Stokes’ settling diameter

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)]. 

Dpo Initial diameter of drop, meter DpS/a Apparent Stokes settling diameter of floe, meters (18 ft u,J[kc g (pP - p)] 1/2 B Diffusion coefficient, meters2/sec Bp Diffusion coefficient for particles, of diameter Dp, meters2/sec Dpi Diffusion coefficient for particles of diameter >pl, meters2/sec Bp2 Diffusion coefficient for particles of diameter Dp2, meters2/sec e Natural logarithmic base, 2.718 E Potential or potential difference, V Ex Ionization potential, V... 

Settling Stokes s diameter Drag diameter Free-fall diameter... 

The case of Stokes settling velocity is considered as an illustrative example. If we assume that the stationary settling velocity, Wg, of a small particle flowing into a liquid or gaseous medium is a function of its diameter, d, specific weight, gAp, and the viscosity of the gaseous or liquid medium, it follows that ... 

Since the drag diameter is otherwise indeterminable, it is usual in practice to assume that, in the Stokes region, it is equal to the surface diameter. This holds at low Reynolds numbers but as Reynolds number increases, d, > d, For irregular, compactly shaped particles the range of settling diameters is limited (Figure 6.3). 

For the case of free sedimentation, a sphere with diameter cf, and density differing from that of the medium by A p moves at the Stokes settling velocity. 

The Andreasen pipette introduced in the 1920s is perhaps the most popular manual apparatus for sampling from a sedimenting suspension. Determination of the change in density of the sampled particle suspension with time enables the calculation of size distribution of the particles. As Stokes law applies only to spherical particles, the nonspherical particles give a mean diameter referred to as Stokes equivalent diameter. The size range measurable by this method is from 2 to 60 pm (8). The upper limit depends on the viscosity of liquid used while the lower limit is due to the failure of very small particles to settle as these particles are kept suspended by Brownian motion. 

The terminal velocity in the case of fine particles is approached so quickly that in practical engineering calculations the settling is taken as a constant velocity motion and the acceleration period is neglected. Equation 7 can also be appHed to nonspherical particles if the particle size x is the equivalent Stokes diameter as deterrnined by sedimentation or elutriation methods of particle-size measurement. 

Wall Effects When the diameter of a setthng particle is significant compared to the diameter of the container, the settling velocity is reduced. For rigid spherical particles settling with Re < 1, the correction given in Table 6-9 may be used. The factor k is multiplied by the settling velocity obtained from Stokes law to obtain the corrected set-... 

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. 

Since the size range of the particles to be removed is less than the critical diameter, we are confident that the particles will follow Stokes law. Hence, the settling velocity for a 50 ft size particle is ... 

Free-falling diameter Also known as sedimentation or Stokes diameter, the diameter of a sphere with the same terminal settling velocity and density as a nonspherical or irregular particle. 

Thus the Stokes diameter of any partiele is that of an equivalent sphere having same terminal settling veloeity and is a useful additional partiele eharaeteristie for partieulate systems involving fluid motion. 

Elutriation differs from sedimentation in that fluid moves vertically upwards and thereby carries with it all particles whose settling velocity by gravity is less than the fluid velocity. In practice, complications are introduced by such factors as the non-uniformity of the fluid velocity across a section of an elutriating tube, the influence of the walls of the tube, and the effect of eddies in the flow. In consequence, any assumption that the separated particle size corresponds to the mean velocity of fluid flow is only approximately true it also requires an infinite time to effect complete separation. This method is predicated on the assumption that Stokes law relating the free-falling velocity of a spherical particle to its density and diameter, and to the density and viscosity of the medium is valid... 

Thru a combination of sedimentation and transmission measurements, a particle size distribution can be found. Tranquil settling of a dispersion of non-uniform particles will result in a separation of particles according to size so that transmission measurements at known distances below the surface at selected time intervals, will, with Stokes law, give the concn of particles of known diameter. Thus, a size frequency distribution can be obtained... 

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. 

Free settling means that the particle is at a sufficient distance from the boundaries of the container and from other particles, and that the density of the medium is that of a pure fluid, as for example, water. If two different mineral particles of densities pj and p2 and diameters d1 and d2 respectively fall in a fluid of density p3 with the same settling rate, then their terminal velocities must be the same. From Stokes law this gives for the laminar range... 

Example 11-1 Unknown Velocity and Unknown Diameter of a Sphere Settling in a Power Law Fluid. Table 11-1 summarizes the procedure, and Table 11-2 shows the results of a spreadsheet calculation for an application of this method to the three examples given by Chhabra (1995). Examples 1 and 2 are unknown velocity problems, and Example 3 is an unknown diameter problem. The line labeled Equation refers to Eq. (11-32) for the unknown velocity cases, and Eq. (11-35) for the unknown diameter case. The Stokes value is from Eq. (11-9), which only applies for - Re,pi < 1 (e.g., Example 1 only). It is seen that the solutions for Examples 1 and 2 are virtually identical to Chhabra s values and the one for Example 3 is within 5% of Chhabra s. The values labeled Data were obtained by iteration using the data from Fig. 4 of Tripathi et al. (1994). These values are only approximate, because they were obtained by interpolating from the (very compressed) log scale of the plot. 

Solid particles can be removed from a dilute suspension by passing the suspension through a vessel that is large enough that the vertical component of the fluid velocity is lower than the terminal velocity of the particles and the residence time is sufficiently long to allow the particles to settle out. A typical gravity settler is illustrated in Fig. 12-2. If the upward velocity of the liquid (Q/A) is less than the terminal velocity of the particles (Ft), the particles will settle to the bottom otherwise, they will be carried out with the overflow. If Stokes flow is applicable (i.e., NRe < 1), the diameter of the smallest particle that will settle out is... 

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... 

For settling in the Stokes law region, the velocity is proportional to the diameter squared and hence the time taken for a 40 pm particle to fall a height li m is ... 

In a hydraulic jig, a mixture of two solids is separated into its components by subjecting an aqueous slurry of the material to a pulsating motion, and allowing the particles to settle for a series of short time intervals such that their terminal falling velocities are not attained. Materials of densities 1800 and 2500 kg/m3 whose particle size ranges from 0.3 mm to 3 mm diameter are to be separated. It may be assumed that the particles are approximately spherical and that Stokes Law is applicable. Calculate approximately the maximum time interval for which the particles may be allowed to settle so that no particle of the less dense material falls a greater distance than any particle of the denser material. The viscosity of water is 1 mN s/m2. 

According to Stokes Law, the resistance force F acting on a particle of diameter d, settling at a velocity u in a fluid of viscosity p is given by ... 

Two spherical particles, one of density 3000 kg/m3 and diameter 20. im, and the other of density 2000 kg/m3 and diameter 30 (im start settling from rest at the same horizontal level in a liquid of density 900 kg/m3 and of viscosity 3 mN s/m2. After what period of settling will the particles be again at the same horizontal level It may be assumed that Stokes Law is applicable, and the effect of added mass of the liquid moved with each sphere may be ignored. 

Fig. 7.1 gives a size spectrum of water-borne particles. Particles with diameters less than 10 pm have been called colloids. In soils, the clay-sized and fine silt-sized particles are classified as colloids. Colloids do not dissolve, but instead remain as a solid phase in suspension. Colloids usually remain suspended because their gravitational settling is less than 10 2 cm s 1. Under simplifying conditions (spherical particles, low Reynolds numbers), Stokes law gives for the settling velocity, vs... 

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