Diffusion, Brownian

The rates of Brownian diffusion can be quantified by considering a box of cross-sectional area 1 cm2 having N particles cm 3 in one section (A) on the left, while the right section (B) is initially empty. Particles (or gas molecules) will always tend to diffuse from a region of higher concentrations to one of lower concentrations. 

FIGURE 9.16 Settling velocities in still air at 0°C and 760 Torr pressure for particles having a density of 1 g cm 3 as a function of particle diameter. For spherical particles of unit density suspended in air near sea level, the Stokes law applies over a considerable range of particle sizes, where the line is straight, but a correction is required at the particle size extremes (adapted from LBL, f979). 

The rate at which they diffuse depends on the concentration gradient, dN dx the larger the gradient, the faster the rate of diffusion. This is the basis of the well-known Fick s first law of diffusion  

J is the flux of particles crossing a 1-cm2 plane in 1 s (i.e., number cm 2 s ). The constant D is known as the diffusion coefficient and is simply the proportionality constant relating the flux to the concentration gradient. (Fick s first law applies, of course, not only to particles but also to gas and liquid molecules.) 

Intuitively, one might expect that the rate of diffusion would increase with temperature and decrease with increasing gas viscosity and particle size. Indeed, this is observed to be the case because all these parameters contribute to the diffusion coefficient D, given by 

Consider deposition of particles on an obstacle due to Brownian diffusion in a slow flow whose velocity far from the obstacle is U [36]. As an obstacle, we take a solid sphere of radius a. 

We assume that deposition on the sphere is ideal, that is, each collision of a particle with the sphere results in the particle being captured. The factor of Brownian diffusion Dj,r = kTwhere Oj, is the particle s radius, is much smaller than the factor of molecular diffusion, therefore the Peclet diffusion number is Peo = Ua/Dhr 1- By virtue of this inequality (see Section 6.5), the diffusion flux of particles toward the sphere can be found by solving the stationary equation of convective diffusion with a condition corresponding to a thick or thin diffusion-boundary layer. Particles may then be considered as point-like, and the diffusion equation will become  

The first condition corresponds to the ideal absorption on the sphere, and the second one - to a constant concentration far away from the sphere. 

Recall that Eq. (10.67) is written in the boundary-layer approximation, and Ur and ug are the components of the Stokes flow velocity near the sphere. Since the Schmidt number is Sc 1, the thickness of the viscous boundary layer is much 

The components of velocity at a slow flow near the sphere are  

The thermal motion of particles in a suspension opposes their sedimentation by creating a counteracting diffusive flux. This thermal motion is called Brownian motion. 

Robert Brown in 1828 was the first to note the random movement of pollen particles dispersed in water with a microscope. The average displacement, x, in three dimensions for a period of time, t, was found 

The diffusion coefficient for a suspension of monosized particles can be measured directly by photon correlation spectroscopy [12] (quasielastic light scattering). For distributions of different particle sizes, the average diffusion coefficient is determined by photon correlation spectroscopy. 

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The tendency for particles to settle is opposed by tlieir Brownian diffusion. The number density distribution of particles as a function of height z will tend to an equilibrium distribution. At low concentration, where van T Ftoff s law applies, tire barometric height distribution is given by... 

If condensation requires gas stream cooling of more than 40—50°C, the rate of heat transfer may appreciably exceed the rate of mass transfer and a condensate fog may form. Fog seldom occurs in direct-contact condensers because of the close proximity of the bulk of the gas to the cold-Hquid droplets. When fog formation is unavoidable, it may be removed with a high efficiency mist collector designed for 0.5—5-p.m droplets. Collectors using Brownian diffusion are usually quite economical. If atmospheric condensation and a visible plume are to be avoided, the condenser must cool the gas sufftciendy to preclude further condensation in the atmosphere. 

Photon Correlation Spectroscopy. Photon correlation spectroscopy (pcs), also commonly referred to as quasi-elastic light scattering (qels) or dynamic light scattering (dls), is a technique in which the size of submicrometer particles dispersed in a Hquid medium is deduced from the random movement caused by Brownian diffusion motion. This technique has been used for a wide variety of materials (60—62). 

Table 1. Effect of Spherical Particle Size on the Relative Brownian Diffusion and the Sedimentation Distances After 1 h in Water at 293 K...

If 0 fluid streomline passes within one particle rodius of the collecting body, o particle traveling olong the streamline will touch the body and may be collected without the influence of inertia or brownian diffusion. 

Brownian diffusion (Brownian motion) The diffusion of particles due to the erratic random movement of microscopic particles in a disperse phase, such as smoke particles in air. 

Diffusion The mixing of substances by molecular motion to equalize a concentration gradient. Applicable to gases, fine aerosols and vapors. (See Brownian diffusion.)... 

Impaction Collection mechanism where the contaminants collide with the surface of the filter by inertia, interception, or Brownian diffusion. 

Rapidly fluetuating stoehastie forees assoeiated with Brownian diffusion. 

From the image sequences, information on the velocities of nano-particles can be extracted. The statistical effect of Brownian motion on the flowing speed of the mixed liquid is found small enough to be ignored as shown in Fig. 37 where most of the particles trajectories in the liquid are straight lines and parallel with the wall basically. Therefore, Brownian diffusive motion is ignorable. 

Fluid density and component brownian diffusivity D are also assumed constant. A steady-state component mass balance can be written for component concentration c ... 

Figure 20-48 shows Wijmans s plot [Wijmans et al.,/. Membr. Sci., 109, 135 (1996)] along with regions where different membrane processes operate (Baker, Membrane Technology and Applications, 2d ed., Wiley, 2004, p. 177). For RO and UF applications, Sj , < 1, and c > Cl,. This may cause precipitation, fouling, or product denatura-tion. For gas separation and pervaporation, Sj , >1 and c < ci. MF is not shown since other transport mechanisms besides Brownian diffusion are at work. 

Larger components with smaller brownian diffusivity polarize more readily. They can exclude smaller components, reducing their concentration at the membrane surface and increasing their retention by the membrane. 

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. 

For displacements shorter than the mean pore dimension, (z2) < a, where flow velocities tend to be spatially constant and homogeneously distributed, Brownian diffusion is the only incoherent transport phenomenon that contributes to the hydrodynamic dispersion coefficient. As a direct consequence, the dispersion coefficient approaches the ordinary Brownian diffusion coefficient,... 

In this section, we consider flow-induced aggregation without diffusion, i.e., when the Peclet number, Pe = VLID, where V and L are the characteristic velocity and length and D is the Brownian diffusion coefficient, is much greater than unity. For simplicity, we neglect the hydrodynamic interactions of the clusters and highlight the effects of advection on the evolution of the cluster size distribution and the formation of fractal structures. 

This chapter describes methods of deriving the exact time characteristics of overdamped Brownian diffusion only, which in fact corresponds to continuous... 

Consider a process of Brownian diffusion in a potential profile ip(x, t) =... 

In the frame of the present review, we discussed different approaches for description of an overdamped Brownian motion based on the notion of integral relaxation time. As we have demonstrated, these approaches allow one to analytically derive exact time characteristics of one-dimensional Brownian diffusion for the case of time constant drift and diffusion coefficients in arbitrary potentials and for arbitrary noise intensity. The advantage of the use of integral relaxation times is that on one hand they may be calculated for a wide variety of desirable characteristics, such as transition probabilities, correlation functions, and different averages, and, on the other hand, they are naturally accessible from experiments. 

We consider the process of Brownian diffusion in a potential cp(x). The probability density of a Brownian particle is governed by the FPE (5.72) with delta-function initial condition. The moments of transition time are given by (5.1). 

In the general case, when arbitrary interaction profiles prevail, the particle deposition rate must be obtained by solving the complete transport equations. The first numerical solution of the complete convective diffusional transport equations, including London-van der Waals attraction, gravity, Brownian diffusion and the complete hydrodynamical interactions, was obtained for a spherical collector [89]. Soon after, numerical solutions were obtained for a panoplea of other collector geometries... 

The polymer radius has to be larger than 80% of the particle radius to avoid adsorption limitation under orthokinetic conditions. As a rule of thumb a particle diameter of about 1 pm marks the transition between perikinetic and orthokinetic coagulation (and flocculation). The effective size of a polymeric flocculant must clearly be very large to avoid adsorption limitation. However, if the polymer is sufficiently small, the Brownian diffusion rate may be fast enough to prevent adsorption limitation. For example, if the particle radius is 0.535 pm and the shear rate is 1800 s-, then tAp due to Brownian motion will be shorter than t 0 for r < 0.001, i.e., for a polymer with a... 

FIGURE 2-10 Tracking a gold particle attached to a single molecule of phosphatidyl ethanolamine. What appears to be simple Brownian diffusion at a time resolution of 33 ms per video frame (A) is revealed to actually consist of fast hop diffusion by recording 300 times faster (B) at 110 ps per video frame. In (A) each color represents 60 frames = 2 seconds. In (B) each color indicates an apparent period of confinement within a compartment and black indicates intercompartmental hops. The residency time for each compartment is indicated. The hypothetical explanations are illustrated in part (C) and discussed in the text. Adapted from [29]. 

Brown Coal Liquefaction process, 6 849 Brown cyclization product, 21 147 Brownian diffusion, 13 151, 152 Brownian diffusion, in depth filtration theory, 11 339... 

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