Particles motion through fluids

A well known result from the theory of inertia free flows is that the effect of the tube wall on the particle motion through a capillary is to slow the particle down relative to the fluid in the neighborhood of the wall. The mean particle velocity given by Eq. (5.7.2) must therefore be too large. From their moment analysis, Brenner 5c Gaydos found for small values of A that to terms of lowest order in A... 

Of course, using the correlations (8)—(13) under circumstances different from those cited above is not allowed. This means, first of all, that the particle motion through the fluid or the fluid flow around the particle should be steady and strictly ID over a sufficiently long distance to allow for the development of boundary layer and wake toward a steady state and to permit the use of the standard drag curve. Lateral forces due an asymmetrical flow field should be absent. The development of both the boundary layer around and the wake behind a particle is affected by local accelerations or decelerations of the immersed particle and/or of the embedding fluid, by a steady shear field in the surrounding fluid, by the dynamics of free-stream turbulence, by particle rotations (either externally and deliberately imposed, or as the result of shear flow), by adjacent walls of a container, and by the presence... 

The natural process of bringing particles and polyelectrolytes together by Brownian motion, ie, perikinetic flocculation, often is assisted by orthokinetic flocculation which increases particle coUisions through the motion of the fluid and velocity gradients in the flow. This is the idea behind the use of in-line mixers or paddle-type flocculators in front of some separation equipment like gravity clarifiers. The rate of flocculation in clarifiers is also increased by recycling the floes to increase the rate of particle—particle coUisions through the increase in soUds concentration. 

Hence, the application of these formulas only applies to very dilute systems. At high particle concentrations, mutual interference in the motion of particles exists, and the rate of settling is considerably less than that computed by the given expressions. In the latter case, the particle is settling through a suspension of particles in a fluid, rather than through a simple fluid medium. 

Crystals suspended in liquors emerging from crystallizers are normally passed to solid-liquid separation devices such as gravity settlers or thickeners that may subsequently feed filters to remove yet more liquid prior to drying. Here the transport processes of particle motion and the flow of fluids through porous media are important in determining equipment size, the operation of which may be intensified by application of a centrifugal force. 

Thus as pointed out above, further treatment on the mechanics of particle motion remains confined only to one-dimensional motion of particle through fluid. A particle of mass m moving through a fluid under the action of an external force Fe is considered. The velocity of the particle relative to the fluid is taken to be v. The buoyant force on the particle is taken to be Fb, and the drag force be FD. Then, the resultant force on the particle is Fe - Fb - Fd, the acceleration of the particle is dv/dt, and the resulting equation of motion is given by... 

A hydrodynamic model of fluidization attempts to account for several essential features of fluidization mixing and distribution of solids and fluid in a so-called emulsion region, the formation and motion of bubbles through the bed (the bubble region ), the nature of the bubbles (including their size) and how they affect particle motion/distribution, and the exchange of material between the bubbles (with little solid content) and the predominantly solid emulsion. Models fall into one of three classes (Yates, 1983, pp. 74-78) ... 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

For a complete review on low Re motion in bounded fluids, see Happel and Brenner (H3). Some general results are of immediate interest. For a particle moving through an otherwise undisturbed fluid, without rotation and with velocity U parallel to a principal axis both of the body and the container,... 

The constant/ of Eq. 3.7 is known as the friction coefficient and, as used here, applies to a single object or particle. It is a fundamental parameter reflecting the magnitude of drag forces through fluids. It depends on the dimensions of the body in motion as well as upon the viscosity of the medium through which it moves (Chapter 4). 

In a top-spray fluidised bed, air is introduced through a uniform air distribution plate. Compared to the particle motion in the bottom-spray fluidised bed, the fluidisation pattern is less controlled. Particular to the top-spray method, the nozzle is positioned above the particle bed, and coating liquid is sprayed coimtercurrently, or down, into the fluidising core particles. Compared to a Wurster fluid bed, it is more... 

Consider turbulent flow in a horizontal pipe, and the upward eddy motion of fluid particles in a layer of lower velocity to an adjacent layer of higher velocity through a differential area ri4 as a result of the velocity fluctuation v, as shown in Fig. 6-21. The mass flow rate of the fluid panicles rising through dA is pu dA, arid its net effect on the layer above dA is a reduction in its average flow velocity because of nioraentum transfer to the fluid particles with lower average flow velocity. This momentum transfer causes the horizontal velocity of the fluid particles to increase by and thus its momentum in the horizontal direction to increase at a rate of pv dA)u, which must be equal to the decrease in the momentum of the upper fluid layer. 

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... 

The analysis is similar to that used in Chapter 2 to derive the Stokes-Einstein relation for the diffusion coefficient. Again we consider only the one-dimensional problem. Particles originally present in the differential thickness around, v = 0 (Chapter 2) spread through the fluid a a result of the turbulent eddies. If the particles are much smaller than the size of the eddies, the equation of particle motion for Stokesian particles, based on (4.24) (see associated discussion), is... 

First the reacting molecule. A. diffuses to the external surface of the particle. Motion of A through the fluid outside the particle is governed by externa or bulk diffusion. The reader should consult standard references for additional discussion. Useful correlations have been found between the mass transfer factor. / >. and the dimensionless particle Reynolds number ... 

Interactions with the surrounding transparent medium include the heating of the medium through conduction fi-om the absorbing particle, and the generation of fluid motions through stirring and convection currents. 

MECHANICS OF PARTICLE MOTION. The movement of a particle through a fluid requires an external force acting on the particle. This force may come from a density difference between the particle and the fluid or it may be the result of electric or magnetic fields. In this section only gravitational or centrifugal forces, which arise from density differences, will be considered. 

Three forces act on a particle moving through a fluid (1) the external force, gravitational or centrifugal (2) the buoyant force, which acts parallel with the external force but in the opposite direction and (3) the drag force, which appears whenever there is relative motion between the particle afld the fluid. The drag force acts to oppose the motion and acts parallel with the direction of movement but in the opposite direction. 

SEPARATIONS BASED ON THE MOTION OF PARTICLES THROUGH FLUIDS... 

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