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These three equations can be solved simultaneously (by iteration) for rj /R, r2/R, and ri/R. It is assumed that the size of the suspended drops is known as well as the density and viscosity of the liquids and the overall dimensions and speed of the centrifuge.

### IV. CYCLONE SEPARATIONS A. General Characteristics

Centrifugal force can also be used to separate solid particles from fluids by inducing the fluid to undergo a rotating or spiraling flow pattern in a stationary vessel (e.g., a cyclone) that has no moving parts. Cyclones are widely used to remove small particles from gas streams ("aerocyclones") and suspended solids from liquid streams ("hydrocyclones").

A typical cyclone is illustrated in Fig. 12-7 (this is sometimes referred to as a "reverse flow'' cyclone). The suspension enters through a rectangular or circular duct tangential to the cylindrical separator, which usually has a conical bottom. The circulating flow generates a rotating vortex motion that imparts centrifugal force to the particles which are thrown outward to the walls of the vessel, where they fall by gravity to the conical bottom and are removed. The carrier fluid spirals inward and downward to the cylindrical exit duct (also referred to as the "vortex finder''), from which it travels back up and leaves the vessel at the top. The separation is not perfect, and some solid particles leave in the overflow as well as the underflow. The particle size for which 50% leaves in the overflow and 50% leaves in the underflow is called the cut size.

Oust out

Figure 12-7 Typical reverse flow cyclone.

Oust out

### Figure 12-7 Typical reverse flow cyclone.

The diameter of a hydrocyclone can range from 10 mm to 2.5 m, cut sizes from 2 to 250 ^m, and flow rate (capacities) from 0.1 to 7200 m3/hr. Pressure drop can range from 0.3 to 6 atm (Svarovsky, 1984). For aerocy-clones, very little fluid leaves with the solids underflow, although for hydrocyclones the underflow solids content is typically 45-50% by volume. Aerocyclones can achieve effective separation for particles as small as 2-5 mm.

Advantages of the cyclone include (Svarosky, 1984)

1. Versatility. Virtually any slurry or suspension can be concentrated, liquids degassed, or the solids classified by size, density, or shape.

2. Simplicity and economy. They have no moving parts and little maintenance.

3. Small size. Low residence times, and relatively fast response.

4. High shear forces, which can break up agglomerates, etc.

1. Inflexibility. A given design is not easily adapted to a range of conditions. Performance is strongly dependent upon flow rate and feed composition, and the turndown ratio (range of operation) is small.

2. Limited separation performance in terms of the sharpness of the cut, range of cut size, etc.

3. Susceptibility to erosion.

4. High shear prevents the use of flocculents to aid the separation, as can be done in gravity settlers.

An increase in any one operating parameter generally increases all others as well. For example, increasing the flow rate will increase both separation efficiency and pressure drop, and vice versa.

B. Aerocyclones

### 1. Velocity Distribution

Although the dominant velocity component in the cyclone is in the angular (tangential) direction, the swirling flow field includes significant velocity components in the radial and axial directions as well, which complicate the motion and make a rigorous analysis impossible. This complex flow field also results in significant particle-particle collisions, which cause some particles of a given size to be carried out in both the overhead and underflow discharge, thus affecting the separation efficiency.

Cyclone analysis and design is not an exact science, and there are a variety of approaches to the analysis of cyclone performance. A critical review of the various methods for analyzing hydrocyclones is given by Svarovsky (1996), and a review of different approaches to aerocyclone analysis is given by Leith and Jones (1997). There are a number different approaches to the analysis of aerocyclones, one of the most comprehensive being that of Bhonet et al. (1997). The presentation here follows that of Leith and Jones (1997), which outlines the basic principles and some of the practical "working relations.'' The reader is referred to other works, especially those of Bhonet (1983) and Bhonet et al. (1997), for more details on specific cyclone design.

The performance of a cyclone is dependent upon the geometry as described by the values of the various dimensionless "length ratios'' (see Fig. 12-7): a/D, b/D, De/D, S/D, h/D, H/D, and B/D. Typical values of these ratios for various "standard designs'' are given in Table 12-1.

The complex three-dimensional flow pattern within the cyclone is dominated by the radial (Vr) and tangential (Ve) velocity components. The vertical component is also significant but plays only an indirect role in the separation. The tangential velocity in the vortex varies with the distance from the axis in a complex manner, which can be described by the equation

For a uniform angular velocity (a = constant, i.e., a "solid body rotation''), n = — 1, whereas for a uniform tangential velocity ("plug flow'') n = 0, and for inviscid free vortex flow a = c/r2, i.e., n = 1. Empirically, the exponent n has been found to be typically between 0.5 and 0.9. The maximum value of Ve occurs in the vicinity of the outlet or exit duct (vortex finder) at r = De/2.

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