Solid particles suspended in stirred liquids

Slurry reactors are usually equipped with stirrers, to prevent the solid particles from settling. Often propellers are employed in slurry reactors, but turbines can also be effective (see section 4.2.2, figure 4.1). The net action by gravity results in a solids concentration that gradually decreases with liquid height. The pumping action of the propeller transports equal volumes of liquid upward and downward however, the former contains more solids than the latter. In the steady state, the concentration profile is such that the difference of these upward and downward flows balances the flow of sinking particles. 

For calculating the minimum stirrer speed necessary to keep a given swarm of solid particles in suspension, a number of semi-empirical correlations have been proposed in literature. They are of two t es either the minimum required Froude number (n dlg) is specified, or the minimum required local energy dissipation e, close to the bottom of the tank, r sufficiently small particles these have the following general form (for d 10 d)  

Note that these relations are similar, jince e n. The local specific energy dissipation is found from e = q e see sections 4.2.2.1 and 4,2,2,3, Tike coefficients a and P are dependent on geometrical factors. Some authors found that P s 0.5, p s 0.33 and q s 0.5. WiA eq. (4.3) this would mean that, e.g. for standard turbine impellers (with q s 0.1), a s 7.7. There is however quite some controversy concerning the exponents p and q. Some authors found that the exponent p approached 1 for very smsJl particles (Molerus and Latzel, 1987), and for large particles p approached 0, which would seem contrary to expectation (see review by Voit and Mersmann, 1986). Others found p s 0.33 for intermediate sizes (see review by Geisler, et al, 1991). Some authors found little influence of the solids fraction, particularly for low values, others found that q = 0.5 in the range of solids fractions between 0.01 and 0.3 (see review by Geisler, et al., 1991). Obviously, the exponential relation cannot be extrapolated to e 0. 

It was found that the minimum specific energy dissipation for standard turbines and downward pumping propellers, in baffled vessels, is approximately the same. 

It would appear logical to replace the factor in relations like eq. (4.43a) by 

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