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Figure 14-4 Generalized correlation of settling and fluidizing velocities. (From Barnea and Mizrahi, 1973.)

or compressed region containing the settled particles. The particle settling rate in this zone is very slow.

In the top (clarifying) zone the relatively clear liquid moves upward and overflows the top. In the middle zone the solid particles settle as the displaced liquid moves upward, and both the local solids concentration and the settling velocity vary from point to point. In the bottom (compressed) zone, the solids and liquid both move downward at a rate that is determined mainly by the underflow draw-off rate. For a given feed rate and solids

Figure 14-5 Schematic of a thickener.

loading, the objective is to determine the area of the thickener and the optimum underflow (draw-off) rate to achieve a specified underflow concentration ('u), or the underflow rate and underflow concentration, for stable steady state operation.

The solids concentration can be expressed in terms of either the solids volume fraction (') or the mass ratio of solids to fluid (R). If 'f is the volume fraction of solids in the feed stream (flow rate Qf) and 'u is the volume fraction of solids in the underflow (flow rate Qu), then the solids ratio in the feed, Rf = [(mass of solids)/(mass of fluid)]feed, and in the underflow, Ru = [(mass of solids)/(mass of liquid)]u, are given by

These relations can be rearranged to give the solids volume fractions in terms of the solids ratio:

Now the total (net) flux of the solids plus liquid moving through the thickener at any point is given by q = Q = q + qL = 'Vs + (1 - ')vl (14-40)

where qs = 'Vs is the local solids flux, defined as the volumetric settling rate of the solids per unit cross-sectional area of the settler, and qL = (1 — ') VL is the local liquid flux.

The solids flux depends on the local concentration of solids, the settling velocity of the solids at this concentration relative to the liquid, and the net velocity of the liquid. Thus the local solids flux will vary within the thickener because the concentration of solids increases with depth and the amount of liquid that is displaced (upward) by the solids decreases as the solids concentration increases, thus affecting the "upward drag'' on the particles. As these two effects act in opposite directions, there will be some point in the thickener at which the actual solids flux is a minimum. This point determines the conditions for stable steady-state operation, as explained below.

The settling behavior of a slurry is normally determined by measuring the velocity of the interface between the top (clear) and middle suspension zones in a batch settling test using a closed system (e.g., a graduated cylinder) as illustrated in Fig. 14-3. A typical batch settling curve is shown in Fig. 14-6 (see, e.g., Foust et al., 1980). The initial linear portion of this curve usually corresponds to free (unhindered) settling, and the slope of this region is the free settling velocity, V0. The nonlinear region of the curve corresponds to hindered settling in which the solids flux in this region depends upon the local solids concentration. This can be determined from the batch settling curve as follows (Kynch, 1952). If the initial height of the suspension with a solids fraction of 'o is Zo, at some later time the height of the interface between the clear layer and the hindered settling zone will be Z(t), where the average solids fraction in this zone is '(t). Since the total amount of solids in the system is constant, assuming the amount of solids in the clear layer to be negligible, it follows that

Thus, given the initial height and concentration (Zo, 'o), the average solids concentration '(t) corresponding to any point on the curve Z(t) can be determined. Furthermore, the hindered settling velocity and batch solids flux at this point can be determined from the slope of the curve at that point, i.e., Vsb = —(dZ/dt) and qsb = 'Vsb. Thus, the batch settling curve can be converted to a batch flux curve, as shown in Fig. 14-7. The batch flux curve exhibits a maximum and a minimum, because the settling velocity is nearly constant in the free settling region (and the flux is directly proportional to the solids concentration), whereas the settling velocity and the flux drop rapidly with increasing solids concentration in the hindered settling region as explained above. However, the solids flux in the bottom

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