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Analysis of drop formation at conical tips
Analysis of Drop Formation at Conical Tips 2. Experimental S. R A M E S H BABU l Department of Metallurgy, Indian Institute of Science, Bangalore-560012, India
Received September 18, 1985; accepted June 6, 1986 Drop formation at conical tips has been experimentallystudied using the apparatus designed for this purpose and with the aid of movie photography. Experimentshave been performed using the low vapor pressure liquids, chlorobenzene and 1,1,2,2-tetrachloroethane, to study the effect of cone angle, rod diameter, and physical properties of the dispersed liquid on drop formation. The experimental results served to verifythe assumptions made in the theoretical analysis (S. Ramesh Babu, J. Colloid Interface Sci. 116, 350 (1987) on the one hand and to test the validity of the predictions of the models on the other. Based on the present experimentalstudies, it is concluded that the phenomenon of drop formation under infinitelyslowformation rates at conical tips is well understood and the proposedmodel adequately represents the phenomenon up to the instability stage. © 1987AcademicPress,Inc. INTRODUCTION The p h e n o m e n o n of drop formation at conical tips has been analyzed using a theoretical approach and the results are presented in the preceding paper (1). The design and development of the experimental techniques for drop formation studies at conical tips are considered in this paper. The objective of developing these techniques is to verify the assumptions made in the development of the models on the one hand and to test the validity of the predictions of the models on the other. EXPERIMENTAL The design of the apparatus essentially consisted of providing a suitable arrangement to allow liquids to flow down a long metal cone, under controlled flow rates, and to form drops at its tip. An exploded view of the apparatus designed and built is shown in Fig. 1. The apparatus consists of an a l u m i n u m cone (A), fit1Currently with the Research Institute of Mineral Dressingand Metallurgy,Tohoku University, 1,1-2Chome Katahira Sendai-980, Japan.
ring perfectly into an a l u m i n u m socket (B) which is extended into a hollow cylindrical portion (C) of about 3 cm in height and 7 cm in inner diameter to hold the experimental liquid. An a l u m i n u m lid (D) is screwed over the top outer surface of C. A n u m b e r of openings into the reservoir (C), above the cone, are provided. One end of an a l u m i n u m rod (E) is screwed onto the cone and the other threaded end passes through the central hollow cylindrical portion of the lid (F). A spring (G) inserted over the free end of the a l u m i n u m rod (E) is held in position by a nut (H) and is positioned against a small projection on the inner surface of F. An a l u m i n u m knob (I) is then screwed over the outer surface of F. By a clockwise rotation of the knob, the spring is loaded and its tension enables the core to be positioned inside the socket perfectly concentric with it. Therefore, by operating the top knob (I), the width of the gap between the cone and the socket could be varied to obtain controlled flow rates of the liquid. By means of a step provided on the outer surface of the alum i n u m socket, the assembled apparatus could
359 0021-9797/87 $3.00 Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
Copyright © 1987 by Academic Press, Inc. All fights of reproduction in any form reserved.
360
S. RAMESH BABU
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FIG. 1. Exploded view of the metal cone apparatus for drop formation studies. be seated on a 1-in.-thick aluminum plate with a central opening, mounted on an a l u m i n u m tripod stand with a spring-loaded arrangement for finer adjustments of levelling. This entire assembly is housed inside a glass chamber to minimize the external disturbance on the drop Journal of Colloid and Interface Science, Vol. 116,No. 2, April 1987
formation. To study the effect of each of the cone angles, 60, 90, 120, 140, and 150 °, a matching cone and socket assembly was built, each assembly for a specific cone angle. With a slight modification in the design of only the conical portion, as shown in Fig. 2a,
361
DROP FORMATION AT CONICAL TIPS, 2
(a)
(b) FIG. 2. Schematic sketches of (a) a finite cone and (b) a truncated cone.
the same apparatus could also be used for drop formation studies at finite conical tips. For each of the cone angles, 60, 90, 120, 140, and 150 °, a minimum of two finite cones were made to study the effect of rod diameter on drop size. Also, by using a truncated cone of an arbitrary cone angle (Fig. 2b), drop formation at a fiat surface could be studied. When the cross-sectional area of the fiat tip is far greater than the predicted critical contact area of the drop at a flat surface, even multiple drop formation and coalescence of drops could be studied. For the present studies the following two low vapor pressure liquids were chosen: (a) chlorobenzene (L.R. grade, BDH supplies); and (b) 1,1,2,2-tetrachloroethane (L.R. grade, SD supplies). The properties of these liquids relevant to the present studies are summarized in Table I. The values of viscosity, boiling point, and vapor pressure are taken from the
"Handbook of Physics and Chemistry" (2). The percentage evaporation per minute per square centimeter of the surface area of evaporation was determined experimentally for both liquids. Since the values of the density and surface tension are required to convert the experimental results to suitable dimensionless numbers, these were determined experimentally by the loss of weight method and the capillary drop-weight method, respectively. Taking advantage of the representation of the results through suitable dimensionless groups, a minimum number of experiments was performed to test the validity of the theoretical predictions over a wide range of variables. Experiments were performed with each of the two liquids chosen at different conical tips and at different flow rates. Drops were considered to have formed at infinitely slow formation rates when the drop mass was practically independent of the drop formation time. A drop formation time of ~ 4 min was found to be adequate for this purpose. The phenomenon of drop formation was followed through movie photography using a Pailard Bolex camera with a close-up lens attachment, at a speed of 64 frames per second (FPS). RESULTS AND DISCUSSION
Infinite Conical Tip Infinitely slow formation rate. The movie photographs were analyzed using a microfilm TABLE I Properties of Liquids Used in Metal Cone Experiments
Liquid
Density (g/c.c) Surface tension (dyn/cm) Viscosity (centipoise) Boiling point (°C) Miscibility with water Vapor pressure (mm) % evaporation/rain/ cm 2
1,1,2,2Tetrachloroethane
Chlorobenzene
1.583
1.097
34.6 1.844 (15°C) 146.0 Slightly soluble 10 (33°C)
32.3 0.900 (15°C) 132.0 Insoluble 10 (22°C)
4.174 X 10-4
8.418 X 10-4
Journal of Colloid and Interface Science, Vol. I 16, No. 2, April 1987
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S. RAMESH BABU
reader to determine the critical profile of drops enclosing the maximum drop volume. The drop profile corresponding to that frame beyond which the drop exhibited rapid change in drop shape with the initiation of necking was considered to be the critical profile (see Plate 1). This critical frame was determined with an accuracy of _ 1 frame at a camera speed of 64 FPS. The reproducibility of the critical profiles following this procedure was found to be extremely satisfactory by superimposing the recorded profiles of independent experiments. The experimentally determined critical profiles superimposed on the corresponding theoretically predicted drop profiles are illustrated in Fig. 3. Clearly, the matching of profiles is excellent. Using the recorded critical drop profiles, the corresponding maximum drop volumes were obtained by numerical integration, by approximating the drop segment as the frustum of the cone. The numerically computed drop volumes are compared with those of the theoretical results in Table II. Considering the uncertainty in tracing the critical profile, the matching of drop volumes is good. The analysis of the movie photographs during the final stages of detachment made it possible to follow the rapidly changing sequence of drop shape characterized by the shrinking and elongation of the neck, ultimately resulting in drop detachment (see Plate 1). Also, an estimation of the instability time was made. The order of magnitude of the instability time was practically independent of the cone angle and was about one eighth of 1 s, which is negligibly small in comparison with the drop formation time of 4 min. This result provides a justification for the assumption that the drop volume, at the onset of instability enclosed by the critical profile, corresponds to the maximum drop volume, under conditions of infinitely slow drop formation times. In each of the experiments, about 10 drops were collected and the individual drop mass was determined. The consistency of the individual drop mass was characterized by a standard deviation of --~0.0003 g. The experimenJournal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
tally determined dimensionless detached drop volumes are compared with the theoretical predictions in Table IlL It can be seen that for each of the core angles for both of the liquids, the dimensionless detached drop volumes tally very well, with a maximum deviation of 1.2%, well within the limits of experimental accuracy. In the preceding paper we showed that for a specified cone angle, the equilibrium model predicts a unique dimensionless equilibrium drop volume at its tip, enclosed by a unique critical drop profile. The present experimental results show that for a specified cone angle, the detached drop volume is system independent. Combining these two results, it follows that the assumption, "similar drops at conical tips during instability, detach in similar fashion," is valid in the range of physical properties of the liquids used in the present studies. It may be recalled that Brown and McCormick (3) used a similar assumption to show through their dimensional analysis that for similar drops, the dimensionless detached drop volumes at conical tips will be identical. This result can be utilized to determine the surface tension of liquids by a comparative method using the relation,
3'2 = \ V d J
\01/
3'1,
[1]
where, % vd, and p represent the surface tension, the detached drop volume, and the density, respectively, and the subscripts 1 and 2 refer to the calibrating liquid and the experimental liquid, respectively. The relative merits of this method over the conventional capillary drop-weight method have been discussed elsewhere (3). Table III shows that the predicted detached drop volumes are smaller than the corresponding experimental values for cone angles 60 to 150 ° and greater than the experimental value at a flat surface, the maximum deviation being 27%. Further, the percentage deviation decreases with increasing cone angle from 120 ° upward. It may be recalled that the detached drop volumes are predicted using a
DROP FORMATION AT CONICAL TIPS, 2
363
PLATE 1. Sequence of drop shapes during the detaching stage at metal conical tips. Liquid, 1,1,2,2tetrachloroethane; camera speed, 64 FPS. Journal of Colloid and Interface Science. Vol. 116,No. 2, April1987
S. RAMESH BABU
364
90 °
FIG. 3. Comparison of the experimental critical drop profiles with the theoretical profiles, at infinite conical tips. Liquid: 1,1,2,2-tetrachloroethane;--,theoretical;C), experimental.
semiempirical approach based on the assumption that drops of similar shape at conical and capillary tips detach in similar fashion. Independently considering similar drops either at capillary or at conical tips, it should be noted that similar drops also satisfy similar boundary conditions and physical configurations. On the other hand, although similar drops (characterized by B) could be formed at a capillary and at a conical tip, the boundary condition at the maximum drop height (z0) will be dif-
TABLE II Comparison of the ExperimentalDrop Volumeswith the PredictedValues at Infinite Conical Tips Cone angle
V~r
Wr~
Percentage deviation
60° 90° 120° 150° 180°
6.14462 9.53750 13.35954 17.00502 18.96394
5.989 9.495 13.983 17.283 19.147
2.53 0.45 -4.7 -1.64 0.97
Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
ferent: x = x0 at a capillary tip, and dx/dz = tan/3 at a conical tip. Further, the presence of the conical portion inside the forming drop might offer an additional constraint when the drop undergoes rapid changes in shape during instability until the detachment takes place. It should also be noted that in the case of drops at capillary tips, although the drop undergoes rapid change in shape during instability, the contact area with the capillary tip will not vary. However, for drops at conical tips, the contact area keeps varying not only during the drop formation, but also during the detachment, as revealed through photographic records (see Plate I). Therefore, the changing sequence of drop profiles may not be identical to those at capillary tips. There have been arguments that in the case of capillary drops the necking initiates in the plane of inflection of the critical drop profile (4). The neck partitioning the detaching frac= tion of the drop from the residual drop elongates, simultaneously diminishing in diameter until the detachment takes place. However,
DROP FORMATION AT CONICAL TIPS, 2
365
TABLE III Dimensionless Detached Drop Volumes at Infinite Conical Tips Dimensionless detached drop volume Experimental Cone angle
I,I,2,2-TCE
60° 90° 120° 140° 150° 180°
3.3262 4.6613 6.5084 7.8377 8.7309 12.2885
Chlorobenzene
3.3475 4.7128 6.5548 7.9130 8.7315 12.1391
the volume below the plane of infection does not correspond to the detaching drop volume, but is always less. This suggests that during the process of necking, some amount of the liquid above the plane of inflection also gets squeezed into the detaching fraction below the neck. Therefore, the location of the point of inflection in the critical profile is expected to play a key role in controlling the detaching fraction of the drop volume. By following the drop growth through photography, it was observed that for all cone angles, up to instability, the drop shows only one point of inflection and once it enters the instability stage, the drop exhibits a neck and no other point of inflection (see Plate 1). This observation indicates the possibility of the neck initiating in the plane of inflection. For cone angles of 140 and 150 °, the point of inflection is below the plane of the conical tip. Therefore, from the point of view of physical configuration, the portion of the drop below the plane of inflection will be exactly identical to the corresponding drop at a capillary tip. The deviation from the experimental results shows a decrease from 27 to 17% when the cone angle is varied from 120 to 150 °. Nevertheless, a 17% deviation is very high. This can be explained considering the fact that although the portion below the plane of inflection is identical to that at a capillary tip, the portions above are not identical. Therefore, the volume of the liquid above the plane of inflection getting squeezed into the
Mean
Theoretical
Percentage deviation
3.3369 4.6870 6.5316 7.8754 8.7312 12.2138
2.4962 3.4191 4.7445 6.1550 7.1877 12.2279
25.19 27.05 27.36 21.84 17.68 -0.12
detaching portion could be different. On the other hand, for a drop at a flat surface, (the conical portion is not constrained) inside the drop. Therefore, one could expect the detaching mechanism to be identical to that at capillary tips. The result shows that the predicted value tallies very well with the experimental value, with a deviation of only 0.12%, which is well within the limits of experimental accuracy. Based on the above discussion, it may be concluded that both the boundary condition at z = z0 and the presence of the cone inside the forming drop affect the mechanism of detachment and therefore, the assumption that similar drops at a capillary tip and at a conical tip detach in a similar fashion is not valid. Based on the present experimental results, correction factors representing the fractional detachment (FD) of the maximum drop volume at different conical tips were calculated. The results are given in Table IV, which shows that the fractional detachment consistently decreases with an increase in cone angle. Using these factors, the detaching drop volumes can be predicted for other experimental systems. The analysis of the capillary drop-weight data of Harkins and Brown (5) has yielded an interesting empirical result, namely, the volume below the plane of the critical profile at a distance of Z = 2.0 represents the detaching drop volume for a wide range of capillary tips (6). The application of this empirical result to Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
366
S. RAMESH BABU TABLE IV Correction Factors Representing the Fractional Detachment at Infinite Conical Tips
Cone angle
VL
V~a
F D = V~n/VL
60 ° 90 ° 120 ° 140 ° 150 ° 180 °
3.52421 5.37767 8.02936 10.52856 12.10466 18.96415
3.3369 4.6870 6.5316 7.8754 8.7312 12.2885
0.947 0.872 0.813 0.748 0.721 0.648
drops at conical tips was attempted. Figure 4 illustrates the comparison of the experimental results with the predicted detached drop volumes using the above approach. The predicted values based on the semiempirical approach discussed earlier are shown for comparison in the same figure. The predictions based on the empirical approach can be seen to be closer to the experimental values up to a cone angle of 150 ° , but thereafter, the deviation is no better than that of the semiempirical approach.
To verify the speculation of a multiple drop formation, experiments were performed using a 150 ° cone truncated at a diameter of 6 cm. More than 10 drops were observed to form simultaneously. Plate 2 illustrates the multiple drop formation using 1,1,2,2-tetrachloroethane. While drops nucleated at a distance less than a certain critical distance, which is characteristic of the liquid properties, coalesced into one stable drop, and grew, drops that nucleated farther apart tended to grow as independent drops. This observation is in accordance with the studies on drop formation under sieves by Saradhy and Kumar (7). Finite formation rate. Experiments were performed to study the effect of flow rate on drop formation over a wide range of tests for each of the two liquids chosen, at conical tips of 60, 90, and 120 °. The reproducibility of the results was well within the experimental limits of accuracy. For each of the cone angles, the drop mass was experimentally found to increase consistently with flow rate. A similar
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367
DROP FORMATION AT CONICAL TIPS, 2
PLATE 2. Multiple drop formation at a fiat surface.
observation on the variation of drop mass with flow rate at capillary tips has been reported in the literature (8). A possible explanation for this has been suggested by Rao and co-workers (9), who argue that during the detaching stage, a greater volume of liquid is pumped into the detaching portion of the drop due to the increased flow rate, thereby increasing its mass. .Considering the detaching drop volume, v~, to be dependent on (a) the interfacial tension at the drop-continuous phase interface, 3'; (b) the density difference between the drop liquid and that of the continous phase, AO; (c) the acceleration due to gravity, g; (d) the semicone angle, r; and (e) the fluid flow rate, q, it can be shown through a simple dimensional analysis that there exists a correlation between the dimensionless detached drop volume, Vd~, and the dimensionless flow rate, (~:
In V~ was plotted as a function ofln (~. Figures 5-7 illustrate these plots for cone angles of 60, 90, and 120 ° , respectively. It may be seen that the experimental data for both 1,1,2,2-tetrachloroethane and chlorobenzene fall on the same dimensionless plot, which suggests that the dispersed phase viscosity does not significantly affect the drop formation in the range of flow rate covered in the present studies. t.6
° CHLOROBENZENE
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FIG. 5. Effect of flow rate on drop volume--60 ° cone angle. Journal of Colloid and Interface Science, Vol. 116,No. 2, April 1987
368
s. RAMESH BABU
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The variation of In V~ vs In O for each of the cone angles can be seen to exhibit two distinct characteristics: (a) for moderate flow rates ( - 8 ~< In 0 ~< -4), the variation is smooth and approximately linear; and (b) for very high rates (In O > -4), the variation is very rapid and nonlinear.
The linear variation of In V~ vs In O over a sufficiently wide range of flow rates provides experimental verification to the form of the relationship deduced using the dimensional analysis approach. The experimental data in this range were subjected to least-squares analysis, and the constants c and d were calculated. The results are summarized in Table V. Using these results, for any other specified system, the detaching drop volume can be predicted as a function of flow rate in the range - 8 ~< In Q ~< -4. The random scatter and negligibly small deviation of the experimental points about the correlation lines for each of the cone angles 60, 90, and 120 ° are illustrated in Fig. 8. The nonlinear variation ofln V~ vs In O in the high flow rate regime indicates that perhaps the mechanism of drop formation and detachment is altered in this range. Therefore, unless the parameter responsible for this is incorporated, the above-mentioned dimensional analysis cannot explain the experimental resuits in the high flow rate regime. The velocity of the drop during growth might be one of the important factors to be considered. Clearly, this parameter has no significant effect under low flow rate conditions.
2.20
7
O 1,1,2,2 - TETRACHLORO ETHANE
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2.15
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2.00 I 1-95
1,90 i -10
I -8
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I -4
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FIG. 7. Effect of flow rate on drop v o l u m e - - 1 2 0 ° cone angle. Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
1-85
369
D R O P F O R M A T I O N AT CONICAL TIPS, 2 TABLE V S u m m a r y of the Least-Square Analysis of the Curves In V~d vs In Q for Different Cone Angles In V~d"= In c + d l n (~ Cone angle
n
Range of In (~
PEM a (%)
60 ° 90 ° 120 °
12 10 10
- 4 . 9 to - 7.7 - 4 . 6 to - 8.7 - 4 . 6 5 to - 7.4
0.65 0.23 0.26
CV
c
d
0.379 0.153 0.152
3.76757 5.26409 7.23885
0.01470 0.01262 0.01484
PEM: M a x i m u m percentage error.
FINITE CONICAL TIP
Experiments were performed to study the effect of rod diameter and cone angle on the drop formation at finite conical tips, using the two liquids chosen, at infinitely slow drop formation rates. The consistency in individual drop formation time and drop mass was ensured in all the experiments. The basic assumption made in the theoretical analysis of drop formation at finite conical tips is that the drops are pendant at the conical base. The photographs taken at different stages of drop formation for each of the experiments provided the experimental evidence for this
assumption. A few representative photographs are shown in Plate 3. The photograph at the 60 ° conical tip (Plate 3a) shows that the drop apex is above the conical tip, even though the drop has grown considerably. This observation was true for drops at all conical tips. Further, it was observed that the ratio of drop volume corresponding to a situation in which the drop apex just touches the conical tip and the maximum drop volume increases with decreasing cone angle. This was also true at infinite conical tips, which is in accordance with the theoretical predictions. The photographs taken during the last few seconds of drop formation were analyzed using
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FIG. 8. Deviation of the experimental points about the least-square fitted lines, for different cones. Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
370
S. RAMESH BABU [a]
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[b]
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FIG. 9. Comparison of the experimental critical profiles of drops at finite conical tips with theoretical predictions.
1 20 °
1500
PLATE 3. Typical pendent liquid drops of 1,1,2,2-tetrachloroethane at finite conical tips during the early stages of growth.
a microfilm reader and the critical drop profiles were traced. Figure 9 illustrates some of these profiles superimposed over the theoretical predictions. The close matching of these profiles provides experimental support to the validity of the predicted drop profiles. Also, using these critical profiles, the maximum drop volumes were calculated by numerical integration. In Table VI, the experimentally determined maximum drop volumes are compared with the corresponding theoretical values. The predictions can be seen to be good, with a maximum deviation of 6%. In each of the finite cone experiments, about l0 drops were collected and the individual drop mass was determined. The dimensionless drop volume plotted as a function of dimensionless radius of the rod, at different conical tips, is shown in Fig. 10. The plots show that Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
the experimental values of the two liquids fall on a smooth curve, although there is a twofold variation in their viscosity values (see Table I). This indicates that the drop formation at finite conical tips is also independent of the viscosity of the liquids, at least in the range covered by the present studies. The experimental results at infinite conical tips are also plotted in the same figure. The results show that the detached drop volume increases with rod diameter for a chosen cone angle up to the critical rod diameter, and for any chosen finite rod diameter the drop volume increases with cone angle, the differences in the drop volume between any two specified cone angles increasing with rod diameter. These experimental trends are exactly concurrent with the theoretical predictions, discussed in the preceding paper. A notable feature of Fig. 10 is TABLE VI Comparison of the Experimental Drop Volumes with the Predicted Values at Finite Conical Tips Percentage Cone angle
)to
V~r
V~x
deviation
60 ° 90 ° 120 ° 150 °
0.5724 1.3257 2.0555 1.6990
2.7456 7.5464 13.4344 10.3591
2.8425 7.9632 14.0109 9.7380
-3.53 -5.52 -4.29 6.00
371
DROP FORMATION AT CONICAL TIPS, 2 14 FLAT PLATE
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1,1,2,2 -TETRACHLOROETHANE
x
CHLOROBENZENE
10
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PARAMETER: CONE ANGLE
8
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140 ° 120°
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FIG. 10. Effect of rod diameter on drop volume.
the distinctive difference in the experimental results obtained using cones of 140 and 150 °, which indicates the sensitivity of the detached drop volumes to even small variations in the cone angle. In Table VII the theoretically predicted dimensionless detached drop volumes are compared with the experimental results. The tabulated results indicate that the predicted values are underestimated throughout the range of rod diameters studied, up to a cone angle of 150 ° . However, at 180 ° finite cones, the deviation is close to the limits of experimental accuracy. The maximum deviation of the pre-
dieted values in the experiments performed at 22 different finite conical tips is 22%. The nature of the deviation of the predicted values is illustrated in Fig. 11. It can be seen that the predictions are closer to the experimental values at smaller finite rods for all cone angles, and the predictions consistently deviate with an increase in the rod diameter. It may be recalled that while discussing the causes for a similar deviation at the infinite conical tips, the presence of the cone inside the forming drop and the boundary condition at the maximum drop height were considered. However, in the case of finite cones, the
TABLE VII Dimensionless Detached Drop Volumes at Finite Conical Tips Liquid
1,1,2,2-Tetrachloroethane
Rod dmmeter (mm)
Xo
Va
V~d
60 °
1.55 1.71
0.5197 0.5724
2.1524 2.2377
2.0017 2.1190
90 °
1.97 2.92 3.14 3.96
0.6595 0.9775 1.0512 1.3257
2.6682 3.7093 3.7586 4.2346
120 °
3.99 6.14
1.3357 2.0555
150 °
3.90 5.08
180 °
8.00
Cone an~e
Chlorobenzene Percentage deviation
Percentage aeration
Xo
Va
V~a
7.00 5.30
0.4480 0.4934
1.8876 2.0190
1.8193 1.9378
3.62 4.02
2.4663 3.0821 3.1793 3.3904
7.57 16.91 15.41 19.94
0.5684 0.8426 0.9061 1.1427
2.3415 3.3212 3.3229 4.1274
2.2312 2.8598 2.9716 3.2760
4.71 13.89 10.57 20.63
4.7466 6.2353
4.0978 4.8368
13.67 22.43
1.1513 1.7717
4.2939 5.8730
3.7603 4.6554
12.43 20.73
1.3039 1.6990
4.8566 6.3377
4.5310 5.5790
6.70 11.97
1.1239 1.4644
4.2263 5.4278
4.0239 4.9676
4.79 8.48
2.6782
10.6026
10.7039
0.96
2.3084
8.8857
9.0427
-1.77
Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
372
S. RAMESH BABU
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angle of 150 °. To test the applicability of this to drop formation at finite conical tips, the liquid volume below the plane at Z = 2.0 of the critical profiles of drops at each of the 22 finite conical tips used in the present studies was calculated. These calculated values are also plotted in Fig. 11. It can be seen that throughout the experimental range, for all the cone angles, the predicted values not only are better than those predicted using the semiempirical approach, but also are quite close to the experimental values, with a maximum deviation of 10%.
I
v~ ~
FIG. 11. Comparison of the predicted detached drop volumes with experimental values, at finite conical tips. ©, Chlorobenzene; A, 1,1,2,2-tetrachloroethane; [3, Vlz=2.0.
boundary condition at z = z0 is identical to that at a capillary tip. Therefore, the possible cause for the deviation has to be due to the presence of the cone inside the forming drop. Based on this, it follows that since a drop at a 180 ° finite conical tip is exactly identical to the corresponding drop at a capillary tip, the fractional detachment can be expected to be identical. The deviation at 180 ° conical tips, which is close to the experimental limits of accuracy, supports this argument. The same comments made earlier on the possible effect of the presence of the cone inside the drop on the mechanism of detachment at infinite conical tips also hold good for the present situation. The extension of the empirical result at capillary tips, viz., VLIz=2.o ~ V d , to infinite conical tips showed that the predictions were closer to the experimental results, up to a cone Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
CONCLUSION
The present experimental results suggest that the proposed model adequately represents the phenomenon of drop formation at conical tips under infinitely slow formation rates until the onset of instability. However, experiments are planned to test the validity of the predictions over a much wider range of physical properties of the liquids. ACKNOWLEDGMENTS I am grateful to Professor A. K. Lahiri for useful discussions in the design and experiments. I am thankful to Mr. R. Muniramappa for his skillful technical assistance in the construction of the apparatus and to Mr. S. Nagaraj for his aid with photography. I thankfully acknowledge the financial assistance received from the Department of Atomic Energy, government of India, during the course of this investigation. REFERENCES 1. Ramesh Babu, S., J. Colloid Interface Sci. 116, 350 (1987). 2. "Handbook of Physics and Chemistry," 48th ed. Robert C. Weast, Ed. Chem. Rubber Co., Cleveland, 1967. 3. Brown, R. C., and McCormick, H., Philos. Mag. 39, 420 (1948). 4. Iredele, T., Philos. Mag. 45, 1088 (1923). 5. Harkins, W. D., and Brown, F. E., J. Amer. Chem. Soc. 41, 499 (1919). 6. Ramesh Babu, S., Ph.D. thesis, Indian Institute of Science, Bangalore, 1985. 7. Saradhy, Y. P., and Kumar, R., Ind. Eng. Chem. 15, 75 (1976). 8. Kumar, R., and Kuloor, N. R., Adv. Chem. Eng. 8, 225 (1970). 9. Rao, E. V. L. N., et al., Chem. Eng. Sci. 21, 867 (1966).
Received September 18, 1985; accepted June 6, 1986 Drop formation at conical tips has been experimentallystudied using the apparatus designed for this purpose and with the aid of movie photography. Experimentshave been performed using the low vapor pressure liquids, chlorobenzene and 1,1,2,2-tetrachloroethane, to study the effect of cone angle, rod diameter, and physical properties of the dispersed liquid on drop formation. The experimental results served to verifythe assumptions made in the theoretical analysis (S. Ramesh Babu, J. Colloid Interface Sci. 116, 350 (1987) on the one hand and to test the validity of the predictions of the models on the other. Based on the present experimentalstudies, it is concluded that the phenomenon of drop formation under infinitelyslowformation rates at conical tips is well understood and the proposedmodel adequately represents the phenomenon up to the instability stage. © 1987AcademicPress,Inc. INTRODUCTION The p h e n o m e n o n of drop formation at conical tips has been analyzed using a theoretical approach and the results are presented in the preceding paper (1). The design and development of the experimental techniques for drop formation studies at conical tips are considered in this paper. The objective of developing these techniques is to verify the assumptions made in the development of the models on the one hand and to test the validity of the predictions of the models on the other. EXPERIMENTAL The design of the apparatus essentially consisted of providing a suitable arrangement to allow liquids to flow down a long metal cone, under controlled flow rates, and to form drops at its tip. An exploded view of the apparatus designed and built is shown in Fig. 1. The apparatus consists of an a l u m i n u m cone (A), fit1Currently with the Research Institute of Mineral Dressingand Metallurgy,Tohoku University, 1,1-2Chome Katahira Sendai-980, Japan.
ring perfectly into an a l u m i n u m socket (B) which is extended into a hollow cylindrical portion (C) of about 3 cm in height and 7 cm in inner diameter to hold the experimental liquid. An a l u m i n u m lid (D) is screwed over the top outer surface of C. A n u m b e r of openings into the reservoir (C), above the cone, are provided. One end of an a l u m i n u m rod (E) is screwed onto the cone and the other threaded end passes through the central hollow cylindrical portion of the lid (F). A spring (G) inserted over the free end of the a l u m i n u m rod (E) is held in position by a nut (H) and is positioned against a small projection on the inner surface of F. An a l u m i n u m knob (I) is then screwed over the outer surface of F. By a clockwise rotation of the knob, the spring is loaded and its tension enables the core to be positioned inside the socket perfectly concentric with it. Therefore, by operating the top knob (I), the width of the gap between the cone and the socket could be varied to obtain controlled flow rates of the liquid. By means of a step provided on the outer surface of the alum i n u m socket, the assembled apparatus could
359 0021-9797/87 $3.00 Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
Copyright © 1987 by Academic Press, Inc. All fights of reproduction in any form reserved.
360
S. RAMESH BABU
@
I1
j
I
[
i
,
]
1,
I
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50
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//
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/ A L L DIMENSIONS ARE IN MM
FIG. 1. Exploded view of the metal cone apparatus for drop formation studies. be seated on a 1-in.-thick aluminum plate with a central opening, mounted on an a l u m i n u m tripod stand with a spring-loaded arrangement for finer adjustments of levelling. This entire assembly is housed inside a glass chamber to minimize the external disturbance on the drop Journal of Colloid and Interface Science, Vol. 116,No. 2, April 1987
formation. To study the effect of each of the cone angles, 60, 90, 120, 140, and 150 °, a matching cone and socket assembly was built, each assembly for a specific cone angle. With a slight modification in the design of only the conical portion, as shown in Fig. 2a,
361
DROP FORMATION AT CONICAL TIPS, 2
(a)
(b) FIG. 2. Schematic sketches of (a) a finite cone and (b) a truncated cone.
the same apparatus could also be used for drop formation studies at finite conical tips. For each of the cone angles, 60, 90, 120, 140, and 150 °, a minimum of two finite cones were made to study the effect of rod diameter on drop size. Also, by using a truncated cone of an arbitrary cone angle (Fig. 2b), drop formation at a fiat surface could be studied. When the cross-sectional area of the fiat tip is far greater than the predicted critical contact area of the drop at a flat surface, even multiple drop formation and coalescence of drops could be studied. For the present studies the following two low vapor pressure liquids were chosen: (a) chlorobenzene (L.R. grade, BDH supplies); and (b) 1,1,2,2-tetrachloroethane (L.R. grade, SD supplies). The properties of these liquids relevant to the present studies are summarized in Table I. The values of viscosity, boiling point, and vapor pressure are taken from the
"Handbook of Physics and Chemistry" (2). The percentage evaporation per minute per square centimeter of the surface area of evaporation was determined experimentally for both liquids. Since the values of the density and surface tension are required to convert the experimental results to suitable dimensionless numbers, these were determined experimentally by the loss of weight method and the capillary drop-weight method, respectively. Taking advantage of the representation of the results through suitable dimensionless groups, a minimum number of experiments was performed to test the validity of the theoretical predictions over a wide range of variables. Experiments were performed with each of the two liquids chosen at different conical tips and at different flow rates. Drops were considered to have formed at infinitely slow formation rates when the drop mass was practically independent of the drop formation time. A drop formation time of ~ 4 min was found to be adequate for this purpose. The phenomenon of drop formation was followed through movie photography using a Pailard Bolex camera with a close-up lens attachment, at a speed of 64 frames per second (FPS). RESULTS AND DISCUSSION
Infinite Conical Tip Infinitely slow formation rate. The movie photographs were analyzed using a microfilm TABLE I Properties of Liquids Used in Metal Cone Experiments
Liquid
Density (g/c.c) Surface tension (dyn/cm) Viscosity (centipoise) Boiling point (°C) Miscibility with water Vapor pressure (mm) % evaporation/rain/ cm 2
1,1,2,2Tetrachloroethane
Chlorobenzene
1.583
1.097
34.6 1.844 (15°C) 146.0 Slightly soluble 10 (33°C)
32.3 0.900 (15°C) 132.0 Insoluble 10 (22°C)
4.174 X 10-4
8.418 X 10-4
Journal of Colloid and Interface Science, Vol. I 16, No. 2, April 1987
362
S. RAMESH BABU
reader to determine the critical profile of drops enclosing the maximum drop volume. The drop profile corresponding to that frame beyond which the drop exhibited rapid change in drop shape with the initiation of necking was considered to be the critical profile (see Plate 1). This critical frame was determined with an accuracy of _ 1 frame at a camera speed of 64 FPS. The reproducibility of the critical profiles following this procedure was found to be extremely satisfactory by superimposing the recorded profiles of independent experiments. The experimentally determined critical profiles superimposed on the corresponding theoretically predicted drop profiles are illustrated in Fig. 3. Clearly, the matching of profiles is excellent. Using the recorded critical drop profiles, the corresponding maximum drop volumes were obtained by numerical integration, by approximating the drop segment as the frustum of the cone. The numerically computed drop volumes are compared with those of the theoretical results in Table II. Considering the uncertainty in tracing the critical profile, the matching of drop volumes is good. The analysis of the movie photographs during the final stages of detachment made it possible to follow the rapidly changing sequence of drop shape characterized by the shrinking and elongation of the neck, ultimately resulting in drop detachment (see Plate 1). Also, an estimation of the instability time was made. The order of magnitude of the instability time was practically independent of the cone angle and was about one eighth of 1 s, which is negligibly small in comparison with the drop formation time of 4 min. This result provides a justification for the assumption that the drop volume, at the onset of instability enclosed by the critical profile, corresponds to the maximum drop volume, under conditions of infinitely slow drop formation times. In each of the experiments, about 10 drops were collected and the individual drop mass was determined. The consistency of the individual drop mass was characterized by a standard deviation of --~0.0003 g. The experimenJournal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
tally determined dimensionless detached drop volumes are compared with the theoretical predictions in Table IlL It can be seen that for each of the core angles for both of the liquids, the dimensionless detached drop volumes tally very well, with a maximum deviation of 1.2%, well within the limits of experimental accuracy. In the preceding paper we showed that for a specified cone angle, the equilibrium model predicts a unique dimensionless equilibrium drop volume at its tip, enclosed by a unique critical drop profile. The present experimental results show that for a specified cone angle, the detached drop volume is system independent. Combining these two results, it follows that the assumption, "similar drops at conical tips during instability, detach in similar fashion," is valid in the range of physical properties of the liquids used in the present studies. It may be recalled that Brown and McCormick (3) used a similar assumption to show through their dimensional analysis that for similar drops, the dimensionless detached drop volumes at conical tips will be identical. This result can be utilized to determine the surface tension of liquids by a comparative method using the relation,
3'2 = \ V d J
\01/
3'1,
[1]
where, % vd, and p represent the surface tension, the detached drop volume, and the density, respectively, and the subscripts 1 and 2 refer to the calibrating liquid and the experimental liquid, respectively. The relative merits of this method over the conventional capillary drop-weight method have been discussed elsewhere (3). Table III shows that the predicted detached drop volumes are smaller than the corresponding experimental values for cone angles 60 to 150 ° and greater than the experimental value at a flat surface, the maximum deviation being 27%. Further, the percentage deviation decreases with increasing cone angle from 120 ° upward. It may be recalled that the detached drop volumes are predicted using a
DROP FORMATION AT CONICAL TIPS, 2
363
PLATE 1. Sequence of drop shapes during the detaching stage at metal conical tips. Liquid, 1,1,2,2tetrachloroethane; camera speed, 64 FPS. Journal of Colloid and Interface Science. Vol. 116,No. 2, April1987
S. RAMESH BABU
364
90 °
FIG. 3. Comparison of the experimental critical drop profiles with the theoretical profiles, at infinite conical tips. Liquid: 1,1,2,2-tetrachloroethane;--,theoretical;C), experimental.
semiempirical approach based on the assumption that drops of similar shape at conical and capillary tips detach in similar fashion. Independently considering similar drops either at capillary or at conical tips, it should be noted that similar drops also satisfy similar boundary conditions and physical configurations. On the other hand, although similar drops (characterized by B) could be formed at a capillary and at a conical tip, the boundary condition at the maximum drop height (z0) will be dif-
TABLE II Comparison of the ExperimentalDrop Volumeswith the PredictedValues at Infinite Conical Tips Cone angle
V~r
Wr~
Percentage deviation
60° 90° 120° 150° 180°
6.14462 9.53750 13.35954 17.00502 18.96394
5.989 9.495 13.983 17.283 19.147
2.53 0.45 -4.7 -1.64 0.97
Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
ferent: x = x0 at a capillary tip, and dx/dz = tan/3 at a conical tip. Further, the presence of the conical portion inside the forming drop might offer an additional constraint when the drop undergoes rapid changes in shape during instability until the detachment takes place. It should also be noted that in the case of drops at capillary tips, although the drop undergoes rapid change in shape during instability, the contact area with the capillary tip will not vary. However, for drops at conical tips, the contact area keeps varying not only during the drop formation, but also during the detachment, as revealed through photographic records (see Plate I). Therefore, the changing sequence of drop profiles may not be identical to those at capillary tips. There have been arguments that in the case of capillary drops the necking initiates in the plane of inflection of the critical drop profile (4). The neck partitioning the detaching frac= tion of the drop from the residual drop elongates, simultaneously diminishing in diameter until the detachment takes place. However,
DROP FORMATION AT CONICAL TIPS, 2
365
TABLE III Dimensionless Detached Drop Volumes at Infinite Conical Tips Dimensionless detached drop volume Experimental Cone angle
I,I,2,2-TCE
60° 90° 120° 140° 150° 180°
3.3262 4.6613 6.5084 7.8377 8.7309 12.2885
Chlorobenzene
3.3475 4.7128 6.5548 7.9130 8.7315 12.1391
the volume below the plane of infection does not correspond to the detaching drop volume, but is always less. This suggests that during the process of necking, some amount of the liquid above the plane of inflection also gets squeezed into the detaching fraction below the neck. Therefore, the location of the point of inflection in the critical profile is expected to play a key role in controlling the detaching fraction of the drop volume. By following the drop growth through photography, it was observed that for all cone angles, up to instability, the drop shows only one point of inflection and once it enters the instability stage, the drop exhibits a neck and no other point of inflection (see Plate 1). This observation indicates the possibility of the neck initiating in the plane of inflection. For cone angles of 140 and 150 °, the point of inflection is below the plane of the conical tip. Therefore, from the point of view of physical configuration, the portion of the drop below the plane of inflection will be exactly identical to the corresponding drop at a capillary tip. The deviation from the experimental results shows a decrease from 27 to 17% when the cone angle is varied from 120 to 150 °. Nevertheless, a 17% deviation is very high. This can be explained considering the fact that although the portion below the plane of inflection is identical to that at a capillary tip, the portions above are not identical. Therefore, the volume of the liquid above the plane of inflection getting squeezed into the
Mean
Theoretical
Percentage deviation
3.3369 4.6870 6.5316 7.8754 8.7312 12.2138
2.4962 3.4191 4.7445 6.1550 7.1877 12.2279
25.19 27.05 27.36 21.84 17.68 -0.12
detaching portion could be different. On the other hand, for a drop at a flat surface, (the conical portion is not constrained) inside the drop. Therefore, one could expect the detaching mechanism to be identical to that at capillary tips. The result shows that the predicted value tallies very well with the experimental value, with a deviation of only 0.12%, which is well within the limits of experimental accuracy. Based on the above discussion, it may be concluded that both the boundary condition at z = z0 and the presence of the cone inside the forming drop affect the mechanism of detachment and therefore, the assumption that similar drops at a capillary tip and at a conical tip detach in a similar fashion is not valid. Based on the present experimental results, correction factors representing the fractional detachment (FD) of the maximum drop volume at different conical tips were calculated. The results are given in Table IV, which shows that the fractional detachment consistently decreases with an increase in cone angle. Using these factors, the detaching drop volumes can be predicted for other experimental systems. The analysis of the capillary drop-weight data of Harkins and Brown (5) has yielded an interesting empirical result, namely, the volume below the plane of the critical profile at a distance of Z = 2.0 represents the detaching drop volume for a wide range of capillary tips (6). The application of this empirical result to Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
366
S. RAMESH BABU TABLE IV Correction Factors Representing the Fractional Detachment at Infinite Conical Tips
Cone angle
VL
V~a
F D = V~n/VL
60 ° 90 ° 120 ° 140 ° 150 ° 180 °
3.52421 5.37767 8.02936 10.52856 12.10466 18.96415
3.3369 4.6870 6.5316 7.8754 8.7312 12.2885
0.947 0.872 0.813 0.748 0.721 0.648
drops at conical tips was attempted. Figure 4 illustrates the comparison of the experimental results with the predicted detached drop volumes using the above approach. The predicted values based on the semiempirical approach discussed earlier are shown for comparison in the same figure. The predictions based on the empirical approach can be seen to be closer to the experimental values up to a cone angle of 150 ° , but thereafter, the deviation is no better than that of the semiempirical approach.
To verify the speculation of a multiple drop formation, experiments were performed using a 150 ° cone truncated at a diameter of 6 cm. More than 10 drops were observed to form simultaneously. Plate 2 illustrates the multiple drop formation using 1,1,2,2-tetrachloroethane. While drops nucleated at a distance less than a certain critical distance, which is characteristic of the liquid properties, coalesced into one stable drop, and grew, drops that nucleated farther apart tended to grow as independent drops. This observation is in accordance with the studies on drop formation under sieves by Saradhy and Kumar (7). Finite formation rate. Experiments were performed to study the effect of flow rate on drop formation over a wide range of tests for each of the two liquids chosen, at conical tips of 60, 90, and 120 °. The reproducibility of the results was well within the experimental limits of accuracy. For each of the cone angles, the drop mass was experimentally found to increase consistently with flow rate. A similar
16 ¸
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9 0°
120 °
150°
180°
~"CONEANGLE FIG. 4. Comparison of the predicted detached drop volumes with experimental values, at infinite conical tips. Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
367
DROP FORMATION AT CONICAL TIPS, 2
PLATE 2. Multiple drop formation at a fiat surface.
observation on the variation of drop mass with flow rate at capillary tips has been reported in the literature (8). A possible explanation for this has been suggested by Rao and co-workers (9), who argue that during the detaching stage, a greater volume of liquid is pumped into the detaching portion of the drop due to the increased flow rate, thereby increasing its mass. .Considering the detaching drop volume, v~, to be dependent on (a) the interfacial tension at the drop-continuous phase interface, 3'; (b) the density difference between the drop liquid and that of the continous phase, AO; (c) the acceleration due to gravity, g; (d) the semicone angle, r; and (e) the fluid flow rate, q, it can be shown through a simple dimensional analysis that there exists a correlation between the dimensionless detached drop volume, Vd~, and the dimensionless flow rate, (~:
In V~ was plotted as a function ofln (~. Figures 5-7 illustrate these plots for cone angles of 60, 90, and 120 ° , respectively. It may be seen that the experimental data for both 1,1,2,2-tetrachloroethane and chlorobenzene fall on the same dimensionless plot, which suggests that the dispersed phase viscosity does not significantly affect the drop formation in the range of flow rate covered in the present studies. t.6
° CHLOROBENZENE
/
1.5
T 1.4 ~
V~= cOd, L3
where .
Aog
Vp--d~--~-]
gp.
,
.
. / ],,,o,
Q:q/---~)
,,4
o
;
[2] I
-8
c and d are constants to be determined. To test the validity of the form of the above correlation using the present experimental data,
I
-7
I
-6
I
-5
I
-4
triO.
I
-3
I
-2
-I
1.2
-
FIG. 5. Effect of flow rate on drop volume--60 ° cone angle. Journal of Colloid and Interface Science, Vol. 116,No. 2, April 1987
368
s. RAMESH BABU
1.95 o CHLOROBENZENE
/
1.90 1.85 1.80
T
•
1.75~ 1,70
t
l.65 1.60
11.55 - 9
I - 5
I
I
-7
-6
I -5
I -4
I -3
I
11.50
I
-2
-1
tnCt FIG. 6. Effect of flow rate on drop v o l u m e - - 9 0 ° cone angle.
The variation of In V~ vs In O for each of the cone angles can be seen to exhibit two distinct characteristics: (a) for moderate flow rates ( - 8 ~< In 0 ~< -4), the variation is smooth and approximately linear; and (b) for very high rates (In O > -4), the variation is very rapid and nonlinear.
The linear variation of In V~ vs In O over a sufficiently wide range of flow rates provides experimental verification to the form of the relationship deduced using the dimensional analysis approach. The experimental data in this range were subjected to least-squares analysis, and the constants c and d were calculated. The results are summarized in Table V. Using these results, for any other specified system, the detaching drop volume can be predicted as a function of flow rate in the range - 8 ~< In Q ~< -4. The random scatter and negligibly small deviation of the experimental points about the correlation lines for each of the cone angles 60, 90, and 120 ° are illustrated in Fig. 8. The nonlinear variation ofln V~ vs In O in the high flow rate regime indicates that perhaps the mechanism of drop formation and detachment is altered in this range. Therefore, unless the parameter responsible for this is incorporated, the above-mentioned dimensional analysis cannot explain the experimental resuits in the high flow rate regime. The velocity of the drop during growth might be one of the important factors to be considered. Clearly, this parameter has no significant effect under low flow rate conditions.
2.20
7
O 1,1,2,2 - TETRACHLORO ETHANE
/ /
O CHLOROB
2.15
2.10
2.05
2.00 I 1-95
1,90 i -10
I -8
I
I -6
I ---'-~ In (~
I -4
I
i
[
-2
FIG. 7. Effect of flow rate on drop v o l u m e - - 1 2 0 ° cone angle. Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
1-85
369
D R O P F O R M A T I O N AT CONICAL TIPS, 2 TABLE V S u m m a r y of the Least-Square Analysis of the Curves In V~d vs In Q for Different Cone Angles In V~d"= In c + d l n (~ Cone angle
n
Range of In (~
PEM a (%)
60 ° 90 ° 120 °
12 10 10
- 4 . 9 to - 7.7 - 4 . 6 to - 8.7 - 4 . 6 5 to - 7.4
0.65 0.23 0.26
CV
c
d
0.379 0.153 0.152
3.76757 5.26409 7.23885
0.01470 0.01262 0.01484
PEM: M a x i m u m percentage error.
FINITE CONICAL TIP
Experiments were performed to study the effect of rod diameter and cone angle on the drop formation at finite conical tips, using the two liquids chosen, at infinitely slow drop formation rates. The consistency in individual drop formation time and drop mass was ensured in all the experiments. The basic assumption made in the theoretical analysis of drop formation at finite conical tips is that the drops are pendant at the conical base. The photographs taken at different stages of drop formation for each of the experiments provided the experimental evidence for this
assumption. A few representative photographs are shown in Plate 3. The photograph at the 60 ° conical tip (Plate 3a) shows that the drop apex is above the conical tip, even though the drop has grown considerably. This observation was true for drops at all conical tips. Further, it was observed that the ratio of drop volume corresponding to a situation in which the drop apex just touches the conical tip and the maximum drop volume increases with decreasing cone angle. This was also true at infinite conical tips, which is in accordance with the theoretical predictions. The photographs taken during the last few seconds of drop formation were analyzed using
2.0 120o
1.9-
1.8-
T
1.7-
•o:>~ ¢: 1.6
-
90°
~
1-5--
1.4--
1"3-60 °
I
1,2
-9
-8
-7
InQ ~
-6
-5
-4
FIG. 8. Deviation of the experimental points about the least-square fitted lines, for different cones. Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
370
S. RAMESH BABU [a]
1,71mm
[b]
3-96 mm
o
~
900
[c]
[d]
o
o
6.14mm
600
90°
)
5.08mm
FIG. 9. Comparison of the experimental critical profiles of drops at finite conical tips with theoretical predictions.
1 20 °
1500
PLATE 3. Typical pendent liquid drops of 1,1,2,2-tetrachloroethane at finite conical tips during the early stages of growth.
a microfilm reader and the critical drop profiles were traced. Figure 9 illustrates some of these profiles superimposed over the theoretical predictions. The close matching of these profiles provides experimental support to the validity of the predicted drop profiles. Also, using these critical profiles, the maximum drop volumes were calculated by numerical integration. In Table VI, the experimentally determined maximum drop volumes are compared with the corresponding theoretical values. The predictions can be seen to be good, with a maximum deviation of 6%. In each of the finite cone experiments, about l0 drops were collected and the individual drop mass was determined. The dimensionless drop volume plotted as a function of dimensionless radius of the rod, at different conical tips, is shown in Fig. 10. The plots show that Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
the experimental values of the two liquids fall on a smooth curve, although there is a twofold variation in their viscosity values (see Table I). This indicates that the drop formation at finite conical tips is also independent of the viscosity of the liquids, at least in the range covered by the present studies. The experimental results at infinite conical tips are also plotted in the same figure. The results show that the detached drop volume increases with rod diameter for a chosen cone angle up to the critical rod diameter, and for any chosen finite rod diameter the drop volume increases with cone angle, the differences in the drop volume between any two specified cone angles increasing with rod diameter. These experimental trends are exactly concurrent with the theoretical predictions, discussed in the preceding paper. A notable feature of Fig. 10 is TABLE VI Comparison of the Experimental Drop Volumes with the Predicted Values at Finite Conical Tips Percentage Cone angle
)to
V~r
V~x
deviation
60 ° 90 ° 120 ° 150 °
0.5724 1.3257 2.0555 1.6990
2.7456 7.5464 13.4344 10.3591
2.8425 7.9632 14.0109 9.7380
-3.53 -5.52 -4.29 6.00
371
DROP FORMATION AT CONICAL TIPS, 2 14 FLAT PLATE
12 o
1,1,2,2 -TETRACHLOROETHANE
x
CHLOROBENZENE
10
~
PARAMETER: CONE ANGLE
8
T6 4 2
I
0
N
90 °
i
I
1.0
0.5
140 ° 120°
I
1.5
2.0
I
2.5
I
3.5
3.0
Xo
FIG. 10. Effect of rod diameter on drop volume.
the distinctive difference in the experimental results obtained using cones of 140 and 150 °, which indicates the sensitivity of the detached drop volumes to even small variations in the cone angle. In Table VII the theoretically predicted dimensionless detached drop volumes are compared with the experimental results. The tabulated results indicate that the predicted values are underestimated throughout the range of rod diameters studied, up to a cone angle of 150 ° . However, at 180 ° finite cones, the deviation is close to the limits of experimental accuracy. The maximum deviation of the pre-
dieted values in the experiments performed at 22 different finite conical tips is 22%. The nature of the deviation of the predicted values is illustrated in Fig. 11. It can be seen that the predictions are closer to the experimental values at smaller finite rods for all cone angles, and the predictions consistently deviate with an increase in the rod diameter. It may be recalled that while discussing the causes for a similar deviation at the infinite conical tips, the presence of the cone inside the forming drop and the boundary condition at the maximum drop height were considered. However, in the case of finite cones, the
TABLE VII Dimensionless Detached Drop Volumes at Finite Conical Tips Liquid
1,1,2,2-Tetrachloroethane
Rod dmmeter (mm)
Xo
Va
V~d
60 °
1.55 1.71
0.5197 0.5724
2.1524 2.2377
2.0017 2.1190
90 °
1.97 2.92 3.14 3.96
0.6595 0.9775 1.0512 1.3257
2.6682 3.7093 3.7586 4.2346
120 °
3.99 6.14
1.3357 2.0555
150 °
3.90 5.08
180 °
8.00
Cone an~e
Chlorobenzene Percentage deviation
Percentage aeration
Xo
Va
V~a
7.00 5.30
0.4480 0.4934
1.8876 2.0190
1.8193 1.9378
3.62 4.02
2.4663 3.0821 3.1793 3.3904
7.57 16.91 15.41 19.94
0.5684 0.8426 0.9061 1.1427
2.3415 3.3212 3.3229 4.1274
2.2312 2.8598 2.9716 3.2760
4.71 13.89 10.57 20.63
4.7466 6.2353
4.0978 4.8368
13.67 22.43
1.1513 1.7717
4.2939 5.8730
3.7603 4.6554
12.43 20.73
1.3039 1.6990
4.8566 6.3377
4.5310 5.5790
6.70 11.97
1.1239 1.4644
4.2263 5.4278
4.0239 4.9676
4.79 8.48
2.6782
10.6026
10.7039
0.96
2.3084
8.8857
9.0427
-1.77
Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
372
S. RAMESH BABU
4.0 f
60° CONE /
4'5f 90°CONE
I 2.5
I 2.0 V
I
V I
I r I I I 2.0 3'0 4'0
I
I
3,5
I
I
4'5
Vffx
s'sI ~so°CONE /
6"5f 120°CONE /
I
I
2.5
I 4-5
4.5
o
~ ~'5o ~ ~ L// r °1 I I I I 4.5 5-5 6-5
//
4.5
°~ 5.5 i 6,5
~ v d =x
180° CONE
~9 8
I
I
angle of 150 °. To test the applicability of this to drop formation at finite conical tips, the liquid volume below the plane at Z = 2.0 of the critical profiles of drops at each of the 22 finite conical tips used in the present studies was calculated. These calculated values are also plotted in Fig. 11. It can be seen that throughout the experimental range, for all the cone angles, the predicted values not only are better than those predicted using the semiempirical approach, but also are quite close to the experimental values, with a maximum deviation of 10%.
I
v~ ~
FIG. 11. Comparison of the predicted detached drop volumes with experimental values, at finite conical tips. ©, Chlorobenzene; A, 1,1,2,2-tetrachloroethane; [3, Vlz=2.0.
boundary condition at z = z0 is identical to that at a capillary tip. Therefore, the possible cause for the deviation has to be due to the presence of the cone inside the forming drop. Based on this, it follows that since a drop at a 180 ° finite conical tip is exactly identical to the corresponding drop at a capillary tip, the fractional detachment can be expected to be identical. The deviation at 180 ° conical tips, which is close to the experimental limits of accuracy, supports this argument. The same comments made earlier on the possible effect of the presence of the cone inside the drop on the mechanism of detachment at infinite conical tips also hold good for the present situation. The extension of the empirical result at capillary tips, viz., VLIz=2.o ~ V d , to infinite conical tips showed that the predictions were closer to the experimental results, up to a cone Journal of Colloid and Interface Science, Vol. 116, No. 2, April 1987
CONCLUSION
The present experimental results suggest that the proposed model adequately represents the phenomenon of drop formation at conical tips under infinitely slow formation rates until the onset of instability. However, experiments are planned to test the validity of the predictions over a much wider range of physical properties of the liquids. ACKNOWLEDGMENTS I am grateful to Professor A. K. Lahiri for useful discussions in the design and experiments. I am thankful to Mr. R. Muniramappa for his skillful technical assistance in the construction of the apparatus and to Mr. S. Nagaraj for his aid with photography. I thankfully acknowledge the financial assistance received from the Department of Atomic Energy, government of India, during the course of this investigation. REFERENCES 1. Ramesh Babu, S., J. Colloid Interface Sci. 116, 350 (1987). 2. "Handbook of Physics and Chemistry," 48th ed. Robert C. Weast, Ed. Chem. Rubber Co., Cleveland, 1967. 3. Brown, R. C., and McCormick, H., Philos. Mag. 39, 420 (1948). 4. Iredele, T., Philos. Mag. 45, 1088 (1923). 5. Harkins, W. D., and Brown, F. E., J. Amer. Chem. Soc. 41, 499 (1919). 6. Ramesh Babu, S., Ph.D. thesis, Indian Institute of Science, Bangalore, 1985. 7. Saradhy, Y. P., and Kumar, R., Ind. Eng. Chem. 15, 75 (1976). 8. Kumar, R., and Kuloor, N. R., Adv. Chem. Eng. 8, 225 (1970). 9. Rao, E. V. L. N., et al., Chem. Eng. Sci. 21, 867 (1966).