Particle transport Brownian diffusion

Transport of particles by Brownian diffusion depends on the Schmidt number, Sc = v/D, where D is now the Brownian diffusivity of particles, given by (9.73) ... 

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... 

Heterodisperse Suspensions. The rate laws given above apply to monodisperse colloids. In polydisperse systems the particle size and the distribution of particle sizes have pronounced effects on the kinetics of agglomeration (O Melia, 1978). For the various transport mechanisms (Brownian diffusion, fluid shear, and differential settling), the rates at which particles come into contact are given in Table 7.2. 

According to Eq. (4) (Table 1.2), the time required to halve the concentration of the virus particles in the suspension containing the virus particles only would be almost 200 days. In the presence of bentonite (kb = 3.1 10 10 cm3 sec1 and Nd2 = 7.35 106 cm 3) we find after integrating that the free virus concentration after 1 hour of contact is only 2.6 particles cm 3. This example illustrates that the presence of larger particles may aid significantly in the removal of smaller ones, even when Brownian diffusion is the predominant transport mechanism. 

Air movement indoors is much slower than outdoors, but it is usually enough to ensure that concentrations are fairly uniform in a room. Convection from heating appliances gives air speeds typically in the range 0.05-0.5 m s-1 (Daws, 1967). However, to undergo deposition, vapour molecules or particles must be transported across the boundary layer, typically a few millimetres thick, of almost stagnant air over surfaces. This may be achieved by sedimentation, molecular or Brownian diffusion, or under the action of electrostatic or thermophoretic forces. 

Fig. 6.1. Transport of gases and particles to and from flat plates and leaves by Brownian diffusion , l3lI vapour, flat plate (Chamberlain, 1953) O, 212Pb vapour, bean leaves (Chamberlain, 1974) , water vapour, bean leaves (Grace Wilson, 1976) +, x, 0.17-jum particles to pine, oak leaves (Belot, 1975) A, V, , 0.03- m particles to nettle, beech, white poplar leaves (Little Wiffen, 1977) line A, theory, laminar flow,...

Experiments on transfer of submicrometre radioactive particles to smooth surfaces (Wells Chamberlain, 1967 Chamberlain et al., 1984) have shown that the dependency of vg on D213 holds over many orders of magnitude of D. This means that the transport by Brownian diffusion becomes progressively less effective as the particle size increases. For example a particle of 0.1 pm diameter has a diffusivity of 6.8 x 10 10 m2 s 1, a factor 1.2 x 104 smaller than that of I2 vapour. Since D does not depend on the particle density, it is appropriate to discuss transport by Brownian motion in terms of the particle diameter. The aerodynamic diameter, dA, is equal to dppp2 where pp is the particle density in c.g.s. units (g cm-3) not SI units (kg m-3), and is the appropriate parameter for particles with dp> 1 pm, for which impaction and sedimentation are the mechanisms of deposition. 

Airborne particles may be delivered to surfaces by wet and dry deposition. Several transport mechanisms, such as turbulent diffusion, precipitation, sedimentation, Brownian diffusion, interception, and inertial migration, influence the dry deposition process of airborne particles. Large particles (dNIOAm) are transported mainly by sedimentation hence, large particulate PAHs tend to be deposited nearer the sources of emission Small particles (dblAm), which behave like gases, are often transported and deposited far from where they originated (Baek et al., 1991 Wu et al., 2005). 

Transport control of flocculation is realized in an especially direct way in the process known as diffusion-limited cluster-cluster aggregation5 (aggregation as used in this term means flocculation in the present chapter). In this process, which is straightforward to simulate and visualize on a computer, particles undergo Brownian motion (i.e., diffusion) until they come together in close proximity, after which they coalesce instantaneously and irreversibly to form floccules (or clusters ). The clusters then diffuse until they contact one another and combine to form larger clusters, and so on, until gravitational... 

Particles smaller than 0.1 tim diameter are able to diffuse through the laminar boundary layer by Brownian diffusion, the efficiency of the mechanism increasing as particle size decreases below 0.1 (J-m. In general, rates of Brownian diffusion, which are small even by comparison with molecular diffusion, and do not therefore, represent an efficient process for the transport of sulphur and nitrogen containing particles across the laminar boundary layer. Another mechanism for transport of particles through this layer is inertial impaction. For this process the particle must... 

Eor a given crossflow filtration, the dominant particle back transport mechanism may depend on the shear rate and the particles size [4]. Brownian diffusion is only important for particles smaller than only a few tenths of a micron in diameter with relative low shear, whereas inertial lift is important for particles larger than several tens of microns with higher shear rates. Shear-induced back transport appears to be important for intermediate particle sizes and shear rates. Li et al. [7] reported that the shear-induced mechanism was able to predict fluxes comparable with the critical fluxes identified by the DOTM. 

This example illustrates that the presence of larger particles may aid significantly in the removal of smaller ones, even when Brownian diffusion is the predominant transport mechanism (cf. Figure 14.4). 

Here (3Br(ij), Psll(i,j), and PDS(ij) are the transport coefficients for interparticle contacts between particles of diameters d, and dj by Brownian diffusion, fluid shear, and differential sedimentation, respectively kB is Boltzmanns constant T is the absolute temperature p, is the viscosity of the liquid G is the mean velocity gradient of the liquid g is the gravity acceleration and pp and p, are the densities of the particles and the liquid, respectively. 

In this chapter, we consider Brownian diffusion, sedimentation, migration in an electric Reid, and thermophoresis. The last term refers to particle movement produced by a temperature gradient in the gas. We consider also the London-van der Waals forces that are important when a particle approaches a surface. The analysis is limited to particle transport in stationary —that is. nonllowing— gases. I ransporl in flow systems is discussed in the chapters which follow. 

We consider particle transport from a gas to a body with a flat bounding surface by Brownian diffusion under the influence of van der Waals forces exerted by the body. The relative contributions of the two mechanisms can be estimated as follows The total flux normal to the surface is given by the x component of the flux... 

When the pipe Reynolds number is greater than about 2100, the velocity boundary layer that forms in the entry region eventually turns turbulent as the gas passes down the pipe. The velocity profile becomes fully developed that is. the shape of the distribution ceases to change at about 25 to 50 pipe diameters from the entry. Small particles in such a flow are transported by turbulent and Brownian diffusion to the wall. In the sampling of atmospheric air through long pipes, wall losses result from turbulent diffusion. Accumulated layers of particles will affect heat transfer between the gas and pipe walls. 

Brownian diffusion is neglected compared with turbulent transport. The left-hand side represents Dpc/Dt, the Stokes or substantive derivative of p--. The first term on the right-hand side is the turbulent diffusion of The second term —2v p j Vp7 is generally positive and represents the generation of p by transfer from the mean How. The third term. 2p//fl, is the contribution of variations in the mte of gas-to-particle conversion by chemical reaction to the rate of production of p . The la.st term is the decrease of mean square fluctuations pj due to the action of small scale diffusion (dissipation). Thus three types of terms appear on the right-hand side of (13,16), the balance equation for Pi (i) turbulent diffusion of p, and tnmsfer from the mean (low to p.. which alTeci... 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

The coefficients a(p, c) and tj(p, c) describe chemical and physical effects on the kinetics of deposition. The transport of particles from the bulk of the flowing fluid to the surface of a collector or media grain by physical processes such as Brownian diffusion, fluid flow (direct interception), and gravity are incorporated into theoretical formulations for fj(p, c), together with corrections to account for hydrodynamic retardation or the lubrication effect as the two solids come into close proximity. Chemical effects are usually considered in evaluating a(p, c). These include interparticle forces arising from electrostatic interactions and steric effects originating from interactions between adsorbed layers of polymers and polyelectrolytes on the solid surfaces. 

The first group of terms on the right-hand-side of Eq. 4 describes particle transport to a collector surface by Brownian diffusion. NPe is the Peclet number, a ratio of particle transport by fluid advection to transport by molecular or viscous processes. The term As is introduced to account for the effects of neighboring collectors or media grains on the fluid flow around a collector of interest. The results here assume Happel s model (Happel, 1958) for flow around a sphere in a packed bed 4S depends only on the porosity of the bed (Table 1). The derivation for diffusive transport is based on the early work of Levich (1962). 

Having a model that has a good theoretical basis, that has been validated in laboratory experiments, and that is consistent with field observations, it is advisable to make some predictions about particle deposition in systems of interest. An example is presented in Figure 3, adapted from the work of Tobiason (1987). The travel distance in an aquifer required to deposit 99% of the particles from a suspension is termed Lgg and is plotted as a function of the diameter of the suspended particles for two different values of a(p, c), specifically 1.0 (favorable deposition) and 0.001 (deposition with significant chemical retardation of the particle-collector interaction, termed unfavorable deposition ). Assumptions include U = 0.1 m day"1, T= 10°C, dc = 0.05cm, e = 0.40, pp= 1.05 gem"3, and H=10 2OJ. These results indicate the dependence of the kinetics of deposition on the size of the particles in suspension that has been predicted and observed in many systems. Small particles are transported primarily by convective Brownian diffusion, and large particles in this system are transported primarily by gravity forces. A suspended particle with a diameter of about 3 /im is most difficult to transport. Nevertheless, in the absence of chemical retardation, a travel distance of only about 5 cm is all that is needed to deposit 99% of such particles in a clean aquifer, that is, an aquifer that has not received and retained previous particles. 

Three particle transport processes that bring about interparticle contacts are considered here Brownian diffusion (thermal effects), fluid shear (flow effects), and differential settling (gravity effects). Following Smoluchowski s approach, the appropriate individual transport coefficients for these three processes arc as... 

The effects of Brownian diffusion, fluid flow, and gravity on the transport of particles in aquatic systems have been described with some success, at least under... 

Because of its large velocity, a freely rising bubble has a diffusion layer much thinner that in the described experiments. This effect can manifest itself only if the particles are small enough so that their thermal motion becomes significant. Thus, electro- and diffusiophoresis should be taken into account in describing the Brownian diffusion of sub-micron particles towards the bubble s mobile surface under the conditions of a sufficiently low electrolyte concentration. The influence of diffusiophoretic transport to the surface of a rising bubble through its diffusion layer is theoretically proved by Zholkovsky et al. (1983). 

For submicron particles (for example, at a size of 0.1 - 0.3 pm) even the use of decimicron bubbles does not provide a sufficiently high collision efficiency. Particles of still smaller dimensions are not discussed here since their transport is influenced by Brownian diffusion. These difficulties are overcome by particle aggregation (cf Section 10.9). 

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