Influence of Impeller Type on the Bubble Breakup Process in Stirred

Ind. Eng. Chem. Res. 2008, 47, 6251–6263

6251

Influence of Impeller Type on the Bubble Breakup Process in Stirred Tanks Mariano Martín,* Francisco J. Montes, and Miguel A. Gala´n Departamento de Ingeniera Qumica y Textil, UniVersidad de Salamanca, Pza. de los Cados 1-5, 37008 Salamanca, Spain

A phenomenological model based on the collision theory and on population balances has been used to study the effect of different impeller geometries on the breakup mechanisms governing bubble dispersion in a stirred tank. Five different types of impellers, including standard and nonstandard ones (the Rushton turbine, the propeller, two different pitched blade turbines, and a modified blade), located at different positions along the vertical axis were used to generate dispersions of air bubbles in water. Air flow rates from 0.6 to 2.8 × 10-6 m3/s were used. The dispersions generated and the breakup processes were recorded by means of a high speed video camera. The different behavior of the impellers, based on its physical effect on the bubbles as well as on the effect of the flow pattern developed in the tank on them, determine the breakup mechanisms responsible for the generation of bubbles dispersions. The Weber number and the energy applied in a region surrounding the impeller quantify the contribution of the different breakup mechanisms to the mean diameter of the dispersion. Good agreement was found between the experimental Sauter mean diameter of the bubbles and those predicted by the model. Consequently, the model provides also physical explanation for the main breakup mechanisms of each impeller based on the critical Weber number and the energy dissipated. The model also allows a good theoretical prediction of the mass transfer rate. 1. Introduction Stirred tanks are widely used as gas-liquid contactors in the chemical and biochemical industry to provide reactants from the gas phase to the liquid bulk in which a reaction is taking place. The gas phase is dispersed by means of spargers and impellers to improve contact between both phases. Mass transfer is many times the limiting stage,1 determining the design of the equipment.2 Further knowledge on the breakup process will provide valuable results on the interfacial area and bubble size, key parameters in mass transfer operations, leading to the optimization of equipment design. Studies on particle stability along with population balances have been used to model the dispersions generated in turbulent systems, either in gas-liquid or liquid-liquid dispersions. The theoretical study of bubble breakup is based on their stability in the flow field developed in the tank. The typical deformation mechanisms leading to breakup are lenticular or cigar shape deformation or bulgy break.3 Bubble deformation depends on the relationship between the forces that try to maintain the bubble shape, related to the surface tension, and the inertial forces resulting from the flow inside the tank, which can be rearranged as the Weber number.3–11 Bubble stability has also been studied from the oscillation frequencies of the deformable surface.12 When the generation of dispersions is controlled by breakup processes, the Sauter mean diameter can be related to the maximun stable size,3,13 widely used for empirical correlations since 1967.14 However, if coalescence controls the generation of dispersions, the Sauter mean diameter of the dispersion is proportional to the minimum stable diameter.15 In the first case, when breakup controls the generation of dispersions, the exponent relating the Sauter mean diameter and the dissipated energy is -0.413 (Kolmogorov’s theory). Meanwhile, in the second case, this exponent turned out to be -0.25.16 Several models have been proposed to predict gas-liquid or liquid-liquid dispersions using the results from stability and * To whom correspondence should be addressed. Phone: +34923294479. Fax: +34923294574. E-mail: [email protected].

coalescence studies.17–24 The models differ mainly in the definitions used for the breakup and coalescence efficiencies and the frequency functions. Furthermore, many of them depend on adjustable parameters. The comparison between the results from different models reveals a certain degree of discrepancy.25 Some of these models, Coulaloglou’s17 and Prince’s,19 have been implemented together with computational fluid dynamics (CFD) codes for the gas-liquid systems in stirred tanks; see refs 26 and 27 for work on Prince’s model or refs 28 and 29 for work on Coulaloglou’s model. However, despite the amount of work on modeling the breakup of gas bubbles in stirred tanks, the use of standard impellers, basically only the Rushton turbine, limits the conclusions regarding the parameters used in the models. This study aims for determining the contribution of different breakup mechanisms in the development of gas dispersions. The region under consideration will be the surroundings of the impeller, where the breakup process determines the bubble mean size.9,30–34 Five different impellers located at two different positions along the vertical axis from the dispersion device were used in order to study the effect of different breakup mechanisms governing the dispersions generated by each one. 2. Theoretical Model The model determines the bubbles in a dispersion using a population balance which accounts for the generation of bubbles, either due to coalescence or at the perforated plate, and the disappearance due to breakup in steady state. The coalescence and breakup rates are based on the collision theory of the gases. In this way, the parameters of the model have physical meaning, which will allow explaining the foundations of the phenomenon. 2.1. Coalescence Rate. Bubble deformation enhances the coalescence rate.35 Therefore, coalescence rates are taken into account because the flow inside the tank deforms the bubbles heavily. In accordance with Prince’s model,19 the coalescence rate can be calculated as the product between the collision velocities

10.1021/ie800063v CCC: $40.75  2008 American Chemical Society Published on Web 07/17/2008

6252 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008

due to several mechanisms, labeled with a superscripted index turbulence (T), buoyancy (B), laminar stress (LS), and the effectiveness of those collisions: Cij ) (θTij +θBij +θLS ij )λi

(1)

Among the collision mechanisms, θijT corresponds to the collision frequency between two bubbles in the turbulent flow field due to their relative motion. It is based on the collision theory of ideal gases, so that bubble collisions are considered to occur similarly as molecular collisions. Then, the collision frequency of bubbles depends on their concentration and size as well as on their velocity. All these variables can be arranged as in the expression given by eq 219 θTij )ninjSij(uti2 + utj2)0.5

(2) 19

Where the cross-sectional area, Sij, is defined by eq 3:

π (d + dbj)2 (3) 16 bi In a turbulent regime, the velocity of the bubbles is determined by eq 4:19 Sij )

ut ) 1.4ε db 1⁄3

1⁄3

(4)

db is the bubble diameter and, ε, the dissipated energy is considered to be the power input per unit mass ε ) Pg/(VFL)

(5)

ni represents the number of bubbles of class i (bubbles of the same size) per unit volume. Another typical collision mechanism is that which occurs when a bubble reaches another bubble during its rising movement, θBij . In a stirred tank, this term is not going to be considered because bubble rising is determined by the flow pattern developed by the impeller and not by their buoyancy, unless the rotational speed is low and the bubbles are big enough. The collision rate due to laminar stress, θiLS, occurs when a bubble overtakes another. This term is negligible and not implemented in the model since it is only of interest when the collision takes place between two equal bubbles in regions where the dissipated energy is lower than the mean. However, the region under study is the one surrounding the impeller where the dissipated energy is higher than the mean. Not all of the collisions result in coalescence. Collision effectiveness is given by the ratio between the contact time and the time needed for draining the liquid film:19

( )

λij ) exp -

tij τij

(6)

The contact time, τij, can be calculated using eq 7.36 τij )

(0.5dij)2/3 ε1/3

(7)

with dij )

(

2 2 + dbi dbj

)

-1

(8)

Coalescence time is considered to be the time necessary to drain the liquid film developed between a pair of bubbles. When two deformable bubbles get close, surfaces become plain. As a result of the deformation, the pressure over the liquid film rises with respect to that of the liquid surrounding the bubbles. If a channel is developed, this overpressure can be alleviated

draining the liquid, assuming that the pressure of the bubble remains constant. To determine this time, eq 9 is used:19 tij )

(

(0.5dij)3FL 16σ

) () 0.5

ln

h0 hf

(9)

According to Kim and Lee,37 the limiting film thicknesses are the following: h0 ) 1 × 10-4 m hf ) 1 × 10-8 m

(10)

2.2. Breakup Rate. Bubble breakup is related to turbulent eddies. Eddies bigger than bubbles guide them throughout the tank, while those very small do not affect the bubbles.19 The breakup rate is defined by the product between the collision rate between bubbles and turbulent eddies, and it is a function which quantifies the fraction of eddies whose energy is big enough to break the bubbles. According to Prince,19 the breakup rate is given by the following: Bi)θieκi

(11)

The collision rate of the bubbles and the turbulent eddies can be expressed as if the eddies were physical entities:38 θie ) nineSie(uti2 + ute2)0.5

(12)

This collision rate, θie, is also called the approaching frequency or the bombing frequency among bubbles and eddies.21 The turbulent velocity corresponding to the eddies, ute, is similar to the turbulent velocity of the bubbles and it can be expressed as follows:19 ute ) 1.4ε1⁄3de1⁄3

(13)

The eddy size, de, can be calculated as the characteristic turbulence length (η), obtained from Kolmogorov’s theory of isotropic turbulence:39

( )

νL3 0.25 (14) ε Prince and Blanch19 reported from their experiments that only those eddies bigger that 0.2db are able to break the bubbles of diameter db; meanwhile, eddies bigger than the bubbles can only move them. So, η should be in the range of db to break the bubbles. Then, it has been considered in this work that, as far as the breakup process is concerned, eddy size has been taken to be 0.6db. The collision area between bubbles and eddies has an expression similar to eq 3, but instead of using the bubble diameter, the size of an eddy is used: η)

π (d + de)2 (15) 16 bi The eddy concentration of a particular size de per unit mass is calculated from the relationship given by eq 1640,41 Sie )

dNe(k) k2 ) 0.1 dk FL

(16)

The wavenumber, k, is defined as follows: k)

2 de

Equation 16 is solved with the following limits:

(17)

Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 6253

Ne ) 0,

0 ) G(z, db, t)

k)

2 N ) Ne , 0.2db e

k)

2 db

Function G represents a balance between the coalescence and breakup processes and determines the number of bubbles of each class in the dispersion.40,41 In our case, G is as follows: (18)

Eddies concentration per unit volume of liquid can be expressed as follows: ne ) NeFL

(19)

The second part of the definition of breakup frequency is the efficiency of the collisions to break the bubbles. The breakup efficiency (κ) is given by the energy distribution (χ), which provides a turbulent velocity higher than the necessary for the bubble breakup, uci. The equation is due to the work of Angelidou:42 χ(Ee) )

( )

-Ee 1 exp Ee Ee

(20)

The effect of an isolated eddy during a short period of time and the gathering of energy due to stochastic turbulence should be taken into account. When turbulence is weak, there is no reported breakup of the bubbles due to turbulent eddies and the storage of energy cannot break the bubbles. If turbulence is moderate, isolated eddies cannot break the bubbles but a continuous collision can make it. In the case of high turbulence, turbulent eddies can break the bubbles without previous deformation. Bubble breakup is due to the force balance between those which maintain bubble shape and those which tend to disintegrate it. Breakup efficiency is reported to be related to the kinetic energy applied on the bubble surface in order to break it.21 Considering that the eddy energy is proportional to the square of the velocity, the fraction of eddies whose energy is big enough to break the bubbles can be written as19

( )

κi ) exp -

uci2

(21)

ute2

The critical velocity of any eddy to break a bubble of diameter dbi is defined as19 uci )

( ) Wecσ dbiFL

(24)

2

Gi )

2

2

∑

∑∑

1 C C + 2Bi-1 - Bi 2 k)1 l)1 i,kl j)1 ij

(25)

Equation 25 will be applied for ten bubble classes (ten bubble sizes). Bubbles of class 1 are those generated at the orifice. Bubbles of class 2 are those which result from breakup of one class 1 bubble to generate two class 2 bubbles of the same volume considering constant volume. Class 3 bubbles are generated as a result of the breakup of bubbles of class 2 in two bubbles of the same size assuming constant volume.

( )

( )

4 dbj 3 4 dbj+1 3 )2Vb,j+1)Vb,j ) π (26) 2 π 3 2 3 2 Coalescence is considered to take place only between two bubbles of the same class to generate a bubble of the superior class. Furthermore, bubbles are periodically generated at the dispersion device (primary bubbles or bubbles of class 1), so that eq 25 must be completed by eq 27, when applied for this particular bubble class, to account for their presence in the tank due to the bubbling process. n1Gen )

1Bubble VsPd

(27)

Where, 1Bubble is the number of bubbles generated at the perforated plate, Vs corresponds to the liquid volume of study, and Pd is the formation period of the bubbles for each experimental condition. The model consists of a system of ten equations using Gi for bubble class i. The Sauter mean diameter of the simulated dispersion was calculated using its definition, eq 28. The experimental number and size of bubbles in the studied region was used instead of measuring the total gas hold up in the tank,19,41 to solve the model. 2.4. Bubble Dispersion. Simplified studies on bubble dispersion relate the bubble Sauter mean diameter, d32, to the input power. The Sauter diameter is defined as:

0.5

(22) d32

∑nd ) ∑nd

3

i eqi 2

(28)

There are several reported values for critical Weber number. The typical numbers in modeling breakup process have been Wec ) 143 or 2.3.19 Other authors take that value to be 0.585,3 0.6,44 1.017,45 1.05,8 1.2,46 or Wec ) 1.26.47 Sevik47 also proposed a resonant mechanism to model bubble breakup. Risso and Fabre48 exposed Weber numbers between 2.7 and 7.8 considering a mechanism for the bubble breakup based on a force balance and the oscillation resonance. Yet no agreement is found in the literature. 2.3. Breakup Probability. The number of bubbles in the dispersion is calculated by means of a balance of bubbles. A population balance is used:40,41

The typical empirical expression commonly used for the Sauter mean diameter as function of the power input is

∂ ∂ ∂ n(z, db, t) + [n(z, db, t)ur(z, db)] + ∂t ∂z ∂db ∂ n(z, db, t) db(z, db) ) G(z, db, t) (23) ∂t In a stationary regime and according to Pohorecki,40,41 eq 23 becomes:

The experimental set up can be seen in Figure 1. Air bubbles are generated in a tank made of glass and laser sealed (15 cm × 15 cm × 15 cm). In the middle of the tank, a gas chamber is built. It consists of a cubic structure 5 cm × 5 cm × 5 cm above which another cube of 2 cm × 2 cm × 2 cm is fixed so that the bubbles are generated at a distance from the bottom and thus be recorded properly. The gas chamber is divided into

[

]

i eqi

where deq is calculated as deq)(a’2b’)(1⁄3)

d32 ) kd

( ) Pg V

(29)

δ

(30)

3. Materials and Methods


Figure 1. (1) High speed video camera. (2) Optical table. (3) Stirred tank. (4) Illumination source. (5) Compressed air. (6) Rotameter. (7) Computer. (8) Impeller.

two. For this work only one-half of the gas chamber was used.54 Above the second substructure, a perforated plate with a 2 mm hole will be located. The tank is illuminated by means of optic fiber. Three different gas flow rates are fed to the gas chamber, 0.6 × 10-6, 1.4 × 10-6, and 2.8 × 10-6 m3/s. Five different impellers were used; from left to right in Figure 2, they are a pitched blade turbine with sharp and narrow curved blades, a modified blade, the typical Rushton turbine, another pitched blade turbine, and a propeller. Three rotational speeds (180, 280, and 430 rpm) for each one of the five impellers are used. Bubble breakup takes place in a region surrounding the impeller where the dissipated energy is several times bigger than the mean energy. Verschuren et al. proposed a region of 0.65DC wide and 0.2DC high.49 Energy values in the environment of the impeller are between 8 and 80 times the mean energy.50–53 This work is focused on this region, which determined the selection of Vs, eq 27. The initial position of the impellers was 2 cm above the perforated plate. After that, the impellers were located 3.5 cm above the sieve plate. Deionized water was the liquid phase, (20 °C, FL ) 998 kg/m3, σ ) 0.073 N/m, µL ) 1.037 × 10-3 Pa s). In this particular configuration, bubbles are injected below the impeller, like standard stirred tanks. In contrast, the presence

of an elevated gas chamber allows a good recording of the growing and detaching bubbles as well as gives a chance for monitoring the effect of the impeller on the rising bubbles and for the identification of different bubble breakup mechanisms leading to the generation of gas-liquid dispersions. However, we acknowledge the fact that surrounding the gas chamber the mixing is poor compared to other dispersion devices such as perforated rings, commonly used in stirred tanks, reducing the mixing and mass transfer rates. The setup was not built for high mass transfer efficiency but to be able to relate the hydrodynamics and the mass transfer rate studying the effect of the dispersions generated on the mass transfer rates measured. The generated dispersion was recorded by means of a high speed video camera Redlake Motionscope PCI able to tape up to 1000 frames/s. The data needed for the model are the initial bubble diameter as well as the mean size of the bubbles in the dispersion, calculated using eqs 28 and 29 by monitoring at least 100 bubbles in the videos, and the elapsed time between the generation of two bubbles in every one of the experimental conditions, as an average of 25 growing bubbles. The software used to edit the videos was MOTIONSCOPE 2.21.1. The dispersed energy into the liquid and the power number for the impellers in all the experimental conditions were simulated using CFX 5.7 based on Ansys 8.1 and verified in a previous work54 where the details of the impellers’ geometry can also be found. 4. Results and Discussion First, the experimental data are reported and commented on. After that, the results of the model will be presented and explained based on the recordings. The experimental Sauter mean diameter for H ) 2 cm for all the impellers versus the power input can be seen in Figure 3. Bubbles can easily be broken by the power input applied by the impeller, which will allow studying the breakup process leading to the generation of the dispersions recorded. 4.1. Breakup Mechanisms. In the first place it is interesting to identify the breakup mechanisms found for each of the

Figure 2. Types of impellers used. (a) Pitched blade turbine: T ) 6 cm; Ti ) 0.6 cm, Wa ) 1.5 mm; Wb ) 1 mm; Re ) 5.1 cm, Rb ) 6.5 cm, Di ) 5 mm. (b) Modified blades: T ) 4.8 cm; Ti ) 0.35 cm; Ti ) 0.4 cm, R ) 0.25 cm; Tk ) 0.5 cm; D ) 1 cm. (c) Rushton turbine: T ) 5.2 cm; Td ) 3.2 cm; Ti ) 1.5 cm; Ti ) 1 cm. (d) Pitched blade turbine: T ) 5.6 cm; Ti ) 0.7 cm 45°, Di ) 0.5 cm, R ) 11.15 cm, W ) 0.6 cm; (e) Propeller: T ) 5 cm; Tlobulus ) 1.7 cm 30°.


Figure 3. Effect of the power input on the bubble mean size. The impellers are located 2 cm above the sieve plate.

Figure 4. Dissipated energy pitched blade turbine: N ) 430 rpm, H ) 2 cm, P/V ) 6.61W/m3.

impellers and relate them to the energy dissipation and the geometry of the blades in order to shred light into the development of gas dispersions. 4.1.1. Pitched Blade Turbine a. The way the bubbles are broken and discharged matches the region of high energy shown in Figure 4. Moreover, this region almost agrees to the dimensions proposed by many authors.49–53 However, the region of high dissipated energy also reaches the walls of the tank due to the direct impact of the flow against them. For this impeller, the sharpness of the blades allows for the cutting of rising bubbles in a medium position of the blades; see Figure 5a. Moreover, due to the impeller geometry, the main breakup mechanism consists of leading the bubbles to the end of the blades where they are cut; see Figure 5b. However,

Figure 5. Dispersion generated and breakup process by a pitched blade turbine: N ) 430 rpm, Qc ) 0.6 × 10-6 m3/s, P/V ) 6.61 W/m3

bubbles are not gathered at the blades because of their geometry. Thus, bubbles can also avoid the blades reducing the breakup efficiency of the impeller and increasing the bubble mean size of the dispersion; see Figure 5a. 4.1.2. Two Modified Blades. The main breakup mechanisms for his type of impeller are the following two: the direct cut of the bubbles once they reach the blade in their rising movement (Figure 6a) and the deformation of the bags of gas accumulated at the end of the blades, followed by the breakage of the bubbles during their discharge (Figure 6b). The inertia of the bubbles is responsible for the force of the collision of the bubble against the blades. Therefore, it has an important contribution to the


Figure 7. Simulated dissipated energy with a Rushton turbine: N ) 430 rpm, H ) 2 cm, P/V ) 4.74 W/m3.

Figure 6. Breakup process for a modified blade: N ) 180 rpm, Qc ) 0.6 × 10-6 m3/s, H ) 2 cm, P/V ) 0.15 W/m3.

direct cut of the bubbles. Furthermore, bubble breakup due to direct cut does not require input energy by means of the impeller. The energy required for the second type of breakup mechanism is not high either, because the bags of gas are easily deformed and broken. Although we do not show it, the dispersed energy pattern calculated is similar to the one shown for the previous impeller. 4.1.3. Rushton Turbine. For the case of the Rushton turbine, the region shown in white in Figure 7 corresponds to the highest dissipated energy. The tank, due to its nonstandard geometry used for making the recording seasier, shows big energy losses in the walls. The dissipated energy presents a pattern which verifies the experimental ones developed by Wu.53 Bubbles break mainly due to the effect of the disk on the bubbles. Bubbles are retained below the disk; see Figure 8a. Once there, the impeller rotation moves the bubbles to the blades where they are broken in the discharge. It can be seen how the broken bubbles left the impeller moving upward (Figure 8b) instead of downward as in the pitched blade turbine. Small daughter bubbles are discharged following the flow pattern predicted by the simulation. Apart from this main mechanism of breakup, big bubbles can be deformed by the flow. Figure 9 shows a satellite breakup process found below the impeller as a result of the cross-flow field created below the impeller. 4.1.4. Pitched Blade Turbine b. The second pitched blade turbine has wider and more hydrodynamic blades than pitched

Figure 8. Breakup process for a Ruston turbine: N ) 280 rpm, Qc ) 1.4 × 10-6 m3/s, H ) 2 cm, P/V ) 1.31 W/m3.

blade turbine a. Consequently, the impact of the blades cannot cut the bubbles, and the main breakup mechanism is due to the deformation of the bags of gas accumulated at the end of the


Figure 9. Satellite breakup process for a Rushton turbine: H ) 0.05 m, N ) 430 rpm, Qc ) 0.6 × 10-6 m3/s, P/V ) 4.31 W/m3.

Figure 10. Breakup process for pitched blade b: N ) 430 rpm, Qc ) 0.6 × 10-6 m3/s, H ) 0.02 m, P/V ) 2.25 W/m3.

blade; see Figure 10a. Bubble discharge determines their breakup; see Figure 10b and c. Bubbles are displaced to the end of the blades and there they are deformed until their breaking point as the blades rotate. As in previous impellers, the zone of high dissipated energy developed by this impeller is of similar size as that reported before. 4.1.5. Propeller. The propeller presents lower dissipated energy than any other impellers. Therefore, only next to the impeller blade, the dissipated energy is very high. However, there is a region of low energy under the impeller where bubbles remain for a while. Under the impeller, shapes like the one shown in Figure 11, where the bubble is turned upside down while rising but remaining stable, can be obtained. This bubble

Figure 11. Bubble deformation before reaching the propeller: H ) 0.02 m; N ) 180 rpm, Qc ) 1.4 × 10-6 m3/s, P/V ) 0.16 W/m3.

shape, instead of representing the beginning of the breakup process, maintains the integrity of the bubble. The absence of sharp parts and the limited contact of the impeller with the bubbles, make the breakup process be mainly due to the deformation of bubbles just under the impeller (Figure 12a), which eventually break (Figure 12b).


Figure 12. Breakup process for a propeller: N ) 280 rpm, Qc ) 0.6 × 10-6 m3/s, H ) 2 cm, P/V ) 0.62 W/m3.

Wec

εmax

classes in a dispersion using the population balance model exposed in the previous section.

1.1 0.5 0.6 0.5 0.9 1

21.3 7.7 7.4 8.2 17.7 20

The experimental gas phase in the studied region, calculated using the observed bubbles and their size, was compared to the calculated one to determine the mean characteristic time of the bubbles in the studied region, which depends on the flow developed by each impeller.

Table 1. Optimized and Referenced Results for the Model

pitched blade turbine (a) modified blade Rushton turbine pitched blade turbine (b) propeller air-water system

4.2. Model Results. It can be considered a two stage model which will determine the concentration of bubbles of different

Figure 13. Model prediction vs experimental values of the Sauter mean size for each impeller: H ) 2 cm.

ni ) (Gi)tc

(31)


Figure 14. Model prediction vs experimental values of the Sauter mean size. The impeller location was at H ) 2 cm.

Figure 15. Model prediction vs experimental values of the Sauter mean size. The impeller locations were at H ) 2 and 3.5 cm.

Together with that, in order to determine the proportion of bubbles of each class, the model depends on two parameters, the Weber critical number and the maximum dissipated energy, εmax, which define the breakup process and are typical of each impeller. An error function was defined as the difference between the calculated Sauter mean diameter using the model and the experimental values. Minimizing the error function determines the characteristic values for the Weber critical number and εmax for each of the impellers. The optimization of the model for H ) 2 cm leads to the results shown in Table 1. Figure 13 shows the good agreement between the experimental and the calculated values for each impeller. The different values of Wec obtained for each of the impellers (Table 1) can be explained based on the reported dependence of the Weber critical number on the breakup mechanisms.3 When the bubbles are easily broken by the impellers, the values of the critical Weber number are small, 0.5 (impellers 2, 3, and

4). This happens mainly because the bubbles are gathered at the impeller and deformed up to their breaking up in the rotational movement of the impeller. However, if the bubbles must be deformed in the flow, either because the effect of the impeller on them is small or because the bubbles can avoid it, the values of the Weber critical number increase (impellers 1 and 5). The more stable the bubbles are in a dispersion, the more difficult they are to break and the more the Sauter mean diameter depends on the dispersion device and on the physical properties of the liquid. Furthermore, the values for the Weber critical number found are within the range reported by Hinze and Sevik in the literature.3,47 Another fact to point out is that the values of Wec are also directly related to δ, eq 30, obtained in a previous work.54 Bigger values of the critical Weber number are obtained for bigger values of δ. Thus, it can be concluded that the physical meaning of this empirical coefficients relies on the contribution of the breakup and coalescence mechanisms to the bubbles mean size of the dispersion. Then, δ has its foundations on the Weber critical number. In addition, the impeller geometry has also an important effect on the energy needed for the breakup, which cannot be entirely coped by the power input or the power number, as can be seen in Table 1. Sharp impellers are capable of cutting the bubbles with no previous deformation and thus the energy involved in the breakup process is small. In this case, the inertia of the bubbles has also a large contribution in the breakup mechanisms. It is also worth mentioning that bigger dissipated energy is needed to break the bubbles in the flow. Examples of this are the three bladed turbine, impeller 1, and the propeller, whose blades do not retain the bubbles and bubble deformation in the flow is the main breakup mechanism. However, if the bubbles are collected by the impeller, bubble breakup is relatively easy as a result of their deformation at the end of the blade. The two bladed and three concave bladed turbines are examples of this behavior. The disk effect of the Rushton turbine, retaining the bubbles, allows applying lower energy to break the bubbles since the bags of gas are developed easily. The calculated values of εmax in the breakup region range from 7 to 23 times the mean value of dissipated energy, see Table 1, next to those proposed by several authors. Verschuren et al. give a typical value of 7.3 times the mean energy ε in the breakup region.49 Okamoto55 shows a value of 11ε, whereas Zhou reports several values for different impellers from 8 to 12ε.51 Wu et al. and Lee et al. show values up to 25ε.52,53 The use of the maximum energy instead of the mean energy to study the bubble mean size of dispersions has already been supported by the results from Parthansarathy et al.9,31 who found that the bubble mean size of the bubbles generated at the swept volume of the impeller was maintained across the tank. That is the reason for several authors to suggest the study of the bubble mean size as function of εmax instead of ε.56 What can be concluded from these results is that the breakup mechanisms and their contribution to the development of gas dispersions depend on the particular type of impeller, particularly, on the effect of the impeller on the bubbles. The variability of We with the impeller agrees with the conclusions of Hinze.3 The different dominant breakup mechanism taking place inside the tank for each impeller determines Wec. Furthermore, for design purposes, it is interesting to determine mean values for both parameters for the air-water system. Solving the model using all the results for H ) 2 cm, it turned out that the Wec was equal to 1ssee the last row in Table


Figure 16. Effect of the critical We and the energy on the breakup zone on the δ coefficient.

Figure 17. Bubble size distribution of a dispersion generated by means of a Rushton turbine: N ) 430 rpm.

1ssimilar to values proposed by Hesketh, Shimizu, and Killen,8,43,45 meanwhile the level of energy in the breakup zone for all the impellers is 20 times the mean dissipated energy of the tankssee the last row in Table 1slike the one proposed by Wu.53 Using Wec ) 1 and εmax ) 20ε, the model agrees reasonably well with the experimental values not only for H ) 2 cm (Figure 14) but also for all H ) 2 and 3.5 cm (Figure 15). As it has been mentioned before, δ can be physically explained based on Wec. Therefore, the model has been used to study the effect of Wec and the dissipated energy on δ of

equations like (30) to provide with certain theoretical basis for this empiric coefficient. Let us assume a bubble with an initial diameter of 8 mm and with a formation period on 0.1 s. With the mean values obtained, Wec ) 1 and εmax ) 20ε, the model solves δ to be -0.4. This is the theoretical value derived from the Kolgomorov theory, which can also be seen in Calderbank’s equation for the Sauter mean diameter.57 The term δ, eq 30, although empirical, represents the contribution of different breakup mechanisms to the developing of dispersions as a result of the input power of the impeller.


Figure 18. Comparison between the calculated kLa using Kawase’s and Calderbank’s equations or coupling Kawase’s kL with population balances.

So, it is possible to use the values of Wec and εmax proposed by some authors51,53,55 to explain the physical meaning of δ. Figure 16 plots δ vs Wec Two limiting cases can be described regarding the breakup of the bubbles and its contribution to the development of dispersions. In the case of bubble columns, where there are no energy sources in the tank, the main breakup mechanism is bubble deformation in the flow. High energy is required to break the bubbles. It has been reported41 that there is slight or no dependence between the bubble mean diameter of the dispersion and the power input, due to the gas flow rate. In that case, δ ) 0 in eq 30. As can be seen in Figure 16, the critical Weber in vessels with a homogeneous distribution of energy for obtaining for δ close to 0 is 2.3, the same value as that used by Prince in his model.19 The breakup mechanism is due to bubble deformation in the flow which requires more energy and results in a larger critical Weber number. The opposite case is when the bubbles are broken in the discharge of the blades. Then, δ was reported to be close to the one predicted by Kolmogorov’s theory, -0.4. In this case, the Weber critical number ranges from 0.5 to 1 for different levels of dissipated energy in the impeller region. From the contribution of both situations, bubbles broken at the blades or in the flow, which depends on the effect of the impeller on the bubbles, the Weber critical number seen in Table 1 can be explained. The model is also able to predict bubble size distributions. These are normal, in accordance with some reported results.11,25,58 Figure 17 shows an example for bubble size distribution generated by a Rushton turbine located at 2 cm from the dispersion device. In spite of the difficulty of experimentally determining bubble classes, the experimental results agree well with the simulated dispersions.

Finally, one of the most important applications of stirred tanks is as contact equipment for mass transfer processes. So far the complex hydrodynamics of the tank has led to use theoretical models for the liquid film resistance kL but empirical correlations for the specific area; Calderbank’s one57 is the most used. Therefore, we are going to use the model to determine the volumetric mass transfer coefficient, kLa, as the product of the liquid film resistance using the Kawase59 approximation and the contact area given by the bubbles of the dispersion predicted by the model for each impeller.

∑ n 4π( 2 ) 10

kLa ) kL

db,i

i

2

(32)

i)1

These results are compared to the equation proposed by Kawase59 using Calderbank’s equation57 for predicting kLa. Figure 18 shows good agreement between Kawase’s model and this work. The differences observed are related to the fact that bubble breakup and the available area depend on the effect of the impeller on the bubbles, and Calderbank’s equation provides a mean value for the specific area. 5. Conclusions Bubble breakup processes have been studied for the five different types of impellers located at two different heights in the vertical axis of the orifice of the dispersion device. The breakup mechanisms due to bubble deformation, satellite, and lenticular bubbles have been reported graphically. The different breakup mechanisms observed depend on the impeller geometry and the flow developed in the tank. They determine the Weber critical number for each case. Impellers capable of breaking the bubbles show lower Weber critical numbers. Furthermore, the maximum energy responsible for the breakup process not only ranges within the values proposed by


the references but also depends on the breakup mechanisms: sharp impellers need less energy to break the bubbles. For bubble breakup in the impeller Wec ) 0.5 which increases as the bubbles are broken in the flow which results in Wec ) 2.3. From the contribution of the different breakup mechanisms, Wec is obtained. In general, a couple of values, Wec ) 1 and εmax ) 20ε, previously reported separately turned out to be the optimum values for the air-water system. The model, based on that presented by Prince,19 not only agrees well with the experimental results following the values for the physical parameters reported in the literature but, at the same time, agrees with Kolmogorov’s theory as it is able to predict the volumetric mass transfer coefficient. Acknowledgment The support of the Ministerio de Educacin y Ciencia of Spain by providing an FPU fellowship to M.M. is greatly welcomed. The funds from the project reference CTQ 2005-01395/PPQ are also appreciated. Nomenclature a′ ) Bubble’s length (m) b′ ) bubble’s height (m) Bi ) breakup rate of bubbles of class i (1/(m3 s)) Cij ) coalescence rate between class i bubble and class j bubble (1/(m3 s)) de ) eddy diameter (m) deq ) equivalent diameter (m) db ) bubble diameter (m) d32 ) Sauter mean diameter (m) dbo ) bubble initial diameter (m) DC ) tank diameter (m) Do ) orifice diameter (m) Ee ) eddy energy (kg m2/s2) g ) gravity (m/s2) G ) generation function (m3/s) h ) drained film thickness (m) H ) separation between dispersion device and impeller (m) k ) wave number (1/m) kd ) constant kL ) liquid film resistance (m/s) kLa ) volumetric mass transfer coefficient (1/s) ni ) class i elements concentration (no. bubbles/m3) ne ) eddy concentration (no. eddies/m3) Ne ) eddy concentration per unit mass (no. eddies/kg liquid) N ) rotational speed (1/s1) P ) impeller power input (W/m3) Pd ) period of bubble generation (s) Pg ) aerated power input (W/m3) Qc ) gas flow rate (m3/s) Sij ) contact area (m2) t ) time (s) tc ) characteristic time (s) tij ) drained time of the liquid film (s) T ) impeller diameter (m) uc ) critical breakup velocity (m/s) ut ) turbulence velocity (m/s) ur ) rising velocity (m/s) V ) volume of liquid in the tank (m3) Vb ) bubble volume (m3) Vs ) liquid volume of study (m3) (0.05 × 0.15 × 0.15) Wec ) critical Weber number WeBubble ) dequr2FL/σ ) FLN2T3/σ

Greek Symbols R, β ) exponents δ ) exponent for the correlation between the bubble mean size and the power input ε ) turbulence energy (W/kg) εmax ) energy in the impeller region (W/kg) κ ) breakup effectiveness χ ) energy distribution function of the eddies θ ) collision rate (m-3/s) µL ) liquid viscosity (Pa s) η ) characteristic length of the turbulence (m) µL ) kinematic viscosity of the liquid (m/s2) FL ) liquid density (kg/m3) σ ) surface tension (N/m) τij ) contact time between bubbles (s) τ ) shear stress (Pa) Subindexes i ) bubble phase o ) continuum phase

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ReceiVed for reView January 15, 2008 ReVised manuscript receiVed May 19, 2008 Accepted May 30, 2008 IE800063V

Influence of Impeller Type on the Bubble Breakup Process in Stirred

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