Terminal velocity sphericity

Fig. 4. Terminal velocities in air of spherical particles of different densities settling at 21°C under the action of gravity. Numbers on curves represent tme (not bulk or apparent) specific gravity of particles relative to water at 4°C. Stokes-Cunningham correction factor is included for settling of fine particles.

FIG. 6-61 Terminal velocities of spherical particles of different densities settling in air and water at 70°F under the action of gravity. To convert fhs to m/s, multiply by 0.3048. (From Lapple, etal.. Fluid and Particle Mechanics, University of Delaware, Newark, 1951, p. 292. )... 

David W. Taylor Model Basin, Washington, September 1953 Jackson, loc. cit. Valentin, op. cit.. Chap. 2 Soo, op. cit.. Chap. 3 Calderbank, loc. cit., p. CE220 and Levich, op. cit.. Chap. 8). A comprehensive and apparently accurate predictive method has been publisned [Jami-alahamadi et al., Trans ICE, 72, part A, 119-122 (1994)]. Small bubbles (below 0.2 mm in diameter) are essentially rigid spheres and rise at terminal velocities that place them clearly in the laminar-flow region hence their rising velocity may be calculated from Stokes law. As bubble size increases to about 2 mm, the spherical shape is retained, and the Reynolds number is still sufficiently small (<10) that Stokes law should be nearly obeyed. 

From the standpoint of collector design and performance, the most important size-related property of a dust particfe is its dynamic behavior. Particles larger than 100 [Lm are readily collectible by simple inertial or gravitational methods. For particles under 100 Im, the range of principal difficulty in dust collection, the resistance to motion in a gas is viscous (see Sec. 6, Thud and Particle Mechanics ), and for such particles, the most useful size specification is commonly the Stokes settling diameter, which is the diameter of the spherical particle of the same density that has the same terminal velocity in viscous flow as the particle in question. It is yet more convenient in many circumstances to use the aerodynamic diameter, which is the diameter of the particle of unit density (1 g/cm ) that has the same terminal settling velocity. Use of the aerodynamic diameter permits direct comparisons of the dynamic behavior of particles that are actually of different sizes, shapes, and densities [Raabe, J. Air Pollut. Control As.soc., 26, 856 (1976)]. 

When a spherical particle of diameter d settles in a viscous liquid under earth gravity g, the terminal velocity V, is determined by the weight of the particle-balancing buoyancy and the viscous drag on the... 

Figure 4-7. Terminal velocities for solid spherical particles in standard air. Courtesy of American Blov er Div. American Radiator and Standard Sanitary Corporation.

A wide range of particle terminal velocities for various Reynolds numbers have been investigated by Kunii and Levenspiel.43 They suggested that if the particles were assumed to be spherical and operated at low particle Reynolds number (Rep < 0.4), the Stokes equation was found to be acceptable (see Figure 17.4). Therefore, the terminal velocity Ut can be expressed as ... 

Alternatively, the model of Shiller and Naumann is commonly used for the prediction of terminal velocity of a spherical particle ... 

Waslo and Gal-Or (Wl) recently generalized the Levich solution [Eq. (65)] by evaluating the effect of and y on the terminal velocity of an ensemble of spherical drops of bubbles. Their solution is... 

The weight of the particles builds up with time and is proportional to 1/d. If we assume spherical particles, then we can convert the above curve to particle diameter from Stokes Law. Although we have added the pcurticle suspension to a "water cushion" as shown above, it might not seem that the settling of the particles would strictly adhere to Stokes Law, which assumes the terminal velocity to be constant. 

It is desired to determine the size of pulverized coal particles by measuring the time it takes them to fall a given distance in a known fluid. It is found that the coal particles (SG = 1.35) take a time ranging from 5 s to 1,000 min to fall 23 cm through a column of methanol (SG = 0.785, p. = 0.88 cP). What is the size range of the particles in terms of their equivalent spherical diameters Assume that the particles are falling at their terminal velocities at all times. 

Many particles are not spherical and so will not have the same drag properties as spherical particles. The effective diameter for such particles is often characterized by the equivalent Stokes diameter, which is the diameter of the sphere that has the same terminal velocity as the particle. This can be determined from a direct measurement of the settling rate of the... 

By definition, the terminal velocity of a particle (ut) is the superficial gas velocity which suspends an isolated particle without translational motion—i.e., the terminal free fall velocity for that particle. From force balance on the particle, the terminal velocity for an approximately spherical particle can be shown to be... 

Time for a solid spherical particle to reach 99 per cent of its terminal velocity when falling from rest in the Stokes regime... 

Consider a spherical particle of diameter dp and density pp falling from rest in a stationary fluid of density p and dynamic viscosity p.. The particle will accelerate until it reaches its terminal velocity a,. At any time t, let a be the particle s velocity. Recalling that the drag force acting on a sphere in the Stokes regime is of magnitude iirdppu, application of Newton s second law of motion can be written as... 

Show that the time to reach 50 per cent of the terminal velocity for a spherical particle falling from rest in laminar flow in a fluid is... 

For a single spherical particle settling at its terminal velocity hq, a force balance gives ... 

The rate of mass-transfer, unlike the terminal velocity, may reach its lower limit only when the whole surface of the drop or bubble is covered by the adsorbed film. In the absence of surface-active material, the freshly exposed interface at the front of the moving drop (due to circulation here) could well be responsible for as much mass transfer as occurs in the turbulent wake of the drop. The results of Baird and Davidson 67a) on mass transfer from spherical-cap bubbles are not inconsistent with this idea, and further experiments on smaller drops are in progress in the author s laboratory. In general, if these ideas are correct, while the rear half of the drop is noncirculating (and the terminal velocity has reached the limit of that for a solid sphere), the mass transfer at the front half of the drop may still be much higher, due to the circulation, than for a stagnant drop. Only when sufficient surface-active material is present to cover the whole of the surface and eliminate all circulation will the rate of mass-transfer approach its lower limit. 

Following a suggestion made by Davies (D2, D4), we define a degree of circulation Z such that the terminal velocity of a spherical bubble or drop in slow viscous flow is given by... 

Wadell (Wl, W2) proposed that the sphericity ij/, defined in Chapter 2, could be used to correlate drag on irregular particles. The appropriate dimension for definition of Re and is then the diameter of the sphere with the same volume as the particle. Figure 6.14 shows velocity correction factors calculated on this basis (G5). This approach has found widespread acceptance, although there is experimental evidence that terminal velocity does not correlate well with sphericity (B8, S8). 

For Eo < 0.15, drops are closely spherical and terminal velocities may be calculated using correlations given in Chapter 5 for rigid spheres. For larger... 

Equation (8-4) cannot be satisfied over the entire spherical-cap surface, but if it is satisfied for 0 -> 0 to terms of order iF, the terminal velocity reduces to... 

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