Brownian motion turbulent

Particle deposition from a moving fluid involves two aspects. First the individual particles have to be transported to the surface by one or a combination of mechanisms, including Brownian motion, turbulent diffusion, and by virtue of the momentum possessed by the particle, as it is carried in the fluid stream. It will be appreciated that the size of the particle will exert a great influence on the dominant mechanism. Larger particles would be expected to move toward a surface, as a result of the momentum they possess. Finer particles with relatively low momentum can only approach the surface across the boundary layers, by Brownian or eddy diffusion. Having reached the surface to form a part of the foulant layer, the particle has to stick , but it may be removed from the surface by the shear forces produced by the flowing fluid [see Eq. (1)]. 

Mixing processes involved in the manufacture of disperse systems, whether suspensions or emulsions, are far more problematic than those employed in the blending of low-viscosity miscible liquids due to the multi-phasic character of the systems and deviations from Newtonian flow behavior. It is not uncommon for both laminar and turbulent flow to occur simultaneously in different regions of the system. In some regions, the flow regime may be in transition, i.e., neither laminar nor turbulent but somewhere in between. The implications of these flow regime variations for scale-up are considerable. Nonetheless, it should be noted that the mixing process is only completed when Brownian motion occurs sufficiently to achieve uniformity on a molecular scale. 

Fig. 2 compares collision kernels calculated for a 250 nm particle as function of the collision partner size for Brownian motion, laminar and turbulent shear flows as well as sedimentation at 25 °C in water based on the equations given... 

For dilute suspensions, particle-particle interactions can be neglected. The extent of transfer of particles by the gradient in the particle phase density or volume fraction of particles is proportional to the diffusivity of particles Dp. Here Dp accounts for the random motion of particles in the flow field induced by various factors, including the diffusivity of the fluid whether laminar or turbulent, the wake of the particles in their relative motion to the fluid, the Brownian motion of particles, the particle-wall interaction, and the perturbation of the flow field by the particles. 

Diffusion Random migration of particles in a favored direction resulting from Brownian motion or turbulent eddy motion of the suspending gas. 

Once particles are present in a volume of gas, they collide and agglomerate by different processes. The coagulation process leads to substantial changes in particle size distribution with time. Coagulation may be induced by any mechanism that involves a relative velocity between particles. Such processes include Brownian motion, shearing flow of fluid, turbulent motion, and differential particle motion associated with external force fields. The theory of particle collisions is quite complicated even if each of these mechanisms is isolated and treated separately. 

The diffusion of small particles depends upon many factors. In addition to Brownian motion, we must consider the effect of gravity and the motion of the fluid in which the particles reside. Ordinary diffusion as understood in colloid chemistry must be modified considerably when we deal with turbulence. However, we still retain the usual definition of diffusion, namely that it is the number or mass of particles passing a unit cross section of the fluid in unit-time and unit-concentration gradient. That is, if dw particles (or mass) move through an area / in time dt and dC/dx is the concentration increase in the jc-directior then... 

Thus, particle concentration in the water column follows an exponential decay upward from the bottom. If the decay constant, ay/D, is small enough, the particle concentration might be nearly uniform over a substantial distance above the bottom of the water column. This could occur for extremely small particles in quiescent waters, or for larger particles if D is much larger than the value inferred from Brownian motion. Because surface waters are rarely quiescent, D can become many orders of magnitude larger in the presence of even mild turbulence. Under these conditions, Fickian transport may be sufficient to keep particles several micrometers in diameter suspended in the water indefinitely. 

With this formulation, chemical effects on coagulation are included in a and physical effects in Particle contacts are usually considered to be caused by three mechanisms differential sedimentation, shear (laminar and turbulent), and Brownian motion. Differential sedimentation contact occurs when two particles fall through the water at different rates and the faster particle overtakes the slower one. Shear contact occurs when different parts of the fluid environment move at different speeds relative to each other, and thus a particle that is moving with one fluid patch overtakes and collides with a particle in a slower fluid patch. Brownian motion contact occurs when two particles move randomly through their fluid in Brownian motion and collide... 

Wiener s process (i.e., Brownian motion) and Kolmogorov s turbulence (i.e., a nonsmooth vector field) may be cited as examples of phenomena which can be described by continuous, nowhere differentiable functions (fractal functions). 

Small particles in a turbulent gas dilfuse from one point to another as a result of the eddy motion. The eddy diffusion coefficient of the particles will in general differ from that of the carrier gas. An expression for the particle eddy diffusivity can be derived for a Stokesian particle, neglecting the Brownian motion. In carrying out the analysis, it is assumed that the turbulence is homogeneous and that there is no mean gas velocity. The statistical properties of the system do not change with time. Essentially what we have is a stationary, uniform turbulence in a large box. This is an approximate representation of the core of a turbulent pipe flow, if we move with the mean velocity of the flow. 

Aerosols are unstable with respect to coagulation. The reduction in surface area that accompanie.s coalescence corresponds to a reduction in the Gibbs free energy under conditions of constant temperature and pressure. The prediction of aerosol coagulation rates is a two-step process. The first is the derivation of a mathematical expression that keeps count of particle collisions as a function of particle size it incorporates a general expression for tlie collision frequency function. An expression for the collision frequency based on a physical model is then introduced into the equation Chat keep.s count of collisions. The collision mechanisms include Brownian motion, laminar shear, and turbulence. There may be interacting force fields between the particles. The processes are basically nonlinear, and this lead.s to formidable difficulties in the mathematical theory. 

The various collision mechanisms are compared in Fig. 7.7 which shows the collision frequency function for l- m particles interacting with particles of other sizes. Under conditions corresponding to turbulence in the open atmosphere ( j ss 5cm /sec- ), either Brownian motion or differential sedimentation plays a dominant role. Brownian motion controls for particles smaller than 1 jam. At lower altitudes in the atmosphere and in turbulent pipe flows, shear becomes important. 

Brownian Coagulation Dynamics of Discrete Distribution for an Initially Monodisperse Aerosol 192 Brownian Coagulation Effect of Particle Force Fields 196 Effect of van der Waals Forces 197 Effect of Coulomb Forces 200 Collision Frequency for Laminar Shear 200 Simultaneous Laminar Shear and Brownian Motion 202 Turbulent Coagulation 204... 

The choices come in defining M for this mixture of gases. We might define M, for each gas separately, or we might define a mean value M = 2, X, M where X, is the mole fraction of component /. The use of M, would hold in the absence of any physical mixing (e.g. by turbulence or Brownian motion), while M would be used in the case of perfect mixing. 

There are a number of criticisms to this approach. First, the model is incomplete, since once growth begins it continues without limit. Nonlinear saturation and interactions with predators would be needed to stop this. The diffusion coefficient D certainly does not originate from the Brownian motion of the organisms, since this would be irrelevant to these processes above, say, on the millimeter scale. It is rather a turbulent eddy-diffusion coefficient aimed to... 

Typical measured values of (8 /v) 2 are on the order of 10 s-1, so turbulent shear coagulation is significantly slower than Brownian for submicrometer particles, and the two rates become approximately equal for particles of about 5 pm in diameter (Figure 13.A.2). The calculations indicate that coagulation by Brownian motion dominates the collisions of submicrometer particles in the atmosphere. Turbulent shear contributes to the coagulation of large particles under conditions characterized by intense turbulence. 

Particles suspended in the fluid are carried by the fluid as it flows across the surface. If the fluid is flowing under laminar conditions the transport of the particles across the fluid layers to the surface will be by Brownian motion. Under turbulent conditions particles will be brought to the laminar sub-layer by eddy diffusion, but the remainder of the journey to the surface, across the laminar sublayer is generally ascribed to Brownian motion. Under these conditions for the small particles involved, they may be treated as molecules. In other words mass... 

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