WHY PARTICLES SEPARATE I N SEDIMENTATION PROCESSES
BRYANT F I T C H
A the09 useful f o r determining operating conditions as well as understanding the nature of
fluid bed classijcation
made by sedimentation is necessarily A separation made on the basis of settling rate. This is not the same thing as particle size. Separation b) sedimentation falls into two categories-one relates particle size to settling rate, the other treats the mechanism b) which differences in settling rate are translated into physical separations. In sedimentation processes, a particle is assigned a size equal to the diameter of an equal-density sphere which would settle at the same rate. This is called sedimentation size. But this size can vary with Reynolds number range in which it is caused to settle. I n Stoke‘s range it can also vary with particle orientation, and in suspensions it may vary with particle concentration. Sedimentation size is used when particles are too small t o be screened. Such particles in gravity fields are restricted to Stoke’s range. Orientation is accounted for by using the averaged value of L. for random orientation, and it is found safe to neglect the effect of concentration. T h e screen size of any given particle can be changed into sedimentation size by means of a shape factor (Equation 1). T h e settling rate of single spherical particles through a large body of quiescent fluid is expressed by Equation 2 and, from dimensional considerations, by 44
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Equation 3. T h e first group of figures in Equation 3 is Reynold’s number. T h e second, when multiplied by a historically derived constant, 4/3> is the friction factor. Figure 1 shows a plot of the friction factor us. Reynold’s number. If the particles have some shape other than spherical, the plot lvould remain the same as long as sedimentation size is used for calculation. i\’hen screen size is used, the curves will be displaced with a shape factor as an added parameter. If the particles are chosen from a population of various shapes, the curve would also be blurred because of shape factor differences. Practical sedimencation processes almost always are concerned with particles in such concentration that they interact hydraulically. They may also cohere. T h e simplest case of interaction exists \\Then there is only one class of particles present, and one particle behaves like another. T h e only further variable which has been added is solids concentration (Equation 4) which by dimensional considerations yields Equation 3 . A correlation of settling data for suspensions containing only one class of particles would comprise a series of Reynold’s number us. friction factor plots for various values of C as parameters. Wilhelm and Kwauk (72), working with spherical and almost spherical particles, developed an empirical relationship for settling velocity (Equaticn 6 ) . T h e data of Steinour ( 7 7 ) on much finer particles, and that of Kermack (6) are well fitted to the equation with appropriate values for the exponent k .
Bryant Fitch, a Chemical Engineer, i s Research Director f o r the Dorr-Oliaer Go.
AUTHOR
reduces to Stoke's range to Equation 7a.
I
I
I
I
I
Effect of Interparticle Forces
Where such forces exist, particles at low dilution are close enough to cohere into a plastic structure (5, 7). Particles not exceeding the yield value are linked into the structure and arc not free to subside independently. T h e suspension thickens but does not classify. As dilution is increased, the yield value of the structure decreases, and at some point the largest particles can crash through it independently. With increasing dilution ever smaller classes of particles are released. Transition from consolidated to independent settling takes place over a relatively narrow dilution range. T h e concentration at which any given class of particles is released is referred to as its hydroseparation point. Just where this point is depends on the fineness and nature of the particles and sometimes occurs only a t amazingly high dilutions. But whatever the material, the pulp must be diluted at least beyond the point where the oversized particles are released to get classification. At dilutions above the hydroseparation point, particles no longer cohere into a n over-all structure. But they can still collide and stick together in individual floccules. If the particles are large, their settling velocity engenders erosive or shear forces which exceed those of cohesion. Most particles in the screen range of 200-mesh or coarser settle individually, if dilution is sufficient. Sedimentation techniques, particularly useful in the subsieve range, are applied commercially for separations down to about 2 microns. Dispersing agents must usually be added in such cases to destroy interparticle cohesion.
TO Find Settling Rates for Particles
D
= =
$Bs
f(D,d , p, Y,
g) (3)
ZI
=
(1-
C)k
(Continued on next page) VOL. 5 4
NO. 1 0
OCTOBER 1962
45
G R A V I T Y SETTLING POOLS
stant in practical cases. Therefore there will be little error in assuming that all members of a given size class settle at some fixed constant rate v . T h e quantity of any given size class originally present above the separation level is AAC. T h e rate a t which the class settles past the separation level is ACvt. T h e fraction of the class which settles out in time t is given by Equation 8. T h e quantity h:t has the dimensions of velocity and is commonly called the overflow rate. It is also equal to the settling rate of the separation size particles (Equation 9’). There are also particles in the slurry which occupied the underflow space originally. A part of this slurry will be displaced by settling solids. Some will remain to fill the spaces between and around the settled solids. This void filling carries into the underflow unclassified feed solids. I t is also responsible for all the fluid discharging with the underflow. T h e underflow from a batch pool classification carries oversize. criticals, and void filling. By calculating the contribution of each to the underflow, using suitable F values for the critical?, it is possible to predict the composition of the products which will be obtained at any separation size (10)
Particles to be classified are uniformly suspended in a fluid a n d are passed through a pool such that only a fraction of the solids have time to settle out. Particles which do not settle are overflowed. Those which do settle are collected and convey-ed out of the pool as underflow or rake product. T h e many different classifiers of this type differ only in the means used to convey the settled solids. T h e conveying mechanisms do no classifying themselves. All settling is done in the pool. A settling pool gives the same kind of result whether operated continuously or batchwise. Batch Settling
Consider a cylinder of suspension with particles uniformly distributed. Immediately after the start of sedimentation, particles start moving toward the bottom, each a t its characteristic rate. Particles reaching the bottom drop out of suspension. After some time t , the slurry is drawn off from above the deposited solids. T h e coarse fraction, containing deposited solids, entrained slurry-, and any unremoved supernatant, is equivalent to the underflow or rake product in a classifier. T h e slurry removed is the overflow. There is some certain size of particles which during sedimentation had just time to settle from the surface of the slurry to a depth zi where the underflow and overflow are parted. This is called the separation size; its settling velocity is h i t . All particles with a faster settling rate would have dropped below the separation level, and so will be absent from the overflow. Particles with a slower settling rate would not reach this level. A certain fraction F of any size class smaller than the separation size will also settle into the underflow cut. These particles are called criticals. Values of F for different size classes can be calculated on the basis of theory which would be strictly valid only for dilute suspensions. But results are approximately correct for the concentrations normally encountered. I n concentrated suspensions, particles of a given class would not continue to settle at identicall). the same rate throughout the suspension. Concentration changes develop due to progressive depletion of larger size classes from any given level. But the concentration at which most of the separation is made remains relatively con-
Influent
I
Contin uous Settling
In an ideal pool (Figure 2), the direction of flow is assumed to be horizontal, and the velocity is the same in all parts of the settling zone ( I ) . Assume that a vertical cylinder of feed suspension extending from top to bottom of the basin is marked at the influent end. I t would maintain its shape and identity as it moved across the basin. T h e results would be the same as though a batch settling operation were moved sideways. T h e only difference would be that the settled solids would be strewn along the bottom of the basin rather than piled up at the bottom of the element. Classification results would be identical in such an ideal basin and in an equivalent batch operation. T h e time it would take for the element to cross the basin and the fraction of any size removed can be calculated from Equations 10 and 11. T h e expression QIA corresponds to the overflow rate v,. It has the dimensions of velocity and equals in magnitude the rate at which the suspension would rise through the pool if it were fed at the bottom and removed at the top. This relation has often led to the misconception that particles are actually required to settle out against a
Part way through basin
[ 1
Effluent
I 1
I
I *I
Figuie 2. illonideal settling pools give the same tesults 46
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
as an ideal
pool shown here
IO
rising current in the pool. Such is not the case in the ideal pool shown. It is also not the case in any pool. A practical classifier is far from a n ideal pool. Agitation is produced by the solid-rake mechanism and by feed turbulence. Flow patterns are nothing like horizontal at all points-liquid must climb u p to go over the discharge weir. I t is also not apparent why some elements should not reach the overflow by a path which is shorter than that taken by others. Classification would seem to take place at different sizes in different elements. Curvature of flow paths will give an upward component of velocity at some points, a downward component at others. And the flow may curve laterally. I t is difficult to calculate trajectory in space of a settling particle, and it is not really necessary to do so. T h e problem can be avoided by transforming to a new coordinate system relating to flow lines or flow net rather than to spatial dimensions ( 4 ) . T h e vertical axis now measures the quantity of flow g which passes above any given point, regardless of how the flow lines may incline or vary. T h e horizontal axis represents the pool area A across which the flow has swept. A vertical section of the flow net bounded by flow lines is considered to extend from inlet to outlet (Figure 3). Its width dzv may vary from inlet to outlet, since the flow lines may be expanding or contracting laterally as in a circular basin. Its depth is considered not to vary from top to bottom of the tank. T h e settling factor F is developed in Equations 12 through 18. Equation 17 states that the flow g down through and out of which a particle can settle is a function only of the particle settling rate and of the projected or pool surface area of the volume swept through by the section of flow. Direction, magnitude, and variation with depth of the flow itself are disregarded. Equation 18 demonstrates that the removal of any size class of particles from a section of flow will remain the same, regardless of such departures from ideal basin conditions. Short-circuiting of flow due to changes in velocity with depth does not harm classification. But it is possible that flow per unit of surface area or overflow rate might differ from one section to another. There is a selfstabilizing effect at work which tends to minimize mixing and short-circuiting. Density Stabilization
I n a pool classifier, a pulp density gradient develops.
As solids settle out, pulp destined for the overflow becomes ever lower in specific gravity. A significant amount of energy would then have to be added to the system to lift the higher gravity layers and remix them with the supernatant lighter ones. If any disturbance short of an intimate mixing is introduced, gravity and density differentiation will tend to return the system subsequently to its predisturbance condition. In continuous operation, the incoming feed, having a pulp specific gravity substantially higher than that of the overflow pulp layer at the surface of the pool, plunges immediately below it and spreads out at its level of hydrostatic equilibrium just above the settled solids, or
/
Horizontal velocity Component v
Figure 3. Elenzents of a flow section showing plan, section, and velocity componenls
In Gravity Settling Pools
V
u
vo
vs
F = - = Ah t=Q
dq u
=
=
dx
dr
dy/dt
=
A=
Vdw
Vdt
Q
ua (areal eJciency) VOL. 5 4
NO. 1 0 O C T O B E R 1 9 6 2
47
above higher density layers if such exist. While it is spreading out horizontally, classification takes place. If there were any lack of uniformity in the classification, the overflow fraction from short-circuiting would be of higher per cent solids than that from others and would be higher in specific gravity. Since higher gravity pulps tend always to stratify below lower gravity ones, the short-circuiting elements would tend to be retained preferentially, increasing their effective sedimentation time and correcting the nonuniformity. T h e degree of agitation normally introduced by the solids-raking mechanism and feed turbulence does not seem enough to affect significantly the sharpness of classification. It is always possible to predict accurately from a batch test the classification xihich will be obtained in a Dorr classifier ( 70).
Control of Separation Sharpness
Overflow from a pool classifier will, at least ideally, contain no particles coarser than the mesh of separation. Underflow will contain both criticals and void filling. Although the criticals are responsible for the bulk of the slightly undersized material in the rake product, void filling accounts for nearly all the slimes. Little can be done about the criticals without changing the type of classification, but 17oid filling, and hence slimes, are under some functional control. If the slimes have negligible settling rate, the); behave pretty much as though they were in solution-i.e., they follow the water. Anything done to decrease the fraction of feed water taken into the underflow will similarly decrease the fraction of feed slimes reporting to the underflow. There are two ways to reduce this fraction; either decrease the water going into the underfloir, o r increase the amount of water in the feed. Most common classifiers discharge the settled solids up a deck to dewater them, thus minimizing water to the underflow. T h e quantity of water in the feed also can be increased, but if we dilute the feed we get a dilute overflow, which is frequently undesirable. O n the other hand, it must be recognized that with this type of classification, because of void filling, a heavier feed necessarily means a dirtier underflow. Separation cannot be truly sharp, because of criticals and void filling. This limitation is inherent in the type of classification. By repeated reclassification, the sephration can be sharpened indefinitely. Each reclassification eliminates a fraction of the residud criticals and dilutes the void filling. But this requires a lot of water, a disadvantage kvhich can be largely overcome through feedback of the classifier overflows from stage to stage in a countercurrent decantation sequence. Normally in such a countercurrent classification sequence, all classifiers are arranged to make the same mesh of separation. In this case there is a certain amount of recirculation of critical sizes within the system. While the elimination of slimes in such a system may be complete, the elimination of those criticals near to the mesh of separation is not as good as might be desired (70) 48
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
5
10 ’.5 Overflow dilution
m
25
Figure 4. A t very low dilutions, the material won’t settle at all. T h e hydroseparation point has not been reached. W i t h increasing dilution, separation becomes j h e r since the settling rate of the particles increases faster than the overflow rate. As free settling conditions are approached, the effect of additional dilution on settling rate diminishes, and the overjow rate takes ooer to coarsen the separation again. In between, the separation passes through a minimum. Most class8ers are oberated somerthere near this minimum in the interest of economy
and not nearly as good as it would be if separate stages of reclassification were used. Elimination of criticals from the ultimate sand product is greatly improved if the reclassification is done at a coarser mesh of separation than the primary classification. Control of Separation Size
T h e separation size particle has a settling rate us equal the overflow rate of the pool, which in a continuous classifier is Q!A. Separation sizes are accordingly controlled by suitably adjusting A , Q, or v,. I t is obvious that A can be adjusted by changing the classifier, or Q can be changed by changing its feed rate. Assuming a given classifier and a fixed rate of solids feed, however, the only way to change overflow rate is to change the dilution of the feed. A4sthe feed is diluted, the overflow rate is increased, and this would tend to coarsen the separation. At the same time, the greater dilution increases the settling rate of the particles, making the separation finer. These opposing tendencies interact to give a curve something like Figure 4. Overflow dilution, tons of solids per ton of water, is used for correlation because critical particles are settling in a zone much nearer to overflow dilution than to feed dilution. Another \vay to control the settling rate and separation of the particles is to use controlled agitation. .4 reasonable amount of agitation does not seem to affect sharpness of separation but does ha\Te the effect of reducing the areal efficiency of the pool. A4realefficiencies range from 100yo in the relatively quiescent pools of a hydroseparator down to about 307, for a raking classifier set to give the maximum permissible agitation. to
Design Criteria
Gravity pool classifiers are designed on the basis of pool area and underflow handling capacity. Theoretical overflow rate needed to produce the desired separation can be determined from appropriate batch tests. The actual pool area needed can be determined with the aid of Equation 1 9 .
FLUIDIZED BED CLASSIFICATION In a fluidized bed classifier a flow of wash fluid is maintained upward through a column of suspension. Solids are fed into the column, usually at the top, and underflow is removed at a controlled rate from the bottom. Particles not removed with the underflow are displaced to the overflow. Such devices have long been known in the ore dressing field as hindered-settling classifiers or sizers. They are capable of making far sharper separations than are obtained in a pool classifier. T h e mechanism of fluidized bed classification, and its relation to pool separation, can be demonstrated qualitatively by assuming that a height h, of mixed suspension is caused to float over a great depth of quiescent fluid (Figure 5). For simplicity, it will be assumed that the suspension contains two sizes of particles only. In the mixed suspension the coarse particles have a settling rate us, the fine particles v. T h e coarser particles are taken as the separation size. Then u / u , is equal to the settling factor F for the fines. T h e starting condition is indicated at A in Figure 5 . T h e upper boundary of the cloud of coarse particles settles in a region of mixed suspension, as do all fine particles below this boundary. The upper boundary of the coarse cloud, therefore, passes fines at a linear rate (us - u ) . At time t it will have passed a fraction E of the fines which have thus been separated from the mixture, and can be calculated by Equations 20 through 23. At values of L, greater than L, the clouds of fine and coarse particles would be separated by a layer of clear fluid. A partition or bottom across the column at L = h, would not alter the analysis if separation is made at a depth L sufficiently less h, so that solids collapsing against the bottom do not build up to separation level. The model is suitable for batch pool classification. The hypothetical experiment could be observed, taking the upper boundary of the coarse cloud as a reference level. In this case the coarse particles would appear as in teeter, the fines would be elutriated, and the clear fluid would have an appropriate upward approach
velocity v,. T h e experiment now appears as a model for batch fluidized bed classification. One further step could be taken. As fast as fines separated above the mixed cloud, they could be removed. Underflow could be withdrawn and the upper boundary of the mixed cloud maintained at a constant level by continuous feed. The operation is now continuous. Free settling or pool classification is limited to settling distances L which are less than h, because some of the original height has to be left to receive the settled solids. From Equation 22 no class of fines is completely eliminated from the underflow. Fluid bed classification should permit complete removal of any size class finer than the mesh of separation, providing enough teeter water is used, corresponding to a large distance L in Equation 22. T h e hypothetical experiment cannot, in fact, be carried out. T h e feed suspension of settleable solids is necessarily of higher specific gravity than the suspending fluid. I t would not float over a column of fluid. T o get an operation which is stable density-wise, the suspension has to be isolated from any subjacent pool of clear fluid by a suitable constriction plate or its equivalent. The coarse particles, restricted from freely settling away as a light phase (8) by the constriction plate, will collect into a dense-phase teeter bed of such concentration as to be in equilibrium with the approach velocity. From Equation 7, it can perhaps be inferred that this teeter bed of coarse particles must have a higher solids concentration and a higher specific gravity than the feed zone. If many size classes are present in a batch teeter test, they will stratify in teetered layers of decreasing particle size and decreasing concentration, one above the other. In a continuous operation, however, such progressive stratification cannot exist. Concentrations in Continuous Operation
An idealized continuous elutriation or sizing operation is represented in Figure 6. At some point in the column, feed is introduced suspended in an amount of water corresponding to a water flux (water flow/column area) of
Eisure 5. Hypothetical experiment illustrates the separation of two sizes nf particles in a suspension
Eigure 6. Elutriation or teeter umn showing Jux contributed various tt $aterJ o pows
601-
VQL. 5 4
by
NO.
10
OCTOBER
1962
49
W > . At the bottom of the column, underflow i? removed comprising a water flux W,. Also an amount of teeter water is added corresponding to W,. T h e net flux of water up this part of the column is Wt - W,. Rut since under ideal conditions all particles below the feed zone are moving toward the underflow, the net flux of water with respect to solids is greater than W, - W,,. T W O conditions are imposed upon the teeter bed below the feed point. The downward flux G of any class of solids is at all levels equal to that feeding into it from the feed zone. And the subsidence velocity U of each class of solids is related to that of the separation size through its settling factor F. Conditions placed on downward flux, subsidence velocity. and concentrations are given in Equations 24 through 28. Particles in the critical range, for which E is between zero and unity, would be relatively most concentrated in the teeter zone. Larger particles are relatively less concentrated because their subsidence velocity C is greater. Smaller particles, at least ideally, are absent because for them (1 - E ) = 0. Note that above the feed point, the subsidence rate U is negative, since particles in the overflow zone are risiug. Particles in the critical range are relatively concentrated. Finer particles are less concentrated because they have a lower settling or slip velocity u. Coarser panicles are absent because for them E = 0. A significant fact is that the solids flux of any size class either downward through the teeter zone or upward through the overflow transport zone is constant from top to bottom of the zone. T h e concentration of any size class in the zone is therefore essentially uniform. The teeter bed is expected and is observed to have an approximately uniform total solids concentration throughout. T h e same should be true for the overflow transport zone. limitations on Separation Sharpness
The ideal mechanism assumes that particles settle at a uniform velocity through and with respect to the teetering water. Observation shows that particles accually have an apparently random motion superimposed on their settling velocity. Also there are relatively large-scale currents which form from time to time. The random dancing of the particles can presumably be attributed to small-scale turbulence arising from the flow. The grosser movements must result from density current effects, or other hydraulic disturbances such as inter position of obstacles in the flow. These factors effect remixing which acts counter to the desired separation. That part of the particle migration caused by turbulence would be random, and each particle would qo through a random walk process. What happens is almost completely analogous to diffusion of gases and would be subject to the same mathematical descriptions. The diifusive flux GD and net flux to the underflow are developed in Equations 29 and 30. The diffusion flux of any class of fines into the underflow will be a function of its concentration gradient and of its separating tendency - U. Particles in or near the critical size range have high concentrations in the upper 50
INDUSTRIAL A N D ENGINEERING C H E M I S T R Y
100
-
L u
r‘
0 ._ c 2 60 ._ E ._
UJ
+
5 40 u L
a a,
20 -
Figure 7 . Plot shows typical elimination E of j r s t , second, and third critical size batids in teeter column as a functioti of relative teeter waier rate
part of the column and a large over-all negativ,: gradient from top to bottom of the teeter zone. They also have low- separating tendencies - U i n the teeter zone. Their net diffusion flux to the underflow will tend to be large. Finer particles are less concentrated above the teeter zone and have higher separating velocities. Diffusive flux should fall off rapidly as particle size becomes smaller. Consider now more systematic motions or convections which may take place. A most obvious cause of convective mixing would be a feed stream. If the velocity of the feed stream is not dispersed, it will set up circulation in the teeter bed. Since there is no density stratification within the teeter zone proper, there exists no stabilizing effect. Circulation currents are easily formed. They can sweep stray criticals and fines directly to the underflow, thereby augmenting the diffusive flux. Separations Actually Produced
The theory presented here is useful for understanding the nature of fluid bed classification and as an aid in providing optimum separating conditions. It is not so useful for predicting separations actually produced. I n the idealized case, the feed was considered to be a t the exact teeter concentration, and only two sizes were present. Actually the feed will contain a range of sizes, both within and outside the critical range, and will be at other than teeter dilution. Diffusivity or mixing also appears as an important and complicating effect. Consequently, actual design data comprise plots of average elimination E for successive bands of particle size below the mesh of separation us. W J G . The “first critical” The second critiextends from separation size s to ~/‘d?. cal extends from s i 4 2 to s.12. Figure 7 shows typical elimination plots for first, second, and third criticals. Conditions on Operation
Certain conditions must be fulfilled for proper operation of a sizer. The teeter bed has to be fluidized.
In Fluid Bed Classification
E= E=
v)t
(us -
O
