Stokes number cyclones

The scale-up of cyclones is based on a dimensionless group, the Stokes number, which characterizes the separation performance of a family of geometrically similar cyclones. The Stokes number Stk o is defined as ... 

Figure 9.5 Geometries and Euler and Stokes numbers for two common cyclones.

This Stokes number is different from that used in cyclone scale-up in Chapter 9. The cyclone scale-up Stokes number Stkgo incorporates the dimensionless ratio of particle size to cyclone diameter, i.e. ... 

We also met the Stokes number in Qiapter 9, where it was one of the dimensionless numbers used in the scale up of gas cyclones for the separation of particles from gases. There are obvious similarities between the collection of particles in a gas cyclone and collection of particles in the airways of the respiratory system. The Stokes number we met in Chapter 13, describing collision between granules, is not readily comparable with the one used here.)... 

Often the designer or investigator is interested only in the cut size X50, when applying scaling rules to cyclones. Then r/ x) in (8.1.6) can be set equal to 0.5 and, denoting the Stokes number corresponding to X50 by Stkso, Eq. (8.1.6) gives ... 

For Rein < 2 x 10, Stkinso can be seen to increase, showing that small cyclones are less efficient than one would expect from Stokesian scaling. Our defining equation, (8.1.4), for the Stokes number. 

This is the formal requirement for dynamic similarity, and is consistent with the results of the classical dimensional analysis in the main text. As we mentioned there, experience teaches us that over a wide range of operating conditions Reynolds number similarity is not all that critical for Stokes number similarity between cyclones, and this indicates that, in this range, it is not all that critical for dynamic similarity. 

The former of these two cases may well arise in cyclone research. If the maximum and/or minimum particle size and the particle density are not known a priori, however, it is impossible to tell whether one of these conditions ap>-ply, and in general it is advisable to sample isokineticaUy whereever possible. Hangal and Willeke (1990) discuss the issue of sampling from a gas stream, and present a model for the capture efficiency of particles as a function of the particle s Stokes number, the ratio of gas to sampler inlet velocities and the angle of the sampler relative to the gas flow. 

The traditional (Rosin, et al., 1932) mechanistic approach equates the time necessary for a particle to settle at a Stokes law velocity across the width of a cyclone s inlet duct, to the available residence time of the carrier gas stream in its number of spiral traverses within the barrel. With reference to Fig. 1, this permits solving for the smallest particle size able to cross the entire width and reach the wall in the available time. 

Thus if the feed flow rate to such a cyclone is altered the new diaracteristic velocity can be calculated iom Equation (8.38), Reynolds number Rom (8.39), Euler number ftom (8.42) and pressure drop from (8.40). The new cut size can be calculated from Equation (8.37) after using (8.41) to give the new Stokes-50 number. However, Equations (8.41) and (8.42) are vaUd only for cyclones of Reitema s optimum geometry. 

A similar case may be made for the use of density in Stokes law, the buoyancy of particles in the separation zone must be taken into account. The fine particles displace the continuous phase and hence it is the density of the liquid that is used in the model. In any case, the suspension density in the zone is not known but is likely to be much less than that of the feed. The second use of fluid density is in the resistance coefficient, Eu. The density to be used there depends on how we define the Euler number the dynamic pressure in the denominator (equation 6.9) is simply a yardstick against which we measure the pressure loss through a cyclone. We have used the clean liquid density in the dynamic pressure alternatively, the feed suspension density may be used. It is immaterial which of the two densities is used (they are both equally unrealistic) provided the case is clearly defined conversion from one to the other is a simple matter. 

A simple two-dimensional simulation of the Earth s atmosphere involves numerical integration of the Navier-Stokes equations. The computer generates a ballet of vortices that mimic the motion of cyclones and anti-cyclones, much like the circulation observed by meteorologists. Unfortunately, even with enormous simplifications and very coarse results, the simulation is too complex for all but the most powerful and expensive computers. One basic problem is that the number of... 

Very central to cyclone technology is the dynamically equivalent particle diameter. This is the diameter of an equi-dense sphere that has the same terminal velocity as the actual particle. Calculating this can be difficult in the range of intermediate Reynolds numbers, or when the Cunningham correction is significant. In the region where Stokes drag law applies, we call it the Stokesian diameter. 

In most applications of interest absolute pressure is of sufficient magnitude to cause the mean free path of the gas molecules to be much smaller than the particles feeding the cyclone. This mean free path is the average distance a gas molecule travels between collisions with another molecule. Under such conditions the gas behaves as a continuum and, if the particle Reynolds number is sufficiently small (less than 1 in any case), the familiar Stokes law may, as discussed in Chap. 2, express the drag force acting on a particle moving through the gas... 

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