Spheres settling velocity

Hindered Settling When particle concentration increases, particle settling velocities decrease oecause of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered setthng is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore Br. J. Appl. Fhys., 9, 477—482 [1958]) give, for uniformly sized spheres,... 

The settling velocity of a nonspherical particle is less than that of a spherical one. A good approximation can be made by multiplying the settling velocity, u, of spherical particles by a correction factor, iji, called the sphericity factor. The sphericity, or shape factor is defined as the area of a sphere divided by the area of the nonspherical particle having the same volume ... 

Note - In designing a system based on the settling velocity of nonspherical particles, the linear size in the Reynolds number definition is taken to be the equivalent diameter of a sphere, d, which is equal to a sphere diameter having the same volume as the particle. 

Aerodynamic diameter The diameter of a unit-density sphere that has the same settling velocity in air as the particle in question. 

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. 

The terminal settling velocity for single spheres can he determined using the contrasts for the flovv regime. 

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. 

From Eqs. (14-16), (14-18), and (14-20), it is seen that the ratio of the settling velocity of the suspension (Fs) to the terminal velocity of a single freely settling sphere (F0) is... 

It may be noted that an iterative solution to equation 3.70 is required for the calculation of the unknown settling velocity u0, since this term appears in all three dimensionless groups, Re m, Bi, and C D, for a given combination of properties of sphere and fluid. 

A suspension of a mixture of large particles of terminal falling velocity uol and of small particles of terminal falling velocity uqs may be considered, in which the fractional volumetric concentrations are Cl and Cy, respectively. If the value of n in equation 5.76 is the same for each particle. For each of the spheres settling on its own ... 

It is common practice to define a hydraulic equivalent sphere as the sphere with the same density and terminal settling velocity as the particle in question. For a spheroid in creeping flow, the hydraulic equivalent sphere diameter is 2a- E/A and thus depends on orientation. 

Another type of diameter commonly used is the Stokes diameter, If. This is defined as the diameter of a sphere that has the same density and settling velocity as the particle. Thus Stokes diameters are all based on settling velocities, whereas the aerodynamic diameter (Da) also includes a standardized density of unity. 

For quartz spheres, the maximum particle radius compatible with Eq. 23-10 is only 24 pm. In contrast, biogenic particles have a much smaller excess density, ps - pw, and complicated shapes (a small). Typical settling velocities for these particles are 0.1 md 1 for 1-pm particles and lOOmd-1 for 100-pm particles (Lerman, 1979). The 100-pm particles just meet the laminar flow limit (Re = 0.1). 

It would take a 3 micron diameter particle of wajer in air two days to fall the same distance that a sphere of water the size of a baseball can fall in one second after it reaches its maximum velocity of fall (settling velocity). 

Figure 5.1. Settling velocities of spheres as a function of the ratio ot densities of the two phases. Stokes law applies at diameters below approximately 0.01 cm (based on a chart of Lapple ei at-, Chemical Engineering Handbook, McGraw-Hill, New York, 1984, p. 5.67).

This result has been generalized (Batchelor, 1982 Batchelor and Wen, 1982) to a polydisperse suspension of settling spheres, obtaining for the average settling velocity us, of species i. 

Aerodynamic diameter Diameter of a unit density sphere (density = 1 g/cms) having the same aerodynamic properties as the particle in question. This means that particles of any shape or density will have the same aerodynamic diameter if their settling velocity is the same. 

Stokes diameter Diameter of a sphere of the same density as the particle in question having the same settling velocity as that particle. Stokes diameter and aerodynamic diameter differ only in that Stokes diameter includes the particle density whereas the aerodynamic diameter does not. 

Example 6.4 A 30- xm-diameter unit-density sphere (t = 2.75 x 10-3 s) falling at a terminal settling velocity of 2.7 cm/s is captured by a horizontal airflow of 100 ft/min which is flowing into a hood. Find its velocity 1 ms later, relative to the point at which it was captured. 

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