IN D U S T R I A L A N D E N G IN E E R I N G C H E M I S T R Y
840
Vol. 36, No. 9
rently amorphous substances, were found on the surface of beads of vitreous compositions of P205/CaO mole ratio 1.4-1.9; these compositions had been made by reaction of rock phosphate with Pz05in excess and had been stored in capped bottles in the laboratory for five years. Autoclaving of pure, finely ground calcium metaphosphate with small amounts of water for a few minutes a t 180" C. and then cooling gave practically complete conversion of metaphosphate to monocalcium phosphate. In boiling dilute acids, complete hydrolysis of calcium metaphosphate to orthophosphates requires several hours, and in aqueous extracts a t room temperature the hydrolysis may continue for months (IO). I n the presence of limestone the hydrolysis of calcium metaphosphate is followed by formation of dicalcium phosphate, which was identified in the solids obtained by boiling an aqueous slurry of the stoichiometric proportions of pure calcium carbonate and calcium metaphosphate.
physical Laboratory, Carriegie Institution of Washington. The generous cooperation of 8. B. Hendricks, K. I>. Jacob, and W. L Hill of the-U. S. Department of Agrirulture is acknowledged.
ACKNOWLEDGMENT
(8) Hill, W. L., Faust, G . T., and Reynolds, D. S., A m . J . Sci., 212
The authors acknowledge the encouragement and aid of R. L, Copson, J. W. H. Aldred, E. H. Brown, and other members of the TVA Chemical Engineering Staff in obtaining the data and preparing this paper. They are grateful also for the helpful advice of J. F. Schairer and other members of the staff of the Geo-
(9) Jacob, K. D., and Ross, W. H., J. Agr. Research, 61, 539 bU
LlTER4TURE CITED
(1) Adams, J. R., and Ross, 1%'. H., IND.ENG.CHEM.,33, 121-7 (1941).
(2)
Bariett, R. L., and McCaughey, W. J., Am. Mineral.,
27, 680
95 (1942). (3)
Bartholomew, R. P., and Jacob, K. D., J . Assoc. Oficiai A ~ T
(4)
Copson, R. L., Pole, G. R., and Baskervill, W. H., IND. E ~ G CHEM.,34,26-32 (1942). Curtis, H. A,, Copson, R. L., and Ahrams, A. J., Chem. & Met
Chem., 16, 698-611 (1933). (5)
Ena.. 44. 140-2 (1937). (6) CurtG,'H. A., Copson, R. L., Abrams, A. J., and Junkins, J X..
Ibid., 45,318-22 (1938). (7) Frear, G . L., and Hull, L. H., IND. ENG.CHZM.,33, 1560 6 (1941). 457 (1944). (1940).
(10) MacIntire, W. H., Hardin, L. J., and Oldham, F. D., IND ENG. CHEM.,29, 224-34 (1937). (11) Tromel, G., Mitt. Kaiser-Wilhelm Inst. Eisenforsch. Dusseldorf 14,25-34 (1932); Stahk u. B k n , 63, 21-30 (1943).
RATE OF SEDIMENTATION Suspensions of Uniform-Size Angular Particles
HAROLD €3. STEINOUR Portland Cement Association, Chicago, 111.
Rates of sedimentation are reported for suspensions of closely sized emery particles, both flocculated and nonflocculated. Except for the value of an experimental constant, one rate equation applies to both states, provided the flocculated suspensions are highly concentrated. Comparison with previous tests on uniform spheres indicates that a portion of the liquid suspension medium is carried down with the angular emery particles during their fall, whether the suspensions are flocculated or not. The question as to 3s hether this liquid is bound to the particles or simply stagnant is studied, and evidence is shown to support the latter view.
I
N T H E first article of this series (15) equations were developed for the sedimentation rates of dispersions of uniform spheres. Under conditionswhere the Stokes law gives the velocity of a single isolated sphere, the velocity at anx concentration of spheres is given by Equation 24 of the first paper: Q = vieeZ10--%.82(1--e)
where the function 10-1.82(1-e) is empirical. At values of c between 0.3 and 0.7, Equation 24 is practically equivalent to Equation 23 (16): €3 Q = 0.123Va -_ 1 . 5
which was derived, except for the value of the proportionality constant, by using the hydraulic radius to denote the size of the flow space around the spheres and by assuming that no additional variable shape factor was needed. I n the present article the sedimentation of angular particles is considered, starting with the equations for spheres. The experiments on which the study is based embraced both the disperse and flocculated states. Low Reynolds numbers (Table 1V) ensured viscous flow in the displaced liquid. Particles of very small size were used td permit flocculation; they were closely sized in order to obtain uniform settlement in the disperse or nonflocculated state. Flocculated suspensions in which many sizes are present will be treated in a third article.
SEDIMENTATION TESTS ON EMERY POWDERS
A commercial emery powder was fractionated by air separation; two fractions called A and B were used in sedimentation testa. Their densities were 3.79 and 3.77 grams per cc.,. respectively. The appearance of emery A is shown by Figure 1. Size analyses (Table I) were obtained for both roducts by an adaptation of the Wagner turbidimeter method 6 8 ) . The sedimentation tests Rere made in water. At first, in tests in which flocculation was to be avoided, no dispersing agent was added, for the fresh powders did not flocculate. However, as a safeguard 0.1274 sodium hexametaphosphate was used in later tests. Absence of flocculation was shown by direct observations with a binocular microscope, and by the firmnevs and constant density of the sediments produced by different initial concentrations of solid. I n all tests in which flocculation was desired, zinc sulfate was used, chiefly a t 0.12%. All suspensions were mixed for 2 minutes with an electrical mixer and, except as noted in the tables, were tested in a cylindrical glass jar about 100 mm. in diameter, filled to a height of 40 to 60 mm. These dimensions were chosen in order that effects of Ta!l friction would be negligible at the center of the jar (14). A micrometer microscope reading to 0.001 mm. was used to follow the change in level of the suspended solids. Readings were timed 1' C. to 1 second. Temperatures were 24' In all tests on flocculated suspensions the readings were taken on a float placed centrally in the jar (Figure 2). The float was like one used by Powers (f4),a thin disk of Bakelite with a thread-like glass stem attached to its upper face. The densities were such that the float rode a t the plane of separation of the suqpension and a layer of vvater which was placed on top in
September, 1944
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Figure 1. Photomicrograph of Emery A (X 500)
quantity sufficient to cover the disk. Readings were taken by sighting on the tip of the stem. I n suspensions that were not flocculated, wall friction was not appreciable; hence, in tests on these suspensions the float methad was eventually abandoned and readings were taken directly on the level at the wall of the vessel. I n all cases the suspensions settled with satisfactorily sharp upper boundaries. Curves of subsidence of the surface of a suspension against time are shown in Figure 3. Their shapes are fairly typical for most of the tests, but a t values of B below 0.65 the nonflocculated suspensions gave curves which sagged a little in the central portion (Figure 4). The effect was distinctly noticeable in tests on emery A at B = 0.60 and 0.577, and those tests were omitted from further study. Liquid content E = 0.577 was the lowest at which a good mixture could be made. Results of sedimentation tests are shown in Table 11, arranged in order of increasing dilution. The rates of settlement are for the initial constant-rate periods.
841
The manner of its occurrence suggested that some of the water was not taking part in the general flow of liquid relative to the solid particles which occurs during sedimentation. Hence, Powers assumed that a small quantity of water remained with each falling particle. Symbol wawhich he applied to the subtractive term in his equation stood for “immobile water”; the same symbol will be used here. Kozeny (11) and Carman (3,6) also found that a term equivalent to wIwas needed in a few instances in an equation, similar in principle to 23 ( l b ) , which Kozeny (11,19) derived for laminar flow through granular beds. The need for the term arose only in a few water permeability tests on clays. The three authors cited above concluded that, in tests reguiring the w,term, the need probably arose because of liquid which in some way became bound to the particles. However, Fair and Hatch (7) who derived an equation like that of Kozeny, believed that stagnation prevented some of the liquid from contributing to the flow. Such an occurrence would be analogous to the separation from laminar flow observed by Johansen (9) at sharp-edged orifices in circular pipe. I n nonflocculated suspensions any liquid that fails to take part in the general flow must be in separate increments associated with the individual solid particles, irrespective of whether it is bound to the particles or stagnant at angularities in their contours. If the amount of such liquid per particle is inde-
TABLE I. SIZEANALYSESOF EMERIES A AND B FROM SEDIMENTATION
IN
Diameter, Microns
50 -40 40 -30 30 -20 20 -18.5 18.5-17 17 -16 16 -15 16 -14
WATER, USING SODIUMHEXAMETAPHOSPHATE AS DISPERSANT
Emery A Emery 33 % b y Wi. % b y Wh.
2.0 1.7 . 2.2
1.7
0.8 1.6 3.3 1.1 1.4 0.5 0.3 0.9 2 _ 2_ 4.2 3.5 24.9
Diameter, Microns
10-9 9-8 8-7 7-43 6-5 5-4 4-3 3-0
1.4 19 3.3 6.8 14 -13 12.6 . is -is 22.0 12 -11 24.1 11 -10 14.4 Sp. surface. calod. &e for spheres sq. cm./oc. Av. diam. crtlcd. fromsp. s u r f a d , microns
Emery A, Emery B, % b y Wt. % b y Wt.
5.2 0.1 0.1 0.0
0.1 0 0
0.1 0.4 Emery
A 4930 12.2
23.3 17.6 11.1 2.1 0.4 03 0.0 0.5 Emery €3
6260 9.6
RATES OF SEDIMENTATION O F NONFLOCCULATED SUSPENSIONS
Rate data that conform to Equation 23 of the first paper fall E ) ] % is along a straight line through the origin when [&(1 plotted against E. In Figure 5 this method of plotting has been applied to the data for the nonflocculated suspensions of emeries A and B. The dashed lines in Figure 5 represent Equation 23 (16) as applied to the systems under study. I n determining the values of V athe average diameters reported in Table I were used. It is evident that the experimental points do not conform to Equation 23. However, up to E = 0.75 the data for both emeries are represented by
-
& = 0.176 Va- (e- 1-0.168)3 € Ltn equation similar to 23 of the first paper but differing, in the term on the right, both in the magnitude of the initial factor and in the nature of the cubed quantity. The function of e shown by Equation 1 is the same form as that found by Powers (14) for flocculated pastes of portland cement and water. For those pastes the subtractive term was considerably larger than 0.168 and varied with the cement used.
Figure 2.
Sedimentation Test i n Progress
Vol. 36, No. 8
INDUSTRIAL AND ENGINEERING CHEMISTRY
$42
pendent of the nearness of other particles, then OF EMERIESA A N D B IN ~OO-MM.-DIAMETER TABLE 11. SBDIMENTATI~N for a given powder the total amount should be proVBSSEL
Fluid Porosity Content R a t e of Settleof %J of Fluid Settlement ment, Sediment. in Total Test NO. Initial "rp of % of (Cm./Seo.) (ChronoEieight, Vol. x 106 Initial Settled Height Vol, (& X 108) logical) Mm. ( 6 X 102) Emery A, Nonflocoulated Float 55 65.0 1300 32 48 2 0 2155 42 48 56 70.0 1 0 Float 3390 52 48 8 0 41 75.0 Float 4 0 Sjde 57 75.0 3355 53 47 5410 63 46 3 0 56 80.0 Side Emery .L Flocculated R 56 B 0.12 51 60.0 566 Float 56 53 62.5 14 12 0.12 736 Float 7 0.12 806 ;. 63.7 Float 932 68 85.0 9 0.12 , 55 Float 27 58 1452 52 8 0.12 59.6 Float 26 59 1520 13 0.00 58 70.0 Float 35 61 2500 42 75.0 10 0.12 Float 64 4700 43 11 0.12 80.0 Float 60 62 9930 14 0.12 73 44 85.0 Float En-tery A . Nonflocculalted 32 14 23 0.12 57 62.5 1108 Float 36 21 0.12 45 56 65.0 1371 Float 46 45 2185 20 0.12 56 70.0 Float 46 22 0.12 46 2190 Float 50 70.0 54 45 Side 15 0.12 42 76.0 3280 55 45 16 0.00 75.0 3270 Side 45 54 42 43 75.0 3210 17 0.20 Side 63 45 42 80.0 19 0.12 5130 Side 46 72 42 86.0 18 0.12 7550 Side Emery R . Nonflocculated Sides 40 60.0 621 27 45 6 0.12 Bideb 58 62.5 675 34 44 8 0.12 7 0.12 Side" 40 85.0 847 38 44 Ride 38 70.0 1350 44 47 I 0.12 Float 37 70.0 149OC 43 47 2 0.12 Side 45 7 5 . 0 2055 56 44 24 0.12 Side 56 80.0 3140 64 44 4 0.12 6 0.12 46 Side 58 85.0 4700 72 Emery 5 , Flocculated Float 32 67.5 658 20 59 11 0.12 Float 36 70.0 841 24 61 9 0.12 ~. Float 28 75.0 1475 35 61 10 0.12 Vessel 68 mm. i n diameter. & Vessel 63 mm. in diameter. e This value waB omitted from the figures. I t is widely difi'erent from the other value ~ Q eI 0.70 and appears t o involve a n error sufficiently large t o warrant rejection. Conon. of Re nlating &gent, % of ReadMater W t . ing
ii
~
-
portional to the volume of the powder. Hence, if CY is the proportionality constant, the immobile liquid per unit of total volume is a(1 e). To correct Equation 23 (16) for such immobile liquid, a(l e) should be subtract,ed from each c in the equation. The limiting velocity, V R needs , no correction since it is found experimentally and should therefore embody the effect of any immobile liquid. The modified equation after some rearrangement is:
-
-
Q = 0.123 (1
-+
(€ CY)*
V,
- "-I)a i - a
-
e
+
If wi is used in place of a / ( l a), a - wi), and the equation becomes:
w/(l
When wi = 0.168 as in Equation 1,0.123/( 1 equals 0.178, and Equation 3 becomw practically identical with Equation 1. Hence the experimental results may be regarded as wholly consistent with the theory that a quantity of liquid proportional t o the volume of the emery remains with the particles during their fall. For both emeries A and B the quantity of thie: liquid is indicated to be 0.202( I E), or one fifth of the volume of solid. According to this theory, E a(1 - E ) is tjhe general expression for the mobile liquid. Therefore, since the linear relation illustrated by Figure 5 was previously indicated (15) to apply up to
-
-
" 0
e, I L
3 M
0
4
8
12
14
16
18
20
Time, Minuter
Figure 3.
Sedimentation Curves f o r E m e r y A at e =
0.65
Figure 4. Sedimentation C u r v e s for Nonflocculated Suspensions of Emery A in Plain Water at e = 0.60
September, 1944
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
843
e = 0.70 when a was zero, it should apply up to e
= 0.75 when with the data.
CY
is 0.2. This agrees satisfactorily 0.10
STAGNANT LIQUID
The fact that the two emery powders of different size particles have the same value of a helps determine whether the immobile liquid is bound or merely stagnant. If this liquid is stagnant, then 0.08 a, the quantity of such liquid per unit volume of solid, should be determined by the shapes of the particles and be independent of particle size, as was -In found. But if the liquid is bound to the particles, m the thickness of the layer should probably be about 0.06 the same for particles of different sizes, and the total quantity would not then remain a constant proportion of the volume of solid, 1 E, when the siae of particle waa changed. It would be more nearly proportional to the exposed surface, the specific surface in sq. om. per cc. times (1 - E). On 0.04 this latter basis wi should be 1.2 times as large for emery B as for emery A. The data in Figure 5 conform much better to the one common value for wb; indeed, as will be shown later, equality of the w i values is also indicated by a more precise method 0.02 of evaluation. However, to obtain more evidence, tests were made on another emery powder of much largec particle sise. A relatively large-grained product of fairly-uniform particle siae waa obtained by sieving a coarse emery powder on screens having square openings, 0.oc 0.0 0.2 0.4 0.6 0.8 62 and 74 microns on a side, respectivcly. The prodE uct, emery E, had a density of 3.93 grams per cc. and was tested for sedimentation characteristics in Figure 5. [Q(l-c)] l / a vs. E for Sedimentation of Nonflocculated diethyl phthalate (density 1.115 grams per cE. a t Suspensions of Emeries A a n d B 22' C., viscosity 0.091 poise at 27.5' C.). No dispersing agent wae needed. The first sedimentation tests were made at 27.5' C., later ones at 23.6' C. All The rates obtained at higher values of E are shown in TabTe IfI; readings were taken on the level of the suspension at the wall thorn determined at 27.5" C. were used in Figure 6. of the vessel. Curves of height of suspension against time were The value of ws indicated by Figure 6 is 0.22, a figure distinctly larger than the value 0.168 found for emeries A and B. This resimilar to those for the finer emeries. The porosities of the sult is the opposite of what would be expected of bound liquid, if sediments were practically constant at 58% of the settled volume. Rates of settlement obtained a t E = 0.60 were exceptionally low the diethyl phthalate is assumed to build about the same thickness of adsorption layer as the water. The large particles of and are not reported; the fluid content was judged to be too close to its lower limit to permit free orientation of the particles. emery E would require much lower values of a and w( than do the particles of A and B if the adsorption layers wereof thesame thickness. It is estimated that the thickness would have to be TABLE 111. SEDIMENTATION OF EMERY E IN DIETHYL PHTHALATE about twenty times as great on emery E particles to account ftw the results obtained. Against the possibility of this occurreaacer Tests in 6&Mm.-Diam. Jar a t Tests in 92-Mrn.-Diam. Jar at 97 K O 17 23.6' C.0 is evidence obtained by the writer in tests on portland cement Fluid Fluid in various liquids, including diethyl phthalate; this evidemm Rate of content Rate of ~ ~ settlement ~ ~ ~ $ of fluid d settlement indicates that there is little change in w i upon changing liquids. Teat No. (om./sec.) Test No. total (am./aeo.) (chrono(chronovol. x 105 vol. x 10s Furthermore, even the estimated layer thicknesses for emeriea logiral) (& X 109 logical) (e X 109 ( 6 x 10') (4x 105) A and B are so great as to comtitute strong independent evidence 347 1-1 62.6 7- 1 76.0b 1153 1-2 62.6 356 7-2 75.0'~ 1191. against the existence of the layers. Assuming that the v o l w 7-3 1203 75.0'~ of bound liquid is approximately equal to the thickness of layer 3-1 66.0 461 7- 4 75.0b 1220 2-2 66.0 455 times the area of surface exposed to liquid, thicknesses of 0.30 8- 1 77.5 1574 8-2 670 3-1 67.6 77.6 1430 and 0.24micron are indicated for emeries A and B, respectively. 3-2 67.5 572 8-3 77.6 1406 [The surface areas were calculated from air permeability data 3-4 1418 77.6 4-1 70.0 739 (13) which indicated values about 1.4 times those obtained by 4-3 70.0 767 9-1 80.0 1806 9-2 4-4 70.0 730 80.0 1868 sedimentation analysis.] Yet BulMey (9) found that in flow 9-8 80.0 $848 of oils the thickness of stationary film does not exceed 0.02 to 926 5- 1 72.5 9-4 80.0 1876 926 6-2 72.6 4 0.03 micron; Bastow and Bowden (1)studied the flow of various 10-1 82.6 2290 1181 0-1 76.0 10-2 82.6 2310 liquids and found that liquid 0.1 micron from a solid surface 1198 6-2 75.0 10-3 82.6 2300 takes part in the movement. 10-4 82.6 2310 a Rates adjusted to 27.5' C. Still further evidence appears in the previous article (16) in by multiplying by 1.16 (factor il-1 85.0 2890 based on tests at e = 0.76). 11-2 85 0 2840 which Equation 23 was found to be adequate for suspensions of * Testa made in 68-mm.-diam. 11-3 86.0 2910 jar. 11-4 86.0 2870 glass spheres that averaged 13.6 microns in diameter. Thorn spheres were only a little larger t$hanthe particles of emery A
?
Y
a
-
% '
INDUSTRIAL AND ENGINEERING CHEMISTRY
average diameters reported in Table I. The ur values as originally estimated were adjusted slightly to make the slopes strictly 1.82; this mas the basis for the statement made earlier that the equality of the w, values for emeries A and B did not rest solely on the evidence from Figure 5. Figure 7 shows that when these WA values are used, the data conform closely to the straight lines even a t the high dilutions; as corrected for stagnant liquid, they conform to the equation found to hold for spherical particles. The stagnant liquid may not be strictly immobile, for a difference simply in the patterns of the streamlines near the surfaces of angular and spherical particles could conceivably cause the difference in results. However, since the flow patterns must be similar in their more general aspects, the close agreement of the theory with the data seems to indicate that any motion in the liquid called “stagnant” must be relatively small. Miscellaneous data are assembled in Table IV to aid in (‘omparing conditions in the suspensions of the various emery powders, The Reynolds numbers are given only for infinite dilution and are calculated from the formula commonly used for spheres. They are not strictly comparable when the particles differ in shape. It should be realized also that a Reynolds number representative of the general flow will decrease with E (4), since both the velocities and the distances between particles decrease.
0.16
0.14
0.12
0.10
.
*/
T= W
--!
Vol. 36, No. 9
008
e 0 06
0 04
0 02
0 00
G COMPARISON WITH WORK OF OTHERS
Figure 6,
[Q(l-e)]’/e 21s. E f o p Sedimentation of Norlflorculated Suspensions of Emery E
and might have been expected t o bind a comparable quantity of water, yet no witerm was needed. The theory that the wzvalues found For the nonflocculated suspensions of emery particles are indicative of adsorbed liquid must be rejected; the theory that there is stagnant liquid which accompanies the angular particles in amounts determined by their shapes is consistent with all data considered thus far. The quantity per unit volume of solid is evidently greater for emery E than for emeries A and B, but the difference in particle shape which this indicates is in accord with the fact that emery E gives sediments of relatively high porosity. RESULTS FOR A L L CONCENTRATIONS
Up to this point the concept of stagnant Iiquid has been tested only with data obtained a t concentrations high enough so that Equation 3 can be applied. The general Equation 24 (16) presented at the beginning of this article provides the opportunity to make a more extensive test of the concept. If the quantity of stagnant liquid remains a fixed proportion of the volume of solid at all concentrations, then Equation 24 should represent the data for the emery if E i s replaced by a quantity representing actual mobile liquid per unit of b t a l volume. This quantity (1 - E), equivalent to (E - wJ /(1 - wJI Substitution is E in Equation 24 gives:
-
1--c where -= volume of solid plus immobile liquid per unit of 1 - WE total volume. If the equation represents the data, a straight line having the slope 1.82 should be obtained when loglo&
(;$)*is
plotted against
(e
- w8)/(1 - w,),the complement
of (1 - e ) / ( l - Wl). This method of plotting was used in Figure 7 which presents the data for the three preparations of emery. The points shown €or emeries A and B at (E - wt)/(l - wL)= 1 were based on the
Equation 4 has been applied to data obtained by Kermaclr, M’Kendrick, and Ponder (10) on the sedimentation of red blood cells, discoidal in shape, apparently uniform in size, and not flocculated. The tests were made on the cells of man, rabbit, ox, and sheep. Gaudin (8) averaged the results obtained for the different types of cells, after converting them to percentages of the estimated velocities a t infinite dilution, and then compared the averages witfi values calculated from several different equations. Table V reproduces part of Gaudin’s table, including the values calculated from the equation of Kermack et al. and those from Gaudin’s best equation. Values calculated from Equation 4 are shown in comparison. The method of applying Equation 4 was graphical, and no weight was given to Gaudin’s 100% rate
TABLE Iv. DATAFROM TESTS ON EMERIES A, B, ANI1 E XONFLOCCULATED SUSPENSIONS A 0.026
Emery Stokes velocity V , cm./sec. Reynolds No. a t inhnite di1n.Q
0.0033 0.168 0.202
Wi
a
B
E:
0.016 0.0016 0.168 0.202
0,098 0.0091a
iN
0.22 0.28
Defined as: particle diam. X liquid density X Stokes velocity viscosity b Particle diameter of emery E was taken as 0.0076 om., corresponding to limiting velocity indicated by Figure 7. a
I__-
TABLE V. APPLICATIONOF VARIOUSEQUATIONS TO DATAOF KERMACK, “ENDRICK, AND PONDER (10) ON SEDIMENTATION OF REDBLOOD CELLS
-
1 e 0.0025 0.006
0.01 0.02
0.04 0.08 0.16 0.32
Exptl. R e t e of Sedimentation, % of ValueEstd. for Infinite Diln. R~~~~ fo; Average tests on calcd. b various Gaudin cells” 97-97 97 93-95 94 86-92 89 78-87 83 67-79 73 46-60 51 22-34 29 7-22 13
6)
Theoretical R a t e of Sedirnentatiou Corresponding to Equation of: Kermack, This article, M’Kendrick, Eq. 4 with Ponder Gaudin wi = 0.167 93 06 97 96 94 95 93 93 90 86 84 86 72 76 72 43 60 63 Neg. b 35 28 Neg. b 7 8
Calculated i n terms of Gaudin’s average percentages by arbitrarily aasuming t h e observed r a t e a t 1 6 = 0.0025 to,be 97% Va in each case. b The equation was derived for low concentrations only.
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
September, 1944
845
why Equation 3 applies as well as it does, for the equation is based on the premise of flow past the individual particles. The increase in W i may mean simply that the contacts or virtual contacts between particles which are evidently a feature of the flocculated state cause the amount of stagnant liquid per particle to b? somewhat greater than that in a dispersion. ABSENCE OF STRUCTURAL RESISTANCE -1.6
-1.8
-2.0
-2.2
-2.4
-2.6
-2.8
0.3
0.5
0.4
0.6
07
0.8
0.9
1.0
(2) -
-
-
Figure 7. Loglo Q [( 1 Wi)/(e Wi) I* us- ( E wi)/(l- W I ) for Sedimentation of Nonflocculated Suspensions of Emeries A, B, and E
since it was not experimental. It is concluded from Table V that the data can be represented as well by Equation 4 as by either of the others with which comparison is made. SEDIMENTATlON RATES OF $FLOCCULATED SUSPENSIONS
Emeries A and B were also tested when flocculated. Those tests made possible a direct comparison of effects in the two states, but flocculated suspensions of a much finer emery powder were also tested to extend the basis for determining the effect of particle size. This finer emery (D) was a commercial preparation. Its density was 3.38 grams per cc. The results of a size analysis are shown in Table VI. I n the tests for rate of sedimentation the 0.12% solution of zinc sulfate was used as flocculant. The tests were made a t 25" C. in the same way as those on emeries A and B, except that they were discontinued before completion of settlement. The data are presented in Table VII. In Figure 8, [&(1 e ) ] % is plotted against E for the flocculated suspensions of emeries A, B, and D. As was true of the data for the nonflocculated suspensions, the points for each product can be represented by a single straight line, except for a few points a t the highest values of E . The slopes of the lines were calculated from Equation 3, the best values of w,being determined by trial. On the whole, there is fair agreement with the equation. Since Equation 3 was derived for nonflocculated suspensions, this agreement indicates a considerable degree of similarity in the sedimentation of the flocculated and nonflocculated suspensions. Nevertheless, it is evident from Table I1 that the flocculation materially reduced the rate of settlement at most concentrations. This is reflected in increased values of wL. This effect of flocculation on the rate of settlement in concentrated suspensions is the opposite of the effect in dilute ones. I n contrast to dilute suspensions, highly concentrated ones may develop so continuous a floc structure that all liquid displaced by the settlement must make its way through the actual floc texture where the resistance is high. Such an assumption helps explain
-
The floc formation assumed above for concentrated suspensions, is essentially a three-dimensional network extending fairly uniformly throughout the whole volume of the suspension, but offering initially almost no mechanical structurd resistance to collapse under the action of gravity. The assumption of negligible structural resistance seems necessary because flocculation causes no change in the numerical factor of Equation 3; support for the 'assumption is provided by determinations of hydrostatic pressure. That the hydrostatic pressure is an index to the nature of the resistance is readily shown. When particles in a liquid form a completely self-supporting structure, they do not, contribute to this pressure which is therefore determined by t h e density of the liquid in the usual way. But in the absence of structural resistance, particles which settle without accelerationl are supported entirely by the liquid, and the hydrostatic pressure is determined by the density of the mixture. No tests for hydrostatic pressure were made on suspensions of the sized emery powders, but fifteen determinations were made on other concentrated flocculated suspensions, most of which were portland cement pastes in which E was 0.58. Structural resistance was thought to be more probable in such suspensions than in those of inert, closely sized particles. The hydrostatic pressures were determined by pouring the suspensions into a vessel in which a manometer tube, partly filled with water and stoppered, was supported with its lower end well above the bottom of the vessel. This manometer was a single tube having an enlarged lower end fitted with a fritted glass disk. Each suspension was brought to a level such that the head of water in the manometer was a little less than that calculated for equilibrium. I n a few minutes after the manometer was unstoppered, the water rose to a height which remained practically constant for some minutes and then declined a t a rate that suggested development of boundary effects at the tube. However, the maximum heights usually prevailed as long as 5 or more minutes after the suspensions were in place, which was long enough to ensure that the constant rates of sedimentation had been es-
.
TABLE VI. SIZEANALYSIS OF EMERY D FROM SEDIMENTATION IN WATER,USINGSODIUM HEXAMETAPHOSPHATE AS DISPERSANT Diameter, Microns
weyt.ht,
Diameter, Microns
15 2.3 2.5 1.1 0.8
weght'
12-11 11-10 10-9 9-8 8-7
0.0 0.3 1.0 1.4 1.1 Sp. surface. calcd. as for spheresa, sq. cm./cc. Av. dism., calcd. from sp. surface, microns
50-40 40-30 30-20 20-15 16-12
Diameter, Microns
7-6 6-5 5-4 4-3 3-O5 12,950 4.0
Weight,
% 21.0 14.8 21 2 24 8 0 2 I
*
Average particle diameter of this portion taken a8 2 microns since rnioroscopio examination indicated absence of extreme fines. (I
TABLE
SUSPENSIONS VII. SEDIMENTATION OF FLOCCULATEI)
Fluid Content,,
Test No. (Chronological) 1 2
3 7
OF
EMERY D
in Total (Cm./Sec,j X 108 Vol. (e X 10%) (Q X 106)
67.6 70.0 72.5 72.7
Fluid Content,
Rate of
yoof Fluid Settlement 95 112 183 171
Test No. (Chronological)
8 4 5 0
% of Fluid
in Total Vol. (e X 10%)
75.0 77.5 80.0 85.0
Rate of Settlement (Cz.(@.)
( Q X 109 246 326 438 956
846
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Val. 36, No. 9
leave (E - w p ) / ( l - wp)as the proportion of the actual fioc space occupied by liquid. This quantity has the same form as ( E - w.)/ (1 - w,) which was previously shown to represent the rnohile liquid when wLresults from stagnant liquid. Liquid in pockets is mentioned merely as a possibility, for t b rate equation is explained more satisfactorily entirely on the basis of the stagnant liquid which, in any case, appears to be responsible for most of wi. The basic assumpiion that there are no channels for liquid to escape without passing through the floc texture ipi in accord with direct observations. Powers, in his work on cement pasteu, usually found no evidence of significant breaks or channels in the paste structure in tests in which his data were in agreement with his rate equation; a t higher dilutions, channels and “boils” could easily be seen, but then the rates were higher than those predicted by the equation. These observations have been confirmed by the writer. Also, many indications of boundary resistance which have been obtained in the work on cement pastes are regarded as further evidence of the continuous nature of the floc structure. I n none of the tests on the emery suspensions were any channels seen. However, it is believed that the poinb in Figure 8 that were obtained a t high dilution and that trend u p ward from the straight lines representing the rest of the data do so because the less concentrated suspensions were too dilute to prevent breaks in the structure, breaks too fine to be seen readily. EFFECT OF PARTICLE
E
Figure 8. [Q(l-e)]‘h os. e for Sedimentation of Flocculated Suspensions of Emeries A, B, and
tablished. In general, the equilibrium heights were within * 1-570 of those calculated by assuming no structural resistance. As the probable experimental error w w estimated to be about the the absence of appreciable structural reeame magnitude (l.W%), sistance was considered to have been demonstrated. FLOC S’rRUCTURE
The assumption that a rather uniform network of p a r t i c k tills all the space in concentrated flocculated suspensions makes no allowance for any liquid outside the floc space proper. It i s presumed that concentrations for which the data can be represented by Equation 3 are so great that there is no tendency for the floe t o occupy less than the total available volume. Howevw9 the data might still be in approximate agreement with Equation 3 if the fine structure were broken by pockets of liquid sf relatively large dimemiom. The existence of random isolated pockets would not conflict with the apparent requirement that liquid escaping from the settling mass should pass thrwgh &bo floo meshwork and contact the individual particles. The pwkets would tend to produce nonuniformity in the flow, but there would probably be readjustments in floc texture, and the c a b of d i m e n t a t i o n might not be far from that of a suspension wbollgp occupied by floc of the same concentration as that in the do@ space proper. Accordingly, it is of interest that liquid in pockets could contribute to wa in the same way as liquid stagnant at angularities in the individual particles. That is, if liquid in psokets mounted to wl, per unit of total mixture, this would
srm IN FLOCCULATED
SUSPFLNSIONS
Particle size affects the value of V,; that it also affects the value of wi when the powders are flocculated is suggested by Figure 8. However, judgment as t o the generality of t h m relation should be reserved until additional data are presented in the third article of this series. The fine emery D, for which ‘IBP is much the largest, had relatively low density and was evidently less pure than the other emeries. It was also less uniform in particle size, on a percentage basis. The w ,values for flocculated suspensions of the various emery powders follow:
Emery A
B
D
Sp. Surface b % Sedimentatio; Method sq. Cm./&r
Diameter. Vicrons
A4V0rag0
i a for Flocculated Suspension8
4,930 6,250 12.950
12.2 9.6 4.0
0.268 0.288
0.360
Among recent investigations of Hocculated suspensions, thorn of Egolf and BIcCabe (6), Ward and Kammermeyer ( l y ) , and Work and Kohler (18) relate to lower concentrations than those dealt with here. The work of Powem (14) on the sedimentation or “bleeding” of portland cement pastes is apparently the most
extensive previous study of highly concentrated flocculated sue pensions with which comparison can be made. Powers9 rate equation differs in some respects from Equation 3, but the differences can be reconciled, as will be shown in the third article OB this series. Although the data of Egolf and NcCabe on 16-micron silica are for more dilute suspensions, they can also be represented against e. Apparently, after a linearly by plotting [&(1 break in slope like that shown for emery A in Figure 8, another linear relation often obtains. CONCLIJSIONS
The effect of parbicle concentration on rate of sedimentation in nonflocculated suspensions settling under conditions of viscoua flow isnot the same for angular particles as it is for spheres. However, the results for the two kinds of particles can be brought into complete agreement by assuming that the angular particlea carry with them a volume of liquid which is a constnnt proportion of the volume of the solid at all concentrations. The quantity
September, 1944
INDUSTRIAL AND ENGINEERING CHEMISTRY
per unit volume of solid appears to be a function only of particle shape. For different preparations of emery the indicated quantities were 20 and 28% of the volume of solid. Several lines of evidence are presented to show that this liquid is not adsorbed by the particles. Apparently it is simply liquid that has remained stagnant at angularities in the particles. Flocculation of concentrated suspensions of the same emery powders that were tested in the nonflocculated state materially reduced the rates of sedimentation. However, the same rate equation could be applied with fair success when modified only by an increase in wd which, for the nonflocculated suspensions, corrects for the liquid that is assumed to be stagnant. It is concluded that in highly concentrated flocculated suspensions there is no opportunity for liquid to by-pass floc structure in escaping from the mass during sedimentation, but that in general this liquid must flow past the individual particles in much the same way as in a nonflocculated suspension. Two possibilities are seen for the increase in wr: (1) The quantity of stagnant liquid may be increased by reason of interparticle contacts caused by flocculation. (2) There may be isolated pockets of liquid distributed through the mass, pockets whose dimensions are materially greater than those of the meshes in the floc structure itself. How such pockets can produce an increase in wi has been Hhown. The possibility that structural (nonviscous) resistance produced by the flocculation may have been a factor in decreasing the rates of settlement was investigated by determining hydrostatic pressures in flocculated suspensions of various powders, but principally in aqueous pastes of portland cement. It was found that there is no appreciable structural resistance during the initial stage of the sedimentation when the constant rate is established. ACKNOWLEDGMENT
The writer was assisted by Richard G. Brusch and Herbert We Schulta in the experimental work reported in this artiole.
847
NOMENCLATURE
Q = initial rate of settlement of top surface of a suspension,
cm. /sec. V . = rate of fall of an isolated sphere as given by Stokes law, cm./sec.; used also to represent rate of fall of an isolated particle when Reynolds number is such tzat a sphere would fall in accordance with Stokea law w, dimensionless constant used by Powers (14) tu, = dimensionless constant analogous to wi a = dimensionless constant denoting volume of fluid per unit volume of solid, that a pears to remain with angular particles during their e = proportion of total volume of suspension occupied by liquid (analogous to porosity in beds of particles)
-
fa8
LITERATURE CITED
Bastow, S. H., and Bowden, F. P., Proc. Roy. SOC.(London), A151, 220-33 (1935). Bulkley, R.. Bur. Standards J. Research, 6,89-112 (1931). Carman, P.C.,J. Agr. Sci., 29, Pt. 2,262-73 (1939). Carman, P.C., J. SOC.Chem. I d . , 57, 225-34T (1938). Carman, P. C . , Trans. Inst. Chem. Engrs. (London), 16, 1138-88 (1938). Egolf, C . B., and McCabe, W.L., Trans. Am. Inst. C h .Engra., 33,620-42 (1937). Fair, G . M., and Hatch, L. P., J. Am. Water Worlcs Assoc., 25, 1551-65 (1933). Oaudin, A. M., “Principles of Mineral Dressing”, 1st d.. Chap. 8,New York, McGraw-Hill Book Co., 1939. Johansen, F. C., Proc. Roy. Soc. (London), A126, 231-45 (1930). Kermack, W. O., M’Kendrick, A. G., and Ponder, Eric, Proc. Roy. SOC.Edinburgh, 49,170-97 (1929). Koaeny, Josef, Kulturtechnilcer, 35,478-86 (1932). Kozeny, Josef, Sitzbe-r. A h a . Wiss.Vien, IIa, 136,271-306(1927). Lea, F. M.,and Nurse, R. W., J . Soo. Chem. Id.,58,277-83T (1939). Powers, T. C . , Research Lab., Portland Cement Aasoo., Bull. 2 (1939). Steinour, H. H., IND.ENQ.C H ~ M36, . , 618-24 (1944). Wagner, L. A., Proc. Am. SOC.Testing Materialrr, 33, Pt. 11, 553-70 (1933). Ward, H. T.,a;nd Kammermeyer, Karl, IND.ENO.CHBY.,32, 62243 (1940). Work, L. T.,and KoNer, A. S., Tbid., 3.2,1329-34 (1940).
PRINTING INKS from Colloidal Dispersions of SOYBEAN PROTEIN
ALFRED F. SCHMUTZLER AND DONALD F. OTHMER Polytechnic Institute, Brooklyn, N. Y.
P
ROTEIN dispersions in diethylene glycol a previous article (98). Printing inks ma persions are nondrying on the printing press, but when printed films of the inks are exposed to steam, they harden immediately. They are not sufficiently water resistant for universal use. To eliminate this defect, reactions designed to lower the water absorption of proteins were investigated. I n aqueous dispersions, soybean protein will not precipitate or gel when reacted with formaldehyde (95). Plastics made from it have relatively high water absorption (1-4). I n contrast to soybean protein, casein and blood albumin gel almost immediately upon the addition of even small amounts of aldehydes to their aqueous dispersions, an indication that the behavior of the former differs considerably from that of the latter. Nevertheless it,
was found that some features of the hardening of casein could be applied to soybean protein. I n order to improve the hardness and durability of casein fibers (artificial wool, lanital), in some instances)small amounts of salts of heavy metals are added to the spinning solution (10, 13, 1.9, 1.6, 20); in others the .dispersed casein is reacted with isocyanates (6, 16, 17, 18),isothiocyanates ( 1 1 , 16),and carbon disulfide (6). Although isocyanates may undergo side reactions in the presence of hydroxyl groups, isothiocyanates and carbon disulfide combine solely with amino and amide groups. The initial step is the formation of thiourea derivatives, followed by condensation reactions with aldehydes, with the formation of modified thiourea resins (9, 10, $4). I n the reaction of proteins with carbon disulfide, hydrogen sulfide is liberated and can combine with aldehydes, with the formation of thioaldehydes, which are much more reactive than the aldehydes from which
