Improving Mass Transfer Coefficient Prediction in Bubbling Columns

Their use greatly improves the prediction of the oxygen transfer coefficient using standard correlations. 1. Introduction. Oxygen transfer in the air-...
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Ind. Eng. Chem. Res. 2004, 43, 6527-6533

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Improving Mass Transfer Coefficient Prediction in Bubbling Columns via Sphericity Measurements J. Enrique Botello-A Ä lvarez,* J. Luis Navarrete-Bolan ˜ os, Hugo Jime´ nez-Islas, Alejandro Estrada-Baltazar, and Ramiro Rico-Martı´nez Departamento de Ingenierı´a Quı´mica-Bioquı´mica, Instituto Tecnolo´ gico de Celaya, Ave. Tecnolo´ gico y Garcı´a Cubas s/n, Celaya, Gto 38010, Me´ xico

The prediction of the oxygen transfer coefficients (kL) in bubbling columns is studied. Bubbling columns have important applications as bioreactors. It is found that the coefficient is strongly influenced by the motion and geometric characteristics of the bubbles. In the experiments, it is observed that the maximum transfer coefficient is achieved when a measure of the bubble sphericity reaches a minimum and the slip velocity a maximum. The introduction of characteristic parameters to account for the bubble deformation is proposed. Their use greatly improves the prediction of the oxygen transfer coefficient using standard correlations. 1. Introduction Oxygen transfer in the air-water system is fundamental in biochemical processes involving the action of aerobic microorganisms. Due to the low solubility of oxygen in water, this phenomenon has been found to be the limiting stage in many such processes. Because of this, bioreactors are often designed and/or scaled based on the volumetric oxygen mass transfer coefficient (kLa).1 Notwithstanding its importance, the phenomena involved in oxygen transfer are not yet fully understood and characterized. They have a complex dependence on reactor geometric characteristics, liquid phase physical properties, and operating conditions; therefore, theoretical and experimental studies to improve the knowledge of these phenomena are needed and may lead to better designs and gas-liquid contact reactor operation.2,3 An important step in understanding the mass interfacial transfer phenomena is to allow the calculation of transfer coefficient (kL) and specific interfacial area based on liquid volume (a) separately, thus isolating the contributions of the operation conditions and physical properties of the phases on these parameters.4 The volumetric oxygen mass transfer coefficient kLa can be considered as the result of the mass transfer interfacial area availability a, and the combined effect of the system transport properties and hydrodynamics, enclosed in a mass transfer coefficient kL. Chanson5 indicates that, in a turbulent regimen, kL is practically constant, independent of the bubble size and hydrodynamic changes, while the interfacial area may significantly vary. Vlae et al.6 proposed, in modeling the oxygen transfer in a stirred tank, that both kL and the bubble size may be considered constant, and thus kLa and a were only dependent on the spatial distribution of the gas holdup in the reactor. Maalej et al.7 found that, in stirred vessels operated at high pressure, kLa varied in direct proportion to a while kL was constant. * To whom correspondence should be addressed. Tel.: +52 (461) 611-75-75, ext. 209. Fax: +52 (461) 611-79-79. E-mail: [email protected].

These observations, for stirred vessels, are not necessarily true for bubbling column bioreactors, where the liquid motion is the result of both the momentum transfer at the interface and the bubbles’ ascending motion, involving severe bubble deformation and irregular gas flow. The superficial stretching theory proposed by Angelo et al.8 suggests that the mass transfer coefficient can be characterized as a function of the stretching of the bubbles’ surface and their oscillatory movement with respect to a reference state. Hecht and King9 used similar ideas in interpreting the mass transfer coefficient increase that they observed during spray-drying experiments. As will be discussed below, there exist clear indications of the dependence of the mass transfer coefficient on bubble deformation and bubble motion. These indications will be translated into terms that can be incorporated into the correlations often used to estimate the mass transfer parameters. Wongsuchoto et al.5 suggested that the correlations to estimate Sherwood numbers may be described by a general algebraic form given by eq 1. There the Grashof number (Gr) represents the mass transfer from natural convection to free rising velocity, whereas the Reynolds number (Re) is related with mass transfer via forced convection, and the R1- R4 parameters are fitted from experimental data.

Sh )

dbkL ) R1 + R2GrR3 + R4ReR5 D

(1)

In accordance with eq 1, the mass transfer coefficient mainly depends on molecular diffusivity of the species, bubble diameter, and the state of motion. Many particular forms of eq 1 can be found in the literature. Among them, we have selected a few as the basis of our analysis. These include correlations for predicting the mass transfer for stagnant bubbles in a liquid,11 bubbles ascending due to buoyant forces,11 bubbles modeled as rigid spheres,12 bubbles with moving surfaces,12 isolated bubbles, and swarms of bubbles.13 In all these correlations the bubble slip velocity is used as the indicator of the bubble motion. The slip velocity of the bubble is the relative velocity with respect to the enclosing liquid. As the slip velocity

10.1021/ie034332z CCC: $27.50 © 2004 American Chemical Society Published on Web 08/27/2004

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of the phases increases, the interfacial mass transfer also increases due to renovation of the interface that allows mass exchange between the bubbles and a larger number of elements of the fluid surrounding them. The implications of this variable over the interfacial mass transfer are reviewed by Pangarkar et al.14 The slip velocity is usually calculated13 using eq 2. However, in bubble columns there are no inflows or outflows; thus, the second term of the right-hand side of eq 2 vanishes, rendering a result that can be interpreted as the case of ascendant bubbles through a stagnant liquid.

vslip )

VSG VSL ( G 1 - G

(2)

In bubble columns, the liquid phase exhibits a circulating motion originated by the drag caused by the ascending bubbles, and the nonuniform radial distribution of the gas holdup.15 In addition, Pangarkar et al.14 indicate that the slip velocity of suspended particles in turbulent liquids is affected by different hydrodynamic phenomena and review some methods proposed for suspended particles in agitated tanks. Ueyama and Miyauchi16 proposed eq 3 to calculate slip velocities in a bubble column. There V0L and VWL are the velocities of liquid in the center of the column and in the interface of the liquid laminar sublayer near the wall, respectively. These velocities are calculated using a semiempirical procedure described in ref 16 which is based on a simplified solution of the equations of motion for the gas and liquid phases in a bubbling column.

( )

[| |

]

V0L + |VWL| VSG 1 ( VSL + vslip ) G 1 - G 6

(3)

The effect of bubble shape on the mass transfer phenomena can also be introduced in its calculations via experimental measurements. Numerous techniques allowing measurement of bubble dimensions and shapes in different equipments where gas-liquid transfer is important have been developed. Polonsky et al.17 report that bubbles can have shapes from completely spherical to ellipsoidal. Bozzano and Dente18 show that bubbles take forms that minimize the total energy associated with them, and that bubbles with diameters up to 1.3 mm and Reynolds number larger than 600 present significant distortions. The bubble shape and its state of movement have a direct repercussion over the interfacial mass transfer coefficient.10 Based on the description made, the objective of this study is to evaluate experimentally the oxygen transfer volumetric coefficient, the gas holdup, the specific interfacial area based on liquid volume, the oxygen transfer coefficient, and the geometric characteristics of bubbles, and to use these measurements to establish quantitative relationships among these variables. In what follows the experimental efforts are described, and their significance in demonstrating and quantifying the role of the bubble deformation on the estimation of mass transfer parameters is assessed. In particular, measurements of bubble deformation are coupled with commonly used correlations for estimation of mass transfer parameters, enhancing the predictive capabilities of such correlations.

2. Experimental Description Experiments were carried out, using the air-water system, at 23 °C, and local atmospheric pressure (0.81 atm) in a cylindrical bubble column of Pyrex glass. The water used was triply distilled. The column has 1.0-m height and an inner diameter of 66 mm. The gas sparger is a porous glass plate with a nominal pore diameter of 160-250 µm. A schematic of the experimental system is shown in Figure 1. The initial height of water was fixed at 0.40 m, ensuring that the liquid phase is well mixed. The model used to analyze the experimental data, described below, requires this condition. The homogeneity of the liquid phase was tested via pulse injection experiments of a NaCl saturated solution. The pulse was injected at the bottom of the column, through the air diffuser. Conductivity was then measured at the top of the column as a function of time. These experiments indicate that the liquid phase behaves as well mixed for all the superficial air velocities investigated here. The air superficial velocity was studied as far as 0.0234 m/s, in which the bubble regime is homogeneous, and does not interfere with the image acquisition system for bubble shape monitoring. Also, the velocity range was selected taking into account the typical operation conditions in aerobic fermentations for enzyme production performed in our lab, as will be reported elsewhere. The global gas holdup (G) was measured using the volume expansion method:

G )

∆V ∆V + VL

(4)

in which VL is the liquid volume without gas and ∆V is the volume expansion after the gas dispersion, calculated from the change of the liquid level and the crosssectional area of the column. The global volumetric oxygen transfer coefficient (kLa) was determined via dynamic experiments of oxygen transfer. The column was filled to the required water level. The liquid was degassed by nitrogen bubbling; subsequently, the nitrogen was replaced by air, keeping the gas flow constant in order to maintain the flow hydrodynamics without changes. The air and nitrogen fluxes were controlled with a calibrated rotameter. The oxygen concentration was recorded at regular intervals until saturation of the dissolved oxygen in water. Oxygen content was measured via a dissolved oxygen probe (Applikon Co.) supplied with a polarographic sensor. The kLa value was determined by least squares using a suitable mathematical model, and experimental data of the oxygen concentration in the liquid as a function of time. The model used is based on the model suggested by Smith11 presented in eqs 5-11. Equation 5 describes the mixing taking place between the fresh air fed to the column, with oxygen partial pressure given by P ˆ O2in, and the nitrogen left from the deoxygenation process previous to the start of the experiment. This equation idealizes the mixing process as taking place in a well-stirred tank from which a mixed current of fresh air and nitrogen leaves to enter the column through the diffuser with an oxygen partial pressure given by P ˆ O2out. Equation 6 represents a differential mass balance of the oxygen contained in the gas phase, considering that it passes through the column as plug flow. Equation 7 describes an oxygen balance in the liquid considering perfect mixing. Finally, eqs 8-11

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Figure 1. Schematic of the experimental setup.

represent the initial and boundary conditions. The numerical solution of these coupled equations is obtained by the method of lines.19

dP ˆ O2out dt ∂P ˆ O2 ∂t

)-

)

Qair (P ˆ -P ˆ O2out) Vinf O2in

(

(5)

)

ˆ O2 ˆ O2 VSG ∂P kLaRT P - (1 - G) - CO2L G ∂z G Hp dCO2L dt

(

P ˆ O2

∫0L Hp - CO L

1 ) kLa L

P ˆ O2out ) 0 P ˆ O2 ) 0

2

)

dz

t)0

(6)

(7) (8)

t ) 0, 0 e z e L

(9)

t g 0, z ) 0

(10)

ˆ O2out P ˆ O2 ) P CO2L ) 0

t)0

(11)

The air bubble dimensions were determined by means of an image digital analysis that involves shooting a film with a high-speed video camera (Sony Handycam DVCVTR730) using high-speed capture “modes”, autoexposure, and flash motion digital effects, as well as high contrast black and white images in order to, subsequently, extract selected snapshots using the image digital analysis software SigmaScan Pro 5 (SPSS Inc., Chicago, IL). The software allows us to obtain the dimensions and geometric characteristics of the bubbles

among other functions. To avoid the optical distortion of the bubbles due to the column curvature, a 15-cm side square glass prism was placed on the outer part of the column, filled with water, and located 20 cm above the diffuser, where the digital picture recorder was placed. The distortion was measured and compensated for by looking at rigid spheres of known dimensions and ensuring consistency against the experimental readings. It is important to note that the software and photographic techniques only allow two-dimensional measurements on the bubbles. The observed bubbles have mainly ellipsoidal shapes characterized by their major axis (dH), which represents the largest distance between two points on a bubble, and their minor axis (dL), which represents the largest length of a line, perpendicular to the major axis, that joins two points of the bubble. The software also reports the shape factor, or sphericity, which is a measure of the bubble distortion. With this information, the bubble diameter associated with an equivalent sphere was calculated as follows:12 3

dbi ) xdHi2dLi

(12)

To obtain an adequate average diameter for each performed experiment, we used the Sauter20 average diameter:

d32 )

∑i nidi3 ∑i nidi

2

(13)

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Figure 3. Volumetric oxygen transfer coefficient (2) and gas holdup (9) as a function of superficial air velocity in the bubbling column. Figure 2. Experiments of dynamic oxygen transfer in the bubbling column using the air-water system at different superficial gas velocities. Temperature 23 °C; local atmospheric pressure 0.81 atm.

The specific interfacial area based on liquid volume is obtained from the gas holdup and Sauter20 average diameter:

a)

6G d32(1 - G)

(14)

Finally, the oxygen mass transfer coefficient is obtained from

kL )

kLa a

(15)

3. Results and Discussion Figure 2 shows the average experimental data of the oxygen concentration in the liquid phase as a function of time, for the different superficial air velocities used during the experiments. Each run was repeated three times; the average absolute error was 3.866 × 10-6 mol of oxygen/L. These data were used in order to evaluate the volumetric mass transfer coefficient, via leastsquares fitting of the solution of the model given by eqs 5-11. Figure 2 shows also the calculated curves using the fitted kLa value coupled with the model for each of the superficial air velocities used for the experiments. The average deviation observed between fitted and experimental data is 6.0760 × 10-6 mol of oxygen/L. Figure 3 shows the estimated gas holdup and oxygen transfer volumetric coefficients. Note that the initial trends of the variables (solid lines) are different. This suggests that the increase in the gas superficial velocity has different effects over the oxygen transfer volumetric coefficient and the gas holdup. As the air superficial velocity increases, the positive effect on the mass transfer coefficient, resulting from the increase on the mass transfer area, which is directly associated with the gas holdup, is partially upset for some negative contributions. These negative contributions seem to be directly linked to hydrodynamic effects between bubbles and the liquid surrounding them, as will be described

below. As stated, a better understanding of the oxygen transfer phenomenon may be possible, if the parameters integrated in kLa, transfer coefficient (kL), and specific interfacial area (a) are analyzed separately. For the digital image analysis, at least 100 bubbles were digitally processed for each superficial velocity studied. The average over this number of bubbles was roughly the same for a number of bubbles 5 and 10 times larger. Distortion measurements were performed using glass spheres. These measurements indicated that the distortion, measured as the percentage of enlargement of the principal axis of the sphere, was larger whenever the sphere was located in the extreme of the observation field, and it was negligible whenever the sphere was at the center of the field. The maximum measured distortion was of only 10.11%; however, when averaging the distortion of the axis over 100 measurements randomly placed in the observation field, the average distortion was again negligible. Figure 4 summarizes the results of the bubble shape measurements as a function of the air superficial velocity. The bubbles exhibit quasi-sigmoidal growth with respect to this velocity. Also, note that the maximum bubble elongation is observed at the minimum shape factor value. In the experiments, it was observed that at the superficial air velocity where the minimum observed value of the shape factor is reached (∼0.009 m/s), the bubbles ascend via an appreciable zigzag motion. This observation coincides with the observations by Xie et al.21 They also mention that bubbles have irregular shapes, and exhibit eddy-like movement with spiral displacement, when the flow is near the plug regime at superficial velocities of gas above 0.014 m/s. Figure 5 shows the estimated mass transfer coefficients (kL) and specific interfacial areas (a), based upon the average values for kLa and G from Figure 2. Note that the specific interfacial area increases in the same way as the gas holdup, indicating that the increase in the bubble size with the air superficial velocity, which diminishes the specific interfacial area, is upset by the larger number of bubbles retained. The transfer coefficient shows a maximum coinciding with the largest bubble dimensions, and the minimum in the shape factor, as shown in Figure 4. The slip velocities calcu-

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Figure 4. Geometric characteristics of bubbles as a function of superficial gas velocity: major axis (9), minor axis (0), equivalent diameter (b), and shape factor (/).

Figure 6. Comparison of experimental oxygen mass transfer coefficients, and predictions using common correlations (Table 1).

based on the slip velocity. However, the deviations are significant for all the used correlations. The predictive capabilities of these correlations can be greatly enhanced by introducing the state of the bubble deformation in the calculations. Bozzano and Dente18 suggested that a deformation factor should be included in the estimation of the bubbles’ terminal velocity. In a similar manner, we propose the following modification in the calculation of the slip velocity (eq 3):

( )

vslip′′ ) vslip

Figure 5. Oxygen transfer coefficient (0), specific interfacial area based on liquid volume (9), and slip velocity (/) as a function of superficial gas velocity.

lated according to Ueyama and Miyauchi16 (eq 3) are also reported in Figure 5; their values show an average deviation of 3.78% with respect to the values calculated from eq 2. Their behavior is similar to that of the mass transfer coefficient, leading us to conclude that when the bubbles experience the largest slip velocity they also undergo their largest distortion. These conditions bring about the most efficient mass transfer. Table 1 presents several correlations often used to predict the mass transfer coefficients, as well as the physical situation to which they are applied. Using these correlations, we estimated the oxygen transfer coefficients, as shown in Figure 6, and compared them with our experimental observations. The necessary physical properties at 23 °C were obtained from Perry et al.22 Note that the trend observed in our experimental data coincides with the predictions given by the correlations

dH d32

-2

(16)

With this “corrected” slip velocity, the predictions given by correlations 4 and 5 improve, when compared with our experimental observations, from deviations of 43.0 and 74.6%, to deviations of 21.5 and 24.8%, respectively. Note that the “gap” between these two correlations is also narrowed from 26.3 to 12.8%. A second modification involves introducing corrections to account for the degree of stretching of the bubble surface, according to the stretching surface theory previously reviewed. Equation 17 presents this correction, which when applied to the results obtained from correlations 2 and 3 improves their predictions of kL from deviations of 66.8 and 72.6% to deviations of only 13.4 and 13.6%, respectively.

kL′′ ) kL

(dH dL )

2

(17)

There exist several reasons to apply these corrections selectively to the correlations, as has been done above. First, we note that there is a significant difference between the two corrections: the first involves exclusively the state of motion of the system via the recalculation of the slip velocity, while for the second we directly account for the bubble deformation, involving the hydrodynamics only implicitly. Under these considerations, the first correction should only be applied to those correlations involving the slip velocity. Correlation

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Table 1. Several Correlations for Estimating kL correlation 1

2

definition

( )

3

2/3

krigid ) 0.6 L 4

) 0.31

6

( ) ∆FµLg

bubbles rising through liquid phase11

1/3

FL2

x () x

kmobile ) 1.13 L 5

stagnant bubble surrounded by stagnant liquid11

dbkL )2 Sh ) D µL kL F LD

description

vslip 2/3 µL D db FL vslip 1/2 D db

( (

bubble behaves like a solid sphere12

-1/6

bubble behaves like a fluid particle with mobile surfaces12

) ( ) ( ) ) ( ) ( )

dbFLvslip ShL ) 2 + 0.0610 µL

0.779

dbFLvslip ShL ) 2 + 0.0187 µL

0.779

µL F LD

0.546

µL F LD

0.546

dbg1/3

0.116

isolated bubbles13

D2/3

dbg1/3

0.116

swarm of bubbles13

D2/3

3 is an interesting case. It exhibits a small variation when the corrected slip velocity is used; however, it does not reach the levels of improvement observed for correlations 4 and 5. Both of the proposed corrections can be applied to correlation 3. Since both, to a certain degree, account for the differences in mass transfer brought about by the bubble deformation, they should not be used simultaneously in the same correlation. Clearly, for the physical case described by correlation 3, the most consistent correction is the one that accounts for the bubble deformation directly, i.e., the correction based on the stretching surface theory. The dominant phenomenon accounted for in correlations 4-6 is related to the hydrodynamics; thus the most appropriate correction for these correlations is the one involving the slip velocity. Correlation 6 is based on the presence of a swarm of bubbles; such a physical situation already involves, partially, the dependence of bubble deformation. It is for this reason that the proposed modification does not improve correlation 6. Finally, correlation 1 is a limiting theoretical case that does not apply to the physical situation in a bubbling column; none of the proposed corrections has any significant effect in improving its predictions. Figure 7 compares our experimental observation against the corrected correlations 2-5. The prediction of correlation 2 is also improved by the second modification. The resulting modified correlation 2 becomes analogous to correlations for mass transfer involving solid particles descending through a stagnant fluid, with corrections to account for the prevalent hydrodynamics.14

very important role in the efficiency of interfacial mass transfer, and, although it may be discarded for velocity profile calculations, should be included in any rigorous formulation of this type of phenomenon if one hopes to reach a better understanding and characterization. The experimental image analysis techniques used in our work are relatively straightforward, and could be used to monitor bubble shape for many gas-liquid contact operations. The constant increase of oxygen transfer volumetric coefficient, gas holdup, and transfer specific interfacial area, with respect to the superficial velocity of air, indicates that the oxygen transfer in bubble columns is mainly controlled by the bubble retention and the available interfacial area and secondly by the hydrodynamics between bubbles and the surrounding liquid. The former phenomenon is better characterized via the change on the transfer specific area, a, while the latter should manifest itself in changes on the mass transfer coefficient, kL.

4. Conclusions It has been found that in bubbling columns, frequently used as bioreactors, there exists a strong correlation between the bubbles’ state of motion and their shape. The inclusion of experimental measurements of the bubble distortions to improve correlation prediction is proposed, via the calculation of the slip velocity for some correlations, in the estimation of the mass transfer coefficient kL. These corrections lead to significant improvements in the prediction of the coefficient, and should be employed whenever the shape of the bubbles is measured. Clearly, bubble shape plays a

Figure 7. Comparison of experimental oxygen mass transfer coefficients, and predictions using “corrected” correlations.

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Acknowledgment We gratefully acknowledge Garcı´a Soto Mariano and financial support from CONCYTEG and COSNET. Nomenclature a ) specific interfacial area based on liquid volume (m2/ m3) CO2L ) oxygen dissolved concentration (mg/L) D ) oxygen diffusivity in water (m2/s) db ) bubble diameter (m) d32 ) Sauter mean diameter (m) dH ) major axes of the bubble (m) dL ) minor axes of the bubble (m) G ) acceleration of gravity (m/s2) Gr ) Grashof number Hp ) Henry’s constant (atm/mg/L) kLa ) volumetric mass transfer coefficient (1/s) L ) liquid level R ) universal gas constant (atm L/K mg) Re ) Reynolds number P ˆ O2 ) axial partial pressure of oxygen (atm) P ˆ O2in ) partial pressure of oxygen at inlet of lower compartment (atm) P ˆ O2out ) partial pressure of oxygen at outlet of lower compartment (atm) Qair ) air volumetric flow (m3/s) Sh ) Sherwood number T ) temperature (K) t ) time (s) Vinf ) volume of lower compartment (m3) VL ) degassed liquid volume (m3) ∆V ) volume expansion after gas dispersion (m3) vslip ) slip velocity of a bubble relative to liquid (m/s) VSG ) superficial air velocity (m/s) VSL ) superficial liquid velocity (m/s) V0L ) liquid velocity at the center of column (m/s) VWL ) liquid velocity at the wall of column (m/s) z ) axial coordinate of column (m) Greek Symbols R ) experimental parameters G ) gas holdup µL ) viscosity of liquid (Pa s) FL ) density of liquid (kg/m3) ∆F ) difference of liquid and gas density (kg/m3)

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Received for review December 20, 2003 Revised manuscript received June 24, 2004 Accepted July 14, 2004 IE034332Z