I
MADHUKAR V. MHATRE and R. C. KINTNER Illinois Institute of Technology, Chicago, 111.
Fall of Liquid Drops through Pseudoplastic Liquids Apparent viscosit velocity of liquid
ALTHOUGH nearly all the industrially important cases of fluid-fluid interchange employ clouds or streams of the'dispersed phase, the study of single liquid drops moving in a liquid field may gain a more detailed knowledge of the forces involved and lead to the establishment of more realistic models upon which to base mathematical analyses of the behavior of multiparticle systems (4). A number of attempts to analyze mathematically and express the speed and motion of a liquid drop moving in a liquid field have been made, but an experimental approach seems more profitable except in regions where viscous forces are completely dominant. The shear stress us. shear rate behavior of pseudoplastic fluids is adequately described for our purposes by the powerfunction equation T,
= K
($)n K =
(%)"-I
(g)
(1)
in which n is a fluid behavior index and K is a fluid consistency index. The group K (du/dr)"-l is termed the apparent viscosity. The value of n for a pseudoplastic liquid must be less than unity. Experimental Drop liquids used were of reagent grade, purchased from laboratory supply companies. Castor oil, USP, was from a similar source. Substances added to water to produce solutions of higher con-
Literature Background Subject Ref. Motion of drops moving in a liquid field (1, 3, 6-8,14) Effect of field viscosity (69 14) Oblate and tear drop shapes (1, 10, 12-14) Non Newtonianism, power function equation, and Brookfield viscometer (0) Experimental techniques used (S,14)
-
of the terminal udoplastic liquid
s the key to ps moving thr
sistency (Table I) were Lytron 890 (Monsanto Chemical Co.), sodium carboxymethylcellulose (Hercules Powder Co.), and corn sirup (Corn Products Refining Co.). Glycerol (Armour and Go.) was used in undiluted form. A Brookfield rotational viscometer was used to evaluate K and n of Equation 1, by Metzner and Otto (9). e determined with a specific bottle and interfacial tensions with a Cenco DuNuoy tensiometer. Drops were introduced into a column of field liquid by the proved pour technique (3, 74). Terminal velocity was determined by timing, with a handc timer, the fall of a a measured distance. Drop shape was recorded by photographing a series of drops for each uids that attacked the tank, a round glass inside the square tank istortion was checked spot occupied by the drop during photography. Illumination by a strong electronic flash unit permitted use of the very slow Kodak Contrast Process Ortho film at f 16. Density differences between phases and interfacial tension were held constant to facilitate a valid comparison of the various types of systems. Discussion Values of flow behavior index, index, K , are tabulat the three types of fie
Drop Shape. If the field fluid be Newtonian, small drops will be spherical. As drop size is increased, distortion to a hen to an oblate ell 11 occur. Internal en observed in drops low Reynolds numbers. Very large drops moving in a very
,
viscous field will exhibit an indented rear surface caused by eddy currents. Such shapes have been reported (7) for drops of chloroform in glycerol and have been photographed by the present authors. If the field fluid be pseudoplastic, an entirely different series of shapes will be observed. Very small droplets will be spherical. At that drop size where surface forces may be overcome by viscous drag, the shape will change to an ovate one, with the larger end leading. As drop size is increased, the ovoid shape becomes more pronounced. A size is eventually reached at which the rear stagnation point is destroyed and drop material pulled out into a trailing thread or filament which breaks up into a myriad of tiny droplets. Savic (73) and Garner, Mathur, and Jenson (7) have reported this phenomenon for water drops falling through castor oil. The latter is mildly pseudoplastic (n = 0.965). Because of the small density difference of this system, there is but little distortion from a sphere at the very large drop size (2.25-cm. diameter) at which the rear stagnation point is destroyed. Aluminum powder inside the drop causes the ovate shape to appear at smaller drop sizes. Savic's explanation would predict the tear drop shape for a drop moving in any liquid, Newtonian or not. Saito (72) carried the Hadamard-Rybczinski (2, 77) analysis to the second approximation to derive an expression which could predict a prolate ellipsoidal shape for a drop of specific gravity greater than 10 moving in a highly viscous Newtonian fluid. Phillippoff (70) observed a n inverted teardrop shape for air bubbles rising through non-Newtonian solutions of a time-dependent nature. Still larger air bubbles will assume a spherical cap shape with a well-rounded edge. Phillippoff's qualitative explanation of the shape agrees well with accepted rheological theory. Surface viscosity, like Savic's comparaVOL. 51, NO. 7
JULY 1959
865
7
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EQUIVALENT
EQUIVALENT
DIAMETER, C
Table 1.
890
apparent viscosity of the field fluid will result in two unique curves. They indicate that the drop size at which such transitions occur is inversely proporLiona1 to apparent viscosity. Terminal Velocity. Terminal velocities of nitrobenzene drops falling through stationary Lytron 890 pseudo-
Physical Properties of Test Liquids PO
0.7
I
0.8
0.9
PF
AP
PD
X 10'
U
T
CMC
solutions also
plastic field fluids are plotted in Figure 1 and for nitrobenzene drops falling through ChlC solutions in Figure 2. Small spherical drops fell at a velocity which increased moderately with increasing drop size. As the ovate shape developed, velociry increased more rapidly with increasing drop size When
Table II. Values of Flow Behavior Index, n, and Fluid Consistency Index, K, for Field Fluids Field Fluid
Nitrobenzene-Lytron 890 solution 1.1920 1.1920 1.1930 1.1930
0.9984 0.9997 1.0010 1.0030
0.1936 0.1923 0.192 0.190
1.738 1.738 1.738 1.738
23.70 23.30 23.87 23.78
29.35 29.45 26.55 24.75
Lytron 890 solution
Nitrobenzene-CMC solution 1 2 3
1.1920 1.1940 1.1920
1.0010 1.0070 1.0060
0.191 0.187 0.186
1.738 1.738 1.738
24.47 24.65 24.84
28.50 21.50 29.00
CMC solution
Tetrachloroethane nitrobenzene-corn sirup solution 1 2 3 4 5
1.5150 1.5260 1.5370 1.5420 1.5640
1.3230 1.3310 1.3340 1.350 1.3710
0.192 0.195 0.193 0.192 0.193
1.625 1.518 1.436 1.429 1.368
24.65 24.70 24.87 24.38 24.92
23.80 24.25 25.10 24.20 24.95
866
I
DIAMETER, CM,
Figure 2. U V S . d curves for drops in provide comparison points
cross
(Cgs units)
System (Drop-Field)
0 - - - 4
I
M.
Figure 1. U vs. d curves for drops in Lytron those for drops in corn sirup at significant points
tively rigid surface layer, cannot be used to explain rhe prolate form in one fluid and the oblate form in another, as it does not differentiate between Newtonian and non-Newtonian fluids. A plot of drop size at which transition to ovoid shape occurs and at which the trailing filament appears against the
3
Q
1 2 3 4
1 2 3
n
IC
17
0.562 6.77 0.554 8.31 0.540 10.40 0.516 16.01
29.35 29.45 26.5.5 24.75
3.91 5.11 6.13
28.50 21.50 29.00
1.000 2.80 1,000 3.61 1,000 4.63 1.000 8.45 1.000 17.91
23.80 24.25 25.10 24.20 24.95
0.879 0,855 0.806
Corn sirup solution
INDUSTRIAL AND ENGINEERING CHEMISTRY
1 2 3 4 5
NON-NE WTON I A N FLU1D S
Figure 3. Drag curves correlate all data for falling drops
ward, returning very nearly to the curve, All of the 11 points from Figure 1 fall on curves showing the same characteristics as the curves for corn sirup. The data for drops falling through CMC solutions provide but three points of intersection on Figure 2. The triangles of Figure 3 represent these three points. One may conclude that an apparent viscosity defined in terms of a drop falling through an equivalent Newtonian liquid offers a convenient means of correlating the terminal velocity data for a drop falling through a pseudoplastic liquid. Acknowledgment The authors express their appreciation of the financial assistance of the National Science Foundation, Washington, D. C., in carrying out this work.
R e =fthe transition to tear drop shape with trailing filament was complete, the slope of the curve lessened. Terminal velocities of drops falling through corn sirups varied directly with diameter for the drop sizes used. This is due to a set of compensating factors. Stokes law for rigid spheres indicates that velocity should vary as the square of the diameter. Internal circulation and drop distortion change this relation, but the straight-line relationship is not true for a wider size range. There should be an increase in average shear rate of field fluid near the drop surface as fall velocity is increased. This increase of shear rate will decrease the apparent viscosity of the non-Newtonian fluid. I t is, therefore, desirable to formulate a relation between apparent viscosity or average shear rate and other variables involved in the experiments. For this purpose, reasoning suggested by Metzner and Otto (9)for the evaluation of average shear rate in a mixer was followed. Corn sirups, when tested by the Brookfield rotational viscometer, showed the flow behavior index, n, to be unity. The apparent viscosity of the non-Newtonian field fluid for a particular drop size was assumed to be the same as that of corn sirup through which a drop of the same size, density difference, and interfacial tension would fall with the same terminal velocity, the effects of slightly different drop shapes and possible internal circulation patterns being neglected. Points used for this comparison, therefore, are the points of intersections of terminal velocity curves of corn sirup and non-Newtonian field fluids (Figures 1 and 2). Drag Correlation. One of the most familiar methods of presenting a relationship among the variables involved in the movement of a submerged object through
a fluid field is plotting a drag coefficient against a Reynolds number
(2) on logarithmic coordinates, as in Figure 3. The solid lines show the fall of drops through corn sirup. All fall below the line for rigid spheres. This is considered due to internal circulation in the drops, resulting in an increased velocity and hence a lowered drag coefficient. The analytical solution by Hadamard (2) and Rybczinski ( 7 7 ) for laminar flow in Newtonian liquids indicates a Co value only "3 that of Stokes. As drop size is increased, the drops falling through corn sirup become distorted to oblate ellipsoids, and present a greater projected frontal area and an increased drag coefficient. This is observed as a series of solid lines, one for each specific corn sirup solution, which curve upward to cross the line for rigid spheres. In these regions, the effect of increased form drag due to distortion from a spherical shape has become predominant over the effect of internal circulation. The data indicate that circulation was induced in drops moving at Reynolds numbers of 0.025 or lower. The open circles represent points for drops falling through Lytron 890 solutions. The apparent viscosity was arrived at as described above. Because the drop diameter and terminal velocity are the same in both corn sirup and Lytron solutions, the points would lie exactly on a corn sir;p curve except for the effect of field density. As this item appears in the numerator of the Reynoi& number and the denominator of the drag coefficient, a small increase will cause the former to increase and the latter to decrease. The resulting point will move to the right and down-
Nomenclature C , = drag coefficient (see Equation 2) d = drop equivalent diameter, cm. g = gravitational acceleration, cm./ sec.2 K = field fluid consistency index, (dynes) (sec.)n/sq. cm. n = flow behavior index r = radius, cm. R e = Reynolds number (see Equation 2) u = velocity, cm./sec. U = gross terminal velocity of drop, cm./sec. y = viscosity of field Auid, poise pa = apparent viscosity, poise p D = viscosity of dro p = density, A p = drop density minus field density, gr ams/ml. T( = shear stress at interface, dynes/sq. cm. Literature Cited (1) Garner, F. H., Mathur, K. D., Jenson, V. G., Nature 180, 331 (1957). (2) Hadamard, M. I., Comfit. rend. 152, 1735 (1911). (3) Hu, S., Kintner, R. C., A.I.CI2.E. Journal 1, 42 (1955). (4) Hughes, R. R., IND. ENG. CHEhc. 49, 954 (1957). (5) Johnson, A. I., Braida, L., Can. J. Chem. Eng. 35, 165 (1957). (6) Keith, F. W., Hixson, A. N., IND. END.CHEM.47, 258 (1955). (7) Klee, A. J., Treybal, R. E., A.I.Ch.E. Journal 2, 444 (1956). (8) Licht, W., Narasimhamurty, G. S. R., Ibad., 1, 366 (1955). (9) Metzner, A. B., Otto, R. E., Ibid., 3, 4 (1957). (10) Phillippoff, W., Rubber Chem. and Technol. 10, 76 (1937). (11) Rybczinski, W., Bull. Acad. Sci. Cracovie A, 40 (1911). 112) Saito. S.. Sci. Redt. Tohoku Imb. . Unzv. 2,'179 '(1913). ' ( l 3 ) Savicy p., Natl. Research Labs*, Ottawa, Canada, Rept. MT-22 (July IL oc1\ / d J ) .
(14) Warshay, M., Bogusz, E., Johnson, M., and Kintner, R. C., Can. J . Chem. Eng. 37, 29 (1959).
RECEIVED for review December 1, 1958 ACCEPTED May 14, 1959 VOL. 51, NO. 7
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JULY 1959
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