Local Droplet Diameter Variation in a Stirred Tank MASATO TANAKA

Department of Chemical Engineering, Faculty of Engineering, Niigata University, Niigata City 950-21, Japan

The local variation of droplet diameter in a stirred tank was measured in the vicinity of the impeller and at another region. The degree of difference in droplet diameter between regions increases with the impeller speed.

A correlation equation between the local difference in droplet diameter and the frequencies of coalescence and circulation of the droplet was derived according to the circulation interaction model. The degree of local difference in droplet diameter was found to be controlled strongly by the ratio of coalescence to circulation frequency.

On a mesurC la variation locale du diamttre de gouttelettes dans un rkservoir agit6 au voisinage de I’hClice et dans une autre rkgion. L’importance des diffkrences du diamttre des goutellettes entre les diffkrentes r6gions augmente avec la vitesse de I’hClice. On a Ctabli une corr6lation entre la diffkrence locale dans le diamttre des gouttelettes et les frkquences de coalescence et de circulation des gouttelettes en utilisant le modtle d’interaction de circulation. On a trouv6 que I’ampleur des differences locales du diamttre des gouttelettes Ctait fortement contrBlCe par le rapport entre les Mquences de coalescence et de circulation.

iquid-liquid dispersion is carried out in such operations L as suspension polymerization, heterogeneous liquid- liquid reaction and liquid- liquid extraction. In these operations the dispersed phase is created mainly by a con- ventional type of impeller and dispersed in fine droplets in the continuous phase. In general, it is highly desirable that the dispersion remains stable during the operation and, in some cases, the droplet diameter should be as uniform as possible (O’shima and Tanaka, 1982a, 1982b; Tanaka and O’shima, 1982).

With regard to the distribution of droplet diameter, how- ever, it is difficult to remove completely the variations of local droplet diameter with the position in the stirred tank, when agitation is performed by conventional types of im- peller (Mlynek and Resnick, 1972; Weinstein and Treybal, 1973; Coulaloglou and Tavlarides, 1976; Sprow, 1976). Accordingly, in designing reactors to process the liquid- liquid dispersions and control the local droplet diameter variation, it is necessary to know to what degree the local droplet diameter varies under given operational conditions.

The purpose of this study is to obtain a correlation be- tween the degree of local droplet diameter variation and the dispersing behaviour of the droplet, which is controlled by experimental conditions.

In this work, the steady-state droplet diameter at a given impeller speed and then the transient droplet diameter after a sudden reduction in impeller speed were measured. These measurements were made at the impeller and in the circu- lation region, respectively.

These results were analyzed according to the circulation interaction model (Reitema, 1964).

Experimental

APPARATUS AND PROCEDURE

Figure 1 shows schematic diagrams of the experimental apparatus, the modified light transmission unit and the de- vice for trapping droplets. The stirred tank was a flanged container equipped with four equally-spaced baffles. The impeller was a six blade Rushton turbine. The stirred tank was filled with the dispersed and the continuous phase to a height equal to the tank diameter. The dispersed phase con-

taining sebacyl chloride (0.05 wt%) was a 4.6: 1 mixture of benzene and carbon tetrachloride.

Sebacyl chloride was dissolved to encapsulate the droplet by reacting with hexamethylene diamine, which was added after trapping droplets, and it didn’t undergo any mass trans- fer during the dispersion process.

The continuous phase was ion-exchanged water. Physical properties for the dispersion system are shown in Table 1. The low volume fraction (10%) of dispersed phase was used. The agitation was started at a given impeller speed and continued until steady-state dispersion was established, as determined by the light transmission method (Howarth, 1967; Hong and Lee, 1983). The light transmission value changed with time, reaching a steady value after about 10 to 20 min.

The droplet diameter was measured by the encapsulation method (Mlynek and Resnick, 1972). The device for trap- ping droplets was immersed at the desired position as shown in Figure 1. For an instant, the spring of the device was depressed and then released, trapping the droplets. Simulta-

I li 4

(unit :cm 1 1 2

Figure 1 - Schematic diagrams of experimental apparatus and device for trapping droplets.

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 63, OCTOBER 1985 723

TABLE 1 Physical Properties of Dispersion Sys-

tem (at 20°C)

- (Sprow, 1967) -

Q '13

-0- -0-

(Sprow,1967) I I

Figure 2 - Dependence of droplet diameter on impeller speed.

n

E *- 0 c

X Q U

Nr 7.5- 5.0

-e- point A 4 - -0- point B

3-

2

0 5 10 15 60 90 time (min. )

Figure 3 - Transient droplet diameter after reduction in impeller speed.

neously, a portion of 0.05 wt% aqueous solution of hexa- methylene diamine was added to the device through the charging port. Thus the droplets in the device were encap- sulated by a thin polymer film whose thickness is thought to be lo%, at the most, of the diameter of a small droplet (about 50 p).

These droplets were transferred to a flat glass dish con- taining 1.0 wt% aqueous solution of polyvinyl alcohol to prevent the droplets from adhering and then photographed. From this photograph the measurement of droplet diameter was made.

To obtain the transient droplet number concentration re- quired for calculation of coalescence frequency, the fol- lowing operation was performed. The impeller speed was suddenly reduced. The droplets coalesced due to the reduc- tion in turbulence level. At appropriate time intervals after the sudden reduction in impeller speed, the droplets were trapped at the desired positions, encapsulated, transferred to the dish and then photographed as described above. This procedure was repeated with several sets of lower and higher impeller speeds. From these photographs the transient drop- let diameters were measured. In this work the mean droplet diameters are expressed as the Sauter mean diameter.

MEASUREMENT OF CIRCULATION FREQUENCY OF A PARTICLE

Because the circulation frequency of droplet in the stirred tank is thought to be an important factor affecting the local droplet diameter variation, the following measurement was made using a polystyrene bead.

The time required for a particle to be discharged from the impeller region, to circulate in the tank and then to return to the impeller region, was measured by visual observation. In this run the density of the continuous phase was adjusted to equal that (1040 kg/m') of the particle by dissolving sodium chloride of 6 wt%. The diameters of particles used were 1.4, 2.1, 2.8, 3.3 and 3.8 mm.

Experimental results

Figure 2 shows the dependence of the mean droplet di- ameters at the impeller (point A in Figure 1) and in the circulation region (point B in Figure 1) on the impeller speed. The results by Sprow (1967) are also shown in Figure 2. It is seen that the droplet diameters in the circulation region are larger than those at the impeller region and the dependencies on the impeller speed are different from each other.

Moreover, the difference between droplet diameters at the two regions is found to become larger with increase in impeller speed.

Figure 3 shows the transient mean droplet diameters at both regions. It is found that the droplet diameter shows the different transient feature according to the sampling position.

Figure 4 shows the dependencies of mean circulation time of a particle on the impeller speed and the particle diameter. From this figure it is found that the mean circulation fre- quency is proportional to the impeller speed and is indepen- dent of the particle diameter.

Discussion

According to the circulation interaction model (Reitema, 1964), under the turbulent conditions a droplet is broken up at the impeller region, discharged from this region, grows by coalescence while being dispersed in the circulation re- gion and then returns to the impeller region. In addition, according to Park et al. (1975), beyond distances from the impeller of only 1/6 impeller diameter, droplet breakup is virtually nonexistent. From these concepts, the coalescence and breakup behaviours of droplets in the stirred tank can be modeled as shown in Figure 5 .

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102 -3 particle size(m) - -10

'""1

1 1 I I t I l l ,

1 Nr 10 Figure 4 - Dependence of circulation time of a particle on im- peller speed and particle size.

In this model the following assumptions are made as by Park and Blair (1975). The flow of fluid along the circu- lation loop is plug flow and its velocity is a weak function of distance from the impeller. Thus the changing rate of droplet number concentration, n, along this loop can be expressed simply as follows.

. . . . . . . . . .

If the probability that more than three droplets coalesce at once is assumed to be extremely low, as in the kinetic molecular theory of gases, Equation (1) is reduced to Equation (2).

dn dt - = -kn2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Equation (2) is transformed into Equation (3).

dn 1 n 2 - _ - - - w,dt . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

where w, is the coalescence frequency for a droplet and equal to 2R,./n.

Since the coalescence frequency is thought to be de- pendent on the droplet number concentration (or diameter), namely, on the time, it must change along the circulation loop. In order to avoid complexity on calculation, it is assumed that the coalescence frequency can be expressed in terms of the time, because the droplet number concentration changes with time. Integrating Equation (3) from ni to n, with respect to droplet number concentration and from 0 to r, with respect to time, Equation (4) is obtained.

(4)

where r, is the time required for the droplet to circulate once along the circulation loop.

impeller region I (breakup zone)

circulation region (coalescence zone)

Figure 5 - Model of coalescence and breakup behaviour droplet.

By using the relations such as +,i, = w,.dt and fc l/Tc, Equation (5) is obtained. 2 = exp (- --) 1 w, . . . . . . . . . . . . . . . . . . . . . . . . . . ni 2 fc Also, assuming a uniform dispersed phase holdup in the

stirred tank, namely, nidii = n,di,, Equation (6) is obtained.

Equation (6) means that the ratio of the droplet diameter at the impeller region to that at the circulation region is controlled by the ratio of mean coalescence to mean circu- lation frequency. The coalescence frequency, which is re- quired to check whether the droplet diameters measured at both regions satisfy Equation (6) or not, is calculated by Equation (7) derived from Equation (3).

1 dn . . . . . . . . . . . . . . . . . . . . . . . . . w, = (- 2 - - ) n dt r=o

(7)

The practical calculation was made as follows. The tran- sient droplet number concentration is obtained from the measured transient droplet diameter. Then, a polynomial such as n = Co + C , t + C2r2 + C3t3 is fitted to the curve of droplet number concentration and coefficients in poly- nomial are determined by the least squares method. By first differentiating this equation with respect to time and then introducing t = 0 and n at f = 0 into resultant equation, the coalescence frequency can be calculated. Coalescence fre- quencies obtained at the impeller and the circulation region are averaged to give the mean coalescence frequency.

Figure 6 shows the comparison of the local droplet diameters calculated by Equation (6) with the measured values. From this figure it is found that the former are in

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 63, OCTOBER 1985 725

l . O \ I I I I I I I I I

Eq.( 6

“I Figure 6 - Comparison of local droplet diameters calculated with those measured.

10 -*I 2 3 4 5 6 7 8 9 1 0

Nr Figure 7 - Dependence of coalescence frequency on impeller speed.

good agreement with the latter with a maximum deviation of 14%. From this fact is is concluded that the local difference in droplet diameter is strongly controlled by the ratio of coalescence to circulation frequency.

Also, the degree of local difference in droplet diameter can be estimated by Equation (6). However, it is question- able whether the model can be applied to the dispersion of the high volume fraction of dispersed phase. This point will be investigated in detail.

An increase of the degree of local difference in droplet diameter with the impeller speed can be interpreted as fol- lows. From Figure 7 the mean coalescence frequency is found to be proportional to the 2.4 power of impeller speed. This means that the coalescence frequency is proportional to the 0.8 power of mean power input per unit mass. Con- sequently, the coalescence frequency may be thought to

increase with the impeller speed, because power input gives the droplet the kinetic energy required for droplet to co- alesce in a turbulent field.

Dependencies of the coalescence frequency on impeller speed and power input are in good agreement with the pre- vious works (Coulaloglou, 1976; Howarth, 1964; Madden, 1962; Nakada, 1977).

Accordingly, Equation (8) is obtained. - 2 - Nf’4 . . . . . . . . . , . . . . . . . . . . . , . . . . . . . . . . . . (8) f c

Equation (8) shows that the relative value of coalescence to circulation frequency, GC/fc, increases with impeller speed. Consequency, according to Equation (6), an increase of the circulation frequency decreases the droplet diameter, in other words, the degree of local difference in droplet diameter increases with impeller speed. This trend is in qualitative agreement with Sprow’s result (1967).

Conclusion

The variation of local droplet diameter with position in a stirred tank was investigated. According to circulation inter- action model, an equation was obtained which correlated between the ratio of droplet diameters in the impeller and circulation regions with that of the frequencies of co- alescence and circulation. It was also found that the degree of variation of local droplet diameter was strongly controlled by both coalescence and circulation frequency.

Nomenclature

Co, CI, C2, C3 = coefficients in a polynomial d P C

d p i

j c k N, = impeller speed (s-I) n n,

ni

R, t = time (s) t c = circulation time (s) 1, WC = coalescence frequency (s-’) WC

Greek letters

= droplet diameter at circulation region (m) = droplet diameter at impeller region (m) = mean circulation frequency (sc’) = coalescence rate constant (s-’ ‘number-’)

= droplet number concentration (number. mC3) = droplet number concentration at circulation re-

= droplet number concentration at impeller region

= coalescence rate (number * s-’)

gion (number * m-3)

(number. m-3)

- = mean circulation time (s)

= mean coalescence frequency (s-’) -

Y k P d

P c

pd

= interfacial tension (N m-I) = viscosity of continuous phase (Pa. s) = viscosity of dispersed phase (Pa. s) = density of continuous phase (kg*m-3) = density of dispersed phase (kg - m-3)

References

Coulaloglou, C. A. and L. L. Tavlarides, “Drop Size Distributions and Coalescence Frequencies of Liquid-Liquid Dispersions in Flow Vessel”, AIChE J. 22, 289-297 (1976).

Howarth, W. J. , “Coalescence of Drops in a Turbulent Flow Field”, Chem. Eng. Sci. 19, 33-38 (1964).

Howarth, W. J. , “Measurement of Coalescence Frequency in an Agitated Tank”, AIChE J. 13, 1007-1013 (1967).

Heng, P. 0. and J. M. Lee, “Unsteady State Liquid-Liquid Dis-

126 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 63, OCTOBER 1985

persions in Agitated Vessels”, Ind. Eng. Chem. Process Des. Dev. 22, 130-135 (1983).

Mlynek, Y. and W. Resnick, “Drop Sizes in an Agitated Liquid- Liquid System”, AIChE J. 18, 122-127 (1972).

Nakada, K. and M. Tanaka, “Interpretation of the Coalescence Behaviour of Dispersed Droplets in a Stirred Flow Tank in Terms of Quasi-Chemical Kinetics”, Trans. Instn. Chem. Engrs. 55, 143-148 (1977).

O’shima, E. and M. Tanaka, “Coalescence and Breakup of Droplets in Suspension Polymerization”, Kagaku Kogaku Ronbunshu, 8, 86-90 (1982).

O’shima, E. and M. Tanaka, “Effect of Solid Powders on Stability of Suspension Polymerization of Styrene”, ibid., 18, 188- 193 (1982).

Park, J. Y. and L. M. Blair, “The Effect of Coalescence on Drop Size Distribution in an Agitated Liquid- Liquid Dispersion”,

Chem. Eng. Sci. 30, 1057-1064 (1975). Reitema, K., “Segregation in Liquid-Liquid Dispersion and its

Effect on Chemical Reactions”, in Advances in Chem. Engng. Academic Press, New York (1964).

Sprow, F. B., “Drop Size Distributions in Strongly Coalescing Agitated Liquid-Liquid Systems”, AIChE J . 13, 995-998 ( 1967).

Tanaka, M. and E. O’shima, “Effect of Viscosity of Continuous Phase on Stability of Droplets in Suspension Polymerization of Styrene”, Kagaku Kogaku Ronbunshu, 8, 734-738 (1982).

Weinstein, B. and R. E. Treybal, “Liquid-Liquid Contacting in Unbaffled Agitated Vessel”, AlChE J. 19, 304-312 (1973).

Manuscript received October 31, 1984; revised manuscript re- ceived March 7, 1985; accepted for publication March 8, 1985.

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