Ind. Eng. Chem. Res. 1994,33, 703-711
703
Bubble Size Distribution in a Gas Sparged Vessel Agitated by a Rushton Turbine Rajarathinam Parthasarathyt and Nafis Ahmed’ Department of Chemical Engineering, University of Newcastle, Newcastle, NS W 2308, Australia
The effect of agitation, due to a Rushton turbine, on the bubble size and bubble size distribution has been studied in an aerated vessel. By using a noncoalescing system, it has been possible to study the bubble breakup process in isolation. Bubbles of two different sizes, with d32 of 300 pm and 2.5 mm, are generated using porous spargers. With no agitation, the bubble populations are found to exhibit a log-normal distribution. The 300-pm bubbles are too small to undergo further breakage for the range of impeller speeds studied, and the size distribution is preserved. The larger bubbles are broken by agitation and the d32 decreases. Also with increasing agitation, the size distribution changes from unimodal to bimodal, and again to unimodal, as the bubble population moves progressively down the size scale. A model proposed, based on the energy dissipated in the impeller zone, predicts the continually changing pattern of the bubble size distributions satisfactorily.
Introduction Numerous industrially important chemical reactions take place between a gas and a liquid or a gas and a solid. In the latter case, the solid is usually in suspension in an aqueous phase. A significant percentage of such reactions are carried out in a mechanically agitated vessel, which is an effective device for contacting gas with liquid. A number of the processes are area controlled and depend on transport across the gas-liquid interface. Insufficient transfer may cause failure of the process, reduction in yield, or production of undesired side producb (Tatterson, 1991). The main performance criterion is the maximum mass transfer in such vessels, which is determined by the gas holdup (volume fraction of gas in the dispersion) and the size distribution of bubbles in the dispersion. A comprehensive review on gas dispersion in agitated vessels has been presented by Tatterson (1991). One of the main findings is that the ability to predict bubble size and size distribution as a function of the operating parameters, for example the power input, is necessary before the performance of gas-liquid mass-transfer systems can be estimated from the first principles. This however, has proved to be an elusive proposition so far. It is probably accurate to say that a systematic approach toward understanding gas dispersion and bubble breakup mechanism in stirred aerated vessels began with the work of van’t Riet and Smith (1973) and Bruijn et al. (1974). They clearly elucidated the mechanism of gas dispersion by a Rushton turbine and found that the trailing vortices present behind the impeller blades play an important role in determining the number of bubbles produced and their size distribution. Since the pioneering work by van’t Riet and Smith, considerable progress has been made in identifying the various hydrodynamic flow regimes occurring around the impeller and the tank. A good understanding now exists on the relationship between the gas dispersion characteristics and important parameters like the power draw and the gas rate, in the form of various dimensionless numbers. Flow regime maps have been suggested which indicate the hydrodynamics (and thus the dispersion conditions) prevailing in the tank, as a
* Author to whom correspondence should be addressed. Phone: +61 49 21 6180. FAX: +61 49 21 69 20. E-mail: cgnaa cc.newcastle.edu.au. + Present address: Department of Chemical Engineering, The University of Malaya, Kuala Lumpur, Malaysia.
0888-5885/94/2633-0703$04.50/0
function of various dimensionless numbers (Smith et al., 1987;Smith and Warmoeskerken, 1985;Warmoeskerken, 1985). Comprehensive reviews on this topic are available (Middleton, 1992; Tatterson, 1991). Although the gas dispersion mechanism by impellers has been understood to an extent, limited information is available on the equilibrium bubble size distribution generated during the dispersion process. van’t Riet and Smith (1973)found that the bubbles ejected by the impeller are subjected to further breakup by high-intensity turbulent forces prevailing around the impeller region. The breakup forces are resisted by the interfacial tension, and an equilibrium bubble size distribution results when a balance between the breakup and surface tension forces is reached. If the liquid phase is noncoalescing, the size distribution of the bubbles generated in the impeller region can be expected to be preserved in the other regions of the tank. In coalescingsystems, there will be spatial variations of bubble size distribution due to increase in coalescence in more quiescent regions of the tank. Kawecki et al. (1967) measured bubble size distributions in the impeller region of a stirred vessel using photography. They found that the impeller speed, as well as superficial gas velocity, influenced the bubble size distribution. Also, the distribution of surface area calculated from the number distribution of bubbles was reasonably represented by the Erlang distribution. The treatment of data was limited to a single expression encompassing all the data collected by them. Using a capillary probe technique, Greaves and Barigou (1988) measured bubble size distributions at a number of locations in astirred vessel. Utilizing a Rushton turbine, they found the bubble size distributions to be dependent on the position and the agitation intensity in the tank. A Weibull distribution was found to approximate the bubble size distributions, averaged for the whole tank. The same authors later provided information on the spatial variations in the Sauter mean bubble diameter in the impeller region and other locations in the tank (Barigou and Greaves, 1992a,b). With an increase in impeller speed, the bubble size distribution was found to shift toward the lower end of the bubble size scale due to an increase in bubble breakup. In the impeller region the vertical and radial variations in the Sauter mean bubble diameter were found to be highly nonuniform. Also, the average bubble size in baffle plane was found to be smaller than those measured in a midplane between the baffles, while the bubbles behind baffles were found to be larger than those 0 1994 American Chemical Society
704 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994
at the front. It is obvious that the observations are due to the different levels of turbulence prevailing in a stirred tank. To summarize the above findings, it may be concluded that some qualitative observations have been made regarding the bubble breakup process and the average size in stirred vessels. The equilibrium bubble size distribution in such vessels is determined by two dynamic processes: bubble breakup and coalescence. Bubble breakup occurs due to the high shear generated either by the impeller or by the turbulent fluctuations along the bubble surface. The coalescence rate of bubbles is mainly controlled by the nature of the liquid phase, the turbulence playing a secondary role. Quantitative information, however, is clearly lacking on the change in bubble size or distribution, as a function of, say, the strength of agitation or the energy dissipation rate in the tank. The most probable reason is the inherent experimental difficulties involved in studying the breakup and coalescenceprocesses simultaneously. To overcome one of them, the present paper describes an investigation in which one aspect of the bubble breakup phenomenon, that is, the coalescence, is inhibited, making it possible to look at only the breakage process in isolation. The possibility of obtaining a practically useful relationship between the observed breakup behavior and the power input into the tank is then explored.
Experimental Section All the experiments were carried out in a 0.195-mdiameter (T)Perspex cylindrical vessel, fitted with four standard baffles (T/lO).The stirrer was driven by a variable speed drive, and its speed was measured using a digital tachometer. A six-bladed Rushton turbine impeller of diameters equal to 0.065 m (=T/3)was used, with the blade width W equal to 015. The Rushton turbine was chosen because of its status as the most studied impeller. The clearance from the tank bottom was set at T/3. The liquid height was maintained equal to the tank diameter. The power drawn by the impeller was determined by measuring the torque experienced by the impeller and the motor assembly using a load cell. The motor was attached to a steel shaft passing through two frictionless ball bearings. The motor and shaft unit was suspended from a flexible steel cable which passed over two pulleys, allowing a counterweight to be used to balance the weight of the motor unit. This arrangement left the motor unit free to rotate inside the bearings with minimal friction. To confirm the accuracy of the power measurements, the power numbers were calculated from power input measurements for the impeller rotating in water. The average power number for the Rushton turbine was found to be 5.1. This value was found to be within the range of values obtained by Bujalski et al. (1987). The stirrer speed was varied between 2.08 and 13.33 rps (125-800 rpm) and was limited on the upper side by the onset of surface aeration. The superficial gas velocity, based on the cross-sectional area of the tank, was varied between 2.5 X lo4 and 1.25 X m/s. The gas rate was kept deliberately low to values below which the gas rate had no effect on the bubble size distribution. Tap water and air were used as the liquid and gas phases, respectively. To establish noncoalescing conditions, methylisobutylcarbinol (MIBC), an industrial frother, used extensively in the mineral processing industry, was used at a concentration of 50 ppm in all the experiments. The surface tension of the liquid was determined to be 71.3 mN/m. All experiments were conducted at a temperature of 21 f 0.5 "C in a temperature-controlled room.
Two spargers were used to generate bubbles of different initial sizes. They were two sintered glass disks of porosities 150-250 and 5-15 pm, designated as porosity 0 and porosity 4 by the manufacturer (Pyrex),respectively. Each disk, 90 mm in diameter, could be attached interchangeably to an aerator fixed to the tank bottom, such that it was centered directly under the impeller. The bubble size was measured by photography. Special cells similar to the one used by Ahmed and Jameson (1985) were used for photographing the bubbles. A sample of liquid with associated bubbles was drawn continuously from the tank, passing between two transparent plates either 2 or 5 mm apart, where they could be photographed. The photographs were projected onto a bitpad, and the bubble sizes were measured using a microcomputer. The bubble size was measured at the mid-liquid-height position. To check the bubble size variations, if any, with position in the tank, measurements were made at the impeller region and randomly around the tank and compared with the bulk values. The variation in the bubble size values measured was found to be within f105% of the average, which suggested that the bubble size determined in the impeller zone was preserved in other parts of the vessel. This observation confirmed that the bubbles were not coalescing and supports the view that, in a noncoalescing system, the bubbles created by the impeller and those in the body of the liquid would have the same size (Andrew, 1982).
Results and Discussion Experimental Observations. In the following discussions, the spargers used for bubble generation will be denoted by their respective classification numbers, Le., sparger-0 and sparger-4. The bubble size measured at the impeller speed of 2.08 rps (125 rpm) is denoted as the initial bubble size in this study. This is the minimum speed beyond which the impeller was able to completely divert the flow of the rising bubbles from the vertical, especially for the larger bubbles. Since the bubbles were being sampled away from the impeller zone, this speed represented a value beyond which there was no ambiguity regarding the dispersion of bubbles, of both sizes, to the bulk. Considering that the gas was introduced as discrete bubbles rather than as bulk gas, and that the gas rates used were low, the bubbles were dispersed with ease. An inspection of the dispersion pattern as a function of the impeller speed, however, was made extremely difficult because the fine bubbles made the dispersion "milky" in appearance, so much so that the impeller region could not be visually observed. Table 1 provides detailed information on all the parameters which would be relevant in describing the dispersion mechanism. It is obvious that the gas rates, and thus the gas flow number, are an order of magnitude smaller than would be expected in industrial units, or those used in conventional studies (Smith et al., 1987;Warmoeskerken and Smith, 1985). As mentioned earlier, the gas rates were kept low to maintain narrow bubble size distributions. Since the experimental range is limited, and in the absence of proper visual confirmation, a detailed analysis of the dispersion mechanisms will not be attempted in this paper. It will be noticed, nonetheless, that the gas holdup values are higher than those one would expect for such low gas rates, and this is directly attributable to the ease of dispersion of the small bubbles. Concentrating on the bubble size, the initial size of the bubbles generated by sparger-4was found to vary between 100and 800pm, giving a Sauter mean bubble diameter of about 300 pm (Figures 1 and 2). The frequency number
Ind. Eng. Chem. Res., Vol. 33, No. 3,1994 705 Table 1. Range of Experimental and Derived Parameters
Common Parameters for Both Bubble Sizes ungassed power (P),W Froude number (Fr) gas flow number (FZ)
0.062 0.028 0.013
4.05 0.46 2.26 X 103
16.4 1.17 2.04 X 103
gas holdup, % gassed power (Pa),W Pgl P
0.29 0.06 0.97
2.43 3.84 0.95
7.14 14.95 0.91
gas holdup, % Pg, PglP
0.07 0.06 0.97
1.39 3.81 0.94
7.26 14.18 0.86
0.039
6.09 x 103
0.065
0.016
0.01
8.93 14.46 0.88
1.8 0.06 0.97
5.25 3.59 0.89
10.0 13.89 0.85
7.68 13.89 0.85
0.71 0.06 0.97
2.97 3.64 0.90
9.11 13.73 0.84
9.74 x 103
Sparger-4 Bubbles, 300 pm 1.25 0.06 0.97
4.64 3.73 0.92
SDareer-0Bubbles, 2.5 mm -
w
I
0.18 0.06 0.97
2.11 3.7 0.91 100
ae
lo4
i
2al ET
p!
c L
80
-
0
A
60
+
5.00 rps 8.33 rps 13.33 rps
m n
5
1 o2 0
2
4
8
6
10
12
14
Impeller speed, rps
I
31
P
1.5
Q,
.B c
-8 a
0.5 0 10’
,
,
.,.-A
1 o2
,-
0 ‘ . 10’
-
m
,
+
A , O , l
10‘
1o3
Bubble size, prn
Figure 1. Effect of impeller speed on Sauter mean bubble diameter. Ug = 1.0 x 103 m/s. (A)sparger-0 and (0) sparger-4. IP
40
1 o3
Average bubble size, krn Figure 2. A typical bubble size distribution for sparger-4. Ug= 1.0 X 1Oa m/s. N = 2.08 rps.
distribution of the bubbles, plotted against the logarithm of the average bubble size, is shown in Figure 2 for U,= 1.0 X 10-3 m/s. The solid line is drawn by joining the midpoints of the histogram bars representing the number frequency in each size band. The bell-shaped curve suggeststhat these bubbles have a log-normal distribution. Similarly, the initial size distribution for the sparger-0 bubbles is also log-normal with a d32 of 2.5 mm as shown in Figures 1 and 4a. Figure 1also shows how agitation has minimal effect on the smaller 300-pm bubbles and the d32 remain unaffected. Even the initial bubble size distribution is preserved. This is further confirmed by plotting the size distributions for the different impeller speeds as cumulative curves (undersize) as shown in Figure 3. Although not shown here
Figure 3. Effect of impeller speed on bubble size distribution for sparger-4. Ug = 1.0 x 103 m/s.
in detail, superficial gas velocity did not have a significant effect on the distributions described above. In the case of the 2.5-mm bubbles there is no effect of impeller speed on bubble size up to 3.33 rps (200rpm), but thereafter it begins to decrease (Figure 1). The decrease in d32 is obviously due to the breakup of the larger bubbles in the population and would also involve a change in size distributions with agitation. The best course of examining the changing pattern of the size distributions would be to study the bubble frequency number distribution for each impeller speed. The frequency number distributions for the sparger-0 bubbles as a function of the impeller speed are shown in Figure 4 for U, = 1.0 X 103 m/s. The unbroken lines represent the best fit line through the experimentally determined values. The bubble size distribution at 2.08 rps (initial speed) displays a log-normal distribution with a single mode (Figure 4a). As the impeller speed increases, the distribution becomes bimodal with the appearance of a second mode toward the lower side of the size scale at 3.33 rps (Figure 4b). This would indicate the commencement of bubble breakup a t this speed, a feature which is corroborated by Figure 1. With further increase in impeller speed, the newly formed second mode gradually increases in area while the first one decreases (Figure 4c-e), because of further breakup. A t even higher impeller speeds, the distribution reverts to unimodal but with a slight skewness toward the larger bubble sizes (Figure 40. This skewness disappears almost completely at 11.67rps and the bubble size distribution becomes log-normal again, but on the lower side of the size scale (Figure 4g,h). At this stage, considering Figures 1and 4, it may be presumed that most of the bubbles generated originally by the sparger have undergone breakup.
706 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994
:
la1
5
4
N=
Oerps
2l
... N.13
1-
'1
N z 6 . 6 7 rps
33 rps
Id1
Average
bubble
s i z e , Urn
Average
bubble
size , urn
Figure 4. Effect of impeller speed on frequency number distribution of bubbles generated by sparger-0. U,= 1.0 X 10-9. (-) Experimental values, (- - -) dividing lines for the model, and predicted values. (-e)
The first transition of unimodal to bimodal distribution occurs between 2.08 and 3.33 rps, and the next transition of bimodal to unimodal distribution takes place between 10 and 11.67 rps. It is hard to pinpoint a single impeller speed for the transitions as the changes in the size distributions happen over a range rather than at a particular impeller speed. Gas superficial velocity had no effect on the pattern of the bubble size distributions observed above. The pattern of bubble size distribution for all the superficial gas velocities followed the characteristic 1-2-1 mode. The impeller speeds at which the transitions occurred also did not vary perceptibly with the superficial gas velocity. I t is interesting to note that the observations made above are similar to those made in the study of droplet breakup in immiscible liquid-liquid dispersions. Drop size distributions have commonly been observed to be either normal or log-normal depending on the nature of the dispersion. Godfrey and Grilc (1977) report that the drop size distribution in an agitated tank changed from normal to log-normal as the impeller speed increased. Stamatoudis and Tavlarides (1981) found that an increase in impeller speed resulted in narrowing of the drop size distribution They postulated that the decrease in the variance of the size distribution was due to the preferential breakage of larger drops as compared to the smaller drops. Chatzi et al. (1989) found that the drop size distribution at low impeller speeds was bimodal. As the impeller speed increased, the frequency of the first maximum of the distribution decreased, with a simultaneous increase in the frequency of the second maximum, leading naturally to an overall lowering in the average drop size. They also
proposed a population balance model to predict the observed drop size distributions. Similarly, Nishikawa et al. (1991) showed that the number distribution of drops in a mixing vessel could be approximated by a combination of two normal distribution curves. Since the above observations were made at different conditions, in terms of the system geometry and power input, it is quite possible that they represent various stages of the droplet breakup process. Expressed differently, the various workers observed effects applicable to their own particular range of conditions. This makes it difficult to obtain a complete sequential picture, as is desirable for comparison purposes. Although elements of findings made in droplet breakup studies have been observed in the present study, especially the form and the change in the size distributions with agitation, it must be strongly emphasized that the two systems are widely different as far as the breakup environment is concerned. For a liquid-liquid dispersion, there is a continual recirculation, or no escape, for the dispersed phase, and the residence time may be extended indefinitely. This factor makes the study of droplet breakup relatively simpler, which probably is the reason for more reported studies with liquid-liquid systems. On the other hand, in the case of gas in liquid, the gas residence time is obviously limited, and a probabilistic approach would be necessary to quantify the time spent by different elements of the gas in the tank. The number of times an element passes through the impeller would have a profound effect on the bubble breakage probability and the resultant bubble size. These considerations are worthy of further deliberations. For the present, however, a simpler ap-
Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 707 proach is adopted, occasionally using concepts developed for the liquid-liquid case. Bubble Breakup in Turbulent Flow. The breakup behavior of the bubbles generated by both spargers can be qualitatively explained considering Kolmogoroff's theory of local homogeneous isotropic turbulence (HIT) (1941a-c, 1949). Accordingto the theory, a turbulent flow similar to the one present in a stirred vessel can be regarded as a superposition of eddies of various orders of magnitude. The primary eddies, which have a scale similar to that of the main flow (impeller blade in the case of stirred vessels), contain the bulk of the kinetic energy transferred from the impeller to the fluid. These eddies are unstable and disintegrate rapidly into smaller and smaller eddies, which ultimately dissipate the energy by viscous effects. These eddies, which are termed the "viscous subrange eddies", are statistically independent of the main flow and may be considered to be locally isotropic. The scale of the viscous subrange eddies q was derived by Kolmogoroff from dimensional reasonings as
where u is the kinematic viscosity and eavg is the average energy dissipation rate per unit mass. Kolmogoroff further hypothesized that at infinitely large Reynolds number an "inertial subrange" exists outside the region where viscous dissipation occurs where the energy spectrum is solely determined by eaVp The scale of the inertial subrange eddies is much larger than q, but still sufficiently smaller than the primary eddies according to the expression
L >> 1 >> q
(2)
where 1 is the scale of inertial subrange eddies. Hinze (1955) assumed that, in a turbulent flow where local isotropy prevails, the bubble breakup is caused by eddies of the inertial subrange if the maximum stable bubble diameter d,, is of the same scale of these eddies. Larger eddies just convect the bubbles, while smaller eddies do not have enough energy to rupture them. Substituting d , for 1 in relation 2, we get
L >> d,, >> q
than the majority of the bubbles in the distribution. Thus, the kinetic energy of the inertial subrange eddies is insufficient to break the very small bubbles produced by sparger-4, and therefore the bubbles and their size distribution are unaffected by increase in impeller speed. In stirred vessels, on the basis of the above considerations, for a particular impeller speed we expect to have a maximum bubble size (d,-), beyond which bubbles will break up due to the prevailing level of turbulence. If the impeller speed is increased, the d m , value will decrease and the system will reach a new equilibrium. The sparger-0 bubbles follow this pattern exactly. With commencement of agitation some bubbles break up, leading to the appearance of a new mode in the distribution on the smaller bubble size side. With further increase in agitation level, the size of the maximum stable bubble decreases and therefore more bubbles are broken up. As a consequence, the number of smaller bubbles in the population increases, which is reflected in the progressivley larger areas of the second mode. At the maximum speed studied, most of the bubbles originally generated by the sparger would be expected to be broken up and the resulting bubble size distribution reaches a new equilibrium, which exhibits a unimodal distribution on the smaller bubble size side. Predicting the Bubble Size Distributions. As observed in Figure 2, the frequency number distribution of bubbles generated by sparger-4 displays normal distribution when plotted against average bubble size on logarithmic coordinates, suggesting log-normal distribution. The log-normal probability density function is given by (Orr, 1966)
(3)
which is a necessary condition for bubble breakup by turbulent fluctuations. The two necessary conditions for the application of the isotropic turbulence theory to a given situation are, firstly, a sufficiently high Reynolds number and secondly L >> q, where L is the scale of the primary eddies (Shinnar and Church, 1960). In the present case the impeller Reynolds number Re1 (pNDz/p),was above lo4 for the range of impeller speed studied, indicating fully turbulent conditions in the vessel. The value of L may be approximated by the impeller width (Shinnar and Church, 1960) which in the present case is 13 mm. The numerical values of q, estimated using experimental values of u and tavg,vary from 98 to 25 pm for the range of impeller speed used. The large difference between the values of L and q validates the application of local isotropy to the present situation. The bubbles generated by sparger-0 are much larger when compared with the values of q satisfying condition 3 and are therefore subjected to breakup. As the power inputs measured using spargers 0 and 4 are not significantly different, the same values of q apply for the case of sparger-4 also. A comparison of the q values with the bubble sizes obtained for the sparger-4 would suggest that the scale of inertial subrange eddies are relatively larger
where d b is the bubble diameter, log mgis the log-geometric mean given by the relation i=n
i=n
and log ugis log-geometric mean standard deviation given by the relation i=n
i=n
The dotted line on Figure 2 represents the log-normal distribution function evaluated using experimental values of mean and standard deviation parameters. I t can be seen that the log-normal distribution function provides a reasonably accurate representation of the experimental distribution. As there is no breakup of the bubbles with increase in impeller speed for the sparger-4 bubbles, the log-normal function approximates the bubble size distribution for the whole range of impeller speeds studied. The cumulative frequency data (undersize) for the range of speeds studied are shown in Figure 5 on a log-normal probability graph for Ug = 1.25 X 10-3 m/s. The linear trend of the data confirms the observations noted above. Sauter mean bubble diameter values for the log-normal distribution function can be estimated from the ratio of the third and second moments of the distribution, which follows from its definition given as i=n
i=n
(7)
708 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 -
99.99 99.9 99 95 90
0
3.33
rps
0
5 00
rps
A
6.67
rps
X
8.33
A
rps 1000 rps 11.67 rps
0
13.33 rps
+
98 50
88
10
5
0
1 ,1
.01
'
t I
1 o2
10'
1 o3
Bubble size, pr Figure 5. Effect of impellerspeed on bubble size distribution plotted on log-normal probability coordinates for sparger-4. V, = 1.25 X 103 m/s. 500
__I
l o0 o 0
2
4
6
10
8
12
14
Stirrer speed, rps Figure 6. Comparison of Sauter mean bubble diameter values predicted by eq 8 with experimental values for sparger-4. ExperiV , = 2.5 X 10-4 m/s; 0 , U, = 1.25 X 103 m/s). mental values (0, Predicted values (- - -, V , = 2.5 X lo-' m/s; - -, U, = 1.25 X lo3 m/s).
-
size distribution curve can be resolved into two distinct curves and the size fraction of the two constituents under the two curves can be accounted for by the respective areas. Vedaiyan et al. (1972) used a similar approach for representing the bimodal distribution observed in the drop size distributions obtained from multiple nozzles in a countercurrent spray column. Recently, Nishikawa et al. (1991) reported that the number distribution curves for the drop sizes obtained in a stirred vessel can be approximated by a combination of two normal distribution curves. A model based on the above approach may be applicable in the present case to approximate the bimodal bubble size distribution obtained in the case of bubbles that are breaking up with increase in impeller speed. In the formation of the model, the following observations on the behavior of the bubble size distributions are considered: 1. The frequency number distribution shows a lognormal distribution at low impeller speeds. 2. A t the beginning of bimodality, the second mode appears on the smaller bubble size side and it grows in size with increase in impeller speed. As impeller speed increases further, the second mode becomes equal to the first mode and thereafter becomes progressively more prominent. 3. The first mode, which is narrower and sharper in the beginning of the bimodality, gradually becomes wider and flatter with increase in impeller speed. The peak of this mode slowly shifts toward the second mode, and at the end of bimodality, the curve merges with the second mode. At the highest impeller speed used, the distribution remains either skewed toward the larger bubble sizes or attains a uniform log-normal distribution. The bimodal log-normal distribution can be resolved into two log-normal distributions under the two modes as shown by dashed lines in Figure 4. As all the distribution curves shown in Figure 4 are normalized over the size range (dmin,dmm), according to the relation
The final expression for Sauter mean bubble diameter from log-normal distribution is derived as (Orr, 1966) log d32 = log mg + 5.7565 log2 ug
(8)
As the effect of impeller speed on the distribution is negligible, a single value of d32 can be obtained for each U,. To calculate d32, a straight line is fit to the experimental data plotted cumulatively on log-normal probability coordinates by least squares analysis. The value of mg is obtained by reading the 50% median value, and ug is estimated by dividing the 84% value by the 50% size. The d32 value estimated using eq 8 compares well with the experimental values as shown in Figure 6 for Ug= 2.5 X 10-4 and 1.25 X 10-3 m/s. Similarly, close agreement was observed between the experimental and predicted d32 for other Ugvalues also. Although there are number of mathematical functions available to describe unimodal particle size distribution in the literature, no satisfactory analytical function has been suggested for bimodal distributions. Few attempts have been made to obtain mathematical methods or functions to describe a continually changing pattern of distributions like the ones observed in the case of the bubbles undergoing breakage. In an analogous situation, particle size distributions arising from mixing powders of different individual distributions have been treated as a combination of two or more distributions. Irani and Callis (1963) and Allen (1968) mention that a bimodal particle
the area under the distribution curve should be equal to unity at any stage. Also, the areas under these two distribution curves are a function of impeller speed. Initially, the area under the second mode is nearly zero, indicating that the distribution is completely represented by the curve wholly under the first mode. From the onset of bimodality, the area under the first mode decreases while the one under the second mode increases with increase in impeller speed. At higher speeds, the area under the first mode either becomes negligibly small or disappears totally. Since both areas under the distribution curves represent parts of the same bubble population, the probability density function for the whole distribution can be obtained by the addition of distribution equations for the two curves. This will be the addition of two log-normal distribution function with separate mean and standard deviation parameters given as
(- (log
1 -eyn
l-c~ ex.( 1%
cg2
db-lOg m g l )
(log db - log mg2I2 -
2 log
erg;
where CY and (1 -CY)are the areas under the first and second
Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 709 modes (or the larger and smaller size fractions, respectively); log mgl and log mg2and log ugl and log ug2 are the mean and standard deviation parameters for the two fractions, respectively. All the bimodal size distributions obtained from experimental measurements are split into two lognormal distributions as shown in Figure 4, and the fractions of area under the two curves are calculated. For each distribution, mean and standard deviation parameters are calculated using eqs 5 and 6, respectively. The calculations have been performed for all the superficial gas velocities studied. The value of log mgldecreases with increase in impeller speed from the beginning of agitation to the end of bimodality. Similarly, the log mg2value starts decreasing from the onset of bimodality and continues to do so until the highest impeller speed is reached. The values of log ugl and log ug2are determined to be independent of other variables. Their values varied between 0.1 and 0.2 for all the conditions studied. Following a detailed analysis of all results, it seemed logical to obtain correlations to estimate the distribution parameters a,log mgl,and log mg2. As discussed earlier in this paper, the extent of bubble breakup and equilibrium bubble size distribution is determined by the energy dissipation rate, and therefore it is the most appropriate parameters for correlating the parameters of the size distribution. Chen and Middleman (1967) derived a relation for the probability density function for drop size distribution in a stirred vessel as a function of the local energy dissipation rate per unit mass, t. Nishikawa et al. (1976)and Okomoto et al. (1981) reported that, in astirred vessel, the maximum energy dissipation rate per unit mass, emax, occurs in the impeller region, and is many times larger than the average energy dissipation rate per unit mass, taw Usually parameters such as d32 and d , are correlated by eavg in stirred vessels. eavg is calculated as the power input per unit mass (P/W)based on the total volume of the liquid. However, as the bubble breakup takes place almost exclusively in the impeller region, it would be rational to use the energy dissipation rate appropriate for this region to correlate the bubble size distribution parameters, as used for droplet breakup by McManamey (1979). Moreover, the present system has been shown to be noncoalescingand the bubbles formed near the impeller are found to preserve their size in the rest of the tank. Thus the case for using the power input per unit mass based on the impeller swept volume (P/WI) becomes stronger. On the basis of these considerations, (PlWd was used to correlate the distribution parameters a,log mgl,and log mg2. The correlations obtained are as follows: a = 0.9 exp(-0.025(P/ W,))
(11)
log mgl = 3 . 4 5 ( ~w,)-0.03 /
(12)
log mg2= 2.96(P/ W1)-0.33
(13)
The correlation coefficient R for each of the above equations had a value of 0.95 or more. The frequency number distributions estimated using eqs 10-13 and experimentalvalues of log ug1,log ug2,and (P/ WI) are shown in Figure 4 as the dotted lines. Equation 11 was used to estimate the a values only for the distributions which exhibited bimodality. From the beginning of agitation until the onset of bimodality, the value of a was taken to be equal to 1. Similarly, when the a value became negligiblysmall at the highest impeller speed, it was taken
a?
99.99I 99.91
G
.1
l
I
t
l
0
Experimental
,
1
- Predicted E l N
OD
io3[
~
1 o2
0
,
1
2
,
1
4
,
1
6
8
,
10
,
,
12
1
14
Impeller speed, rps Figure 8. Comparison of Sauter mean bubble diameter values predicted by eq 14 with experimental values for sparger-0. U, = 1.0 x 10-3 m/s.
to be 0. The fairly good agreement between the experimental and predicted distributions validates the suitability of the model. These data were also plotted cumulatively on a log-normal probability graph, as shown in Figure 7 for Ug = 7.5 X 10" m/s. For 2.08 and 13.33 rps, the lowest and highest impeller speeds used, the plots are linear. The characteristic curves at 5.0 and 8.33 rps indicate the bimodal nature of the distribution. The curves can be viewed as a combination of two straight lines of different slopes representing two different bubble distributions. The lengths of the straight lines represent the relative size fractions. An estimate of Sauter mean bubble diameter may be also made from the probability density function. Thus, using the following equation which results from the definition of Sauter mean bubble diameter,
The numerical values for Sauter mean diameter were obtained by solving the integrals in the equation using Simpson's one-third rule between dmi, and dm=. About 50 discretization steps of equal width were used in each calculation. The d32 values predicted using eq 14are shown as a solid line in Figure 8 for Ug = 1.0 X 10-3 m/s. The agreement between the experimental and predicted values is reasonable. Considering that the predicted value at
710 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994
this stage is only dependent on the numerical technique used for the integration, it could most probably be improved further by using more accurate integration methods. It may be mentioned at this stage that although it has been possible to obtain good approximation to the experimentally determined values using the theory of homogeneous isotropic theory (HIT), its use is ultimately questionable in the vigorous theoretical sense. This is because, nontwithstanding the arguments used in developing the model, it must be conceded that the turbulence in a system as described here cannot be considered to be strictly isotropic. However, the concept has been used extensivelyfor establishing practical correlations in mixing studies and its use has been recommended strongly by some workers (Kawase and Moo-Young, 1990). The approach discussed above offers a convenient and practical method of estimating the bubble size and size distribution around the impeller region from a knowledge of the energy dissipation in that zone. For noncoalescing systems, these values are preserved in the bulk of the tank. However, as the power drawn by the impeller and the impeller swept volume is a function of the impeller geometry, so would be the bubble size distribution. More studies need to be carried out, incorporating different types of impellers, in order to confirm the generality of the above analysis. Tatterson (1991) classified the three areas of interest in gas dispersion in agitated tanks as (i) the hydrodynamic flow regimes occurring around the impeller and in the tank, (ii)bubble size and holdup, and (iii)the mass-transfer coefficient. It is obvious that the current investigation falls in the second category. I t would be interesting, and rewarding, to be able to determine the relationship between the observed changes in the bubble size distribution and the hydrodynamic flow regimes prevailing in mechanically agitated vessels. This would allow one to predict design parameters such as gas holdup and interfacial area from the prevailing hydrodynamic regime of operation. This has not been attempted in this paper because, as explained earlier, not all the suggested hydrodynamic regimes for gas-liquid dispersion could be studied due to experimental limitations. Ways of repeating the experiments reported here with operating parameters, especially the gas rate, comparable to those normally used in industrial units are being explored. Increasing the scale of the experimental equipment would be another serious consideration. Conclusions The effect of impeller speed on the equilibrium bubble size and size distribution in a stirred vessel has been studied by using bubbles of known, but different, initial Sauter mean diameters. For bubbles undergoing breakup the size distribution changes as a function of the impeller speed. The original log-normal distribution becomes bimodal due to the generation of smaller bubbles by the breakup of larger ones in the population. At high impeller speeds, the bimodal distribution reverts to unimodal again, but on the lower side of the size scale. This would indicate that the bubbles above a certain size are broken up at each agitation level and attain a new equilibrium distribution, and thus an average size. A method is proposed for estimating the continually changing pattern of the size distribution of bubbles. This is achieved by resolving the bimodal distribution curve into two unimodal components, and by then accounting for the size fractions under the two curves by their respective areas. The distribution parameters are esti-
mated as a function of the energy dissipated in the impeller region, with good agreement between the predicted and actual distributions. The proposed method offers a simple technique of predicting the bubble size distribution in a stirred vessel, especiallyfor a noncoalescing system. The findings should, therefore, be applicable to chemical and biochemical reactors in which the presence of salts or other surface active agents make the medium noncoalescing. In the absence of reliable experimental information on bubble breakup in stirred vessels, the reported observations also provide a convenient backdrop for future deliberations in this area. Further studies, however, incorporating experiments in larger scale would make the results more useful to the industry. Acknowledgment The authors gratefully acknowledge the financial support from the University of Newcastle Senate Research Committee for this project. R.P. also received a postgraduate research scholarship from the University. Nomenclature B = impeller blade width, m D = impeller diameter, m d32 = Sauter mean bubble diameter, pm db = bubble diameter, pm dbj = average bubble diameter of group i, pm d, = maximum bubble diameter, pm d,k = minimum bubble diameter, pm g = gravitational acceleration, m/s2 L = scale of the primary eddies, m 1 = scale of inertial subrange eddies, m mg = geometric mean, pm N = impeller speed, rps ni = number of bubbles in group i P = power input, W Q = volumetric gas rate, m3/s R = correlation coefficient Re1 = impeller Reynolds number, pND2/p T = tank diameter, m V g= superficial gas velocity, m/s W = mass of the liquid in the tank, kg W I= mass of the liquid in impeller swept volume (prD2B/4), kg
Greek Symbols = fractional area under the first mode of the bimodal distribution curve e = local energy dissipation rate per unit mass, W/kg taVg = average energy dissipation rate per unit mass, W/kg em, = maximum energy dissipation rate per unit mass, W/kg p = liquid viscosity, Pes p = liquid-phase density, kg/m3 ug = geometric mean standard deviation u = kinematic viscosity, m2/s Subscripts 1 = larger size fraction in bimodal distribution 2 = smaller size fraction in bimodal distribution avg = average max = maximum min = minimum Dimensionless Numbers Fr = Froude number (WDlg) F1 = gas flow number (Q/ND3) CY
Ind. Eng.Chem. Res., Vol. 33, No. 3, 1994 711
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