Sedimentation Reynolds number

The sedimentation of particles under inclined sur ces may be characterised by two dimensionless numbers Gr, a sedimentation Grashof number which represents the ratio of gravitational forces to viscous forces in a convective flow and R a sedimentation Reynolds number representing the signihcance of inertial forces to viscous forces in a convective flow [Acrivos and Erbholzeimer, 1979, 1981]... 

As discussed in Chapter 3, two dimensionless numbers are used to characterise the settling behaviour Gr, a sedimentation Grashof number and R, a sedimentation Reynolds number defined by the equations ... 

Ratio of Grashof number to sedimentation Reynolds number ... 

The particle size deterrnined by sedimentation techniques is an equivalent spherical diameter, also known as the equivalent settling diameter, defined as the diameter of a sphere of the same density as the irregularly shaped particle that exhibits an identical free-fall velocity. Thus it is an appropriate diameter upon which to base particle behavior in other fluid-flow situations. Variations in the particle size distribution can occur for nonspherical particles (43,44). The upper size limit for sedimentation methods is estabHshed by the value of the particle Reynolds number, given by equation 11 ... 

This expression represents the first form of the general dimensionless equation of sedimentation theory. As the desired value is the velocity of the particle, equation 64 is solved for the Reynolds number ... 

This is the second form of the dimensionless equation for sedimentation. The Reynolds number also may be calculated from this equation ... 

The first critical values of the dimensionless sedimentation numbers, S, and Sj, are obtained by substituting for the critical Reynolds number value, Re = 0.2, into the above expressions ... 

The turbulent regime for Cq is characterized by the section of line almost parallel to the x-axis (at the Re" > 500). In this case, the exponent a is equal to zero. Consequently, viscosity vanishes from equation 46. This indicates that the friction forces are negligible in comparison to inertia forces. Recall that the resistance coefficient is nearly constant at a value of 0.44. Substituting for the critical Reynolds number, Re > 500, into equations 65 and 68, the second critical values of the sedimentation numbers are obtained ... 

Those particles with sizes d > d" at a given set of conditions (v, p, Pp, and a ) will settle only in the turbulent flow regime. For particles with sizes d < d, d" will settle only when the flow around the object is in the transitional regime. Recall that the transitional zone occurs in the Reynolds number range of 0.2 to 500. The sedimentation numbers corresponding to this zone are 3.6 < S, < 82,500 and 0.0022 < S2 < 1,515. 

Da (Dl IDA)m is the Stokes diameter, equal to the diameter of sphere, which in a laminar region (low Reynolds number Re < 0.2), sediments with the same velocity as the considered particle. 

Richardson and Mijkle 63 1 have shown that, at high concentrations, the results of sedimentation and fluidisation experiments can be represented in a manner similar to that used by Carman" 4 1 for fixed beds, discussed in Chapter 4. Using the interstitial velocity and a linear dimension given by the reciprocal of the surface of particles per unit volume of fluid, the Reynolds number is defined as ... 

Because equation 4.18 is applicable to low Reynolds numbers at which the flow is streamline, it appears that the flow of fluid at high concentrations of particles in a sedimenting or fluidised system is also streamline. The resistance to flow in the latter case appears to be about 30 per cent lower, presumably because the particles are free to move relative to one another. 

Thus, for any voidage the drag force on a sedimenting particle can be calculated, and the corresponding velocity required to produce this force on a particle at the same voidage in the model is obtained from the experimental results. All the experiments were carried out at particle Reynolds number greater than 500, and under these conditions the observed sedimentation velocity is given by equations 5.76 and 5.84 as ... 

Finally, sedimentation at Reynolds numbers greater than one is the realm of empirical equations. Several of these have been developed, mostly as fits to experimental data sets. Some of these are discussed and illustrated by Nguyen and Schulze [53],... 

In order to achieve simultaneous suspension of solid particles and dispersion of gas, it is necessary to define the state when the gas phase is well dispersed. Nienow (1975) defined this to be coincident with the minimum in Power number, Ne, against the aeration number, 1VA, relationship (see Fig. 12 [Sicardi et al., 1981]). While Chapman et al. (1981) accept this definition, their study also showed that there is some critical particle density (relative to the liquid density) above which particle suspension governs the power necessary to achieve a well-mixed system and below which gas dispersion governs the power requirements. Thus, aeration at the critical stirrer speed for complete suspension of solid particles in nonaerated systems causes partial sedimentation of relatively heavy particles and aids suspension of relatively light particles. Furthermore, there may be a similar (but weaker) effect with particle size. Wiedmann et al. (1980), on the other hand, define the complete state of suspension to be the one where the maximum in the Ne-Ren diagram occurs for a constant gas Reynolds number. 

We can use the same record of the trajectory of a particle to determine at the same time its hydrodynamic radius and its density. For sedimentation at low Reynolds number, the average vertical velocity is given by the balance between the body force and the friction force ... 

The influence of the second part of the right-hand-side of the equation increases with floe diameter. However the Reynolds number during sedimentation increases and thus n decreases. Calculations for sedimentating hydroxide... 

Using the linear relation between drag coefficient and Reynolds number, a pure laminar flow pattern of the sedimentating particles is implied. For spherical particles, flow is considered as laminar for Reynolds number smaller than 0.1. For Reynolds numbers up to 1, the linear relation with drag coefficient is generally accepted. Several investigators accepted the linear relation as far... 

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