, but as the particles approach each other, the relative diffusion coefficient becomes dependent on their spatial separation. In liquids for large particles this arises from hydrodynamic interactions ( bow waves ), while in the gas phase the particles screen each other from the bath collisions. For small particles the viscoelastic projjerties of the fluid will become important near contact. The solution of Eq. (2.23) applies only for sufficiently large friction where the relative motion on all length scales is diffusive. In the other limit of very low friction, the general result obtained from molecular theory is of the form... 

However, the theory of Section II in this chapter gives the exact criterion that diffusion theory for the encounter rate is valid when a in Eq. (2.33) is large than unity. For potentials without barriers the factor IF(0)/IF(oo) is close to unity, and the criterion for applicability of the diffusion theory is that the mean free path Dj/C8kT/nfi) be much smaller than the particle size (1 + b)/4-However, for large barriers the criterion is similar to that of Verwey and Overbeek, except that SR is the half-width of the potential barrier at an energy of only kT/2 below the maximum. This is much less than the full width of the barrier, and for reasonable values of the parameters in aqueous solution it can be shown that DLVO theory breaks down for equal-size particles with R, > 150 nm. However, when hydrodynamic interaction between particles is introduced in the formula for in terms of and it is concluded ... 

Williams s core-shell theory of particle growth, however, has many unresolved conflicts. Napper [18] pointed out that the diffusion rates of species present within the polymer particle did not support the hypothesis for such large differences in the polymer concentrations between the core and shell. Moreover, Garden [19] showed that the diffusive mean free path of monomer molecules, which was much larger than the radius of the polymer particle, would not favour the core-shell equilibrium theory. Garden, as well as Friis and Hamielec [20], also indicated that Williams experimental results, i.e. a nearly constant polymerization rate, could be attributed to the concurrent decrease in [M]p due to... 

ABSTRACT With the increase of mine exploitation depth and appliance widely of large-scale full-mechanized equipment, coal block gas emission has been one of the most gas effusion source. Base on unsteady diffusion theory and mass transmission fundamental, the mathematical and physical model of gas diffusion through coal particles with third type boundary condition was founded and its analytical solution was obtained by separate variableness method. The characteristics of gas through coal particles was analyzed according as mass transmission theory of porous material. The results show that the Biot s criterion of mass transmission can reflect the resistance characteristic of gas diffusion and the Fourier s criterion of mass transmission can represent the dynamic feature of diffusion field varying with time. 

It is of practical importance to know the number z of precipitate particles as a function of time, z continually decreases with time as a result of the Ostwald ripening process, since the large particles grow at the expense of the small particles. After an infinitely long time, the precipitated phase should, in theory, consist of one single compact particle. In the case of diffusion-controlled mass transport it can be shown that ... 

Diffusion theory 1 /N (or better) convergence Follows rare particles well Large computer memory required User must determine accuracy with repeated calculations... 

---