Settling
of
P.T.SHANNON E. M. TORY
Slurries
New Light on an Old Operation This new method for obtaining thickening curves from batch tests is useful in the design of batch and continuous thickene r s and fills i n gaps i n the l i t e r a t u r e . T h e method a l s o more clearly identifies research still to be done
hickening is the process of propagating concentration Tchanges and is readily analyzed in terms of the
.
solids flux using continuity considerations as was first done by Kynch (77). The analysis accounts qualitatively for the observed behavior of slurries in both batch and continuous thickeners, relates the two, and explains many previously puzzling phenomena. For systems in which secondary variables are quantitatively important, the analysis forms the basis for understanding and correlating the deviations encountered. Both batch settling and continuous thickening should be thought of as the process of propagating density or concentration changes upward, due to the downward movement of the solids. The concentration changes may be either finite or infinitesimal depending on the continuity of the concentration gradient. The slurry itself may be moving downward as in continuous thickening. The downward slurry velocity may be such that the concentration change being propagated up from below remains stationary relative to the thickener walls. This is the case for steady state continuous thickening. The basic approach, which developed over the last 12 years, is to analyze batch and continuous thickening in
4 Figure 7 . Choosing the proper coordinaks fm solids pur dlows the same analysis to be applied to both batch and continuous thickening
terms of the movement of planes of constant concentration-Le., continuity waves. Use of the solids flux as a primary variable greatly facilitates the analysis. A proper choice of coordinates in defining the solids flux, shown in Figure 1, clarifies the relationship between batch and continuous thickening and allows the same basic analysis to be applied to both. The extension of the analysis to continuous thickening has been presented and by Shannon and Tory (27,30,33), Yoshioka H a w t t (76). All three studies were independent and where results overlap, they are in essential agreement. This approach offers a consistent interpretation of experimental results, directly relates batch results to the continuous thickener operation, and provides a logical explanation of many previously unexplained phenomena encountered in batch and continuous thickening. The fundamental assumptions underlying the theory are: the local settling velocity of the solids relative to the slurry is a function only of the local solids concentration; solids are uniform or closely sized; concentration at the bottom of the thickener immediately reaches its maximum value; there are no radial variations; ,settling velocity of solids relative to the slurry is independent of the velocity of the slurry itself. A detailed study (8, 25, 27, 28) of the batch settling behavior for uniform rigid spheres in water has confirmed the theory for this relatively ideal system. The experimental results were in complete qualitative and very close quantitative agreement with the predicted behavior. A rising concentration gradient immediately above the tixed bed was observed for initially uniform slurries of intermediate concentrations. It was the intersection of the rising concentration gradient with the fluid-slurry interface which caused the nonlinearity of settling curves. Thus, the settling behavior was similar in many respects to that of industrial slurries.
(a,
Solids Flux Plot
The concentration just below the slurry-supernatant interface determines its rate of descent. As settling proceeds, concentrations are propagated upward from the bottom of the vessel with constant, Characteristic velocities (77,25). Thus, at all times during settling, VOL 57
NO. 2
FEBRUARY 1965
19
From the flux plot, the settling curves can be constructed the velocity of the solids is the same as if the slurry had originally been at the interface concentration. I n Figure 2, a slurry with an initial concentration c, begins settling from height zi. After a short constantrate period, the settling rate gradually begins to decrease as higher concentrations, propagated upward from the bottom, reach the interface. At time t the interface solids concentration c is that which gives a settling velocity u. Now consider a uniformly mixed slurry which begins settling from z,,~. If it is assumed that velocity is a function of concentration only, a material balance yields where cizi is the weight of solids per unit area. From Figure 2, it is exactly this concentration which gives a settling velocity u . Hence, the concentration at the interface is always given by Equation 1 where zext is the height at which the tangent to the settling curve intersects the z axis. In batch settling, the solids flux in grams per square centimeters per second is
s = cu
(2)
A flux plot is a graph of the solids flux as a function of the solids concentration (Figure 3). I t may be constructed from a series of tests in which the initial settling rate is measured for a wide range of initial concentrations. From the earlier discussion, it is apparent that at least part of a flux plot can be constructed from a single settling test. In view of Equations 1 and 2
s = eizi,/tezt
(3) where the relationship u = zezt/’tezt(Figure 2) has been used to simplify the result. Unfortunately, relatively small deviations from ideality in the settling curve lead to appreciable errors in a flux plot constructed from Equations 1 and 3. Originally it was hoped that continuous thickeners could be designed on the basis of a single batch test using the proposed feed to the thickener (29), thereby eliminating the effect of initial concentration on floc structure. However, the stipulation that the concentration at the bottom of the thickener immediately reach its maximum value (77, 25) is so severe that even slightly compressible slurries are not susceptible to this method (5, 6, 75, 32). I n this respect, the theory has not fulfilled its earlier promise (29), although it has been successful in some instances (70, 29). Construction of Ideal Settling Curves from Flux Plots
The general characteristics of settling curves for a given slurry can be determined by inspection of the flux plot (77, 25, 30, 33). Recently, two rigorous methods have been devised for computing the theoretical settling curves for slurries of initially uniform concentration (8, 25, 26, 34). For slurries whose particle characteristics are unaffected by initial concentration, a comparison of predicted and actual settling curves provides 20
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
insight into the effect of variables other than local solids concentration. Here is a simple, rapid, accurate graphical method for predicting the batch settling curves for any initially uniform concentration. From initial rate data, construct a flux plot. Figure 4 will be constructed from Figure 3 for illustration. I n Figure 3, lay a straightedge along the concentration axis and, using c, as a pivot, swing it upward toward the appropriate initial concentration ci. In Figure 3, a straight line has been drawn from c, tangent to the curve at c3 and intersecting it at c1. These values define Regions 1 and 4 in which it is possible to draw a straight line from c, to ci. I n this case, the settling curve is simply a straight line from the initial to the final height. For a given concentration, a straight line from the corresponding point on the flux curve to the origin has a slope equal to the settling velocity. If the initial concentration lies between c1 and c3, the settling curve is constructed as follows. Draw a straight line on Figure 4 from 23 = cizt/cs with slope -SS,,/C, = -u3 to intersect the final height z, at the time t , at which settling is complete. O n the flux plot, draw a straight line from et (in Region 2) tangent to the curve at c, (in Region 3). O n the z-t diagram (Figure 4) draw straight lines from z i and z , with slopes - u i and -u, respectively. This intersection marks the end of the constant-rate period. It remains only to complete the curved portion of the settling curve. This is easily accomplished by selecting a series of heights z3 and drawing straight lines of slope - u j = -Ssl/’cj, where c, = cizi/z3, As shown in Figure 4, these form an envelope which is the settling curve. If the flux curve is convex downward at ci (Region 3), there is no break from the initial slope. A slurry of initial concentration c,, for example, has a settling curve which follows the straight line from z , and gradually veers off along the envelope. Theoretical settling curves can be constructed from more complicated flux plots such as that shown in Figure 5. A straight line from c1 tangent to the curve at ca is also tangent at c;. For ci 5 c1 or ci 2 c3, draw a straight line from c I tangent to the curve at e,. O n the z-t diagram, draw a straight line from zi with slope -ui to intersect a straight line from 2, with slope - u i . Draw a series of straight lines from z j with slopes - u i , ending with a horizontal line from z,. The resulting envelope is the settling curve. For c1 < ci < c3, the first part of the settling curve is similar to that shown in Figure 4, except that the straight line from z3 intersects that from zj. Straight lines from a series of heights between zj and z,, form an envelope which completes the settling curve. The final result for c1 < ci < cz is shown in Figure 6. Types of Flux Plots and Their Significance
Various types of flux plots can be suggested. Which have actually been observed and what do they reveal
TIME
Cl
t
c2
c4
El
c5
CONCENTRATION, c (M/;)
Figure 2. Determining local solids concentration. See explanation at top offirst column on page 20
n
Figure 5. Flux plots can be more complicated than that of Figure 3. T h i s one is doubly concave, doubly convex
. - . 1 *
-I v,
x'
=
3
P
8 REGION REGION
REGION I RE
1
(2
(1
c3
c-
tcr
CONCENTRATION, c (M/L3)
Figure 3. A doubly concave flux plot. T h i s curve can be constructed from a series of tests in which the initial settling rate is measuredfor a wide range of initial concentrations
+cr
TIME ( t )
Figure 6.
Theoretical settling curves can be constructed from T h i s one was constructed from the doubly concave, doubly convex curve of Figure 5
j u x plots more complicated than Figure 3.
REDUCED SOLIDS CONCENTRATION, I1- ( ) TIME It1
Figure 4. Settling curve constructed from Figure 3. See explanation at top of second column on page 20
Figure 7. An actual Jux plot for 6 7 - p rigid glass spherical beads with a density of 2.45 g . per cc. at 76' F. Note that it is doubly concave VOL. 5 7
NO. 2 F E B R U A R Y 1 9 6 5
21
0.04
- 0.03 I
s s 2 g 0.M d
s
z
0.01
0
0.05
0.10 0 15 SOLIOS C O N C E ~ l I O N ,6 4 C
0.20
Figure 8. Actualfixplot fm calcium c a r b m k
Figure 9. Re&ckd batch seftling m s for t k 67-p sphm’cal glars beads (Iiw) cmnpartd with expdmantol rasulrs (From Figure 7)
about the slurry properties? Until very recently, most people believed the singly concave flux plot applied to most slurries. The singly concave flux plot presents no difficulties to understanding. At the two extremes the flux is zero; in one case the solids concentration is zero and in the other, the settling velocity is zero. However, the actual flux plots for 67-p rigid spheres (27) and calcium cafbonate (30,37) shown in Figures 7 and 8 are both doubly concave. What causes the second “hump” in the doubly concave flux plot? The case of rigid spheres, which have the flux plot shown in Figure 7, is easy to understand. At dilute and intermediate concentrations, only fluid-particle interaction occurs. However, the solids in the packed bed are supported entirely by other particles. The change from fluid-particle to particle-particle interaction accounts for the sudden drop to zero flux. The only question is whether the change is continuous or not (25). A great deal of additional evidence for the existence of doubly concave flux plots can be deduced from initial settling rates found in the literature. The earliest example known to the authors is Nichols’ study (7, 27) of a fine clay slime. From a substantial value at 20% solids, the flux falls almost to zero at 30% (Nichols’ Figure 4). Mishler’s slimes 1, 2: 3, and 4, described in his Table I (79)as flocculant clay, sand, and their mixtures, all give “double-humped” flux plots. The data of Figure 4 in the paper by Coe and Clevenger (4) give a flux plot similar to that in Figure 2. Their Figures 5, 6, 7, 8, 10, 11, and 13 also suggest doubly concave plots, as do the data of Ralston (23) in his Tables 3 and 3a. The data of Egolf and McCabe (9)on silica particles (in their Runs 30 to 47 and 61 to 81, and Figures 2, 3, and 4) also indicate doubly concave flux plots. Their curves show only a slight “hump: that is most noticeable in the work with 5 - p silica at 30’ and 40” C. Coulson and Richardson (7) show a flux plot for precipitated calcium carbonate (their Figure 15.4, page 515). Although the values at very high concentrations are missing, the flux plot is obviously “double-humped.” Complete flux plots of this nature are given by Tory (30, 37) and Mallareddy (78). Evidence for the doubly concave flux plot is also found in the settling curves containing two breaks. Ralston states that “Roughly speaking, most ordinary dilute slimes will settle by subsidence until the thickened material reaches about 1&20% solids, at which the surface of the settling material suddenly becomes clearcut and the settling enters the consolidation phase, which it pursues until the pulp contains an average of about 40% solids, where it enters the compacting phase. Some decrease in velocity is noticed on passing from the subsidence to the consolidation phase, but a great decrease in velocity often takes place on passing into the compacting phase.” Ralston’s description (22), quoted above, is the type of settling behavior shown in Figure 6, and indicates a double concave flux plot such as that shown in Figure 5. On a reduced plot, Examples A and B in Tables I and I1 of Coe and Clevenger (4) suggest two breaks in
"
1W
4w
TIML SK.
Figure 10. Predidtd dimcnrionr and position of cowentration gradient during botch settling of 6 7 -rphcricd ~ glass beads in wafn (line) compared witk cxpnimcntal data
the settling curve. The most positive examples of this type have been noted by Shannon and Tory and associates for rigid spheres (8, 25-28) and calcium carbonate (30, 37) and by Egolf and McCabe (9) for silica slurries (their Figures 3 and 4). Successes and Limitations of Theory
If the weight of solids per unit area is held constant, slurries of different concentrations give a family of settling curves as indicated in Figures 4 and 6. Central to the sets derived from both Figures 3 and 5 is that from z2 = c,z,,'c,. Other curves appear as a spray of straight lines from this curve. In Figure 5, the gap between z8 and 25 is filled by a spray of straight lines from the curve from 24.
The flux plot for 67-i~glass spheres in water, Figure 7, is of the type shown in Figure 3 (27). Theoretical curves for this system are compared with experimental results in Figure 9. The existence of two breaks in the
settling curves for dilute slurries is especially significant in that it implies that a concentration gradient is formed between two concentration discontinuities. The width and position of the concentration gradient were experimentally observed. As can be seen in Figure 10 (25), the experimental and theoretical values were in w m plete accord. A 16-mm. film was made showing the rising gradient and its subsequent collapse into the packed bed (8, 25, 26). In the past, the solids concentration was often varied by changing the weight of solids while holding the slurry volume constant. In such cases, it is not immediately apparent whether the results are in accord with Kynch's theory. Velocityconcentration relationships from settling curves, calculated as shown in Figure 2, may be compared to those based on initial rates and concentrations. Boyd and Whitton (2) noted a slurry of uranium trioxide for which this method could be used to predict accurately the settling rates of various concentrations of the same suspension.
Figure 11. DctmniMtion of t h i c k capm'ty and profile from thpux plot. For detail, of we, see text in middh of f i s t column on page 25 VOL 57
NO. 2
FEBRUARY 1965
23
Work and Kohler (35) noted that when slurries of the same concentration were settled from different heights, the settling curves were related by straight lines from the origin through points at which the tangents have the same slope. From this, expressed by Kynch as u = J ( z / t ) , it follows that settling results for a given slurry can be correlated by plotting either u or z / c i z , vs. t/c,zi where ctzl is recognized as the weight of solids per unit area. Gaudin, Fuerstenau, and Mitchell (73) used the former to correlate results for dilute, flocculated kaolin slurries and found that u decreased continuously with relative time ( t / z l ) for slurries initially of the same solids concentration. Contrary to their statements that this was a new result, it was recognized by Coe and Clevenger ( 4 ) that with certain pulps a constant rate of settling did not occur because a zone of intermediate concentration rapidly reached the surface (their Figure 3, curve B, type I1 settling). Gaudin et al. (73) found that settling rates for thick pulps increased with increasing pulp depth. Both this result and their “new tool” of plotting u vs. t were given in 1912 by Mishler (19). Recently, this method has been used by Tory (30) for calcium carbonate slurries. Plots of z / c 1 z 5 us. t/cizi have been used by Tory (30), Yoshioka (36), and Wallis (34) who used the data of Egolf and McCabe ( 9 ) . The results of Yoshioka and of Wallis showed good agreement with Kynch’s theory (17). With thick calcium carbonate slurries (32), agreement was good until a marked concentration gradient reached the fluid-slurry interface. Tory (30, 3 7 ) has made a very complete study of the batch settling of calcium carbonate, the classical compressible solid which has been used by many investigators (5, 6, 76). The flux plot based on initial settling rates was doubly concave, doubly convex as shown in Figure 8. The settling curves predicted from the flux plot were in complete qualitative agreement with the experimental curves. Concentration gradients and the weight of solids per unit area were quantitatively important. For compressible slurries the actual concentration profiles differ from those predicted (32). The major cause of the deviation is that the maximum solids concentration is not reached immediately at the bottom of the bed. Thus the concentration just below the interface is different from that given by graphical extrapolation. However, the slurry does settle at a velocity corresponding to the true interface concentration (32). Thus the deviation from ideal slurry behavior is due to a variation in interface concentration, not a variation in velocity for a given concentration nor the presence of the concentration gradient per se. Increasing weight of solids per unit area increases ultimate solids concentration at the bottom of the bed. I n the deeper beds, the increase in concentration releases fluid which dilutes the solids at the top. This dilution causes the apparent increase in settling velocity with increasing weight of solids per unit area. The apparent effect of the weight solids per unit area and concentration gradients led to a reappraisal of settling “in compression” (32)which brought the above facts to light. Tory and Shannon (30-32) noted that channeling 24
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
was very extensive at intermediate concentrations but disappeared after the sharp break. A marked reduction in channeling at very high concentrations was mentioned by Mishler (20) in his second paper (his Table 3). In some cases, at least, the second “hump” reflects abnormally high settling rates caused by channeling. I n deep beds of uniform concentration, channeling increases as settling proceeds and hence rates increase (30, 37). Thus, channeling is more important at higher column depths (20) and, if maximum velocity is used as the initial rate, the latter may depend on slurry depth (20). Mishler (20) found that settling rates were independent of depth at very high concentrations (his Table 3). There appears to be a limited range of concentrations where channeling is extcnsive, although it may sometimes occur at higher concentrations if sand is present (20) or if the slurry is stirred. The principal weakness of the theory is that it assumes that the concentration at the bottom reaches its maximum value immediately. While for excellent rigid spheres, this assumption is not valid for even slightly compressible slurries (5, 6, 1 7 , 72, 75, 30, 32) and is far from being true for gelatinous precipitates (3, 14, 24). Xevertheless, the local solids concentration is an important 1,ariable even at the very high concentrations classically considered to be “in compression.’’ Even the deviant factors such as “true compression” and channeling are related to the local solids concentration. The former depends on how much solid has settled, and the extent to which the latter occurs is determined in large part by the local solids concentration. Contin uous Thickening
I n continuous thickening, the downward flux relative to the thickener, G, results from two different phenomena: settling of the solids through the slurry and movement of the slurry itself. Thus
G
=
cu $. cz;
(4)
The downard velocity of the slurry relative to the thickener, u , is defined by the equation
(5) where p = density, subscripts f and J refer to liquid and solid, respectively, Q = volumetric underflow rate, and A = cross-sectional area. The slurry velocity is an operating variable. The batch settling contribution enters in the term cu. Csing the extended definition of solids flux given in Equation 4 and flux plot determined from batch tests, equations can be deduced Lvhich explain both the steady and unsteady state of behavior of slurries in a continuous thickener (76, 30, 33, 37). It is possible to account for a thickener’s “solids profile” AUTHOR P. T . Shannon is Associate Professor of Engineering at Dartmouth College, Hanover, S. H. E. M . Tory is Associ-
ate Chemical Engineer at the Brookhatien ;2’ational Laboratory, Upton, Long Island, ;V.Y.
(including the break attributed to rate action), the sudden appearance at abnormally high feed rates of a new zone of constant concentration, the velocity with which it rises, and the effect of detention time a t constant weight of solids per unit area. One must bear in mind that the same factors which cause deviations in batch settling are operative in the continuous thickener. Persons in the field have continuously searched for a quick and reasonably accurate method for sizing continuous thickeners and predicting the concentrations which will be present under given operating conditions. This may be very easily done with a straightedge for a given solids flux plot. The procedure is illustrated using the flux plot of Figure 11, which is similar to that obtained for calcium carbonate. Let us assume that the feed concentration, c F , is sufficiently high so that it is not a limiting factor. We also assume that a certain underflow concentration, c,, is required. We wish to find the limiting concentration, cL, and the k a x i m u m throughput. 03 the solids flux curve of Figure 11 we simply pivot the straightedge at point c, on the abscissa until the line first becomes tangent to the flux curve. The point of tangency is the limiting concentration, cL (the “point of compression”). The interaction with the ordinate gives the throughput per unit area, Sezt. The thickener area required is then simply
A,,,
W = dezt
The intersection of the straight line with the flux curve gives the concentration occurring between the feed well and the thick bed. Of course, one may just as easily determine the underflow concentration for a given throughput per unit area. A complete explanation of this simple construction is given by Yoshioka (37) and Tory and Shannon (30, 33). This method avoids the necessity of drawing a different flux curve for every value of slurry velocity as in Hassett’s method (76). I t is possible to construct a straight line tangent to the flux curve at two points as shown by the dotted line in Figure 11. Under these operating conditions, the concentrations cA, cN, and cL all may exist in the constant area portion of a continuous thickener. Thus we have a very simple explanation of the sudden appearance of a zone of intermediate concentration, cN (not the feed concentration), in a n overloaded continuous thickener, This phenomenon, first reported by Comings in 1940 (5),was finally explained by Tory and Shannon in 1961 (30, 33). The accuracy of the behavior predicted for a continuous thickener directly from the solids flux plot determined from initial rate data obtained in batch tests depends on the accuracy of the batch data and is subject to all of the second-order effects discussed previously. I t must also be borne in mind that for some slurries there may not be a unique relationship between velocity and concentration. For such slurries, this analysis does not apply. However, even in this case, the analysis may well provide greater insight.
NOMENCLATURE
A c
G
Q
S
t U U
z
W E
(1 I.L
= cross-sectional area, L2 solids concentration, M I L 3 = total solids flux = cu cu, M / L 2 T = volumetric flow rate, L3/ T = batch solids flux = cu, M / L Z T = time, T = batch settling velocity of solids, L / T = slurry velocity relative to thickener walls, L / T = height from bottom of vessel, L = throughput, M / T = liquid volume fraction, dimensionless = solids volume fraction, dimensionless = microns, L =
- e)
+
SUBSCRIPTS extrapolated value initial value-Le., at time zero liquid solids tangent point on flux curve of chord from initial flux interaction of flux curve and chord from final flux tangent to curve (Figure 2 ) or chord which is tangent to the curve a t two points (Figure 4) at first inflection point of flux curve tangent point on flux curve of chord from final flux or chord which is also tangent at another point of higher concentration at second inflection point of flux curve tangent point on flux curve of chord which is also tangent at a point of lower concentration
REFERENCES (1) Ashley, H. W., Mining and Sci. Press 98, 831 (1909). (2) Boyd, A. W., Whitton, J. L., Can. J . Chem. Ens. 36, 217 (1958). (3) Bretton, R. H., “The Design OS Continuous Thickeners” (report), Yale University, New Haven, Conn., 1949. (4) Coe, H. S., Clevenger, G. H., Trans. A.Z.M.E. 5 5 , 356 (1916). (5) Comings, E. W., IND.ENO.Cmht. 32, 663 (1940). (6) Comings, E. W., Pruiss, C. E., De Bord, C., Zbid.,46, 1164 (1954). (7) Coulson, J. M., Richardson, J. F., “Chemical Engineering, Vol. 2,” Pergamon Press, London, 1955. ( 8 ) De Hass, R. D., M.S. thesis, Purdue University, 1962. (9) Egolf, C. B., McCabe, W. L., Trans. A.Z.Ch.E. 33, 620 (1937). (10) Fitch, E. B., Trans. A.I.M.E. 223, 129 (1962). (11) Gaudin, A. M., Fuerstenau, M. C., International Mineral Processing Congress, pp. 115-127, Inst. Mining & Metallurgy, London, 1960. (12) Gaudin, A. M., Fuerstenau, M. C., Trans. A.Z.M.E. 223, 122 (1962). (13) Gaudin, A. M., Fuerstenau, M. C., Mitchell, S. R . , Mining Ens. 11, 613 (1959). (14) George, E. T., Ph.D. thesis, Yale University, New Haven, Conn., 1955. (15) Goldfarb, M., B.S. thesis, University of Illinois, Urbana, Ill., 1939. (16) Hassett, N. J., Ind. Chemist 34, 116, 169, 489 (1958); 37, 25 (1961): 40, 25 (1964). (17) Kynch, G. J., Trans. Faraday SOC.48, 166 (1952). (18) Mallareddy, V., M . Eng. thesis, McMaster University, 1963. (19) Mishler, R. T., Ens. Mining J . 94, 643 (1912). (20) Mishler, R. T., Trans. A.I.M.E. 58, 102 (1918). (21) Nichols, H. S., Mining Sci. Press 97, 54 (1908). (22) Ralston, 0. C., Ens. Mining J. 101, 763 (1916). (23) Ibid., p. 890. (24) Sala, L. M., M.S. thesis, Yale University, New Haven, Conn., 1952. (25) Shannon, P. T., De Haas, R. D., Stroupe, E., Tory, E. M., IND.ENC. CHEM. FUNDAMENTALS 3, 250 (1964). (26) Shannon, P. T., De Haas, R. D., Tory, E. M., Chemical Engineering Symposium, Division of Industrial and Engineering Chemistry, ACS, University OS Maryland, College Park, Md., Nov. 14-15, 1963. (27) Shannon, P. T., Stroupe, E., Tory, E. M., IND.ENCI.CHEM.FUNDAMENTALS 2, 203 (1963). (28) Stroupe, E., M.S. thesis, Purdue University, Lafayette, Ind., 1962. (29) Talmage, W. P., Fitch, E. B., IND.ENO.CHEM.47, 38 (1955). (30) Tory, E. M., Ph.D. thesis, Purdue University, Lafayette, Ind., 1961. (31) Tory, E. M., Shannon, P. T., paper presented a t 13th Chem. Eng. Conference C.I.C., Montreal, Oct. 21, 1963. (32) Tory, E. M., Shannon, P. T., paper presented a t ACS meeting, Chicago, Ill., Aug. 30-Sept. 4,1964, IND.ENQ.CHEM.FUNDAMENTALS, in press. (33) Tory, E. M., Shannon, P. T., paper presented a t A.1.Ch.E. Meeting, New Orleans, La., Feb. 27, 1961. (34) Wallis, G. B., Symposium on Interaction between Fluids & Particles, Instn. Chem. Engrs., London, June 20-22, 1962. (35) Work, L. T., Kohler, A. S., IND.ENC. CHEM.32, 1329 (1740). (36) Yoshioka, N., Hotta, Y . , Tanaka, S., Kagaku Kogaku 19, 616 (1955). (37) Yoshioka, N., Hotta, Y., Tanaka, S., Naito, S., Tsugami, S., Zbid., 21, 66 (1957).
VOL. 5 7
NO. 2 F E B R U A R Y 1 9 6 5
25
