Brownian motion laminar

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... 

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. 

Stokes diameter is defined as the diameter of a sphere having the same density and the same velocity as the particle in a fluid of the same density and viscosity settling under laminar flow conditions. Correction for deviation from Stokes law may be necessary at the large end of the size range. Sedimentation methods are limited to sizes above a [Lm due to the onset of thermal diffusion (Brownian motion) at smaller sizes. 

Studies on orthokinetic flocculation (shear flow dominating over Brownian motion) show a more ambiguous picture. Both rate increases (9,10) and decreases (11,12) compared with orthokinetic coagulation have been observed. Gregory (12) treated polymer adsorption as a collision process and used Smoluchowski theory to predict that the adsorption step may become rate limiting in orthokinetic flocculation. Qualitative evidence to this effect was found for flocculation of polystyrene latex, particle diameter 1.68 pm, in laminar tube flow. Furthermore, pretreatment of half of the latex with polymer resulted in collision efficiencies that were more than twice as high as for coagulation. 

Let us apply the interpolation procedure to a case involving an electric field. It is well known that the efficiency of the granular bed filters can be significantly increased by applying an external electrostatic field across the filter. In this case, fine (<0.5-/rm) particles deposit on the surface of the bed because of Brownian motion as well as because of the electrostatically generated dust particle drift [51], The rate of deposition can be calculated easily for a laminar flow over a sphere in the absence of the electrostatic field [5]. The other limiting case, in which the motion of the particles is exclusively due to the electric field, could also be treated [52], When, however, the two effects act simultaneously, only numerical solutions to the problem could be obtained [51],... 

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. 

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. 

For solutions of nonspherical particles the situation is more complicated and the physical picture can be described qualitatively as follows for a system of particles in a fluid one can define a distribution function, F (Peterlin, 1938), which specifies the relative number of particles with their axes pointed in a particular direction. Under the influence of an applied shearing stress a gradient of the distribution function, dFfdt, is set up and the particles tend to rotate at rates which depend upon their orientation, so that they remain longer with their major axes in position parallel to the flow than perpendicular to it. This preferred orientation is however opposed by the rotary Brownian motion of the particles which tends to level out the distribution or orientations and lead the particles back toward a more random distribution. The intensity of the Brownian motion can be characterized by a rotary diffusion coefficient 0. Mathematically one can write for a laminar, steady-state flow ... 

Chang HM, Robertson CR Platelet aggregation by laminar bear and Brownian motion. Ann Biomed Eng 1976 4 151-183. 

Swift DL, Friedlander SK. The coagulation of hydrosols by Brownian motion and laminar shear flow. J Colloid Sd 1964 19 621-647. 

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... 

Consider the flow of an aerosol through a 4-in. duct at a velocity of 50 ft/sec. Compare the coagulation rate by Brownian motion and laminar shear in the viscous sublayer, near the wall. Present your results by plotting the collision frequency function for particles with dp = 1 /im colliding with particle.s of other sizes. Assume a temperature of 20 C. Hint In the viscous. sublayer, the velocity distribution is given by the relation... 

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... 

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... 

Particles are transported across the quasi-laminar sublayer by Brownian motion analogous to gaseous molecular diffusion. The dependence of the particle Brownian diffusivity on particle size results in a transfer rate that depends on particle size (see (8.73)). Transfer is rapid, and hence resistance is low, for the very smallest particles. As particle size increases, the Brownian diffusivity decreases and transfer is less rapid (see Figure 8.8). The... 

The use of turbulent emulsion flow regime to facilitate integration of drops is justifled by the substantial increase of collision frequency that is achieved in a turbulent flow as compared to the collision frequency during the sedimentation of drops in a quiescent liquid or in a laminar flow. Particles suspended in the liquid are entrained by turbulent pulsations and move chaotically inside the volume in a pattern similar to Brownian motion. Therefore this pulsation motion of particles can be characterized by the effective factor of turbulent diffusion Dj, and the problem reduces to the determination of collision frequency of particles in the framework of the diffusion problem, as it was first done by Smoluchowsld for Brownian motion [18]. A similar approach was first proposed and realized in [19] for the problem of coagulation of non-interacting particles. The result was that the obtained frequency of collisions turned out to be much greater than the frequency found in experiments on turbulent flow of emulsion in pipes and agitators [20, 21]. 

The sedimentation technique is reliable for particle size determination when rf is in a size range of 2-50 pm. The falling rate of smaller particles is affected by Brownian motion resulting from collisions with the molecules of the liquid and other interactions between particles. Stokes law is valid only for laminar or streamline flow (i.e., when there is no turbulence). The Reynolds number (Re) is a measure of when the process transitions from turbulent to laminar flow ... 

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