RISING VELOCITY AND SHAPE OF SINGLE AIR BUBBLES IN HIGHLY VISCOUS LIQUIDS"

RISING VELOCITY AND SHAPE OF SINGLE AIR BUBBLES IN HIGHLY VISCOUS LIQUIDS" EIICHI KOJIMA, TAKASHI AKEHATA, AND TAKASHI SHIRAI Research Laboratory of Resourses Utilization, Tokyo Institute of Technology, Tokyo Introduction The behavior of singlebubbles rising in quiescent liquids has been studied by many investigators both theoretically and experimentally. The previous theoretical investigations on single air bubbles were mainly on the slow motion of small spherical bubbles6"* or on the motion of large spherical cap bubbles3'4). Recently Taylor and Acrivos14) studied the slight deformation of bubble from the sphere at low Reynolds number and Moore10>n) examined the problem at higer Reynolds number by the boundary-layer theory. The experimental works published so far were mostly on the rising velocity of the bubble in liquids of small viscosity. Harberman and Morton7) made an< extensive study of it. Tadaki and Maeda13) proposed the empirical correlations of Cd vs. ReM0'2* for the drag and d/a vs.^remq'23 for the deformation. Kubota et al.9) suggested the, correlations, Cd/Cdf' vs. ReM1/Qand 1-e vs. We for drag and deformation respectively and stated that the bubble in purified liquids behaved quite differently from that in not-purified liquids. The experimental studies on the bubble in highly viscous liquids are very scarce. Some years ago Bond and Newton2) found that while the air bubble in a golden syrup behaved as a gas sphere when the bubble diameter was larger than about 0.4cm, the smaller bubble tended to show the drag of the rigid sphere. Garner and Hammerton55 reported a similar experience. Recently Red field and Houghton125 measured simultaneously the velocity and deformation of carbon dioxide bubbles rising in viscous dextrose solutions and Angelinoi:> made the study of large bubbles. However the bubble behavior in highly viscous liquids is not fully understood yet in terms of physical properties such as surface tension, viscosity and density. It is the purpose of this paper to study experimentally the behavior of single air bubbles in quiescent viscous liquids, to see and determine -the-applicable range of the previously reported correlations, and to develop correlations to predict the rising velocity and shape of the bubble in highly viscous liquids from the knowledge of the liquid properties. Experimental Apparatus and procedure The apparatus used is shown in Fig. 1. The tank A * Received on July 1, 1967 VOL T Nib,1 t9"6'8' was made of glass and was of 20X20cmcross section and 30cm height, wide enough to reduce wall effect and long enough to avoid end effects. The tank was placed in a thermostat D and the temperature of the test liquid was controlled within* 0. 1 C. Air bubble was released from a dumping cup by tilting it. The rising velocity of a small bubble was obtained by measuring the time required to travel the 10 cm-distance by means of a stopwatch and the shape of the bubble was determined from photographs8), When the bubble was large and thus the bubble rose fast, the shape and velocity were both obtained from photographs taken with a stroboscope which flashed one to ten times per second. A camera, a 6x6cm Zenzabronica, was placed lm apart from the object and was equipped with 200mm telescopic lens and extension rings to avoid the distortion of the image. The volume of bubble was measured by capillary C, balance B or photograph, according to the bubble size and the liquid viscosity. It was tested that three methods gave satisfactorily consistent results. The tank E contained a less viscous liquid which resolved the viscous liquid tested,,for examplewater for glycerine. Whena bubble was introduced in the tank E the bubble easily moved into the capillary C. Otherwise the pulling the bubble into C was hard and the reading of the bubble volume was erroneous due to the sticked thick layer of the viscous liquid. Fig. Experimental apparatus 45

Table I Properties of liquids tested Liquid Symbol Temp. [ C] jo[g/cm3] å j"[poise] ff[dyne/cm] M # 16.7 0.962 13.6 39.2 561 Caster Oil - - Q 31.8 0.953 3.76 38.8 3.52 d). 15.9 1.27 19.6 63.6 443 20.0 1.26 13.3 63.4 95.4 Glycerine 35.0 1. 25 3.64 62. 6 0.560 25.3 1.26 3.78 63.2 0.629 O 25A 1-26 1-57 63.2 0.0187 0 13.5 1.38 103.8 95.6 95400 Corn Syrup ~O~ 20.5 1.38 45.8 86.0 4910 P 29.2 1.38 19.4. 81.0 189 Liquids used The properties of the liquids employed, caster oil, glycerine and corn syrup, are given in Table 1. Some of the experimental conditions were chosen in order that two liquids had practically the same viscosity but different surface tension. The viscosity of the liquids was measured by means of a Brook field-type viscosimeter and also by the falling sphere method, the surface tension by the bubble pressure method and the density by pycnometer. Someof the values of surface tension and density were taken from International Critical Tables and Kagaku Benran (Handbook of Chemistry). The values of M, a dimensionless group g^/pa3 widely employed in the field of bubble studies7>9ll3), are-also shown in Table 1. Results and Discussions Rising velocity of single air bubbles Effect of bubble diameter The rising velocity measured are shown in Fig. 2, along with the data by Redfield and Houghton12) and Angelino0. The velocity U increases monotonously with increasing equivalent diameter of the bubble d. The data show that the rise velocity of the bubble of a given diameter is determined mainly by the viscosity of the liquid, In case of small viscosity liquids it has been reported that U reaches a maximumat a certain value of d, for example at d of about 2mmin water. Muchlarger bubbles rise independently of the kind of liquid. But in the high-viscosity liquids such phenomena, were not observed in the extent of this study. And the trajectories of the bubbles were always rectilinear. Correlation between CDand Re Fig. 3 presents the results in terms of drag coefficient CD and Reynolds number Re. Data by Red field and Houghton12) and Angelinoi:> are also, included for comparison. The drag Fig. 2 Rising velocity of air bubbles in high-viscosity liquids as a function of equivalent diameter coefficients decreases independently of the kind of liquid with increasing Reynolds number. At low Reynolds number the data of drag coefficient agreed the Hadamard equation for gas sphere though the liquids used were not purified especially. It -appears that contamination is not very important in case of highly viscous liquid. The drag curve, however, deviates from the Hadamard equation in the range Re>0.5, indicating that the deformation of bubble and the inertial effect begin to dominate in the drag. Taylor and Acrivos145 derived the following equation for the drag of single bubbles at low Reynolds number, c»=i -+2+ihHr+L33^+ ci> 46 JOURNAL OF CHEMICAL ENGINEERING OFJAPAN

Fig. 3 Drag coefficient as a function of Reynolds number Fig. 4 Ratios of drag coefficients (Cd/Cdf) as a function of Reynolds number Fig. 5 Reynolds number as a function of dimensioniess group M for given values of drag coefficient ratio (Cd/Cbf) V NO.1 OL.1 1968 47

The second and third terms of the right hand side of the equation are the contribution of the inertial effect and the fourth, of the deformation of bubble. The present data in the range Re<2 follow the theory. With Reynolds number larger than 10 the drag coefficient tends to become a constant value. Although a further work will be necessary, a small number of data at very low Reynolds number showed a tendency to approach the Cd curve of the rigid sphere. Since Kubota et al.9) proposed the correlation Cd/Cdf vs. ReMinfor low Msystems, the present data are plotted in terms of Cd/Cdf and Re in Fig.4. Cdf is the drag coefficient of gas sphere having the same volume as a given bubble. And it was calculated from the equation9 * i-ga-d" = CDFReF2 (= CDRe2) (2) and the correlation13) Cdf = 18.5i^~0-82 (3) Data in low viscosity liquids by Tadaki and Maeda13) and Kubota et al.9) are also shown in Fig. 4. Examination of Fig. 4 shows that in case of highly viscous liquids Cd/ Cdf appears to be a function of only Reynolds number and becomes 1 when Re<l. While in case of the low viscosity liquids the effect of surface tension force dominates and the curves of Cd/Cdf are parallel, shifting to right in the decreasing order of Mvalue. From these facts Kubota et al.9) obtained the above mentioned correlation. The applicable range of this correlation was found from Fig. 5 that Mis smaller than about 1CT1. A close look at Fig. 5 might show that there may be a transition region in the range 10~2<M<l, where Cd/Cdf depends on viscous, surface tension, and inertial forces. Shape of single air bubbles Effect of bubble diameter In highly viscous liquids the shape of the bubble changes, as the size of bubble increases, from sphere to oblate spheroid and then into spherical cap, whose bottom may be convex, flat or concave depending on the size of bubble. In describing the shape of bubble the ratios such as d/a and b/a have been used in literature. In this paper a deformation index e(=b/a) was employed as a measure of the bubble shape. The experimental results of e vs. d are shown in Fig. 6. Figs. 6-a, 6-b and 6-c are arranged in the increasing order of surface tension of the test liquids. It is noted that both in 19-poise liquids of glycerine and corn syrup the shape of bubble is spherical when d Fig. 6 Deformation index e as a function of equivalent diameter Fig. 8 Reynolds number as a function of dimensionless group M for given values of deformation index e Fig. 7 Deformation index e as a function of Reynolds number 48 JOURNAL OF CHEMICAL ENGINEERINGS JAPAN

Fig.9 number Deformation as a function of Weber is small, but the bubble of d larger than about 0.8crr deforms. Similar facts are apparent for 13-poise and 4- poise liquids of caster oil and glycerine,respectively. å These critical values are determined mainly from viscosity and not from surface tension, and the larger the viscosity, the larger the, critical value of diameter. The relation betweenj4 and e is linear in a logarithmic plot. The gradients of the line are determined alsc mainly from the viscosity but independent of surface tension, and the slope is small when the liquid viscosity is high, in other words, the deformation does not occur easily. In case of the low viscosity liquid95 the corre lation of logd and loge is known to be linear also. The line shifts to left according as the surface tension is small because deformation easily occurs and the gradient of the line is large when the viscosity is high. From these, it is seen that the dependency of the deformation index on the liquid properties is different for the low and high viscosity liquids. It should be pointed out here that if the value of M differs very much the shape of bubble is a little different for the same value of the deformation index. Two bubble shapes are shown in Fig. 7 as examples. Though both have the same value of e of 0.8, the bubble has a flat or concave bottom in the high Mliquid but convex end in the low M liquid. Correlation between e and Re The deformation index e as a function of Reis shown in Fig.7. These curves may be classified into two groups, one being related to the data of high Mliquids and the other to those of low Mliquids. The deformation index is seen to be a function of only Re in the high Mliquids. The deformation index in this case is described by a following equation * = 0.81-0.217(logRe) - 0.084(logRe)2 (4) 0.1<Re<20 Eq.(4) will be useful, though entirely empirical, since there is no available correlation for e of the high Msystems. It was noted that at Re=l the bottom of the bubble became flat. While in case of the low Mliquids the curves appear parallel, shifting to right in the decreasing of M. The effect of M is more easily understood by plotting the data in terms of MandRe, with e being taken as the parameter as shown in Fig. 8. In the region M>3, the deformation index is proportional to M to O-th order, namely e being a function of only Re. And as for the region M<0.1, e is a function of ReMn, n being nearly equal to 1/4. It means VOL.1 NO.1 : 1:9.6:8 that the shape of bubble in the low Msystems is determined from surface tension rather than viscosity and the correlations, d/a vs. ReM0 23 by Tadaki and Maeda13) and (X-e) vs. We by Kubota et al.9\ show this situatioln. Taylor and Acrivos10 calculated the shape of bubbles slightly deformed from the sphere to give 0. 1575 We 6 1+0.0525We (5) Fig. 9 shows the present data interms of eand We. It may be said that Eq.(5) predicts the present data only approximately. The data deviate from the theory as Weber number inceases. The effect of viscosity is very remarkable. As the Cd/Cdf correlation appears to have a transition region, the deformation index seems also to be a function of both viscous and surface tension forces in the range 10~1<M<3. In the above discussions, inertial, surface tension and viscous forces were considered as the factors influencing on the deformation index. To estimate which factor is most dominant, it is necessary to obtain the solution of the equation of motion considering the deformation of bubble. But the most dominant factor to determine the shape of a given bubble can be estimated easily by calculating the order of magnitude of viscous and surface tension forces as dftu and da respectively and compar. ing them. The ratios of pu2d2 and these quantities give either Reynolds and Weber numbers. If a magnitude force is very much larger than the other of the dominant factor is the former. In fact, in case of low M liquids shape of dftu is very much smaller bubble is well described by than da and the the correlation e vs. ably We, while in larger than case da, of high Mliquids and the shape d{*u is is well consider- correlated in the form of e vs. Re. When the bubble diameter large, although very large bubbles are not considered is in this study, gravitational force must be also taken into account in addition to these three factors mentioned above. Summary The shape and rising velocity of single air bubbles were investigated experimentally in quiescent, highly viscous liquids such as caster oil, glycerine and corn syrup. It was found that the bubble behavior in highly viscous liquids is very different from that in low viscosity liquids. The bubble rose rectilinearly and without oscil- 49

lation. The terminal velocity was dependent mainly en the liquid viscosity and the surface tension showed only a minor effect. The Cd curve obtained experimentally agreed well, though no special caution of purification of liquids was taken, with Hadamard equation for gas sphere in the range of Re-about 10~3 to 10"1 and with Taylor-Acrivos equation for gas oblate spheroid in the range of Re up to about 2. The correlation proposed previously for low Msystems, Cd/Cdf vs. ReMl/\ was found to be valid in the range of M<10~1. For the region M>10"1 the Cb/Cdf was a function of only Reynolds number and unity when Re<l. The bubble shape varied as the bubble diameter increased, from sphere to oblate spheroid whose bottom is convex, flat or concave depending on the size of bubble. At Reynolds number of about unity the bottom of the bubble became flat. The e curve obtained was a function of only Reynolds number in the high Msystems, an empirical relationship being given by Eq.(4). The correlation, l~e vs. We, proposed previously for the low Mliquids was found applicable up to M=0.1. Nomenclature a= major axis of bubble b= minor axis of bubble Cd- drag coefficient of bubble, 4dg/3 U2 Cbf= ideal drag coefficient of a spherical gas bubble which has the same volume as a given bubble d- equivalent spherical diameter of bubble e= deformation index, b/a g= gravitational acceleration M- dimensionless group, gfj-a/'paz Re- Reynolds number, pdu/f* U=terminal velocity of bubble We=Weber number, i= liquid viscosity p= liquid density a=surface tension Literature cited pdu2/a [-1 [cm] [cm/sec2] [cm/sec] [g/cm-sec] [g/cm3] [dyne/cm] 1) Angelino, H.: Chem. Eng. Set., 21, 541 (1966) 2) Bond, W.N. and D.A. Newton: Phil. Mag., 5, 794 (1928) 3) Collins, R.: J. Fluid Mech., 25, 469 (1966) 4) Davies, R.M. and G.T. Taylor: Proc. Roy. Soc, A200, 375 (1950) 5) Garner, R.M. and D. Hammerton: Chem. Eng. Sci., 3, 1 (1954) 6) Hadamard, J.: Compt. Rend., 152, 1735 (1911) 7) Harberman, W.L. and R.K. Morton: = Trans. Am. Soc. Civil Engrs., 121, 227 (1956) 8) Kintner, R.C. et al.: Can, J, Chem. Eng., 39, 235 (1961) 9) Kubota, M., T. Akehata and T\ Shirai: Kagaku Kogaku, 31 (1967) 10) Moore, D.W.: J. Fluid Mech.y 16, 161 (1963) ll) Moore, D.W.: ibid., 23, 749 (1965) 12) Red field, J.A. and G. Houghton: Chem. Eng. Set., 20, 131 (1965) 13) Tadaki, T. and S. Maeda: Kagaku Kogaku 25, 254 (1961) 14) Taylor, T.D. anda. Acrivos: J. Fluid Mech., 18, 466 (1964) 50 JOURNAL OF CHEMICAL ENGINEERING OFJAPAN