Settling velocity, single particle

The simplest case to consider is the settling of a single homogeneous sphere, under gravity, in a fluid of infinite extent. Many experiments have been carried out to determine the relationship between settling velocity and particle size under these conditions and a unique relationship between drag factor (C ) and Reynolds number Re) has been found that reduces to a simple equation, known as the Stokes equation, at low Reynolds number. 

Both equations (6.4) and (6.5) contain D and u and need to be expressed in terms of a single variable in order to determine Z) if m is known or m if Z) is known. Stokes neglected the terms due to inertia and obtained a very simple relationship between settling velocity and particle size for particles settling with low velocities. Several attempts at theoretical solutions for the relationship between and Re at higher velocities have been made. Oseen [29] partially allowed for inertial effects to obtain ... 

As for settling of single particles in Newtonian liquids, the fundamental hydrodynamic characteristic for particle motion in non-Newtonian fluids is again the drag coefficient. Its prediction allows calculations of terminal settling velocities. Note that equation 18.10, which applies to low particle concentrations (below 0.5% by volume) in Newtonian liquids at low Reynolds numbers, can, in principle, also be used non-Newtonian fluids where viscosity // then becomes the apparent viscosity but, depending on the type of the non-Newtonian behaviour (= model), its determination may require an iterative procedure. Each model redefines the particle Reynolds number so that, for example, for a power law fluid characterized by constants n and K... 

Measurement of single particle settling velocity in a turbulent field is not easy. However, it is known to be a function of free settling velocity which for spherical particles can be estimated from the following ... 

Fig. 14. Drag coefficient for terminal settling velocity correlation (single particle) where A represents Stokes law B, intermediate law and C, Newton s...

The above analysis applies only to the free settling velocities of single particles and does not account for particle-particle interactions. 

The following semi-empirical equation relates the (hindered) settling velocity of a slurry of particles to the settling velocity of a single particle, known as the Richardson and Zaki (1954) (RZ) equation. The RZ equation is also used for liquid fluidization whereby particles are supported by an up-flow of fluid. 

If a single particle is falling freely under gravity in an infinitely dilute suspension, it will accelerate until it reaches a steady-state velocity. This final velocity is known as the terminal settling velocity (t/t) and represents the maximum useful superficial velocity achievable in a fluidised bed. Thus, the contained particles will be elutriated from the column if the superficial velocity is above Ut, the value of which can be predicted using the Stokes equation... 

V, = settling velocity for single spherical particle, ft/s and nt/s (terminal)... 

The flow problems considered in previous chapters are concerned with homogeneous fluids, either single phases or suspensions of fine particles whose settling velocities are sufficiently low for the solids to be completely suspended in the fluid. Consideration is now given to the far more complex problem of the flow of multiphase systems in which the composition of the mixture may vary over the cross-section of the pipe or channel furthermore, the components may be moving at different velocities to give rise to the phenomenon of slip between the phases. 

The foregoing expressions give the suspension velocity (Fs) relative to the single particle free settling velocity, V0, i.e., the Stokes velocity. However, it is not necessary that the particle settling conditions correspond to the Stokes regime to use these equations. As shown in Chapter 11, the Dallavalle equation can be used to calculate the single particle terminal velocity V0... 

The terminal settling velocity u, for a single spherical particle in a centrifugal separator can be calculated from equation 9.5 with the centripetal acceleration rto2 replacing the gravitational acceleration g to give... 

Venumadhav, G. and Chhabra, R.P. Powder Technol. 78 (1994) 77. Settling velocities of single nonspherical particles in non-Newtonian fluids. 

Sizing procedures to design a mixer for one closely sized particle settling velocity are modified considerably when there are other solids present. Figure 17 shows the effect of settling velocity on power when there are other solids present in the system. The slope is much less pronounced than it is when a single particle size alone is being suspended. 

Cell models constitute a second major class of empirical developments. Among these, only two will be mentioned here as constituting the most successful and widely used. The first, due to Happel (1957,1958), is useful for estimating the effective viscosity and settling velocity of suspensions. Here, the suspension is envisioned as being composed of fictitious identical cells, each containing a single spherical particle of radius a surrounded by a concentric spherical envelope of fluid. The radius b of the cell is chosen to reproduce the suspension s volume fraction

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Cunningham correction factor for, 63 settling velocity of, 81-82 Simple coagulation theory, 308 Simple diffusion, 131-143 Single-particle scattering measurements, 297-299... 

Solid concentration profiles are produced from a balance of gravitational with buoyancy and kinetic energy transfer forces. For a single particle in a stagnant liquid, the settling velocity,... 

Impeller-agitated tanks. These can be operated as continuous leaching tanks, singly or in a series. If the solids feed is a mixture of particles of different settling velocities and if it is desirable that all particles reside in the leaching tank the same lengths of time, design of a continuous stirred leach tank is difficult and uncertain. 

Sedimentation in liquids During gravitational sedimentation particles dispersed in a liquid settle with a velocity that is a function of their size. For a single, spherical particle in an infinite body of liquid the Stokes law is valid at low settling velocity (Reynolds number <0.2) ... 

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