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Figure 14-6 Typical batch settling curve for a limestone slurry.

(compressed) zone is much higher because of the high concentration of solids in this zone. The minimum in this curve represents a "pinch" or "critical" condition in the thickener that limits the total solids flux that can be obtained under steady-state (stable) operation.

Because the batch flux data are obtained in a closed system with no outflow, the net solids flux is zero in the batch system and Eq. (14-40) reduces to VL = — 'Vs/(1 — '). Note that VL and Vs are of opposite sign, because the displaced liquid moves upward as the solids settle. The relative velocity between the solids and liquid is Vr = Vs — VL which, from Eq. (14-20), is Vr = Vs/(1 — '). It is this relative velocity that controls the dynamics in the thickener. If the underflow draw-off rate from the thickener is Qu, the additional solids flux in the thickener due to superimposition of this underflow is qu = Qu/A = Vu. Thus, the total solids flux at any point in the thickener (qs) is equal to the settling flux relative to the suspension (i.e., the batch flux qsb) at that point, plus the bulk flux due

Solids Volume Fraction (cp)

Figure 14-7 Typical batch flux curve with operating lines ( ) underloaded;

Solids Volume Fraction (cp)

Figure 14-7 Typical batch flux curve with operating lines ( ) underloaded;

to the underflow draw-off rate, 'Vu, i.e., qs = qsb + 'qu. Furthermore, at steady state the net local solids flux in the settling zone (qs) must be equal to that in the underflow, i.e., qs = qu'u. Eliminating quand rearranging leads to qsb = qs(l - fy (14-42)

This equation represents a straight line on the batch flux curve (qsbvs.') that passes through the points (qs, 0) and (0, 'u). The line intersects the ' axis at 'u and the qsb axis at qs, which is the net local solids flux in the thickener at the point where the solids fraction is '. This line is called the "operating line'' for the thickener, and its intersection with the batch flux curve determines the stable operating point for the thickener, as shown in Fig. 14-7. The "properly loaded'' operating line is tangent to the batch flux curve. At the tangent point, called the critical (or "pinch") point, the local solids flux corresponds to the steady state value at which the net critical (minimum) settling rate in the thickener equals the total underflow solids rate. The "underloaded" line represents a condition for which the underflow draw-off rate is higher than the critical settling rate, so no sludge layer can build up and excess clear liquid will eventually be drawn out the bottom (i.e. the draw-off rate is too high). The "overloaded" line represents the condition at which the underflow draw-off rate is lower than the critical settling rate, so the bottom solids layer will build up and eventually rise to the overflow (i.e., the underflow rate is too low).

Once the operating line is set, the equations that govern the thickener operation are determined from a solids mass balance as follows. At steady state (stable) operating conditions, the net solids flux is qs=Q=f=Qr (14-43)

This equation relates the thickener area (A) and the feed rate and loading (Qf; 'f) to the solids underfow rate (Qu) and the underflow loading ('u), assuming no solids in the overflow. The area of a thickener required for a specified underflow loading can be determined as follows. For a given underflow solids loading ('u), the operating line is drawn on the batch flux curve from 'u on the ' axis tangent to the batch flux curve at the critical point, (qc, 'c). The intersection of this line with the vertical axis (' = 0) gives the local solids flux (qs) in the thickener that results in stable or steady-state (properly loaded) conditions. This value is determined from the intersection of the operating line on the qsb axis or from the equation of the operating line that is tangent to the critical point (qc, 'c):

If the feed rate (Qf) and solids loading ('f) are specified, the thickener area A is determined from Eq. (14-43). If it is assumed that none of the solids are carried over with the overflow, the overflow rate Qo is given by

Likewise, the underflow rate Qu is given by

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