(12) Lodge, A. S., “Elastic Liquids,” pp. 236-42, Academic Press, New York, 1964. (13) Lodge,’A. S., Proceedings of Second International Congress on Rheology, Oxford, p. 229, 1953. (14) Markovitz, H.. Proceedings of Fourth International Congress on Rheology, Providence, in press. (15) Merrill, E. \V., in “Modern Chemical Engineering,” A. Acrivos, ed., Vol. 1 , p. 145, Reinhold, New York, 1963. (16) Noll, IV., Arch. RationalMech. Anal. 2, 197 (1958). (17) Noll, LV., J . RationalMech. Anal. 4, 3 (1955). (18) Oldroyd, J. G.: Proc. Roy. Soc. (London) A200, 523 (1950). (19) Oldroyd, J. G., Strawbridge, D. J., Toms, B. A , Proc. Phys. Sac. B64, 44 (1951). (20) Philippoff. \V.: Gaskins, F. H., Trans. Soc. Rheol. 2, 263 11958). (2f) Philippoff, LV., Gaskins, F. H., Brodnyan, J. G., J . A p p l . Phvs. 28. 1118 11957). (22) ’Pollett, \V. F. O.,’Brit. J . A p p l . Phys. 6 , 199 (1955). (23) Reiner, M., A m . J . M a t h . 67, 350 (1945). (24) Ri\ lin, R. S., Proc. Roy. Soc. (London) A193, 260 (1948). (25) Roberts, J. E., Proceedings of Second International Congress on Rheology, Oxford, p. 91, 1953. (26) Scott Blair. G. LV.. Folley, S. J., Malpress, F. H., Coppen, F. M. V.. Bzochem. J . 35, 1039 (1941). (27) Stokes, G. G., Trans. Cambridge Phzl. Soc. 8, 287 (1849). (28) Volterra, V., “Theory of Functionals,” pp. 192-94. Dover, New York, 1959. (29) LValters, K., Quart. J . M e c h . A p p l . M a t h . 15, 63 (1962). (30) ii’eber, N., Bauer, i V . H., J . Phys. Chem. 60, 270 (1956).
GREEKLETTERS 6 6(t)
6, 7
io Po
t P
li. T
= displacement, cm. = Dirac’s delta function, s e c . 3
final displacement, cm. viscosity, poise Maxwell relaxation time, sec. retardation time, sec. time constant for decay of pressure gradient, sec. density, grams/cc. = relaxation function, dynes/sq. cm. = time =
= = = = =
literature Cited
(1) Alfrey, T., Jr., Gurnee, E. F., in “Rheology,” F. R. Eirich, ed., Vol. 1, pp. 387-429, Academic Press, New York, 1956. (2) Boltzmann, L., Sitzungsber. M a t h . Naturwiss. Classe Kaiserlich. A k a d . Wiss. 70, 279 (1874). (3) Coleman, B. D., Noll, W., Rev. M o d . Phys. 33, 239 (1961). (4) Etter, I., Schowalter, W. R., Society of Rheology meeting, Pittsburgh, Pa., October 1964. ( 5 ) Ferry, J. D., J . A m . Chem. SOC.64, 1330 (1942). (6) Ferry, J. D., “Viscoelastic Properties of Polymers,” pp. 15-16, Wiley, New York, 1961. (7) Fredrickson, A. G., Chem. Eng. Sci. 17, 155 (1962). (8) Fredrickson, A. G., “Principles and Applications of Rheology,” pp. 168-72, Prentice-Hall, Englewood Cliffs, N. J., 1964. (9) Harple, W. H., Wiberley, S. E., Bauer, W. H., Anal. Chem. 24, 635 (1952). (10) Kalb, J. W., Ph.D. thesis, University of Minnesota, 1965. (11) Kapoor, N. N., M.S. thesis, University of Minnesota, 1964.
RECEIVED for review July 6, 1964 ACCEPTED December 31, 1964 Second in a series on Stress-Relaxing Solids. published in 1962 ( 7 ) .
The first article was
R E A P P R A I S A L OF T H E CONCEPT OF SETTLING IN COMPRESSION Settling Behavior and Concentration Projles for Initial& Concentrated Calcium Carbonate Slurries ELME
R M.
T0
R Y’ A N
D PA U L T
.
SHAN N 0N ,
Purdue Uniuersity, Lafayette, Ind.
(>
In settling of COCO3 slurries initially in compression 145 grams per liter), the rate of elimination of fluid was not always proportional to the amount that could be eliminated up to infinite time. For slurries with initial concentrations from 10 to 190 grams per liter, the descent of the slurry-supernate interface followed the Deerr-Roberts-Yoshioka equation only after a marked concentration gradient had reached the interface. The solids profiles were fairly close to those predicted from Kynch’s theory for rigid spheres, but the concentration a t the bottom reached its maximum gradually rather than instantaneously. Mechanical stress, transmitted b y particle-particle contact, i s believed to account for this gradual increase as well as for the strongly curved final solids profiles and the increase in mean final concentration with increasing weight of solids per unit area. A review of early literature revealed that clear explanations of true compression have been ignored in favor of unwarranted deductions from an empirical equation.
HE distinction between “free” (more accurately, “hindTered”) settling, in which the settling rate depends on concentration. and “settling in compression,” in which elimination of fluid is a function of time, was first made by Coe and Clevenger (3)and has become a fundamental tenet of thickening theory. I n the light of recent developments, including the present work, the basis for this distinction requires a reappraisal.
1 Present address, Brookhaven National Laboratory, Upton, L. I., N. Y . * Present address, Thayer School of Engineering, Dartmouth College, Hanover, N. H.
194
I&EC FUNDAMENTALS
Concept of Settling in Compression
Development of Concept.
Coe and Clevenger ( 3 )stated:
After pulp reaches the consistency where the flocs touch each other, further elimination of water becomes approximately a function of time , . , A large number of comparative tests have been made in vessels of from 1 to 10 feet in d e p t h , , , I n the deep vessels the critical point will be proportionately lower than in the shallow vessels. This may be explained by the fact that the average time of compression in the thickening zone before the critical point is reached is greatest for the pulp in the deep vessel. If the fluid be expelled as a function of time after the flocs enter the zone of compression, it is but natural that this should be the case. ,
In two vessels of unequal depth, filled with the same kind of pulp and started to settle at the same time, the total pulp in the more shallow vessel will begin to compress more quickly and will thereafter for many hours remain thicker than the pulp in the deeper vessel, even though the critical points occur not more than an hour apart. ‘This indicates that thickening in the compression zone is a function of time. I n his experiments with alumina hydrate, Deerr ( 8 ) found settling curves which suggested a logarithmic function for the final stage of settling. I n 1920, he proposed the equation
which can be rearrangrd to the more familiar form 2
-
tu+(, = ( 2 ,
-
z,,o)c-f’(l--IC)
(2)
where z, is the “critical” height and tc is the ‘ 27) have shown that Kynch’s theory holds very well for slurries of closely sized rigid spheres in water. Many other studies ( 7 7, 75, 78, 25, 27-30) suggest that the theory gives a t least a first approximation for the behavior of industrial slurries. Modification of Roberts’ Equation. Recently, Yoshioka et al. ( 2 9 )found that although a semilogarithmic plot of (z - z J us. t gave a straight line, k’ varied with L i z i , the weight of solids per unit area. The equations of Work and Kohler (28) and Robinson (78) suggested that k’ = k / c i z i where k is a true constant for a given slurry-that is,
dZ/’dt = - ( k / c t z t ) ( z
- Zm)
(5)
or z / c g z g - zm/ctzi =
(z,/ctzt
-
z m / c I z t j e - k ( l - l C ) ~ C ~ z (61 ~
Equations 5 and 6 may be called the Deerr-Roberts-Yoshioka VOL. 4
NO. 2
MAY
1965
195
(DRY) equations. They imply a drastic departure from Roberts’ original hypothesis. If k and e,( = c(z,/z,) are true constants, Equation 6 is now a n empirical height-time relationship of the type expected for a slurry whose settling rate is a function only of the local solids concentration-i.e., one which obeys Kynch’s theory. Synthesis of DRY Equation with Kynch’s Theory. T h e definition of settling velocity
propagated through a concentration gradient in a settling slurry is
when the slurry velocity is zero. If the settling rate is a function only of the local solids concentration and if the solids a t the bottom reach their maximum concentration instantaneously, then
p* may be combined with the DRY equation for the velocity of fall of the slurry-fluid interface
where the prime (I)denotes that the terms refer to conditions a t the interface. For a given batch test? z’ = z’ ( u ’ ) and u ’ = u t ( t ’ ) . Hence, Equation 7 can be written for the interface as
-
=
-z’/t’
d(cu)/dc
=
(18)
where t’ is the time at which concentration c reaches the slurry interface a t z’. From Equation 8
and from Equations 11 and 14
dt’ du‘ _ _ du‘ dt‘
When dz’ldu’ is evaluated from Equation 8, Equation 9 becomes
Thus
-k + -1 11’
Z’
Cm
or, in integrated form
t’
=
t,’
+
ClZt’
u,’
k
u
- In
where subscript e refers to experimental values. Kynch‘s equation
z
- A -
t’
+ ut
which is general, applies at the interface. T h e values of z’ and t’ from Equations 8 and 11, respectively, are substituted in Equation 12 to give
= cmzm
1
ut
ctti
c = -
’ has been used.
where once again the relationship c t z y ’ From Equations 15 and 21
&-:)-i 1 1 1
T h e level, z , which is reached by a concentration, c, after a time, t , is simply (23)
z = -p*t
From Equations 8 and 12
Because the settling velocity is a function only of the local solids concentration
-k + -em1 u
z =
1
Making use of Equation 14 and the relationship cdzi’ = cmzm’, Equation 13 becomes
According to Kynch, u = u ( c ) . so that solids a t a particular concentration settle with a given velocity whether they are a t the interface or not, Thus. Equation 15 can be written
and applies throughout the entire slurry \vhose fluid-slurry interface obeys Equation 8. This result agrees with that obtained by Tory (25) using a n equivalent but less direct method. Calculation of Theoretical Solids Profiles for Settling Slurries. T h e velocity Tcith which a given concentration is 196
I&EC FUNDAMENTALS
t
(24)
T h e concentration, c, is obtained from Equation 16 and height, z , from Equation 24. .4s the quantity ( l / u ) ( l , / c - l/cm) is obtained during the calculation of c, the computation of z is particularly simple. For some purposes it may be desired to have parametric equations in u . I n this case, from Equations 18, 21, and 23
z =
k
cm
1
1
t
Physically, the weight of solids per unit area remains constant during settling. Mathematically, this is easily demonstrated using integration by parts and Equations 12: 18, and 23.
Initial H‘t. 97.4 cm.
5 4
Initial Conc. 287 g.p.1
1
days. All samples for settling tests were taken from this master slurry. T h e weight and volume of slurry were measured and the weight of solids was calculated. The apparent density of C a C 0 3 in well mixed slurries of 200 to 300 grams per liter was determined to be 2.684 i- 0.012 grams per ml. (standard deviation of mean). This value differed only 1% from the real density of calcite (2.711 grams per cc.) but was felt to be more appropriate. Procedure. T h e same slurry sample was used for a series of runs, the solids concentration in the cylinder being adjusted by adding or removing water. T h e slurry was mixed thoroughly with a perforated aluminum plunger and the time of removal of the plunger was recorded. T h e level of the interface a t later times was noted; the frequency of readings varied between 2 per minute and 3 per hour for the initial period, depending on the velocity of fall. T o afford a comparison with Comings’s work (4), the first test was done a t 44 grams per liter. A second run a t the same concentration showed that the results were reproducible. A series of tests was undertaken in which each solids concentration was run twice (except 44 grams per liter, which was run once) and the order of runs was determined by lot. The second series was also randomized. T h e third was run in descending and the fourth in ascending order of solids concentration. One-minute radiation counts were taken every 2 minutes except when settling was very slow, in which case longer counts were feasible. T h e count rate was plotted against time to give the change in solids concentration. I n some cases, notably those involving thick slurries, counts were taken at various heights.
t (min)
.
Figure 1 Semilogarithmic height-time relationship during final stage of settling
C‘
[’cdz
= [cz]gL’ -
Jm
zdc = c’z’ =
+t
l‘”’
c’z!
+ c’u’t
Table
d(cu) =
1.
Summary
Weight
(26)
G ~ z ~ ’
Graphical integration may be used to check the numerical values of c and z obtained from Equations 16 and 24. Experimental
Apparatus. Settling rates were measured in a cylindrical borosilicate glass colunin 5 feet long with a n inside diameter of 6 inches. T h e appar,atus for determining solids concentration consisted of a 50-millicurie Cs137 gamma source and a Harshaw scintillation crystal on opposite sides of the column and mounted on a vertically movable platform (25). A lead cylinder with a ‘/*-inch axial hole provided collimation. A lead shield with a 3/*-inch (vertical) by ’/s-inch (horizontal) hole was placed in front of the crystal. This combination gave a rate of about 60,000 counts per minute when the cylinder was filled with water. Counting equipment consisted of R . I . D . L . Model 200T scaler, Model 11 5 pulse-height analyzer, and Model 39-1 electronic sweep and count-rate computer. T h e effect of voltage drift was minimized by using a 1-volt window and by refinding the voltage for maximum count with water before each run. Since it was not feasible to surround the column by a constant-temperature bath, the temperature of the room was kept between 25’ and 27’ C. T h e viscosity of water varies less than 5% between theise extremes. I n addition, special care was taken to maintain constancy of temperature during any run, especially during the fast-settling period. Temperature control was achieved by forced convection of air over a large water-cooled surface. Materials. Comings (4)has shown that prolonged agitation of a slurry stabilizes its settling properties. A thick slurry (about 200 grams of solids per liter) containing 100 pounds of C a C 0 3 (Mallinckrodt Code 4052 Pptd. U.S.P. Light, Lot JB 297, particle diameter .0
A 0
I
z-
(28)
where c ( z ’ ) is the concentration a t the bottom of a slurry which has settled to a height of z ’ . For incompressible sludges, c(z’) is constant; for those shown in Figure 5, the solids profile should be strongly curved. I n Figure 6, the experimental profile for a slurry which has settled almost to its final height is compared to the theoretical profile calculated from Equations 16 and 24. Again the compressibility of these C a C 0 3 slurries is indicated. Comings obtained similar results with a C a C 0 3 slurry having somewhat different characteristics ( 4 ) and with Mississippi pot clay (5). Similarly, Bretton (7), Sala ( 7 9 ) , and George (72) found that the final concentrations of ferric hydroxide and copper hydroxide slurries increased with depth.
C A L C I U M CARBONATE 124) CEMENT ROCK B ( 2 4 ) K A O L I N (101 METALLURGICAL PULP ( 2 4 ) CEMENT ROCK A (24)
0 I .o !-
a 0.8 a
I-
O.€ V
z
0 V
0.4
v)
D
0.:
v,
2_J
0.2
LL
W E I G H T OF S O L I D S , ( g / s q . c m . )
Figure
5.
Dependence of final solids Concentration on weight of solids VOL. 4
NO. 2
MAY 1965
199
I
Confirmation of DRY Equation. As indicated in Table I, the constant, k , in the Deerr-Roberts-Yoshioka equation was remarkably constant when the weight of solids was varied from 504.9 up to 4841.0 grams. Over this range k = 1.197 i 0.102 (standard deviation). Statistical analysis confirmed that between these results and those for 250.4 grams there was a small but significant difference, probably caused by minor differences in slurry properties rather than the lower weight of solids per unit area. If the settling rate were a function of the local solids concentration only. all results at high concentrations within a concentration gradient could be correlated by Equation 6. The "extrapolated" solids concentration in Table I is based on the extrapolation of the semilogarithmic plot to zero time. I n addition to giving the simplest possible expression, this has the advantage of reflecting the settling history of the slurry. The absence of channeling at high concentrations accounts for the low extrapolated concentration. Where channeling was extensive, as in runs 16 and 19, the extrapolated concentration was high. For a given initial solids concentration, it increased with increasing weight of solids per unit area.
I
IO0
EXPERIMENTAL 4 THEORETICAL ----RUN 22 t =8640 min. C i = 190.2 9.p.I. Z i = 141.4 cm. W = 4841 .Og.
40
30
Application of Theoretical Concepts to Experimental Results c (g.p.l.1
Figure 6.
Solids profile when settling i s virtually complete
-
..
R EDU CED
Figure 7.
TIM
E,(mi n./g. )
Variation of reduced height with reduced time
A slurry which is compressed by the weight of solids above obviously does not follow Kynch's theory that the settling rate is a function of the local solids concentration only. However, by calculating results for an ideal slurry, it is possible to evaluate disparities between ideal and actual behavior and to attribute them to rational causes such as mechanical stress. If factors other than local solids concentration have an appreciable effect on the settling rate, this will be shown by plotting z/cl+ us. t / c i z t . This correlation is shown in Figure 7 for slurries with an initial concentration of 146 grams per liter and different weights of solids. For the first part of the curve, the points lie on a single line, indicating ideal behavior. The marked divergence during the last stage of settling indicates that weight of solids per unit area is an important parameter. The reason for this can be discovered only by a study of the propagation of concentrations upward through the settling slurry.
-2 -2 40
z
a E 200 0 l-
z
W
u
z
0 IS0 v)
c)
A 0 v,
120 1000
2000
3000
4000
5000
T I M E , (min.)
Figure 8. 200
I & E C FUNDAMENTALS
Solids concentration at fluid-slurry interface
I
Where the local solids concentration is only one of several variables, Equation 31 can be used as a first approximation, provided that (z - z,) is comparatively small. Curves of height us. time a t constant concentration, constructed from several solids profiles, can be used for a second approximation if necessary. The solids concentration, c, is related to the radiation count, R, by the equation
I
SOURCE HEIGHT 16.4 c m .
Zi' II 5.3crn.
c=-
-u+ oa W
(L
a
30
tL
i L
.
I (XI0
where B is the count for zero solids concentration and k is a constant for the system. When radiation counts are taken over a long period, the base count may shift slightly. If it is assumed that the same concentration always produces the same value of ( B - R ) / B , a simple adjustment is possible. A n estimate, B1, is made, either from the calibration curve or b y putting in an estimate of c for some count R. I t is easily shown that
3000
2000 TIME,(rnin)
Figure 9. Interface concentration predicted from concentration near bottom
Calculation of
Experimental
Concentration
Profiles.
B = Bl(1
T h e experimental apparatus permitted the determination of the solids concentration a t only one level a t any given time. T h e simplest method of constructing a solids profile, linear interpolation between r'radings a t different times a t the same height ( 7 7 ; 25), is incoiisistent with Kynch's theory. Mathema tically
=
-z,/t,
+
(Ze/te)(t
where F 1 is the mean concentration from graphical integration and is the true mean concentration. I n most cases (c - c 1 ) is almost constant for a given profile. I n the present work k = 0.654 cc. per gram and B N 60,000 counts per minute. Ideal Settling and True Compression. The experimental solids profiles were all in fairly good agreement with those calculated, but the differences, though not large, were consistent. Figure 4 is typical. Initially, the concentration at the bottom is less than predicted; further up, the lines cross and the experimental concentration exceeds the theoretical. Figure 8 shows that the concentration at the interface remains close to the theoretical even when settling is well advanced. As illustrated by Figure 6, the experimental concentration must eventually be less than predicted because the final experimental profile is curved rather than flat. Given the actual fall of the interface and the solids concentration a t ( z e ,t,) Equation 30 can be used to predict when that concentration will reach the interface. T h e prediction shown
(30)
- te)
(33)
(34)
Thus z = zt.
- kFi)/(l - kF)
and
Ideally, @ * is constant, but in general the slope of the concentration profile is not. Hence the variation of concentration with time is not linear and could lead to apparent disagreement between theoretical and experimental results for an ideal slurry. When Kynch's theory holds, a mathematically rigorous method of calculation i j available. Between t,, the time the concentration was determined, and t , the time to which all readings must refer, a concentration was propagated with a velocity given by @*= - z / t
B - R kB
(31)
--I
d rn
5 290 -
RUN 21 C i ~ 1 4 6 . 7g.p.1. Z i = I 15.3 cm. W'=3053.2 W=3053.2 g
L
0
m I-
a 2
0
t 280 Fz
I
w V
z
0 V
2 10
I IO00
I 2000
I 3000
I 4000
TIME, (rnin.)
Figure 10.
Solids concentration at bottom of slurry VOL. 4
NO. 2
MAY
1965
201
7 0 1
-E N
"1
4o
3o
2o
c t
EXPERIMENTAL THEORETICAL
\
----
RUN 21 t = 3800 m i n . Ci = 146.7 g.p.1. Zi I I 5.3 cm. W =3053.2 g
\ o
\.
I 00
2 00
300
c(g.p.1.)
Figure 11.
Solids profile when settling is well advanced
in Figure 9 may be compared with Figure 8. T h e premature appearance of the concentration increase in Figure 9 indicates that the lines of constant concentration either are not straight or d o not meet a t the origin. T h e theoretical common meeting is dependent upon the maximum concentration being formed immediately a t the bottom ; the experimental concentration there, shown in Figure 10, indicates that this does not occur. An increase at the bottom of the slurry as settling progresses has also been noted by Comings (4, 5), Goldfarb (73), Bretton (7), Sala (79). George (72), and Gaudin and Fuerstenau ( 7 7 ) . This might be interpreted either classically or as resulting from the increasing stress as more and more solids are supported. Support for the latter viewpoint is the increase in final solids concentration a t the bottom with increasing weight of solids per unit area, as found in this work and by Bretton ( 7 ) . Further support comes from the data on compressibility shown in Figure 5. When the solids concentration increases a t the bottom, the lines of constant concentration will naturally be curves; when it is virtually constant, as from 1100 to 2900 minutes (Figure l o ) , one of the conditions for straight lines is fulfilled. An examination of the solids profiles showed that the lines were indeed straight but did not extrapolate to the origin. Table I1 indicates the propagation velocities and the extrapolated times at which the concentrations are a t zero height. Apparently, initial velocities of propagation were high for concentrations of 160 and 170 grams per liter. This is also shown by the comparison of Figures 8 and 9 noted above. Concentrations greater than 210 grams per liter originated later as the solids piled u p , This is consistent with the compressibility of CaCOz slurries shonn in Figure 5. Hypothetical solids profiles can be constructed by extrapolating the lines of constant concentration based on Table 11. Comparison of these profiles n i t h the experimental ones indicates that the actual concentrations are greater near the bottom and less near the interface. Compression in the lower levels releases liquid. lvhich causes dilution further up. When the solids concentration at the bottom reaches the mean final concentration of the slurry. the theoretical profiles calculated from Equations 16 and 24 agree very closely with the experimental profiles. This is shown in Figure 11. As noted earlier. the experimental concentration will eventually be 202
l&EC
FUNDAMENTALS
greater a t the bottom and less a t the top than the theoretical (Figure 6) because of the compressibility of the slurry. T h e divergence of the reduced curves in Figure 7 can now be understood. Until a marked concentration gradient reaches the fluid-slurry interface. the settling rate there decreases only slightly. With a low weight of solids per unit area, there is little compression near the bottom and thus little dilution near the top. T h e solids concentration at the interface increases and hence the settling rate decreases. With double the weight of solids, a given concentration should theoretically take twice as long to reach the interface. However, the increase in concentration a t the interface is delayed beyond expectations by compression near the bottom and consequent dilution of the upper layers. Because the solids concentration just below the interface is less than in the case of a lower weight of solids, the settling rate is greater. In this view, a given settling velocity means that a particular solids concentration exists just below the fluid-slurry interface. Accordingly, settling velocities were plotted against 25 interface concentrations based on experimental solids profiles constructed for slurries with initial concentrations of 146 and 190 grams per liter and weights of solids from 1057 to 4841 grams. Figure 12 shows that, within experimental error, the settling rate was a function only of the local solids concentration. Although Figure 12 indicates that a unique velocity-concentration relationship exists, the solids concentration is not necessarily that determined theoretically by extrapolating a tangent to the settling curve back to the z-axis (76) or, what is equivalent, using Equation 16. I t is the effect of true compression at the bottom which makes it necessary to use different values of cm in Equation 16 which relates settling rate to local solids Concentration. As the lines of propagation of concentrations are not straight, Equation 16 is an approximation which gives better results for the slurry-fluid interface than for lower levels. By the time a marked concentration gradient has reached the interface, enough solids have settled to compress those a t the bottom to a concentration close to cm. T h e theoretical line in Figure 12 is that calculated from Equation 16 using values for R u n 21, Considering experimental uncertainties in both velocity and concentration, the agreement is good. Thus, the theoretical method is, in fact, useful for calculating the solids concentration just below the fluid-slurry interface. At a given time early in settling. solids profiles for runs with low weights of solids should be identical to the lower part of the profiles for runs with higher weights of solids, provided that the
Table II.
Propagation of Concentrations in a Settling Slurry
R u n 21 w = 3053.2 grams Velocity of Propagation Concn., x 702, G./ L . Cm./Min. 160 2.17 170 2.10 180 1.97 190 1.86 200 1.79 210 1.72 220 1.67 230 1.59 240 1.47 250 1 .27 260 0.99 270 0.60
c, = z, =
Height at 2000 M i n . ,
Cm. 53.8 46.4 41.9 38.3 35.3 32.7 30.0 27.1 23.7 19.7 14.5 7 8
146.7 g./liter 115.3cm. Extrapolated T i m e of Origin, M i n . -485 -210 -125 - 60 25 105 210 300 380 455 5 40 690
solids in the upper layers d o stress o n those below. Figure profiles are virtually identical the very bottom. and are close
not exert a direct compressive 13 shows that the experimental in the lower regions, except a t together for their entire height.
Discussion
T h e present work has shown that settling velocity is determined largely by the local solids concentration even for slurries normally considered to be ”in compression”-Le., slurries which follow the DRY equation. Hence the old concept of detention
I
20 -
I
I
I
I
I
THEORETICAL EQUATION (16) FOR RUN 21-
.-.i E
>
E 10-
-2 9 -
;*x
>-
t v
8-
765-
s u >
4-
cl
3-
5
time is meaningless. I n continuous thickening of a n ideal slurry, the underflow concentration is affected by the drawoff rate (74, 26, 30), which in turn affects the detention time, Low underflow rates are often associated with greater thickener depths and hence greater weight of solids per unit area. T h e importance of the latter is indicated in the present work, Cortinuour thickening will be discussed in detail in a subsequent paper. Recently, Fitch ( 9 ) has suggested that channeling may increase the solids flux a t high concentrations to the level required by material balance in continuous thickening. I n this work, channeling was extensive just before the sharp break in the settling curves of slurries of dilute and intermediate concentrations but was entirely absent thereafter. If a concentration gradient has been established, channeling appears to be much less a factor a t high than a t intermediate concentrations. Such instability as exists appears to be associated with large depths of slurry in \yhich no gradient exists. T h e stability of slurries will be discussed in a subsequent paper. Acknowledgment
T h e assistance of Audrey Tory in the longer experimental runs and S. Nagata in translating reference 29 is gratefully acknowledged. T h e authors also thank Purdue Research Foundation for financial assistance and Brookhaven National Laboratory for facilitating the completion of this work.
-I
RUN 15 17 0 20 0 21 22
A A
I
W(g.1
u 140
Ci(g.p.I.)
1057.2 2063.6 2063.6 3053.2 4841.0 160
145.2 146.3 189.8 146.7 190.2 180
‘1
Nomenclature
200
u
220
240
SOLIDS CONCENTRATION AT INTERFACE, (g.p.I.1
Figure 12. Dependence of settling rate on local solids concentration
A
=
B
= radiation count for zero concentration, counts per min. = solids concentration, grams per cc.
c
t
D
k = k’ =
R
S t u
1 -__
l-
0
ao
= =
= =
z
=
@ *=
Ci= 146 9.p.i
I
= =
ZL’’
u,
1
= =
W(g.1
Z i (cm.)
3053.2 2063.6 1057.2
115.3 78.0 40.05
cross-sectional area of thickener, sq. cm. mean solids concentration, grams per cc. dilution (mass ratio of fluid to solids), dimensionless constant of proportionality, grams per sq. cm.-min. constant of proportionality, min.-’ radiation count, counts per min. batch solids flux = cu, grams per sq. cm.-min. time, min. batch settling velocity of solids, cm. per min. weight of solids, grams weight of solids per unit area, grams per sq. cm. height, cm. propagation velocity of plane of constant concentration, cm. per min.
SUBSCRIPTS of compression experimental value i = initial value m = final value 1 = first estimate
c
= value a t point
70
e
=
50
literature Cited
-5 6 0 N
40 30
2o
t c ( 8 . p. 1.)
Figure 13.
Solids profiles for different weights of solids
(1) Bretton, R. H., “Design of Continuous Thickeners” (report), Yale Universitv. New Haven. Conn.. 1949. (2) Brown, G. G., and associates, ‘;Unit Operations,” Wiley, New York, 1950. (3) Coe, H. S., Clevenger, G. H . , Trans. Am. Inst. Mining Engrs. 5 5 , 356 (1916). (4) Comings. E. \V.- Znd. Ene. Chem. 32. 663 (1940’1. j5j Comings, E. \$;.. Pruiss,-’C. E., D e Bord: C., ’Ibtd., 46, 1164 (1954). (6) Coulson, J. M., Richardson, J. F.; “Chemical Engineering.” Vol. 2, Pergamoii Press, London, 1955. (7) Deane, \V. A , , T r a n s . A m . Elpctrochem. SOC. 37, 71 (1920). (8) Deerr, N., Intern. Sugar J . 22, 618 (1920). (9) Fitch, E. B., 7’rans. A.Z.M.E. 223, 129 (1962). (10) Free, E. E., En!. M i n i n g J . 101, 681 (1916). (11) Gaudin, A. M.. Fuerstenau, M . C., International Minexal Processing Congress, pp. 115-27, Inst. Mining & Metallurgy. London ,1960. VOL. 4
NO. 2
MAY 1965
203
(23) Talmage. W. P., private communication. (24) Talmage, W. P.. Fitch, E. B., Ind. E n g . Chem. 47, 38 (1955). (25) Tory, E. M., Ph.D. thesis, Purdue University, Lafayette, Ind., 1961. (26 Tory, E. M., Shannon, P. T., 44th National Meeting A.I. dh.E., New Orleans, Feb. 27, 1961. (27) Tory, E. M., Shannon, P. T . , 13th Chem. Eng. Conference, C.I.C., Montreal, Oct. 21, 1963. (28) Work, L. T., Kohler, A. S., Znd. E n g . Chem. 32, 1329 (1940). (29) Yoshioka, N., Hotta, Y., Tanaka, S.,Kagaku Kogaku 19, 616 (1 955). (30) Yoshioka, N., Hotta, Y . , Tanaka, S., Naito, S.,Tsugami, S., Ibid., 21, 66 (1957). RECEIVED for review April 27, 1964 ACCEPTED October 6, 1964
(12) George, E. T., Ph.D. thesis, Yale University, New Haven, Conn., 1955. (13) Goldfarb, M.. R.S. thesis, University of Illinois, 1939. (14) Hassett, N. J., Znd. Chemist 34, 116 (1958). (15) Zbtd., p. 489. (16) Kvnch. G. .J.. Trans. Faradav SOC.48. 166 11952). ,~ ., ~~, ~~~~~, (17) Roberts, E. .J:, M i n i n g Eng. 1,-6l (1549). (18) Robinson, C. S., Znd. Eng. Chcrn. 18, 869 (1926). (19) Sala, L. M . , M.S. thesis, Yale University, New Haven, ’ Conn., 1952. 120) Shannon. P. T.. De Haas. R. D.. Strouue. E., Torv, E. M.. ‘ 1 ; ~ . ENG.CHEM. FUNDAMENTALS 3, 250 (1664). (21) Shannon, P. T . , Stroupe, E., Tory, E. M., Zbzd., 2, 203 (1963). (22) Stewart, R. F., Roberts. E. J.; Trans. Znst. Chem. Engrs. 11, 124 (1933). \
I
BIAS IN PARTICLE-SIZE ANALYSES BY T H E COUNT METHOD J . E. G W Y N , ’ E. J.
CROSBY, A N D W.
R . M A R S H A L L , J r .
University of Wisconsin, Madison, W i s . Particle-size analyses b y the count method of samples taken from populations which follow distributions that are infinite in extent-e.g., the log-normal distribution-indicate the existence of maximum size classes by the usual methods of calculation. The resulting forms of the particle-size distributions based on these samples, as a consequence, are biased. This bias was investigated and a simple correction was introduced which yielded unbiased estimates of the particle-size distributions based on number, area, and volume.
NTEREST
in the concept of the existence of a “maximum
I size” for particle-size distributions has increased because of
the improved manner in which experimental data have been fitted by an upper-limit equation (5). This approach reduced the apparent deviations of the tails of plots obtained with experimental data from the postulated size-distribution laws. Because of the controversial nature of a well defined maximum size, the particle-size distributions predicted by small samples taken from large parent populations were investigated (2). This presentation considers the relation between the distribution predicted by a small sample and the parent distribution from which the sample was taken. T h e results are applied to experimental data obtained by microscopic count to illustrate that a distribution with a n infinite upper bound represents such data satisfactorily. T h e apparent maximum size is then increased by increasing the size of the sample tenfold to illustrate the elusiveness of a maximum size. Finally, the experimentally determined distribution on a number basis is transformed to distributions on a basis of area and volume to indicate the goodness of fit obtained by the derived techniques. Theory
T h e count method is a widely used technique for particlesize analysis. I n this technique a dispersed sample is generally magnified to such an extent that individual particles, drops, or bubbles can be measured and classified according to size. T h e particle-size distribution for the material from which the 1
Present address, Shell Oil Co.: Houston. Tex.
204
l&EC FUNDAMENTALS
sample is taken is then determined from such data. T h e number of particles counted is generally within a range of several hundred to several thousand because of the timeconsuming and tedious nature of the counting technique. Such samples are still small enough to allow significant statistical variations, especially a t the “tails” of the distributions. If data obtained by the count method are to be used with confidence, it is necessary to consider the nature of the variations a t the tails for statistically small samples. Evidence of Bias. Whenever a sample of particles is analyzed by the count method, a largest size is always observed. Graphical representation of the particle-size distribution, which is predicted by the sample, yields a curve that is asymptotic to the largest observed size if the coordinate giving the level of occurrence is a probability scale. For example, a sample taken from a population with a n actual lognormal distribution does not predict a particle-size distribution which conforms exactly with this parent distribution. This nonconformity is much more serious for the distributions based on area and/or volume than for those based on number. Consider a sample of n particles which has been classified in order of increasing diameter. Let the diameter of the largest observed particle size be XL. Now XL will usually assume different values for different samples which are taken from the same parent population. T h e probability that X L will assume a given value, x , for any sample is predicted from order statistics ( 4 ) to be
pn(XL =
2)
= n[FN(X)ln--]f&)
(1)
where p n ( X L = x ) is the probability that the largest of the n particles will take on a value x ; F,v(x) is the value of the
