Mass Transfer Rates from Oscillating Bubbles in Bubble Columns

Ind. Eng. Chem. Res. 2008, 47, 9527–9536

9527

Mass Transfer Rates from Oscillating Bubbles in Bubble Columns Operating with Viscous Fluids Mariano Martı´n,* Francisco J. Montes, and Miguel A. Gala´n Departamento de Ingenierı´a Quı´mica y Textil, UniVersidad de Salamanca, Pza. de los Caı´dos 1-5, 37008 Salamanca, Spain

In spite of the work on bubble columns, their design and scale-up is still a difficult task due to the lack of understanding of bubble dispersions and mass transfer mechanisms. Even less known are viscous or nonNewtonian fluids. Therefore, a theoretical model for predicting the volumetric mass transfer coefficient, kLa, in bubble columns operating with viscous fluids has been proposed. The model consists of a population balance coupled with a theoretical equation for the Sherwood number for oscillating bubbles, considering the effect of liquid viscosity on both. Experimental results for Newtonian and non-Newtonian viscous liquids from the literature are used to validate the model. Bubble dispersions have been simulated with good agreement using the Weber critical number, Wec, as a parameter to account for the effect of liquid viscosity, which increases bubble stability. A correlation between the liquid viscosity and Wec has also been proposed. The mass transfer resistance is calculated taking into account the hydrodynamic processes involving bubbles (collisions, breakup, coalescence, detachment) because they provide initial oscillation amplitudes. However, bubble oscillation decays in viscous liquids because the oscillating energy is absorbed as viscous dissipation. Good agreement is found between the experimental and the predicted kLa when considering that bubble oscillations do not decay completely by viscous dissipation due to the continuous bubble collisions, breakup, and coalescence. 1. Introduction Bubble column reactors (BCRs) are widely used in a variety of industries, particularly biochemical ones, because of advantages such as simple maintenance due to lack of moving parts, high gas-liquid interfacial area, good mass/heat transfer rates between gas and liquid phase, and large liquid holdup.1 However, the lack of understanding of generation of dispersions and of mass transfer mechanisms due to the number of variables involved in the performance of bubble columns (gas holdup distribution, bubble breakup, coalescence and dispersion rate, bubble rise velocity, bubble size distribution, gas-liquid interfacial area, concentration distribution, gas-liquid mass/heat transfer coefficients and the extent of liquid phase backmixing)2 makes their design and scale-up challenging. Many times, mass transfer rates are the limiting stage. Therefore, the volumetric mass transfer coefficient, kLa, is the typical design parameter of BCRs determining performance and yield.1-5 The complexity of bubble columns operating with viscous and non-Newtonian fluids has limited their theoretical study. Thus, so far, BCRs have been designed and scaled up using empirical correlations.1,3-5 Therefore, a deeper understanding and accurate prediction of kLa is required to optimize BCRs design. kLa is a function of the contact area between the gas phase and the liquid phase, “a”, together with the resistance to mass transport in the liquid side, kL. It is important to study the effect of the liquid viscosity on both. The contact area “a” is given by the number of bubbles in a dispersion and their size. A population balance model, PBM, is typically used to determine “a”. The population balance is written considering bubble breakup and coalescence processes. Different mechanisms have been considered. Regarding coalescence, the type of collisions between bubbles defines the coalescence mechanisms such as turbulent, laminar stress, and * Corresponding author. Phone: +34923294479. Fax: :+34923294574. E-mail: [email protected].

buoyancy.6 With respect to bubble breakup, turbulence breakup and bubble instability are the most common mechanisms.6,7 Most of the breakup and coalescence closures proposed have been developed for low viscous fluids.6,8-12 However, liquid viscosity reduces the drainage velocity of the liquid film between two bubbles. As a result, the coalescence efficiency decreases.13 Besides, bubbles are more stable in viscous fluids.14 Thus, bubble breakup processes are not so often. Recent studies comparing the results of different closures15-17 reveal a wide range of results and a certain degree of discrepancy. Therefore, the PBM must be carefully defined to cope with the effect of liquid viscosity on the coalescence and breakup rates. With regard to kL, many approaches have been proposed to understand the interfacial mass transfer. The two film model,18 the penetration theory,19 surface renewal,20 and the film penetration model21 are the most widely accepted. More recently, mass transfer has been calculated based on the velocity gradient at the interphase.22 However, the effect of bubble deformation and the effect of liquid viscosity on kL have barely been considered, and most of the models for mass transfer in bubble column reactors have focused on the air-water system (e.g., Shimizu,9 Tiefeng and Jinfu23). So far, bubble deformation has been studied for inviscid fluids, either using correction factors for kL23 or based on the study of the effect of bubble oscillations on the concentration gradients surrounding the bubbles.24 Meanwhile, few are the ones that consider the effect of viscosity on kL, such as Kawase et al.25 Kawase et al.25 combined Higbie’s theory19 with Kolmogorov’s theory of isotropic turbulence, where liquid viscosity plays an important role in the contact time of the bubbles. Therefore, a newly developed Sherwood number26 will be used to define kL. The new model is based on the perturbation theory to implement the effect of bubble oscillations on the velocity gradients surrounding the bubbles in viscous fluids. Understanding bubble oscillations is a key issue to determine the Sherwood number. The characteristics of bubble oscillations, oscillation amplitude and frequency,

10.1021/ie801077s CCC: $40.75  2008 American Chemical Society Published on Web 11/01/2008

9528 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

Figure 1. Scheme of the calculation of kLa.

depend not only on their size but also on the physical properties of the gas-liquid system.27,28 Furthermore, liquid viscosity absorbs oscillation energy as viscous dissipation reducing the oscillation amplitude of the bubbles. Additionally, the oscillation decay is not complete in a bubble column due to the birth and death of bubbles as a result of coalescence and breakup processes.26 Thus, this paper addresses the problem of theoretically predicting kLa in bubble columns operating with viscous fluids by coupling a PMB and a newly developed model for the Sherwood number,26 considering the effect bubble oscillations and liquid viscosity. Experimental results for bubble mean diameter, d32, and kLa from the literature3,29-31 have been used to validate the model proposed. 2. Theoretical Model The model calculates first the bubble dispersion in the system and then kL for each bubble class present in the dispersion. A population balance is used to determine the distribution of bubbles across the vessel based on the bubbles formed at the dispersion device and the processes they experience (coalescence and breakup). The stability of the bubbles in the flow will define the Sauter mean diameter, d32. It is known that the Weber critical number (Wec) defines bubble breakage and depends on the gas-liquid system and the breakup mechanisms.32 Therefore, Wec will be a parameter of the model to define the mean diameter of the dispersion. The calculated d32 will be compared to the experimental one from the literature. Once the dispersion is simulated, the Sherwood number for each bubble class is calculated using the model developed by Martin et al.,26 which considers the effect of liquid viscosity on the development of concentration profiles surrounding the bubbles. By combining both, kLa is calculated. Figure 1 shows a scheme of the calculus of kLa by combining both kL and a. 2.1. Hydrodynamics. The hydrodynamics involve the processes related to bubble formation, collisions, rising, coalescence, and breakup. We are going to propose a model based on equations from the literature taking into account the effect of liquid viscosity in bubble coalescence and breakup. 2.1.1. Bubble Coalescence. Inside a column, the flow field leads to bubble collision. During this process, bubble shape can be deformed to develop a drainage channel and the liquid film between the bubbles is partially or totally drained. Figure 2 shows both possibilities. In the case of coalescence, it is worth mentioning that the bubbles resulting from coalescence are subjected to big oscillations looking for stable shapes, in accordance with their new size and the physical properties of the liquid. Meanwhile, in the absence of coalescence, bubble collisions induce an initial

Figure 2. Collision processes between bubbles.

oscillation amplitude. Bubble oscillations determine the concentration profile surrounding the bubbles defining the liquid film resistance, kL. Furthermore, liquid viscosity partially absorbs the oscillating energy as viscous dissipation.33 Therefore, bubble oscillations and the effect of liquid viscosity on them should be taken into consideration when calculating kL. To simulate coalescence, a modification of the model proposed by Prince and Blanch6 has been used. The model considers that the coalescence rate, Cij (m-3 · s-1), for two bubbles i, j whether they are of the same class (i ) j) or not (i * j), is given by the product between the collision frequency and the efficiency by which those collisions derive in coalescence. The total collision frequency is reported to be the sum of different mechanisms, turbulence, buoyancy, and Laminar stress collisions. Each mechanism is labeled with a superindex, indicating the physical process from which it is derived:6 Cij ) (θTij + θBij + θLS ij )λij

(1)

T

In eq 1, θij corresponds to the collision frequency between two bubbles in the turbulent regime due to their relative motion. It is based on the collision theory for ideal gases where concentration, bubble size, and velocity can be arranged in the expression given by eq 2.6 θTij ) ninjSij(uti2+utj2)0.5

(2)

The collision cross -sectional area, Sij, between bubbles is defined as eq 3, Shimizu et al.:9 π (d + dbj)2 (3) 16 bi The turbulent velocity, ut, for bubbles of diameter db in the inertial subrange of isotropic turbulence is34 Sij )

ut ) 1.4ε1/3db1/3

(4)

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9529

where ε is the dissipation energy; meanwhile ni represents the number of bubbles of class i per unit volume. Another typical collision mechanism is that which occurs when a bubble reaches another bubble:6 θBij ) ninjSij(uri - urj)

(5)

In the case of large bubbles (db > 10 mm) in viscous fluids, the rising velocity does not depend on the fluid and ranges from 0.8 to 1.2, Grund et al.35 Therefore, this mechanism will not have a big influence in coalescence. However, for smaller bubbles, the rising velocity, uri, can be calculated following eq 6, Prince and Blanch:6

(

uri ) 2.14 ·

σ + 0.505gdbi FLdbi

)

0.5

(6)

When a bubble overtakes another bubble of the same size, the collision rate due to laminar shear, θiLS, can be expressed as eq 7, proposed by Frinlander:36

(

)( )

4 dbi dbj 3 dU1 + ninj 3 2 2 dr where the shear rate can be expressed as follows:6 θLS ij )

( )

(7)

(8)

DC is the diameter of the BCR. We are going to consider DC ) 1 m to avoid slug flow in the column. However, not every collision results in coalescence. Coalescence probability depends on the intrinsic contact between the bubbles. Coulaloglou et al.37 define the collision efficiency between bubbles of classes i and j, λij, as a probability function which depends on the relationship between the required time for film drainage, tij, and the contact time, τij

( )

tij λij ) exp τij

(9)

Contact time can be calculated through eq 10, Prince and Blanch:6 τij )

(0.5 · dij)2/3

(10) ε1/3 When two bubbles come close, the overpressure on the liquid film generated between both drives liquid drainage. Liquid viscosity affects liquid film drainage, reducing the movement of the liquid layers and slowing down the thinning of the liquid film. A model proposed by Chesters13 has been implemented to account for the effect of liquid viscosity on bubble coalescence: tij )

(

)()

3π(0.5dij)µL h0 ln 2σ hf

Bubble breakage provides an initial oscillation amplitude for the resulting bubbles, looking for a stable shape according to their new size. Bubble breakup is modeled considering that bubbles break due to their interaction with turbulent eddies as if bubbles collide with eddies. Therefore, an expression of the breakup frequency is given by the product between the collision frequency and the efficiency of that collisions to break the bubbles:6 Bi ) θieκi

(14)

The collision rate of turbulent eddies and bubbles is based on the same mechanism as for coalescence assuming that turbulent eddies behave as entities.6 θie ) nineSie(uti2 + ute2)0.5

1/3

0.787(gDCuG) dU1 U1 ) ≈ drC DC/2 DC/2

Figure 3. Scheme of bubble breakup process.

(11)

(15)

The turbulent velocity of the eddies can be written as eq 16:6 ute ) 1.4ε1/3de1/3

(16)

Particularly interesting for bubble breakup are eddies from 0.2db to db. Smaller eddies do not have enough energy meanwhile, bigger ones drag the bubble without breaking it.6 As a mean value, de ) 0.6 db has been proven to be valid to evaluate bubble breakup.39 And the cross-sectional area is calculated as follows:8 π (d + de)2 (17) 16 bi An expression for the number of eddies of a particular size as a function of the wavenumber is given by eq 19:8 Sie )

dNe(k) k2 ) 0.1 dk FL

(18)

where the wavenumber is8 k)

2 de

(19)

The differential equation has to be solved using as lower and upper limits, given by8 Ne ) 0, Ne ) Ne,

k)

2 0.2db

(20)

2 db

(21)

k)

Vortex concentration per unit volume of liquid is

where dij )

(

2 2 + dbi dbj

)

-1

(12)

In accordance with the results by Kim and Lee,38 the limits for liquid film stability are h0 ) 1 × 10-4 m

hf ) 1 × 10-8 m

(13)

2.1.2. Bubble Breakup. The fluid flow generated inside BCRs deforms and eventually breaks the bubbles. Figure 3 shows a scheme of bubble breakup due to turbulent eddies.

ne ) NeFL

(22)

It is necessary to determine the efficiency of the collisions between eddies and bubbles. breakup efficiency was proposed by Prince and Blanch,6 eq 23, as a function of the ratio between the turbulent energy of the eddies and the critical kinetic energy of the bubbles:

( )

κi ) exp -

uci2

ute2

(23)

9530 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

where the critical vortex velocity capable of breaking a bubble of diameter dbi6,8 is given by uci )

( ) Wecσ dbiFL

0.5

(24)

where the Weber critical number determines bubble stability in the flow. The critical Weber number, Wec, depends on the system and the breakup mechanisms.32,40 Liquid viscosity plays here an important role in bubble stability.41 Therefore, the Weber critical number will be a parameter of the model to determine the stability of the bubbles in different viscous fluids. 2.1.3. Bubble Scheme. In most of the studies on bubble dispersions, a number of bubble classes/sizes is defined in advance. The bubbles resulting from coalescence and breakup processes are reorganized to fit the predefined classes. This fact leads to problems regarding the total gas phase in the tank and results in difficulties when determining the effect of bubble coalescence and breakup on the mass transfer. For instance, coalescence decreases the gas-liquid contact area and increases the oscillation amplitude. These two hydrodynamic processes with opposed effects on mass transfer may reduce the mass transfer rates depending on the bubble size.33 Therefore, we propose a scheme of bubbles shown in Figure 4 so that the bubbles resulting from coalescence and breakup processes exist. Figure 4 represents the distribution of bubble sizes present in the bubble column in a normalized way and how one bubble size is obtained from the others by coalescence and breakup processes. Each circle in Figure 4 represents a bubble class/ size. The initial bubble size generated at the sieve plate is considered to have a normalized volume of 16u3. The number 16 comes up because it can be easily divided into integer numbers for the scheme whatever the actual size. The arrows that link two or three bubble sizes show the coalescence and breakup processes allowed, assuming constant volume in the coalescence and breakup. Thus, we can say the following: (a) We assume that the bubbles generated at the dispersion device have a normalized volume of 16u3. The actual bubble initial size is given by dbini)5do

(25)

(b) The number of bubbles of volume 16u3 generated at the sieve plate per unit of time in the bubble column is that of the number of orifices. The number of orifices in the dispersion device, No, depends on the configuration of the sieve plate. Holes are usually placed on 60° equilateral triangular pitch with the liquid flowing normally. Holes spaced closer than twice the hole diameter lead to unstable operation. Therefore, the recommended spacing is 2.5do to 5do. In this paper, 3.5 will be used.42 Meanwhile do ) 0.005 m as is the industrial preference43 According to this typical configuration the number of orifices in a sieve plate is given by Ludwig:44 Noa ) 1.158C-2

(26)

No ) Noa(π((Dc - clearance)/2)2)

(27)

Clearance will be 5% of DC (DC ) 1 m). From the initial bubble size at the orifice (normalized volume of 16u3), bubble coalescence and breakup processes are allowed within certain limits to obtain different bubble sizes; see Figure 4. The processes allowed are represented as arrows linking the bubble sizes under consideration for each single process, with operators attached to the tip of the arrows defining the processes involving the bubbles: (2x) to identify that two equal bubbles of the smaller bubble will merge to give another of twice the

Figure 4. Scheme of bubble sizes.

volume, (/2) to indicate binary breakup of the bubble into two equal ones, or an operator (+) located to identify coalescence between bubbles of different sizes. After this general description of Figure 4, the representation of coalescence and breakup processes is described: (c) Regarding coalescence, two bubbles of the same size will be allowed to coalesce to obtain a bubble of twice its initial size. For example, a bubble of volume 8u3 can merge with another one of the same size (2x) to obtain a bubble of volume 16u3. The arrow points from the circle with 8 to the circle with 16 with the operator (2x) on it to define this coalescence process. For bubbles of different sizes, the coalescence processes are limited to those in which the circles are linked and an operator (+) is added to the arrows. For instance, a bubble of size 4u3 can merge with another of size 16u3 to obtain another of 20u3. In order to determine the probability of coalescence, different mechanisms are considered. For bubbles of the same size, only laminar and turbulent collisions are used since they are supposed to have the same rising velocity. No wake effect is considered. For coalescence between bubbles of different sizes, buoyancy collision is also considered. (d) Only binary breakup into two daughter bubbles is considered for all bubble sizes, since it is overwhelmingly the major breakup manner supported by experimental observations.14 Furthermore, it will be assumed that a bubble breaks into two equal size daughter bubbles, which has proved to be successful in modeling the hydrodynamics of bubble columns8,9 and stirred tanks.39 Equation 28 allows determining the bubbles generated in the breakup. For example, a bubble of normalized volume of 8u3 can be broken into two equal ones of 4u3. This process is represented in the figure by an arrow pointing from the circle with number 8 to the circle with number 4 and an operator (/2) indicating that the bubble splits into two”.

( )

( )

4 db,j 3 4 db,j+1 3 ) 2Vb,j+1 ) Vb,j ) π (28) 2 π 3 2 3 2 2.1.4. Turbulent Energy. The energy available in a bubble column is usually given by eq 30:9 ε ) uGg

(29)

2.1.5. Dispersion Generated. A population balance in the terms proposed by Fletcher et al.45 will determine the fraction of bubbles of each size: ∂ ∂ n(z, db, t) + [n(z, db, t) ur(z, db)] + ∂t ∂z ∂ ∂ n(z, db, t) db(z, db) ) G(z, db, t) (30) ∂db ∂t

[

]

In a stationary regime eq 31 becomes8 0 ) G(z, db, t)

(31)

Function G represents a balance between the coalescence and breakup processes8 and determines the number of bubbles of each class in the dispersion. In our case, function G is as follows:

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9531 2

2

2

∑∑

∑

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

Gi )

Bubbles are periodically generated at the dispersion device (primary bubbles or bubbles of class 1) so that eq 33) must be completed by eq 34), when applied for this bubble class, to account for their presence in the tank due to the bubbling process. n2Gen )

norifices VsPd

εg ) 1.07n2/3Fr1/3

3εgDc2(4Dc + VOlgas/(π(Dc/2)2)) 2db,j3

(35)

The model for predicting the mean diameter of the bubbles of the dispersion consists of 30 equations like eq 32 where the breakup and coalescence rates have been defined by the equations presented along the paper. The bubble fraction of class i, xj, is obtained minimizing

∑G )0

(36)

i

Once xi is calculated, d32 is obtained. 2.2. Mass Transfer. In the flow developed inside bubble columns, the hydrodynamic processes involving bubbles such as bubble collisions, breakup, detachmen,t or coalescence provide the bubbles with an initial oscillation amplitude. Bubble oscillations modify the velocity profiles surrounding them, the concentration gradients, enhancing the mass transfer rates from the bubbles.24,26 Therefore, the study of bubble oscillations is very important for a better understanding of the performance of BCRs. In the case of viscous fluids, common fluids for many biochemical processes,47 bubble oscillations are absorbed as viscous dissipation. The time bubble oscillations last depend, not only on the energy available in the system, which defines the frequency of coalescence and breakup processes in the vessel, but also on the physical properties of the liquid. Valentine et al.27 proposed a model based on energy conservation which predicted oscillation decay depending on the liquid viscosity. This model was implemented by Martin et al.26 to develop a model for the Sherwood number of oscillating bubbles in viscous liquids. The main equations can be seen as follows:

(

(

[

-1 ∂Ψ(1) ∂Ψ(0) 1 - 2F(1) + Vθ ) Vθ(0) + A 2 n ur ∂η R η sin θ ∂η ∂2Ψ(0) F(1) n ∂η2 *

1 If ) T

∫∫ T

0

π

0

[ ( )]

∂Fn Vθ ′|n)1Fn Fn + ∂θ 2

2 1/2

2

(39)

(40)

Meanwhile, in order to determine bubble oscillation amplitude, A, the maximum and minimum diameters of a bubble class in the dispersion is calculated using its deformability defined by means of Eotvos number49 Ex )

1 1 + 0.163Eo0.707

(41)

1 - Ex 2Ex

(42)

A)

so that kLa in the BCR will be 30

k La )

∑ i)1

xj

1/2 1/2 If

pe

εg)uG/ur

(34)

The number of bubbles of each class is calculated as a fraction of the total gas phase assuming that the column length is 4 times its diameter:43

 π2 N

Bubble rising velocity, ur, depends on bubble size. For small bubbles eq 6 can be used. However, in the case of big bubbles the rising velocity does not depend on the fluid and ranges from 0.8 to 1.2, Grund et al..35 It is possible to calculate ur from the experimental value of εg and uG based on the definition of gas hold up by Sideman et al.48

(33)

where Vs corresponds to the liquid volume of study and Pd is the formation period of the bubbles in each experimental condition. The number of bubbles in the column will be determined by the gas hold up. A theoretical model for the gas hold up can be used46 so that a completely theoretical model for kLa is obtained.

nb,j )

Sh )

(32)

ShDL db,i

∑ n 4π( 2 ) db,i

2

(43)

i

3. Results Once the model has been proposed, different particular cases will be studied using experimental data from the literature for different viscous Newtonian and non-Newtonian liquids. 3.1. Case Study I: Newtonian Viscous Fluid. Different solutions of viscous liquids have been selected from the literature to evaluate the effect of viscosity on kLa and validate the model. The selection of solutions is based on the availability of experimental data, not only for the physical properties but also for the transport properties, DL, due to the strong link between both, kL and DL. Moreover, the Reynolds number of the bubbles must be small so that the model for the Sherwood number can be used. The model proposed so far is completely theoretical. However, in order to reduce the errors when comparing the calculated kLa with the experimental one from the literature, instead of using eq 34 or predicting εg, it is better to use a correlation for εg obtained for the same gas-liquid system as that in which kLa values were measured. Thus, the well-known equation by Akita and Yoshida29 will be used to predict the hold up of a Newtonian viscous solutions.

( )

εg

) 0.2 (1 - εg)4

gDcFL σ

1/8

( )( ) gDc2FL2

1/12

uG

√Dcg

2

µL

(44)

Meanwhile, as a comparison for the mean diameter of the dispersion, the correlation given by eq 4530 is used: d32 ) 26DcFL

(

gDc2FL σ

) ( ) ( ) -0.5

gDc3 (µL/σ)

-0.12

uG2 Dcg

-0.06

(45)

Finally, the simulated kLa is going to be compared with that obtained by Akita and Yoshida’s equation.29

]))

(37)

sin θ dθ dt (38)

kLa ) 0.6

( )( ) (

DL µL Dc DLFL

0.5

gFL σ

0.62

g (µL/FL)2

)

0.31

εg1.1

(46)

The viscous solutions should be within the application range of eqs 44 and 46 (µL ) 0.00058-0.021 Pa · s, σ ) 0.022-0.0742 N/m, FL ) 800-1600 kg/m3) so that it is possible to compare

9532 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 Table 1. Properties of Newtonian Viscous Solutions

solution

σ FL µL (kg/ (mN/ m3) m) (mPas)

DL (m2/s)

ref

glycerol 50% 62% 65% 70%

(w/w) (w/w) (w/w) (w/w)

1127 1152 1165 1183

69.1 68.7 67.5 66.8

6 9.63 14.45 24

5.3 × 10-10 4.5 × 10-10 2.99 × 10-10 eq 49

Voight et al.31 Delaloye et al.54 Kiyomi and Fumitake30 Voight, J. Et al (1980).31

sucrose 50% (w/w) 1230 75.4 51.8% (w/w) 1165 67.5

11.1 19

4.6 × 10-10 3.1 × 10-10

Hikita et al.55 Onken and Weiland56

Table 2. Calculated Weber Critical Number of Different Viscous Newtonian Solutions solution

Wec glycerol

50% 62% 65% 70%

(w/w) (w/w) (w/w) (w/w)

10 12 16 23

Figure 5. Comparison between the experimental and calculated mean diameter of the dispersion. Viscous Newtonian fluid: glycerol solutions.

sucrose 50% (w/w) 51.8% (w/w)

13 18

the experimental results of the hydrodynamics and mass transfer of the gas-liquid systems with the ones obtained by the model. The solutions selected for this study can be seen in Table 1. To determine DL of the 70% (w/w) solution of glycerol, not ¨ ztu¨rk et appearing in the study,31 the correlation proposed by O al.50 is going to be used:

()

µ1 DL,2 ) DL,1 µ2

0.57

(47)

where DL,1 and µ1 are the ones for the solution of 50% (w/w) by the same authors.31 Thus, for the solution of glycerol in water 70% (w/w) is DL ) 2.4 × 10-10 m2/s

(48)

The dispersion of bubbles is obtained by solving eq 36. The bubble dispersion depends on the flow regime.6,8 In order to solve eq 36, a mathematical shape for the bubble dispersion has commonly been proposed. Homogeneous flow regime is considered for the solutions in Table 1. Therefore, a normal distribution for the bubbles in the bubble column is assumed whose parameters (mean and variance) are determined using the Weber critical number in eq 25 as a parameter.6,8 The objective function is the difference between the bubble mean diameter predicted by the model and that calculated from eq 40 for a range of superficial gas velocities. Table 2 shows the Wec for each of the solutions tested in this paper. Figure 5 shows the comparison between the experimental bubble mean diameter for the different solutions of glycerol using Akita’s equation (eq 46), and the modeled ones for the using Wec from Table 2. In general, good agreement is found. However, for the less viscous fluids shows lack of agreement for uG < 0.02 m/s. The energy dissipated in the column under these conditions is small so that the bubbles tend to remain stable. This fact cannot entirely be explained by the closures because breakup and coalescence mechanisms may be between those proposed for inviscid fluids and the ones proposed for viscous ones. Apart from glycerol solutions, sucrose solutions from the literature have also been tested. Figure 6 shows the comparison between the mean diameters predicted by the model

Figure 6. Comparison between the experimental and calculated mean diameter of the dispersion. Viscous Newtonian fluid: sucrose solutions.

and the ones given by eq 46 for the solutions of sucrose in water. Good agreement is also found. For all solutions in Table 1, an increase in liquid viscosity increases Wec; see Table 2. This fact can be explained because viscosity leads to more stable bubbles in the flow.14 Figure 7 shows the profile of Wec with liquid viscosity. An empirical correlation can be obtained so that it can be implemented in the model for a general use of it. Wec ) 734µL + 5.2

(49)

Once the bubble dispersions have been simulated, it is possible to couple the model with the equation for the Sherwood number developed by Martin et al.28 The oscillation amplitude of each bubble size in the dispersion modeled is calculated using eqs 41 and 42. Martin et al.26 proved that the Sherwood number in viscous fluids depends on the decayment of the oscillation amplitude due to the absorption of energy as viscous dissipation. At the same time, the hydrodynamic processes of the bubbles’ experience (collisions, bubble breakup, coalescence, detachment) provides the bubbles with an oscillation amplitude. Therefore, a mean Sherwood number must be obtained representative of the oscillating time of the bubbles in the column. In inviscid fluids, the oscillation energy remains and the characteristic time for bubble oscillation is one complete oscillation because

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9533

Figure 7. Effect of liquid viscosity on the Weber critical number.

oscillations are a periodic phenomenon.24 However, in the case of viscous fluids, the oscillating time depends on the processes taking place in the tank together with the energy available. Therefore, characteristic times of 1, 1.5, and 2 bubble oscillations have been tested as representatives for the bubble lifetime span between bubble collisions, breakup, or coalescence. Figure 8 shows the comparison between the results predicted by eq 46) and those obtained by the model proposed in this work. Bubble oscillation is not completely absorbed as viscous dissipation due to the frequency of collisions, breakup, and coalescence processes taking place inside the bubble column. Good agreement is found for bubbles oscillation 1.5 times. This result leads to the conclusion that bubbles in the tank remain as entities for a period of time equal to 1.5 oscillation times during which bubble oscillation decays but is not completely absorbed as a consequence of the hydrodynamic processes involving bubbles. Therefore, a characteristic time equal to 1.5 oscillation times can be considered as a mean characteristic

value. The discrepancy for uG < 0.02 m/s for the less viscous fluids used in this work is related to the lack of agreement in the predicted bubble mean diameter. 3.2. Case Study: Non-Newtonian Fluids. The validation of the model for non-Newtonian fluids is more complex due to the lack of experimental data available in the literature. The work by Godbole3 has been used because it is interesting to use a correlation for εg that has been obtained for the same system than the measured kLa so that the error in predicting kLa will not be subjected to that of predicting εg, in spite of the good accuracy of Kawase’s eq 34. A solution of 1.4% CMC in water is considered (whose physical properties are experimentally determined to be FL) 1004 kg/m3, σ ) 68 N/m, m ) 0.581, n ) 0.757,33 DL ) 0.69DL,air-water51). In industrial equipment, bubble column diameter will not allow the development of slug flow. Therefore, the churn turbulent regime is considered. According to Haque52 for columns over 1 m, no flow transition is shown. As was exposed before, DC is assumed to be 1 m. Thus, a correlation for the gas hold up in churn turbulent is given by εg ) 0.207uG0.6(0.581(5000(uG))0.757-1)-0.19

(50) 3

As a comparison the mean diameter of the dispersion is d32 )

6εg 6(0.207uG0.6(0.581(5000(uG))0.757-1)-0.19) ) (51) a (19.2(u 0.47/(0.581(5000(u ))0.757-1)0.76)) G

G

The churn turbulent regime is characterized by bimodal distributions. According to Buchholz et al.,53 the mean diameter of solutions of 1.4% CMC is 0.15 m, the same result that can be obtained using eq 51; see Figure 15 of ref 53. The two peaks are found at 12.5 and 22.5. This means that the bubbles are mainly in the range of 12.5. According to the figures of bubble size distribution given by Buchholz et al.53 it is possible to simplify the solution of the population balance model using a

Figure 8. Comparison between the experimental and calculated kLa. Viscous Newtonian fluid.

9534 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

further. The best value turns out to be 1.5. Bubble oscillations are partially absorbed due to liquid viscosity, but the frequency of bubble collisions, coalescence, and breakup processes induces oscillations on the bubbles in a regular basis. 4. Conclusions

Figure 9. Comparison between the experimental and calculated mean diameter of the dispersion. Viscous non-Newtonian fluid.

A theoretical equation for the Sherwood number of oscillating bubbles has been combined with a population balance model in order to provide a deeper understanding regarding the mass transfer mechanisms in bubble columns operating with viscous fluids. Although the model relays on two parameters, Wec and T, they provide a physical explanation for the effect of the viscosity on kLa. The hydrodynamics are controlled by the viscosity of the liquid, defining the breakup and coalescence rates. Bubbles in viscous fluids are more stable in the flow and, as a result, Wec increases with viscosity. A correlation between Wec and liquid viscosity has been proposed. Furthermore, bubble collision, breakup, and coalescence processes provide an initial oscillation amplitude which enhances mass transfer with respect to rigid bubbles. Bubble oscillations are only partially absorbed as viscous dissipation because the processes the bubbles experience (collisions, coalescence, and breakup) provide them with an initial oscillation amplitude. The characteristic bubble oscillation time turned out to be that corresponding to 1.5 bubble oscillations. The model is able to predict the mass transfer rates once the hydrodynamics has been simulated. Acknowledgment The support of the Ministerio de Educacio´n y Ciencia of Spain providing a FPU fellowship to M.M. is greatly appreciated. The funds from the project reference CTQ 2005-01395/PPQ are also appreciated.

Figure 10. Comparison between the experimental and calculated kLa. Viscous non-Newtonian fluid.

broad log-normal distribution whose parameters µdist and σdist will be found. The model determines first a bubble dispersion representative of the experiment. In order to do so, the Weber critical number is used as a parameter to define bubble breakup so that the calculated mean diameter of the dispersion matches the experiment. Solving eq 41 for our particular case, a gas-viscous liquid dispersion working with non-Newtonian rheology, Wec, turned out to be 350 to match the bubble mean diameter obtained experimentally by eq 52. Figure 9 shows the results. Fairly good agreement has been found between experimental and theoretical results. Once the hydrodynamics have been modeled, the equation for the Sherwood number is combined to the population balance model to determine kLa. The experimental data for bubble columns operating with viscous non-Newtonian fluids in churn turbulent regime can be found in Godbole et al.3 kLa ) 0.00315(uG)0.59(0.581(5000(uG))(0.757-1)))(-0.84) (52) As in the case of Newtonian viscous fluids, values of 1, 1.5, and 2 bubble oscillations have been tested. In Figure 10, it can be seen that for bubbles oscillating with one oscillation period, the kLa is bigger than the experimental, so it can be conclude that bubble lifetime is longer and bubble oscillation decays

Nomenclature Bi ) breakup frequency (m-3 · s-1) C ) hole pitch (m) Cij ) coalescence frequency for bubbles of class i, j (m-3 · s-1) db ) bubble diameter (m) do ) orifice diameter (m) d32 ) Sauter mean diameter of the dispersion (m), d32 ) (∑nideqi3)/ (∑nideqi2) DC ) column diameter (m) DL ) diffusivity (m2 · s-1) Eo ) Eo¨tvo¨s number, Eo ) (FL - FG)gdeq2/σ F ) shape function Fr ) Froude number Fr ) uG2/(gDC) g ) gravity (m · s-2) G ) generation function defined by eq 33 h ) thickness of the drainage film (m) If ) integral of shape k ) wave number (m-1) n ) power law index ni ) concentration of class i elements per unit volume (no. bubbles · m-3) Ne ) eddy concentration (eddies · kg-1liquid) Noa ) number of orifices per unit area No ) number of orifices Pe ) Peclet number, Pe ) urdb/DL Pd ) formation period of bubbles (s) Qc ) gas flow rate (m3 · s-1)

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9535 r ) bubble radius (m) rC ) column radius (m) Sij ) surface contact area (m2) Sep ) separation between centers of orifices (m) t ) time (s) tformation ) formation time of a bubble until its detachment (s) tij ) film drainage time (s) tr ) liquid film breakup time (s) tt ) period of bubble formation (s) T ) oscillation time (s) ua ) expansion mean velocity of a bubble (m · s-1) uc ) critical velocity of a bubble (m · s-1) ur ) rising velocity of a bubble (m · s-1) ut ) turbulent velocity (m · s-1) Ul ) liquid velocity (m · s-1) uG ) superficial gas velocity (m · s-1) V ) drainage velocity (cm · s-1) Vs ) water volume seen in the videos (m3) 0.15 × 0.15 × 0.025 Vb ) bubble volume (m3) Vθ ) angular velocity (m · s-1). We ) Weber number, We ) dequr2FL/σ z ) vertical coordinate (m) Greek symbols κ ) breakup efficiency ε ) dissipated turbulent energy (W · kg-1) λ ) coalescence efficiency θ ) collision frequency (m-3 · s-1) η ) radial dimensionless unperturbed coordinate µL ) liquid viscosity (Pa · s) εg ) gas holdup νL ) kinematic viscosity of the liquid (m2 · s) FL ) liquid density (kg · m-3) FG ) gas density (kg · m-3) σ ) surface tension (N · m-1) τij ) contact time between bubbles (s) Ψ ) potential function (m3 · s-1)

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ReceiVed for reView July 14, 2008 ReVised manuscript receiVed September 22, 2008 Accepted October 6, 2008 IE801077S