Motion of particles in fluids

In this first section, the fundamentals of particle-fluid interaction are reviewed, with particular emphasis on the concepts used in scale-up of equipment in particle-fluid separation. 

If a particle moves relative to the fluid in which it is suspended, there exists a force opposing the motion, known as the drag force. Knowledge of the magnitude of this force is essential if the particle motion is to be studied. The conventional way to express the drag force, Fd, is according to Newton  

Dimensional analysis shows that Cd is generally a function of the particle Reynolds number  

This is an approximation which gives best results for Rep 0 the upper limit of its validity depends on the error which can be accepted. The usually quoted limit for Stokes region of Rep = 0.2 is based on the error of approximately 2%. 

Elimination of Fd between equations 18.1, 18.2, and 18.3 gives another form of Stokes law  

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H Heywood. Uniform and non-uniform motion of particles in fluids. In Proc Symp Interaction Fluids and Parts. London Inst Chem Eng, 1962, pp 1-8. 

In order to apply the Smoluchowski equation (Equations (1.3), (2.1), (3.29)), we need values for the least distance of approach (rAn) and the diffusion coefficient (Dab)- The value of tab can be estimated from molecular volumes (Section 2.5.1.2). The diffusion coefficient can be determined by various methods, but experimental values are available only for a minority of the myriad possible situations. A common practice is to use the Stokes-Einstein relation (Section 1.2.3), which rests on the assumption that solute molecules in motion behave like macroscopic particles to which classical hydrodynamic theory can be applied. We shall first outline (a) the relation between the diffusion coefficient D and the mechanics of motion of particles in fluids, leading to the Stokes-Einstein equation relating D to solute size and solvent viscosity and (b) the direct experimental determination of D. We shall then (c) compare the results and note the reservations that are required in relying on the Stokes-Einstein estimates of D in various cases. 

The motion of particles in a fluid is best approached tlirough tire Boltzmaim transport equation, provided that the combination of internal and external perturbations does not substantially disturb the equilibrium. In otlier words, our starting point will be the statistical themiodynamic treatment above, and we will consider the effect of botli the internal and external fields. Let the chemical species in our fluid be distinguished by the Greek subscripts a,(3,.. . and let f (r, c,f)AV A be the number of molecules of type a located m... 

In the equation referred to above, it is assumed that there is 100 percent transmission of the shear rate in the shear stress. However, with the slurry viscosity determined essentially by the properties of the slurry, at high concentrations of slurries there is a shppage factor. Internal motion of particles in the fluids over and around each other can reduce the effective transmission of viscosity efficiencies from 100 percent to as low as 30 percent. 

The flow problems considered in Volume 1 are unidirectional, with the fluid flowing along a pipe or channel, and the effect of an obstruction is discussed only in so far as it causes an alteration in the forward velocity of the fluid. In this chapter, the force exerted on a body as a result of the flow of fluid past it is considered and, as the fluid is generally diverted all round it, the resulting three-dimensional flow is more complex. The flow of fluid relative to an infinitely long cylinder, a spherical particle and a non-spherical particle is considered, followed by a discussion of the motion of particles in both gravitational and centrifugal fields. 

Only a very limited amount of data is available on the motion of particles in non-Newtonian fluids and the following discussion is restricted to their behaviour in shear-thinning power-law fluids and in fluids exhibiting a yield-stress, both of which are discussed in Volume 1, Chapter 3. 

Low Reynolds number flows with boundary integral representation have been used to describe rheological and transport properties of suspensions of solid spherical particles, as well as for numerical solution of different problems, including particle-particle interaction, the motion of a particle near a fluid interface or a rigid wall, the motion of particles in a container, and others. 

FIG. 6-57 Drag coefficients for spheres, disks, and cylinders A = area of particle projected on a plane normal to direction of motion C = overall drag coefficient, dimensionless Dp - diameter of particle Fd = drag or resistance to motion of body in fluid Re = Reynolds number, dimensionless u = relative velocity between particle and main body of fluid (I = fluid viscosity and p = fluid density. (From Lapple and Shepherd, Ind. Eng. Chem., 32, 60S [1940].)... 

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. 

PEPT has become an established technique for studying the motion of particles in granular and fluid systems. Until recently, this technique was confined to use with medically derived detectors, which places constraints on the geometry and scale of process equipment that can be viewed. Demand for greater flexibility in the use of the PEPT... 

MD methods arise primarily from solution of the equations of motion for particles in fluid. Thus, one considers the sum of forces acting on each particle in the simulation, which results in motion over some sufficiently small time step. It is typical to use Newton s equation of motion, which can be written in the general form... 

However, most such work has been concerned with classical fluidization, and relatively little has been devoted to the simultaneous motion of particles in vertically flowing fluids. 

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