BATCH ,AND CONTINUOUS THICKENING Basic Theory.
Solids Flux f o r Rigid Spheres PAUL T. SHANNON1 AND ELWOOD STROUPE
Purdue University, Lafayctte, Ind. ELMER M. T O R Y
McMaster University, Hamilton, Ontario, Canada Batch and continuous thickening are regarded as the process of propagating concentration changes upward from the bottom of the settling vessel owing to the downward movement of the solids. A proper choice o f coordinates in defining the solids flux clarifies the relation between batch and continuous thickening and allows the same basic analysis to b e applied to both. The equations for the movement of planes of constant concentration are derived from continuity considerations. Experimental results for the batch settling of rigid spheres, in water are presented. The plot of solids flux vs. concentration was doubly concave. ~ T C Hsedimentation
and continuous thickening may be
B‘thought of as the process of propagating concentration
changes, due to the downivard movement of the solids, from the bottom of the settling vessel up\vard through the slurry. These concentration changes may be either finite or infinitesimal as in the case of the propagation of a continuous concentration gradient. The slurry itself may be moving do\vnward as in the case of continuous thickening. The doivnward velocity of the slurry may be such that the concentration changes being propagated up from below remain stationary relative to the thickener walls. This is the case for steady-state continuous thickening. It seems appropriate to analyze batch and continuous ;:hickening in terms of the movement of planes of constant concentration within the vessel, as was first done by Kynch (9)in his analysis of batch sedimentation. Many theoretical and semitheoretical equations have been proposed \vhich relate the initial dimensionless settling rate of a to the volumetric concentration of solids. The simplest system \vhich can be envisaged is that for which the dimensionless settling rate at any point in a slurry is a function only of the local solids concentration, and Kynch’s theory of batch sedimentation is based on this postulate. To date. this theory appears to have been judged primarily on the basis of its usefulness in the design of continuous thickeners (4.7 6 ) . It seems that it might better be considered analogous to the ideal gas law in thermodynamics Lvhich. though often not sufficiently accurate for design purposes, p.rovides a standard against which real gases can be compared. ‘The behavior of actual systems can then be attributed to r,itional causes. -4 system which fulfills K y c h ’ s basic postulate will be termed a n “ideal slurry.” The assumption that the local settling velocity of solids relative to the slurry is a function only of the local solids concentration is essentially equivalent to the assumption that the only forces acting on the particles are caused by the local interstitial fluid velocity. The forces caused by fluid and solids acceleration are neglected. Thus, solids in a plane of constant concentration within a concentration gradient, wherein the solids entering and leaving the :plane are being decelerated and at the same time the fluid is being accelerated upward: are assumed to move with exactly the same velocity relative to rhe slurry as they would in a region of uniform concentration. This will be true only for systems Ivhich are highly damped and in which the solids concentration varies slowly. 1 Presrnt address, Tha.yer School of Engineering. Dartmouth College, Hanover, N. H. 2 Present address, U. S. .4rmy, USAECR, Fort Belvoir, Va.
From derived equations for the velocity of planes of constant concentration. the present series of papers develops and applies a more general treatment of settling encompassing batch settling lvith constant or variable area and steady- and unsteady-state thickening and relates the batch and continuous cases. Experimental results for rigid spheres and calcium carbonate are presented and interpreted in terms of the theory. This approach offers a consistent interpretation of the experimental results and provides a logical explanation of many previously unexplained phenomena encountered in batch and continuous thickening. Theory
The theoretical treatment given below is not confined to ‘.ideal slurries” but is limited by the folloiving assumptions : The particles are all of the same size, shape: and density. \Val1 effects are negligible. No radial variation occurs in the solids concentration. No radial variation occurs in the downward component 01 slurry velocity. The settling velocity of the solids relative to the slurry is independent of the velocity of the slurry itself. I n continuous thickening, the downward solids flux relative to the thickener, G: results from two different phenomena : settling of solids through the slurry and movement of the slurry itself. Thus G = c(u
+
(1)
a)
\\.here c = local concentration of solids in the slurry (weight of solids per unit volume of slurry) u = doivnward velocity of the solids relative to the slurry u = downward velocity of the slurry relative to the thickener u u = downward velocity of the solids relative to the thickener
+
All velocities are defined as positive doivnward and height is defined as positive upward. The coordinate system and velocities are shown schematically in Figure 1. The slurry velocity, u, is defined by the equation
L ,G = L G + Q s = - . PL
Pa
‘4
(2)
Lvhere p = density, subscripts 1 and s refer to liquid and solid, respectively, Q = volumetric underflow rate, and A = crosssectional area. By definition, u = 0 for batch thickening. For this case G(v = 0) = S = cu VOL. 2
NO. 3
AUGUST
(3) 1963
203
The continuity equation for the solids in a thickener in which the concentration and velocities varv continuously with time t and height z and the cross-sectional area. A . may vary continuously \\ith z is (4)
equations c = c ( t , z ) and u = u ( t . z ) . implicit in the continuity equation. gives u = u ( c . 1 ) . Hence G = G(6,v.t) and
But
.4t constant c.4. constant \\.eight of solids per unit height. z and t , are related by
and, from Equation 1 (5)
.4 comparison of Equations 4 and 5 shows that so that
where @ is defined by the first equality and is seen to be the velocity with bvhich a plane of constant weight of solids per unit height moves relative to the thickener. If the crosssectional area is constant. then from Equation 6. fi is the velocity of a plane of constant concentration. I n any zone of uniform concentration. the continuity equation cannot be ivritten. Hojvever, the basic approach is the analysis of the propagation of concentration changes upward through the slurry which may be of uniform concentration above the zone of concentration changes. Where there are finite changes of concentration with height. Equation 5 no longer applies. .4s shown by Kynch ( 9 ) :it must be replaced by a n equation stating that the flow of particles into one side of the discontinuity equals the flow out the other side. .4ssume that at time t and height z there exists a finite discontinuity in concentration-i.e., c is a step function of 2. -4ssume further that the variation in A is continuous. If suffix (I denotes the layer above the discontinuity and suffix b the layer below. and 6 is the downward velocity of the discontinuity. then a mass balance on the solids gives C.(Uo
+
0
-
6) =
Cb(Ub
+
L'
- 6)
(7)
and
- Ga 6 = Gb __Cb
-
Co
(8)
LVithin the limitations of the assumptions. Equations 6 and 8 apply to the propagation of concentration changes in any slurry. For 21 = 0. these equations apply to batch settling. If crosssectional area is also constant: Equation 6 reduces to
NOM'8 is simply a n operating variable and u can be determined from batch tests. To apply a value of u determined from initial batch rates directly to the design of a continuous thickener assumes that u = u ( G ) . Even when this is not true. batch tests can be used. provided that the functional dependence of u can be completely defined. If assumption 5 is cormust be possible. at rect-i.e.. u and u are independent-it least in principle. to use batch tests to predict results in continuous thickening. For steady-state continuous thickening, fi = 0 and 6 = 0. Equations 1, 6, and 8. together with a functional relationship for u, will be used to treat this important industrial process in a later paper. T h e design of thickeners on this basis depends upon the validity of the five assumptions and the accuracv with Xvhich u is characterized. I n general. /3 = P ( z , t ) and 6 = 6 ( z , t ) . This is true even for batch settling with constant area. T h e conditions for \vhich fi is constant can be found as follows. Elimination of z from the 204
ILEC FUNDAMENTALS
For the special case ivhere u is a function only of c.
and if also the cross-sectional area is constant
.4ccording to Equation 15. a plane of constant concentration moves relative to the thickener \vith a constant velocity \vhen u = u ( c ) and u and A are constant. Similarly, if u = U ( C ) and cross-sectional area and underfloi\. rate are constant? 6 is constant. provided that c, is maintained. In both cases, if the underflow rate is not zero. the specification of constant L) requires constant -4. Obviously the problem is greatly simplified if u = u ( c ) . ,4s pointed out by Kynch, if L' = 0 and -4 is constant. 6 and 6 are time-independent quantities ivhose values can be calculated from Equations 9 and 8. Initial batch settling rates can be used to establish the flux plot \vhich sholvs the batch flux. S. as a function of local solids concentration-Le., S = S(c). The experimental part of this paper is devoted to finding the fluxconcentration relationship for rigid spheres. T h e prediction of the complete family of settling curves from initial rate data constitutes a test of Kynch's basic hypothesis. and ivill be given in subsequent articles. Determination of Flux Plot
The rate of fall of the fluid-slurry interface is the settling rate of the solids immediately below the interface. If the slurry is thoroughly mixed to a uniform concentration, the initial interface concentration is known. Otherivise the interface concentration is unknown and unequal to the calculated average concentration. Because of the nature of the system, if mixing is not uniform, the upper portion of the slurry is a t a concentration less than that calculated and the settling rate is greater than that expected. Since settling rates are very strongly dependent on concentration, a small variation in interface concentration effects a large change in the observed settling rate. Inadequate mixing also produces initial concentration gradients which have pronounced effects on the fall of the fluid-slurry interface. Attaining a n initially uniform concentration becomes increasingly difficult as initial solids concentration decreases. I n order to reduce the effect of variables other than concentration, one should determine the flux plot by taking a given weight of solids and effecting concentration changes by adding and subtracting fluid. It is good experimental practice to randomize the initial concentrations. T h e settling rate during the initial constant-rate period is plotted against the initial
uniform concentration, which is the concentration at the fluids1urr)- interface during the constant-rate period. If settling rates are reproducible, the data \vi11 form a smooth curve for the given slurry. The corresponding flux plot is then readily obtained by multiplying the concentration by the corresponding settling rate. .4\vide range of initial concentrations should be studied in order to determine accurately the complete flux plot. This method enables one to evaluate systematically the effect of variables other than concentration, by examining the flux curves obtained for a series of slurries. Kynch (9)has pointed out that it is theoretically possible to deduce a portion of the velocity-concentration relationship from the analysis of a single batch settling curve for a slurry conforming exactly to the theoretical assumptions. Such a method requires extremely accurate data. Secondary effects caused by concentration gradients. nonuniform initial concentration. particle segregation effects. etc.: which may represent relatively small deviations from the behavior predicted from concentration considerations alone, may cause a large error in the velocity-concentration relationship deduced by extrapolation procedures applied to one batch settling curve. Nevertheless. it is possible to gain a great deal of information from the analysis of sett'ling curves. If a settling curve is to be interpreted meaningfully. readings must be taken frequently until settling is complete or negligible. Only if the entire batch settling curve is accurately kno\vn can the results be used to test predictions based on the flux plot determined from the initial rate data. In fact. it is the portion of the batch settling curve after the constant-rate period which furnishes the most information regarding the settling characteristics of the slurr>-. Unfortunately. in the past. data for this period often !cere not obtained or \vere not sufficient to establish the settling curve. Obviously. the correct analysis of erroneous batch settling cun'es will not produce correct answers. serve to test the theoretical analysis. or suggest and allo\v the evaluation of the effect of variables other than solids concentration. .4 large portion of the previous experimental work reporred in the literature has been concerned \vith the settling of calcium carbonate slurries. Sumch slurries are compressible and. therefore. additional variables are introduced. Tory (77) has studied this system extensively?and the results will be reported in subsequent papers. To gain a better insight into the behavior of these complex systems, the settling of closely sized, rigid, spherical particles 'glass beads) in a Newtonian fluid (water) \\*as chosen for investigation and analysis. Some work has been done on the batch settling of rigid spherical particles in water (77: 72, 7 1 : 79). The solid and liquid properties of this type of slurry are constant with time and can be measured easily and accurately. A knowledge and understanding of such a system are prerequisite to Understanding and interpreting the settling behavior of compressible slurries and 'or those ivith time-dependent properties. For a.ny slurry. the test of the basic assumption, 21 = u ( c ) . and theory is lchether or not one can predict from the flux plot, based on initial settling rate data, the complete batch settling curve for any initial concentration.
DIFFERENTIAL VOLUME ELEMENT
\
1"
Figure 1. system u.
v. z.
/
I I
I
Schematic of coordinate
Solids velocity relative to slurry Slurry velocity relative to wall Height
.A 600-po\cer Cooke, Troughton. and Simms microscope wirh movable stage and micrometer eyepiece was used to measure particle diameter. The size distribution was obtained by raking fi1.e samples a t random from the population and measuring 200 beads per sample. The micrometer scale was calibrated by measuring six lengths of 60 to 120 microns on a standard millimeter scale. Settling curves were determined in a 4.76-cm. inside diameter borosilicate glass column 1.83 meters in length. Care !vas taken to ensure that the column was vertical. The spheres were allowed to soak to become thoroughly wet before any runs were made. Mixing was accomplished by blowing humidified air into the slurry through a porous bronze disk (Type G: Micrometallic Co.) mounted on the bottom of the column. The rising air bubbles produced a vigorous random mixing of the solids in the column. Complete details of the experimental procedure are given by Stroupe (75)and De Haas ( 3 ) . Each set was run with a constant iveight of solids which was kno\vn to +0.5 gram. The solids concentration was changed by adding or subtracting water. The order of slurries to be run \vas randomized by selecting the initial concentration b!lot. The slurry was mixed for at least 5 minutes until apparently uniform and then allowed to settle. The initial concentrations were not measured but were assumed as uniform for the basis of all calculations. The fall of the interface between the slurry and the supernatant water and the rise of the slurry-fixed bed interface from the bottom of the column were followed by taking readings at 1.5-second intervals. The heights of these interfaces were known to i l to =k3 mm. and the times to within 0.25 second. Settling times were measured from the moment the air was shut off. Tivo or 3 seconds ivere required to dissipate the last of the air bubbles and for rhe slurry to stabilize. Temperature for each run was held as close as possible to 76' F. by using forced air convection over a large Mater-cooled (or heated) surface. The temperature \vas recorded after each run and ranged from 74' to 78' F. Results
Particle Size Distribution. Of the 1000 spheres measured: 8.58 formed a normal distribution with a mean diameter. 2 = 66.49 microns and standard de\iation. u = 6.70 microns as sho\cn in Figure 2. -4chi-square test confirmed that the distribution was normal (P = 0.38). The remaining 142 spheres made up a ..tail" ranging do\cn to 5 microns. The distribution in this tail was fairly even. with a slight peak a t 10 microns. For the normal distribution hiean volume per particle
=
Experimental Procedur(e
Class IV standard microbeeds (Slicrobeads: Inc., Jackson. Yliss.), average particle size 50 microns, were screened to obtain the fraction noniinally between 61 and 74 microns. After the initial screening. the desired cut was rescreened twice. The beads were washed thoroughly in distilled ivater and acerone. Three ..%,eightsof glass spheres were studied: Set X (597.7 grams), Set XI (193.5 grams): and Set X I 1 (1992. grams). The same beads were used in all the sets in order to eliminate differences in particle size distribution.
J
-m
\vhere .Y = ( d - d ) , u . The integral {cas evaluated by expanding the cubic and integrating by parts. The volume of the particles constituting the tail \*'as calculated directly. On this basis the tail was found to make up 1 .48Tcof the total particle volume. VOL. 2
NO. 3 A U G U S T 1 9 6 3
205
c = p*(1
- e)
(18)
I
PART I C L E Figure 2.
DIAMETER ,(microns)
Particle size distribution of glass spheres
Similarly, mean area per particle =
J
u = --m
Table 1.
Initial Reduced
0.15
Initial Height, Z., Cm. 93.6
XII-4
0.30
15.6
XII-8
0.50
9.4
Concn.,
206
(7
Ai(1
- e)>
=
- e,)
I&EC FUNDAMENTALS
The constants, A,, ivere evaluated by a least-mean-square error fit of the experimental results. A digital computer was used to do the calculations. All experimental results for each set \\.ere used. including all the final bed concentrations a t which the flux was zero. Power series in reduced concentration from third to seventh order were evaluated. It was found that a
Typical Experimental Results for Spherical Glass Beads in Water
Mean particle diameter, d = 67 microns Column diameter = 4.76 cm.
Run No. X-3
2
,=o
The apparent surface area of the fines was computed directly and found to be 2.86% of the total. The characteristic or effective mean diameter was taken as that of a sphere with the same volume-surface ratio as that of the distribution of spheres. Including the tail, de = 66.94 microns. Independent of the particle size distribution, the solids concentration is defined by
CilP,
where ps is the solids density (weight of solids divided by their effective volume in solution) and e is the void fraction. The quantity c is not the slurry density. While the solids concentration is always calculable, for many slurries the terms on the right-hand side of Equation 18 cannot be separated. It is at times more convenient to use the reduced concentration or volume fraction (LIP,), which is equal to 1 minus the void fraction. Similarly, the flux divided by the product of the solids density and a n appropriate Stokes velocity is defined as the reduced flux. For slurries having a range of particle sizes, in which case the Stokes velocity is ambiguous, it is convenient to use the flux divided by the solids density. Solids Settling Velocity. Initial settling velocities were obtained from plots of the height-time readings for each experimental run a t the various initial concentrations for all sets. The initial settling rate of the fluid slurry interface was taken as the settling rate of the solids immediately below the interface, which was assumed to be at a known initial uniform concentration. For each run, a straight line was fitted by eye to the experimental data in the initial constant settling rate period. The initial settling velocity of the slurry was determined for at least three runs at each initial concentration. Settling rates of repeated runs varied by 0.5 to 5%, with an average of less than 3%. The initial concentrations, initial bed heights, final concentrations, final bed heights, initial settling rates, rates of rise of the fixed bed: and temperature for Sets X. XI, and X I 1 are given by Stroupe (75) along with other batch settling data. Additional settling data obtained with thr same so!ids in water are given by De Hass (3). Typical results are shown in Table I. The initial settling velocity of the solids was expressed as a simple power series in the reduced initial concentration ( c / p S ) as given by Equation 19 :
Final Height, Z m ,Cm. 21.90 21.95 22.00 22 .oo 7.35 7.35 7.30 7.
7.30 7.25
Solids density, p. = 2.45 cm. per cc. X, weight of solids per unit area = 33.6 grams per sq. cm. XII, weight of solids per unit area = 11.2 grams per sq. cm. Final Reduced Concn., Initial Slurry Rate o/ Rise CmlPa = Settling Rate, of Awed Bed, ( 7 - em) u, Cm./Src. Cm . /Sec. Temp., OF. 0.641 0.218 76.0 0.640 0.191 0.6486 76 . O 0.638 0.206 0.0494 ... 0.638 0.204 0,0498 76 .O 0.637 0.0659 0.0463 75.5 0.637 0.0665 0.0473 75.5 0.641 0.0667 ... 75.5 n 641 0.0133 0.0457 76.0 . . . ~ ~ ~ 0.0450 76 .O 0.0136 0.641 76.0 .. 0.0133 0.646
I \i
third- or fourth-order power series gave the best fit to the data, particularly a t high solids concentration. While the normalized error squared deceased slightly for the higher order power series, the resultling calculated velocity curves oscillated around the experimental points and actually gave a poorer fit to the experimental results. Table I1 summarizes the constants for Equation 19 evaluated from the experimental results for all sets. Figure 3 shows the experimental initial settling velocities for Sets X and X I 1 and the settling velocities calculated from Equation 19 using the experimentally determined constants given in Table I1 The flux curves for Sets X and X I 1 are shown in Figure 4. Both curves are definitely doubly concave. The size of the circles in Figures 3 and 4 is consistent with the precision of the data The average final reduced concentration \\as the same. 0.64. for all sets The values of the reduced concentration a t the inflection points and thr maximum flux are also ver) nearly equal. as shown in Table I1 Results for Set X I were identical. uithin experimental error. to the results of Set XII.
11.2 g./sq.cm.
-
Discussion of Results
Experimental Settling Velocity and Solids Flux. The third- and fourth-order power series for the settling velocity correlated the data for a given set within the experimental variation. Since the power series expressing the initial settling velocity as a function of initial solids concentration were evaluated by fitting the experimental data, the backcalculation of initial seixling rates does not test the theory but merely tests the fit of the curve of the data. T h e test of the theory lies in its ability to predict the total settling curve for any given initial concentration. I t is the measurement and analysis of the settling curves after the initial constant rate period which provide ):he test of the theory and give insight into the basic fluid-solids dynamics of the system. This will be dealt with in subsequent papers. The significant difference between the curves for Set X and those for Sets X I to X I 1 was that the values of the settling velocity and the flux are consistently higher for Set X (weight of solids per unit area. ze = 33.6 grams per sq. cm.) than for Set XI1 ( E = 11.2 granis per sq. cm.). The per cent deviation of the curves increases from 4 to 5% a t the highly concentrated range to approximately 2070 in the dilute range. There were several possible reasons for this variation of results: variation of Jveight of solids per unit area, mixing effects, particle size
segregation. reading variations (which would be important primarily in the coilcentrated range), and also orientation effects. The deviations in the settling velocity and corresponding solids flux curves for Sets X and X I 1 are believed primarily caused by small initial concentration gradients in the slurries. It is important to notice that the flux curves for Sets X and X I 1 were nearly identical in the high concentration range. This is the second concave region of the flux plot and has a pronounced effect on the batch settling curves expected from such a system. .4n indirect effect of the variation of weight of solids per unit area on the solids settling velocity was the inability to mix slurries of Set X ( w = 33.6 grams per sq. cm.) to uniform initial concentrations. For any given initial concentration the slurry height for Set X \vas three times as high as for Set X I I , be-
Table ll, Constants for Power Series Fit of Experimental Initial Settling Data u = A,
+ Al(l
-
e)
+ z-il(l - + A , ( 1 - + A,(1 E)*
e)3
-
e)l.
cm./sec. Errora Squared 0.25 X l o - ? 0.13 X 0 . 3 4 x 10-3
,\‘ormalized dl
S SI XI1 srt
x
XI XI1
0
0.465798 0.378215 0,3384:33 ( 7 - e ) at >Max. Flux 0.162 0.169 0.179
AVormalized error squared
=
-2.23693 -0.683673 - 1 ,37672 ( 7 - e ) at InJection Point5 0.313, 0,546 0.325, 0.542 0.337, 0,545
f3.77187 f2.88495 + I ,62275
- 2.20863 - 1 ,66367 +0.11264
...
-0.90223j
I= 1
M
VOL. 2
NO. 3
AUGUST
1963
207
----
g./sq. c m . 33.0 g./sq.cm. Pa 2.46 g./CC. 11.2
.. &
,
I
+
-
REDUCED SOLIDS CONCENTRATiON.(I-E) Experimental solids flux plot for glass
Figure 4. spheres
w
= 11.2 p.Ibq.crn.
.. o
i
120
050-0.15, 0.10
0.075 0.05
180
TIME,(sec.)
Figure 5.
Rise of fixed b e d for glass spheres
0.6 a4
< ai D8 0" .06
7,
I;
I,
-I
U .04
>
I
t
0.4 O 06 Q8 1.0 L VOIDAQE Figure 6. Comparison of results with Richardson-Zaki correlation D
208
cause the weight of solids per unit area for Set X \\'as three times as great. Mixing effects produced by air agitation apparently Lvere not great enough to lift the solids for Set Y the greater distance required to give a n initially uniform concentration. The concentration of the upper portion of the bed \vas very probably less than calculated. Because the concentration was less. the settling velocity and hence the solids flux (calculated concentration times the actual settling velocity) were greater than they \.iould have been for a n initially uniform concentration. .4s the solids velocity was strongly dependent on solids concentration. small errors in interface concentration due to mixing would produce significant errors in the resulting \Telocity and flux correlations. Particle Segregation and Agglomeration. The theoretical analysis assumes that the particles are all of the same size, shape. and density. For the solids used in this study, this is obviously not the case (see Figure 2). The ratio of the Stokes velocities for particles of diameter (d u) and (d - u) is 1 .SO. For those of (2 2u) and (d - 2u), the ratio is 2.26. Kynch ( 9 ) has suggested that further development of the theory of settling lies in a n attempt to remove the restriction of identical particles. One of the objectives of the present \>.ark was to study the extent to Lvhich this restriction could be relaxed. The theoretical assumption that all particles are identical. \vhich greatly simplifies the mathematics, is equivalent to the assumption that no solids segregate during settling. If little or no segregation occurs? the slurry may be treated as if all particles were identical for the purpose of analysis. Segregation is indicated by a hazy fluid-slurry interface and. for rigid spheres. a nonlinear rate of rise of the fixed bed. Thick slurries settle with a sharp interface despite a range of particle sizes. Kermack. McKendrick, and Ponder ( 8 ) have presented a theoretical analysis of segregation and stability of suspensions. I n the present work, slurries of reduced concentration of 0.30 to 0.55 had distinct slurry-ivater interfaces. If particle size segregation \
