Multiple Particle Systems
3.3 CONTINUOUS SETTLING
Ct ð3:37Þ
hence, substituting forUp from Equation (3.33), we have C ¼CBh0
h1 ð3:38Þ
3.3 CONTINUOUS SETTLING
3.3.1 Settling of a Suspension in a Flowing Fluid
We will now look at the effects of imposing a net fluid flow on to the particle settling process with a view to eventually producing a design procedure for a thickener. This analysis follows the method suggested by Fryer and Uhlherr (1980).
Firstly, we will consider a settling suspension flowing downwards in a vessel.
A suspension of solids concentrationð1 eFÞ or CF is fed continuously into the top of a vessel of cross-sectional areaA at a volume flow rate Q (Figure 3.11): The suspension is drawn off from the base of the vessel at the same rate. At a given axial position,X, in the vessel let the local solids concentration be ð1 eÞ or C and the volumetric fluxes of the solids and the fluid beUpsandUfs, respectively. Then assuming incompressible fluid and solids, continuity gives
Q ¼ ðUpsþ UfsÞA ð3:39Þ
At positionX, the relative velocity between fluid and particles, Urel is given by Urel¼ Ups
1 eUfs
e ð3:40Þ
Our analysis of batch settling gave us the following expression for this relative velocity:
Urel ¼ UTefðeÞ ð3:7Þ
Q
Ups Ufs
X (1-εF)
1-ε
Figure 3.11 Continuous settling; downflow only
and so, combining Equations (3.39), (3.40) and (3.7) we have Ups¼Qð1 eÞ
A þ UTe2ð1 eÞfðeÞ ð3:41Þ or
total solids flux¼ flux due to bulk flow þ flux due to settling
We can use this expression to convert our batch flux plot into a continuous total downward flux plot. Referring to Figure 3.12, we plot a line of slopeQ/A through the origin to represent the bulk flow flux and then add this to the batch flux plot to give the continuous total downward flux plot. Now, in order to graphically determine the solids concentration at level X in the vessel we apply the mass balance between feed and the point X. Reading up from the feed concentrationCF
to the bulk flow line gives the value of the volumetric particle flux fed to the vessel,QCF=A. By continuity this must also be the total flux at level X or any level in the vessel. Hence, reading across from the flux of QCF=A to the continuous total flux curve, we may read off the particle concentration in the vessel during downward flow, which we will callCB. (The subscript B will eventually refer to the ‘bottom’ section of the continuous thickener.) In downward flow the value of CBwill always be lower than the feed concentrationCF, since the solids velocity is greater in downward flow than in the feedðconcentration velocity ¼ fluxÞ.
A similar analysis applied to upward flow of a particle suspension in a vessel gives total downward particle flux,
Ups¼ UTe2fðeÞ Qð1 eÞ
A ð3:42Þ
Continuous total downward flux Ups
Ups = QC F/A
CB
Feed
concentration, CF Batch setting flux
C Slope Q/A
Bulk flow
Figure 3.12 Total flux plot for settling in downward flow
62 MULTIPLE PARTICLE SYSTEMS
or
total solids flux¼ flux due to settling flux due to bulk flow
Hence, for upward flow, we obtain the continuous total flux plot by subtracting the straight line representing the flux due to bulk flow from the batch flux curve (Figure 3.13). Applying the material balance as we did for downward flow, we are able to graphically determine the particle concentration in the vessel during upward flow of fluid, CT. (The subscript T refers to the ‘top’ section of the continuous thickener.) It will be seen from Figure 3.13 that the value of particle concentration for the upward-flowing suspension,CT, is always greater than the feed concentration,CF. This is because the particle velocity during upward flow is always less than that in the feed.
3.3.2 A Real Thickener (with Upflow and Downflow Sections)
Consider now a real thickener shown schematically in Figure 3.14. The feed suspension of concentrationCFis fed into the vessel at some point intermediate between the top and bottom of the vessel at a volume flow rate,F. An ‘under-flow’ is drawn off at the base of the vessel at a volume flow rate, L, and concentration CL. A suspension of concentration CV overflows at a volume flow rate V at the top of the vessel (this flow is called the ‘overflow’). Let the mean particle concentrations in the bottom (downflow) and top (upflow) sections be CB and CT, respectively. The total and particle material balances over the thickener are:
Total:
F ¼ V þ L ð3:43Þ
C CT
CF
Bulk flow slope Q/A
Total
continuous total downward flux Batch setting
Ups flux
Ups = QCF/A
0
Figure 3.13 Total flux plot for settling in upward flow
Particle:
FCF¼ VCVþ LCL ð3:44Þ
These material balances link the total continuous flux plots for the upflow and downflow sections in the thickener.
3.3.3 Critically Loaded Thickener
Figure 3.15 shows flux plots for a ‘critically loaded’ thickener. The line of slopeF/A represents the relationship between feed concentration and feed flux for a volumetric feed rate, F. The material balance equations [Equations (3.43) and (3.44)] determine that this line intersects the curve for the total flux in the down-flow section when the total flux in the updown-flow section is zero. Under critical loading conditions the feed concentration is just equal to the critical value giving rise to a feed flux equal to the total continuous flux that the downflow section can deliver at that concentration. Thus the combined effect of bulk flow and settling in the downflow section provides a flux equal to that of the feed. Under these conditions, since all particles fed to the thickener can be dealt with by the downflow section, the upflow flux is zero. The material balance then dictates that the concentration in the downflow section,CB, is equal toCFand the underflow concentration,CLis FCF=L. The material balance may be performed graphically and is shown in Figure 3.15. From the feed flux line, the feed flux at a feed concentration,CFis Ups¼ FCF=A. At this flux the concentration in the downflow section is CB ¼ CF. The downflow flux is exactly equal to the feed flux and so the flux in the upflow section is zero. In the underflow, where there is no sedimentation, the underflow flux,LCL=A, is equal to the downflow flux. At this flux the underflow concentra-tion,CLis determined from the underflow line.
Figure 3.15 indicates that under critical conditions there are two possible solutions for the concentration in the upflow section,CT. One solution, the obvious one, is CT¼ 0; the other is CT¼ CB. In this second situation a fluidized bed of particles at concentrationCBwith a distinct surface is observed in the upflow section.
Feed F, CF
CT CB
Overflow V, CV
Underflow L, CL
Figure 3.14 A real thickener, combining upflow and downflow (F, L and V are volume flows;CF,CLandCVare concentrations)
64 MULTIPLE PARTICLE SYSTEMS
3.3.4 Underloaded Thickener
When the feed concentration CF is less than the critical concentration the thickener is said to be underloaded. This situation is depicted in Figure 3.16.
Here the feed flux,FCF=A, is less than the maximum flux due to bulk flow and settling which can be provided by the downflow section. The flux in the upflow section is again zero ðCT ¼ CV¼ 0; VCV=A ¼ 0Þ. The graphical mass balance shown in Figure 3.16 enables CB and CL to be determined (feed flux¼ downflow section flux ¼ underflow flux).
3.3.5 Overloaded Thickener
When the feed concentration CF is greater than the critical concentration, the thickener is said to be overloaded. This situation is depicted in Figure 3.17. Here
Below feed
Figure 3.15 Total flux plot for a thickener at critical loading
the feed flux, FCF=A, is greater than the maximum flux due to bulk flow and settling provided by the downflow section. The excess flux must pass through the upflow section and out through the overflow. The graphical material balance is depicted in Figure 3.17. At the feed concentrationCF, the difference between the feed flux and the total flux in the downflow section gives the excess flux which must pass through the upflow section. This flux applied to the upflow section graph gives the value of the concentration in the upflow section, CT, and the overflow concentration,CV(upflow section flux¼ overflow flux).
3.3.6 Alternative Form of Total Flux Plot
A common form of continuous flux plot is that exhibiting a minimum total flux shown under critical conditions in Figure 3.18. With this alternative flux plot the critical loading condition occurs when the feed concentration gives rise to a flux