## The Sedimentation Process In Greater Detail

To examine sedimentation in greater detail, let us examine the events occurring in a small-scale experiment conducted batchwise, as illustrated in Figure 9. Particles in a narrow size range will settle with about the same velocity. When this occurs, a demarcation line is observed between the supernatant clear liquid (zone A) and

The plot shows the difference in interface height plotted against time, which is proportional to the rate of settling as well as to concentration. Examining these data in more detail as a plot of sediment height, Z, versus time, t, in Figure 11, we note that Z <* t, meaning that the sedimentation rate is, and continues to be, constant. Then the sedimentation rate of the heavy sludge decreases with time, which corresponds to the curve on the graph after point K.

Naturally, the higher the concentration of the initial suspension, the slower the sedimentation process. Observations show that the solids concentration in the dilute phase is constant up to the point of complete disappearance of phase A. This is illustrated by the plot in Figure 11, and corresponds to a constant rate of sedimentation in the phase.

Note, however, that the concentration in phase B changes with height Z and time t (as shown by Figure 11) and, hence, each curve in Figure 12 represents the distribution of concentrations at any Oven moment. The initial concentration is C1( which remains in the dilute phase during the process. After a sufficient period of time, the concentration increases to Q, but in zone D. Obviously, if the concentration of the feed suspension is too high, no dilute phase will exist, even during the initial period of sedimentation. Hence, there is no constant sedimentation rate. In this case, concentration, not height, will change with time only.

As follows from an earlier discussion on spheres falling through a fluid medium, sedimentation is faster in liquids having low viscosities. Hence, sedimentation rates are higher at elevated temperatures. In addition to temperature, an increase in the process rate may be realized by increasing particle sizes through the use of coagulation or agglomeration. In the case of colloidal suspensions, this is achieved by the addition of an electrolyte. Instead of using the concentration of the initial suspension to describe the process, we introduce a void fraction for the suspension. The void fraction is the ratio of the liquid volume, Vf, filling the space among the particles, to the total volume which is the sum of the liquid volume and the actual

For a unit volume of slurry, its weight, y. is the sum of the weights of the solid particles, YP(I - e), and of the liquid, Y(€, where y, = specific weight of liquid and Yp = specific weight of particles:

This expression can be used to compute the void fraction from experimentally determined values of the specific weights.

Let's now direct our attention to the sedimentation process in the zone of constant settling velocity, i.e., in the dilute phase. To simplify the analysis w assume spherical particles of the same size. The process may be simplified further by viewing sedimentation as fixed particles in an upward-moving stream of viscous liquid, whose average velocity is Due to the viscosity of the liquid, a certain velocity gradient exists relative to the distance from the surface of spherical particle, du/dx. This velocity also depends on the average distance among particles, which is determined at any moment by a void fraction, e, of the slurry and the particle diameter, d. The average velocity of the liquid may be presented in this case as uf = f (d, e, du/dx). And rewritten on the basis of dimensional analysis in the following form:

where K, = constant

O/e) = dimensionless function of the void fraction

The resistance to liquid flow around particles may be presented by an equation similar to the viscosity equation but with considering the void fraction. Recall that the shear stress is expressed by the ratio of the drag force, R, to the active surface, K27id2. The total sphere surface is ltd2 and K2 is the coefficient accounting for that part of the surface responsible for resistance. Considering the influence of void fraction as a function <J>2(e), we obtain:

Dividing equation 19 by Equation 18, we obtain:

For very dilute suspensions, in which the void fraction does not influence the sedimentation process, the function <3>2(e)/Oj(e) = $(e) reduces to unity. It is known also that in dilute suspensions the sedimentation of small particles follows Stokes' law:

Equating equations 20 and 21, we find that (K^/Kj) is equal to 3. Hence, the resistance of the liquid relative to a spherical particle in the sedimentation process

This resistance is balanced by the gravity force acting on a particle:

where yp is the actual specific weight of a particle, and y is the average specific weight of the sludge, which depends on void fraction e. Using equation 17, we replace (yp - y) with (yp - yf)e, where yf is the specific weight of the liquid:

By comparing the gravity force acting on the particle (equation 24) with the resistance to liquid flow (equation 22), we obtain the average liquid velocity relative to the particles:

In practice, however, the liquid velocity relative to fixed particles, uf, is not very useful. Instead, the velocity of settling relative to the walls of an apparatus, uf - u, is of practical importance. The volume of the solid phase moving downward should be equal to that of liquid moving upward. This means that volume rates of these phases must be equal. Consider a column of slurry having a unit cross section and imagine the liquid and solid phases to have a well defined interface. The column of solid phase will have a base 1 - e, and the liquid column phase will have a base e. Hence, the volumetric rate of the solid column will be (1 - e)u, and that of the liquid column will be (uf - u)e. Because these flowrates are equal to each other, we obtain

Therefore, the settling velocity of the solid phase relative to the wall of an apparatus, depending on the average liquid velocity relative to the sludge with void fraction e, will be u = uf e (27)

Substituting this expression into equation 25, we obtain the actual settling velocity:

Note that the term in parentheses expresses the velocity of free failing, according to Stokes' law:

For a very dilute suspension, i.e., e = 1 and <J>(e) = 1, the settling velocity will be equal to the free-fall velocity. As no valid theoretical expression for the function <D(e) is available, common practice is to rely on experimental data. Note that a unit volume of thickened sludge contains e volume of liquid and (1 - e) volume of solid phase, i.e., a unit volume of particles of sludge contains e/(l - e) volume of liquid. Denoting o as the ratio of particle surface area to volume, we obtain the hydraulic radius as the ratio of this volume, e/(l - e), to the surface, a, when both values are related to the same volume of particles:

For spherical particles, o is equal to the ratio of the surface area, ltd2 , to the volume itd3/6, i.e., o = 6/d. Hence, rh = e/[(l - e)6] (32)

For a specified void fraction, the diameter of the sphere is a measure of the distance between sludge particles (Uf = f (d, e, du/dx)). However, it is more practical to introduce the hydraulic radius, and instead of 3>,(e) and <J>2(€)> according to equation 30, we assume the following value:

where 0(e) is the new experimental function of the void fraction. Hence, the settling velocity equation may be rewritten in the following form:

By representing the velocity in this manner, we can anticipate a small change in the function 6(e) because the influence of the flow pattern is, to a large extent, accounted for in the hydraulic radius. Laboratory and pilot scale testing have shown that the function 3>(e) may be presented by the following empirical equation:

Multiplying equation 35 by (1 - e)/e, we obtain the function 0(e). For e s 0.7, i.e., for thickened sludges, this function is practically constant and equal to 0(e) = 0.123. The settling velocity of spherical particles is therefore:

For more thickened sludges:

Thus far, the analysis has been based on independently settling spherical particles. To relate to the design of the unit operations, we now must consider the kinetics of nonspherical particle settling and the sedimentation of flocculent particles. In contrast to single-particle settling, such systems form a certain structural unity similar to tissue. The sludge is compacted under the action of gravity force, i.e., the void fraction decreases and the liquid is squeezed out from the pore structure. The formation of a regular sediment from a flocculent may be achieved by the addition of electrolytes, as described earlier. The general characteristic of normal settling of nonspherical particles (as well as flocculent ones) is that the sediment carries along with it a portion of the liquid by trapping it between particle cavities. This trapped volume of liquid flows downward with the sludge and is proportional to the volume of the sludge. That is, it can be expressed as a(l - e), where a is a coefficient and (1 - e) is the volume of particles. Consequently, a portion of the liquid remains in a layer above the sludge, and a portion is carried along with the sludge corresponding to the modified void fraction:

This is the difference of the total relative liquid volume and liquid moving together with particles. Substituting e' for e in equation 37, we obtain the settling velocity at e < 0.7:

Denoting a/(l + a) = p, the above expression can be written as follows:

Similarly equation 36 for slurries with nonspherical particles is u = UpKe - P)2/(l - p)]io-i-n<i-«wi-» (41)

Parameter P is equal to the ratio of the liquid volume entrained and the sum of the volumes of this liquid and particles. Values of P are determined experimentally from measured settling velocities. In general, the smaller the effective particle size, the more liquid is entrained by the same mass of solids phase. For example, particles of carborundum with d = 12.2 ¡im have P = 0.268; d = 9.6 ¿mi, p = 0.288; and d = 4.6 ¡j.m, P = 0.35.

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