Measurement of local specific interfacial area in bubble columns via a non-isokinetic withdrawal method coupled to electro-optical detector

Chemical Engineering Science 63 (2008) 1029 – 1038 www.elsevier.com/locate/ces

Measurement of local specific interfacial area in bubble columns via a non-isokinetic withdrawal method coupled to electro-optical detector Sergio García-Salas a, b,∗ , M.E. Rosales Peña Alfaro a , R. Michael Porter c , Frederic Thalasso b a Department of Bioengineering, Unidad Profesional Interdisciplinaria de Biotecnología del IPN. México, Av. Acueducto S/N, Ticomán,

México D.F. 07340, Mexico b Department of Biotechnology and Bioengineering, Centro de Investigación y de Estudios Avanzados del IPN, Apdo. Postal 14-740, México, D.F. 07360, Mexico c Department of Mathematics, Centro de Investigación y de Estudios Avanzados del IPN, Apdo. Postal 1-798, Arteaga 5, 76001 Santiago de Querétaro,

Qro. 76001, Mexico Received 19 April 2007; received in revised form 26 October 2007; accepted 3 November 2007 Available online 12 November 2007

Abstract Precise measurement of gas–liquid interfacial surface area is essential to reactor design and operation. Mass transfer from the gas phase to the liquid phase is often a key feature that controls the overall process. Measurement of gas–liquid interfacial area is often made through a separate measurement of the gas holdup and bubble size with complex and/or sophisticated methods. In this work, an inexpensive method is presented for the simultaneous determination of both local gas holdup and bubble diameter. The method is based on the withdrawal of the air–liquid dispersion under non-isokinetic conditions and on bubble counting via a simple optical device. The method was calibrated in a bubble column with several withdrawal pressures using coalescing and non-coalescing media. During the same calibration experiment, gas holdup was also measured manometrically and individual bubble diameters were measured by a photographic method. With a vacuum pressure of 3 kPa, local interfacial area measured with the withdrawal method produced a relative error below 13%, compared to the manometric/photographic method. The method was then used to characterize local specific interfacial area in a bubble column under several operating conditions with coalescing and non-coalescing media. In coalescing media and with superficial gas velocities (vg ) from 0.25 to 3.5 cm/s, the average interfacial area ranged from 17 to 197 m−1 . With non-coalescing media the average interfacial area ranged from 40 to 560 m−1 . Under the test condition it was observed that gas holdup is a parameter that has a greater distribution (standard deviation from 30% to 70%) than the volume-mean bubble diameter (standard deviation from 6% to 12%). It is shown that a model previously developed for characterizing gas holdup homogeneity is also suitable for characterizing interfacial area homogeneity. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Bioreactor; Bubble column; Hydrodynamics; Mass transfer; Coalescing; Non-coalescing media

1. Introduction Specific interfacial area is defined as the bubble surface area in a gas–liquid dispersion. Under standard agitation and aeration in bioreactors, the mass transfer coefficient (kL a) is mainly determined by the specific interfacial area (a), as the liquid film coefficient (kL ) is not as variable as the interfacial area (Kawase et al., 1992). The specific interfacial area is therefore ∗ Corresponding author. Department of Bioengineering, Unidad Profesional Interdisciplinaria de Biotecnología del IPN. México, Av. Acueducto S/N, Ticomán, México D.F. 07340, Mexico. Fax: +52 55 57 29 60 00x56305. E-mail address: [email protected] (S. García-Salas).

0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.11.003

a key parameter to be determined in mass transfer characterization. To our knowledge, the literature reports only two methods for direct measurement of the specific interfacial area. The first method was published in 1958 (Calderbank, 1958) and is based on optical detection of a light beam transmitted through the gas–liquid dispersion. This relatively simple method requires calibration before each experiment. The second method (Cents et al., 2001) is based on the measurement of a chemical reaction rate. This method has a restricted application range due to the reaction conditions required. Specific superficial area is commonly estimated from separate measurement of the local gas holdup and the bubble

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diameters. Gas holdup can be measured by well described physical methods such as manometric or expansion methods for global gas holdup or methods based on conductivity (Bin et al., 2001), optical detectors (Schweitzer et al., 2001) or gamma ray tomography (Kemoun et al., 2001) for local gas holdup. Bubble diameter can be measured by photographic methods (Akita and Yoshida, 1974; Buchholz and Shügerl, 1979) or using optical (Keitel and Onken, 1982), conductivity (Adler and Schügerl, 1983) or acoustic (Boyd and Varley, 1998) detectors. These methods are time consuming and/or require complex calibration or expensive equipment. Additionally, it has been suggested that specific interfacial area must be preferably estimated from simultaneous gas holdup and bubble diameter measurements, to guarantee no change of the liquid composition and operating conditions during measurement (Joshi, 2001). Most of the techniques described above are based on separate measurement of gas holdup and bubble diameter. To measure gas holdup, Greaves and Kobbacy (1984), Tabera (1990) and Yang and Wang (1991) have proposed a method based on volumetric measurement of gas and liquid phases after they are withdrawn from the gas–liquid dispersion. This method is simple and inexpensive in as much as no specially designed equipment is required. Greaves and Kobbacy (1984) also applied the withdrawal method to measure bubble diameter. When the gas–liquid dispersion is withdrawn through a capillary tube, bubbles are detected by means of an electrooptical probe. The signal obtained is interpreted and used to calculate bubble diameters. Greaves and Kobbacy (1984) and Hofmeester (1988) have stated that the aerated media must be withdrawn under isokinetic conditions, that is, with the same withdrawal speed as the speed of bubbles arriving at the suction probe. The purpose of isokinetic conditions is to guarantee that the gas/liquid ratio extracted is equal to the gas/liquid ratio of the aerated media and to avoid bubble breaking or coalescence. The need to maintain isokinetic conditions is a significant limitation. This requirement implies a preliminary characterization of the reactor hydrodynamics, often at least as complex as the gas holdup or bubble measurement itself. Further, bubble columns, the simplest system, characteristically exhibit variable bubble diameters (Akita and Yoshida, 1974; Pohorecki et al., 2001), bubble speeds (Deckwer, 1992) and liquid velocities (Wu and Al-Dahhan, 2001). A normal distribution of bubble rising velocities is typically observed, and isokinetic or proportional sampling is therefore an unrealistic condition that can, at best, be approximated, as noted by Nevers (1991), Hofmeester (1988) and Greaves and Kobbacy (1984). In a previous paper, García-Salas et al. (2005) showed that non-isokinetic conditions allow an easier implementation of the withdrawal method for gas holdup estimation. They have also shown that the withdrawn air/liquid ratio is different from the actual ratio but the difference can be easily compensated for via proper calibration with the manometric method. The method developed is of interest for the estimation of non-gassed volume and/or gas distribution homogeneity. However, additionally to the local gas holdup, bubble diameter is mandatory for mass transfer quantification.

The aim of this work is the development of a method for simultaneous gas holdup and bubble size determination, based on non-isokinetic withdrawal of the gas–liquid dispersion. As the gas–liquid dispersion is withdrawn, the system measures the gas and liquid volume as well as the number of bubbles withdrawn. These three parameters allow the estimation of the gas holdup and volume-mean bubble diameter. Here the method is described and calibrated in a bubble column by comparison to traditional manometric/photographic methods. This method is then used to estimate gas–liquid dispersion in a bubble column. Results are discussed with a particular focus on the discrepancy between volume-mean and Sauter-mean diameters. A model previously introduced for the characterization of the non-gassed volume and gas holdup homogeneity is also used to characterize specific interfacial area homogeneity in bubble columns. 2. Experimental 2.1. Bubble column and operating conditions A glass bubble column (internal diameter: 0.12 m; height: 1.6 m; working volume 14 L) fitted with a glass porous plate (0.02, 0.04, 0.06 or 0.12 m diameter) was used (García-Salas et al., 2005). The bubble column was aerated at superficial gas velocities (vg ) ranging from 0.25 to 3.5 cm/s. The air supply was controlled by a needle valve and measured with a variable area flow meter (Cole Parmer, USA). During the experiment, distilled water was used as coalescing medium and 0.13 M KCl aqueous solution as non-coalescing medium. The total height Hd of the gas–liquid dispersion was kept constant at Hd /(2R)= 10.4, where R is the column radius. 2.2. Electro-optical apparatus and method The measurement system was composed of (i) a withdrawal device to transport gas–liquid dispersion through a straight end capillary tube (García-Salas et al., 2005), (ii) an electro-optical probe attached to the capillary tube to detect bubbles and (iii) an electronic circuit to count bubbles passing through the capillary tube. As mentioned above, to keep the method as simple as possible, the use of the optical device was limited to bubble counting. The bubble size was estimated by the volume-mean diameter described below. Bubbles were counted when passing through the capillary tube and detected by an infrared opto-sensor and emitter (600 nm, QT optoelectronics, USA) connected to a simple electronic circuit (Fig. 1). An amplifier produced an analogue signal proportional to the radiation intensity detected by the probe and a commercial digital counter was used to count the number of bubbles detected (4 single digit display 8051-21, Led Tech, USA, coupled to integrated circuits 7447 and 7490, National Semiconductor, USA). The photo-sensor probe was located at 8 cm from the capillary tube entrance. The method for determining the withdrawn gas volume corrected to normal atmospheric pressure (VG ) has been previously described (García-Salas et al., 2005). From the number n of bubbles detected and the air volume VG withdrawn, a

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Fig. 1. Electronic circuit of the bubble detector and counter device.

parameter D proportional to the volume-mean bubble diameter was calculated according to Eq. (1). ⎛ V ⎞1/3 G 6 ⎜ n ⎟ D=⎝ ⎠ . 

(1)

Fig. 2. Parameter D versus photographic volume-mean diameter (d 3 ), for a height h/(2R) = 6.75 using sparger diameter of 12 cm and vacuum pressures of 1 kPa (), 3 kPa (), 9 kPa (◦), 20 kPa (—). (A) Non-coalescing liquid (aqueous solution of KCl 0.13 M), D = 1.7924 d 3 − 0.9721, r 2 = 0.9562. P < 0.0001. (B) Coalescing liquid (distilled water), D = 0.6965 d 3 + 0.2147, r 2 = 0.9582. P < 0.0001.

2.3. Photographic method A standard photographic method was used to measure bubble diameters. For this purpose, a digital camera was used (Nikon Coolpix 5700, shutter speed: 1/2000 s, sensitivity: ISO 400), with lateral illumination (500 W light). The curvature of the column caused optical distortion of the bubbles. To avoid this, a glass box (14 cm ×14 cm ×14 cm) filled with water was fixed on the reactor wall at a relative height (h/(2R)) of 6.75. The illumination conditions and photographic method were similar to those reported by Buchholz and Shügerl (1979) and by Buchholz et al. (1981). Each bubble was measured with a digital image analyzer (Digital Kodak Science 1D, USA). Individual bubble volume was calculated from the major (L) and minor (l) diameter, using the prolate spheroid volume formula (Bartsch, 1974):   2 4 L l V =  . (2) 3 2 4 From the bubble volume, an equivalent diameter that corresponds to a spherical bubble of the same volume was calculated according to  6V 1/3 d= . (3)  In each experiment, from 1000 to 2000 bubbles were measured. This number is greater than 500 as recommended by Lübbert (1991) for reliable results. The bubbles’ arithmetic mean diameter (d) was determined using Eq. (4), where di is the diameter of bubble i estimated as

described above and n is the number of bubbles measured.

n di (4) d = i=1 . n The photographic volume-mean diameter (d 3 ) was calculated using Eq. (5) with V obtained by Eq. (6). 1/3  6V d3 = , (5) 

n

V =



3 i=1 di

6n

.

(6)

For comparison the Sauter-mean diameter was calculated by the following:

n 3 d d 32 = ni=1 i2 . (7) i=1 di The calibration of the withdrawal method in terms of photographic method was carried out by fitting the computed values of the photographic volume-mean diameter d 3 to the parameter D, giving a linear relation. After proper calibration, the volume-mean bubble diameter estimated through withdrawal (d W 3 ) can be therefore estimated from D by using the same linear relation as indicated in the following: d W 3 = c1 D + c 0 .

(8)

As will be presented in Section 3 (Fig. 2), c0 and c1 were 0.5579 and 0.5423 (r 2 : 0.9562), respectively, for non-coalescing

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media and 1.4357 and −0.3083 (r 2 : 0.9582), respectively, for coalescing media. The specific interfacial area (a) was calculated from the withdrawal volume-mean diameter and the gas holdup according to a=

6 dW3

.

(9)

The gas holdup () was calibrated using a manometric method according to García-Salas et al. (2005). The volume-mean diameter in Eq. (9) is different from the Sauter-mean diameter, defined by Eq. (7) and typically used to characterize mass transfer. The discrepancy generated by using the volume-mean diameter instead of the Sauter-mean diameter will be discussed later. 3. Results 3.1. Calibration of the withdrawal method The calibration of the withdrawal method in terms of the photographic method is presented in Fig. 2, where the parameter D is graphed against the photographic volume-mean diameter (d 3 ) measured with the photographic method, using non-coalescing (Fig. 2A) and coalescing media (Fig. 2B) with vacuum pressures from 1 to 20 kPa. This vacuum pressure range was previously considered optimal for gas holdup measurement (García-Salas et al., 2005). Fig. 2 shows that the dependence relation of D on d 3 was not influenced by the vacuum pressure used and was linear for all vacuum pressures tested and in both media (r 2 > 0.9800). D values greater than d 3 in non-coalescing medium and vice versa for coalescing medium should be explained by the coalescence and the rupture of bubbles being withdrawn under non-isokinetic conditions (Greaves and Kobbacy, 1984). However, this matter should be investigated. With coalescing medium, the standard deviations for D and d 3 were from 0.7% to 3.0% and from 2.4% to 5.8%, respectively. With non-coalescing medium, the standard deviations for D and d 3 were from 0.5% to 2.0% and from 1.2% to 2.8%, respectively. According to these standard deviations, the precision of the withdrawal and the photographic methods were judged to be similar. According to García-Salas et al. (2005), a vacuum pressure of 3 kPa was the optimum vacuum pressure for the gas holdup measurement; this pressure allows therefore the simultaneous determination of gas holdup and bubble diameter and it was selected for the subsequent experiments. After the non-isokinetic withdrawal method was calibrated by manometric and photographic methods, gas holdup and bubble diameter were simultaneously determined at 35 locations in the bubble column (5 radial and 7 axial points). Gas holdup results were similar to those previously presented (García-Salas et al., 2005). The determination of D at different axial points using Eq. (1) was performed correcting the air volume withdrawn according to the hydrostatic pressure at the withdrawal location.

Fig. 3. Radial and axial distribution of the withdrawal volume-mean bubble diameter (d W 3 , mm), at vg = 0.54 cm/s for distilled water and KCL 0.13 M aqueous solution, with sparger diameters of 2, 6 and 12 cm. x-axis values: r/R =−1, −0.75, −0.50, −0.25, 0, 0.25, 0.50, 0.75, 1 (dimensionless); y-axis values: h/(2R) = 0.16, 1.83, 3.45, 5.16, 6.75, 8.45, 10.10 (dimensionless).

3.2. Distribution of the volume-mean bubble diameter Fig. 3 shows the distribution of the withdrawal volume-mean bubble diameter (d W 3 ) observed in the bubble column, for coalescing and non-coalescing media, with three sparger diameters and for a superficial gas velocity vg of 0.54 cm/s. Fig. 4 shows the same distribution but at superficial gas velocity of 1.74 cm/s. Both figures show larger bubbles in the center of the bubble column as was previously reported by Buchholz et al. (1979). A more homogeneous radial distribution was observed when larger spargers were used. Table 1 presents the average volume-mean bubble diameter shown in Figs. 3 and 4 as well as the standard deviations. Under coalescing conditions, the bubble volume-mean diameter was about twice as large. This was previously described by Zieminski and Whittemore (1971). In both conditions, the standard deviation was similar, 9% under coalescing conditions and 8% under non-coalescing conditions. 3.3. Distribution of specific interfacial area Figs. 5 and 6 show the distribution of the specific interfacial area determined in coalescing and non-coalescing media, with

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different sparger diameters, at superficial gas velocities of 0.54 and 1.71 cm/s. When the superficial gas velocity increased from 0.54 to 1.71 cm/s, the specific interfacial area increased in the same proportion, by a factor of 3–4. The specific interfacial area observed in coalescing medium was about twice that of

in coalescing medium. When the sparger diameter was raised from 2 to 12 cm, the specific interfacial area increased by a factor of 2–2.5. These results were similar to those reported previously by Zieminski and Whittemore (1971), as well as by Godbole et al. (1984).

Fig. 4. Radial and axial distribution of the withdrawal volume-mean bubble diameter (d W 3 , mm), at vg = 1.71 cm/s, for distilled water and KCL 0.13 M aqueous solution, with sparger diameters of 4, 6 and 12 cm. x-axis values: r/R = −1, −0.75, −0.50, −0.25, 0, 0.25, 0.50, 0.75, 1.00 (dimensionless); y-axis values: h/(2R) = 0.16, 1.83, 3.45, 5.16, 6.75, 8.45, 10.10 (dimensionless).

Fig. 5. Radial and axial specific interfacial area distribution (a, m−1 ), at vg = 0.54 cm/s, for distilled water and KCL 0.13 M aqueous solution, with sparger diameters of 2, 6 and 12 cm. x-axis values: r/R = −1, −0.75, −0.50, −0.25, 0, 0.25, 0.50, 0.75, 1.00 (dimensionless); y-axis values: h/(2R) = 0.16, 1.83, 3.45, 5.16, 6.75, 8.45, 10.10 (dimensionless).

Table 1 Global average of withdrawal volume-mean bubble diameter (d W 3 ) Superficial gas velocity

Sparger diameter

(cm/s)

(cm)

Distilled water ∗

KC1 0.13 M ∗

d W 3 (mm)

 (mm)

d W 3 (mm)

 (mm)

054

12 6 2

3.60 4.19 4.85

0.37 0.37 0.34

1.72 1.92 2.12

0.13 0.13 0.17

171

12 6 4

4.10 4.59 4.42

0.29 0.51 0.45

1.97 2.27 2.03

0.09 0.21 0.25

∗

d W 3 : Global average on 35 data of d W 3 , : standard deviation.

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Fig. 7. Bubble diameter distribution measured with photographic method fitted to a normal distribution function. Bubble diameter distribution obtained at superficial gas velocity vg = 2.42 cm/s, for distilled water, with sparger diameter of 12 cm. Bubble diameters (—), normal distribution function (- - - - -). r 2 = 0.9727. (–) Dimensionless.

and  is the standard deviation of the x values. Additionally, the relative error between the normal distribution function and the frequency function of the experimental data f (b) was calculated using Eq. (11). 1 2 f (x) = √ e−(1/2)[(x−)/] ,  2 Fig. 6. Radial and axial specific interfacial area distribution (a, m−1 ), at vg = 1.71 cm/s, for distilled water and KCL 0.13 M aqueous solution, with sparger diameters of 4, 6 and 12 cm. x-axis values: r/R =−1, −0.75, −0.50, −0.25, 0, 0.25, 0.50, 0.75, 1 (dimensionless); y-axis values: h/(2R) = 0.16, 1.83, 3.45, 5.16, 6.75, 8.45, 10.10 (dimensionless).

4. Discussion 4.1. Discrepancy between the Sauter-mean diameter and the volume-mean diameter As mentioned, the withdrawal method presented here is a method to determine the volume-mean bubble diameter which is different from the Sauter-mean bubble diameter, usually used to characterize mass transfer. To determine the Sauter-mean diameter, each bubble diameter must be measured separately, which is not possible with the simple method presented here. The discrepancy between the Sauter and the volume-mean diameter depends on the bubble diameter distribution. Akita and Yoshida (1974), as well as Deckwer (1992), have reported that the distribution of bubble diameters in bubble columns can be fitted to a normal distribution function. To compare the bubble diameters distribution observed in this work to a normal distribution, the bubble diameters measured with the photographic method were grouped into class intervals. The frequency or number of bubbles for each diameter interval were fitted to the normal distribution equation (Eq. (10)), (Montgomery, 1997), using the Sigma Plot software (SPSS Inc, USA). In Eq. (10), f (x) is the normal distribution function, x is the arithmetic average of a class interval,  is the arithmetic average of all x

error rel =

f (b) − f (x) . f (x)

(10) (11)

Fig. 7 shows an example of how bubble diameters measured with the photographic method fitted the normal distribution function. These data were obtained at superficial gas velocity vg = 2.42 cm/s, with distilled water and with a sparger diameter of 12 cm. Table 2 presents, among other parameters, the correlation coefficient and the relative error between the normal distribution function and the frequency function of the experimental data. Table 2 shows the experimental data obtained in this work alongside data published previously by Weiland et al. (1980) and by Buchholz et al. (1981). In general, the relative errors were smaller than 10% which confirms that bubble diameters in bubble columns fitted adequately a normal distribution function, as Akita and Yoshida (1974) and Deckwer (1992) previously mentioned. It is important to note that according to Table 2, even for stirred tank and airlift bioreactors, the bubble diameter distribution can be modeled by a normal distribution function (restricted to positive diameters). The discrepancy between Sauter- and volume-mean diameters depends on the standard deviation of the bubble diameter normal distribution. Indeed, for a standard deviation of zero, the volume-mean bubble diameter would be equal to the Sautermean diameter. In order to estimate the theoretical discrepancy between Sauter- and volume-mean diameters, theoretical normal distribution curves were generated and the error was estimated for several standard deviations. The results of this theoretical analysis confirmed that the discrepancy was dependent only on the standard deviation. Fig. 8 shows the relative error as a function of the standard deviation. Eq. (12) is

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Table 2 Statistical parameters of normal distribution function for bubble diameter distribution at several superficial gas velocities Bioreactor

Liquid

Method

vg (cm/s)

r2

d



/d (%)

Stirred tankb Airliftb

Water

Electro-optic Electro-optic

0.19 0.10

0.9442 0.9712

2.8 2.61

1.066 1.12

37 43

4.94 6.47

Bubble columnc

Water

Electro-optic Photographic

1.07 1.07 2.13 3.20 5.33

0.8566 0.9705 0.9596 0.9341 0.9991

3.98 3.05 3.93 3.76 4.02

1.38 0.97 1.37 1.46 1.52

35 32 35 39 38

20.37 8.06 4.81 −12.09 0.38

Bubble columnd

KCl 0.13 M

Photographic

0.54 1.08 1.71 2.43 3.43

0.9652 0.9423 0.9455 0.9543 0.9637

1.61 1.69 1.72 1.88 2.00

0.22 0.21 0.22 0.28 0.29

14 12 13 15 15

4.77 6.07 −3.13 5.77 6.82

Bubble columnd

Distilled water

Photographic

0.54 1.08 1.71 2.43 3.43

0.9224 0.9295 0.9214 0.9727 0.9531

3.03 3.32 3.44 3.73 3.83

0.64 0.54 0.53 0.60 0.67

21 16 15 16 17

−9.42 6.02 11.20 −1.63 −3.06

Average of relative errora (%)

vg : superficial gas velocity, d : arithmetic average of bubble diameter, : standard deviation. a Average of relative error according to Eq. (11). b Parameters calculated from data of Weiland et al. (1980). c Calculated from Buchholz et al. (1981). d This work.

4.2. Bubble diameter

Fig. 8. Relative error between volume-mean bubble diameter and Sauter-mean bubble diameter as a consequence of standard deviation, /d. Eq. (12) (—).

the mathematical model that best fitted the relation between discrepancy and standard deviation.  100 1.5587 error rel = 0.0327 . (12) d According to Eq. (12) and Table 2, maximum standard deviations of 40% reported by other authors (Weiland et al., 1980; Buchholz et al., 1981) correspond to a discrepancy between the volume- and Sauter-mean diameter of about 10%. Here, the standard deviations observed were less than 20% (Table 2) which corresponds to a maximum discrepancy of 4% between the volume- and Sauter-mean bubble diameter. This total discrepancy is similar to the errors usually introduced when bubbles are considered as perfect spheres (Deckwer, 1992).

As previously mentioned, the non-isokinetic withdrawal method can be used to determine gas holdup and bubble diameter simultaneously. When used to determine gas holdup, the method appears independent of the liquid phase composition as a single correlation for coalescing and non-coalescing liquids was sufficient (García-Salas et al., 2005). When used to determine volume-mean diameter, important differences between coalescing and non-coalescing media were observed (Fig. 2). This fact means that it is probably necessary to calibrate the withdrawal method whenever the media composition or property is changed. Although this has to be confirmed, the application of the withdrawal method in fermentation broths, where chemical composition changes constantly, could require frequent calibration. However, it would suffice to calibrate the method at single location of the reactor to apply the resulting formula at an arbitrary location. 4.3. Specific interfacial area Local determination of gas holdup and volume-mean bubble diameter allows estimating the specific interfacial area. Interfacial area distribution has been reported previously. For example, Kulkarni et al. (2001) used the hot film anemometry technique in bubble columns. Popovic and Robinson (1987) used photography, light dispersion and sulfite oxidation methods to study interfacial area in an external loop airlift. Oyevaar and Westerterp (1989) compared specific area measured with an oxidation method and a photographical method in a

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mechanically agitated reactor and in a bubble column. Alves et al. (2002) have determined the distribution of interfacial area in stirred tank reactors using an electro-optical method under isokinetic conditions. Compared to these methods, the withdrawal method presented here has potential advantages: (i) combined measurement of the gas holdup and volume-mean bubble diameter, (ii) low cost equipment and (iii) non-isokinetic conditions. The discrepancy introduced by considering the volume-mean diameter instead of the Sauter-mean diameter was estimated to be from 1.5% to 4%. Additionally, an error in D of about 0.7–3% is made during measurement. García-Salas et al. (2005) reported also an error on the gas holdup under 6%. According to these partial errors, the combined error estimated by simple sum procedure (Taylor, 1982) on the specific surface area was from 3% to 13%. This error is similar to those reported for other methods by Oyevaar and Westerterp (1989). 4.4. Model for specific interfacial area homogeneity The results presented in Figs. 5 and 6 show a nonhomogeneous specific interfacial area. García-Salas et al. (2005) developed a model that describes the gas holdup homogeneity in terms of two parameters: the non-gassed volume 0 ) of the reactor volume essentially without (portion (Vdead bubbles) and the gas holdup homogeneity parameter ( 1, 1 being a homogeneous distribution). Applied to specific surface area, this model is also based on partitioning the total volume (Vtot ) of the reactor into N subregions of volumes Vj , N being the number of specific surface area measurements made (aj ). Next we reorder the subscripts of Vj and aj so that a1 a2 a3  · · · aN . From the ordered partial volumes and specific interfacial area, a relative cumulative volume (Vi0 ) and a relative cumulative specific interfacial area (ai0 ) can be defined up to the ith subregion, by Eqs. (13) and (14), respectively. In Eq. (14) a0 is the global specific interfacial area observed in the reactor:

i j =1 Vj 0 Vi = , (13) Vtot

i j =1 aj Vj 0 ai = . (14) a0 0 , a 0 a 0 and that when i=N , we have Observe that Vi0 Vi+1 i i+1 Vi = Vtot , Vi0 = 1 and ai0 = 1. The functional relationship ai0 = g(Vi0 ), which can be observed by graphing the pairs (Vi0 , ai0 ), is described ideally by

g(v) = 0  g(v) =

Fig. 9. Two examples of experimental data () fitted to the model of specific interfacial area homogeneity (—). Relative cumulative specific interfacial area (ai0 ) and relative cumulative volume (Vi0 ). Superficial gas velocity of 1.71 cm/s. (A) Sparger diameter of 4 cm using distilled water, r 2 =0.9991. (B) Sparger diameter of 12 cm using KCl 0.13 M aqueous solution, r 2 = 0.9996. (–) Dimensionless.

Fig. 10. Model parameters of gas holdup homogeneity () and specific interfacial area homogeneity (). y = 1.1351x − 0.1453; r 2 = 0.9250. (–) Dimensionless.

0 for 0 v Vdead , 0 v − Vdead 0 1 − Vdead

 for

0 Vdead v 1,

(15)

The exponent  describes the specific interfacial area homogeneity in the column. A value of  = 1 describes a mixture completely homogeneous in the complement of the dead

region, while large values of  correspond to a strongly nonhomogeneous mixture. In all cases, the experimental data fitted adequately the homogeneity model (Eq. (15)) with a correlation factor r 2 > 0.995 (r 2 average = 0.9990). Fig. 9 shows two examples of how experimental data fitted to Eq. (15). The 0 distribution parameters (Vdead and ) allowed the characterization of the specific surface area distribution. Fig. 10 shows the

S. García-Salas et al. / Chemical Engineering Science 63 (2008) 1029 – 1038

relation between gas holdup homogeneity () and specific interfacial area homogeneity () observed during the experiments. This relation exhibited a slope close to 1, which indicates that the local gas holdup distribution is similar to the specific interfacial area distribution. These analogous distributions can be explained easily, considering that gas holdup is a parameter that has a greater distribution (standard deviation from 30% to 70%) than the volume-mean bubble diameter (standard deviation from 6% to 12%). This indicates that when determining the specific interfacial area distribution, the effect of the gas holdup distribution predominates over the bubble diameter distribution. Oliveira and Ni (2001) have previously mentioned that gas holdup has a greater effect on the specific interfacial area than bubble diameter does. 4.5. Generalization of the method The combined measurement of the gas holdup and volumemean bubble diameter with the withdrawal method presented here, has been applied so far to the simplest reactor design: a labscale bubble column. The application of the method to other free rising bubble designs (i.e., bubble columns with other spargers designs or airlift reactors) should be feasible as the same bubble behavior is expected. The application to reactors with mechanical stirring should be considered more carefully. The key point to be considered is the fluid velocity relative to the probe, which could significantly affect the withdrawal conditions. We expect that application of the withdrawal method to larger scale reactors would not be a major concern as the local scale hydrodynamic conditions should be similar. However, the effect of the hydrostatic pressure should be taken into account. The application of the method to other fluids also deserves close attention. In a previous paper, García-Salas et al. (2005) showed that the withdrawal method was not affected by liquid composition changes during yeast fermentation. However, in this work, we observed that the bubble size estimation required a separate calibration for coalescing and non-coalescing media. Therefore the validity of the non-isokinetic method depends at least partially on the fluid properties and composition. This should be confirmed by additional studies with a special emphasis on superficial tension and liquid viscosity in Newtonian and non-Newtonian liquids. 5. Conclusions The withdrawal method presented here is a simple, inexpensive and effective method for the determination of the specific surface area in aerated media. After proper calibration, the method allows the determination of the specific surface area at a specific location of the reactor in a couple of minutes. The standard error on the specific surface area determination is from 3% to 13% which is an error similar to those reported for electro-optical, photographic and chemical oxidation methods. Additionally to the method development, this paper has shown that a simple model describes adequately the specific surface area homogeneity. In all the experiments, two parameters allowed the description of the specific surface

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area homogeneity. According to the model, it was confirmed that the gas holdup is less homogeneous than the bubble diameter and is the main parameter that influences the specific surface area. The non-isokinetic withdrawal method is likely applicable to other free rising reactor designs, but its application to mechanically stirred reactors should be studied experimentally. Notation a a0 c1 c0 d d d3 d 32 dW3 D f (b) f (x) g(v) h Hd l L n R vg V V Vdead 0 Vdead VG Vtot

specific interfacial area, m−1 global specific interfacial area, m−1 slope of Eq. (8), dimensionless ordinate to origin of Eq. (8), m bubble diameter, m arithmetic average of bubble diameter, m photographic volume-mean bubble diameter, m Sauter-mean bubble diameter, m withdrawal volume-mean bubble diameter, m parameter (=[6VG /n]1/3 ), m frequency function of the experimental data dimensionless normal distribution function of x, dimensionless cumulative specific interfacial area function for volume v, dimensionless height from the bottom of the column, m gas–liquid dispersion height, m minor axis of the two-dimensional projection of bubbles, m greater axis of the two-dimensional projection of bubbles, m number of bubbles, dimensionless column radius, m superficial gas velocity, cm/s volume, m3 average volume of bubbles, m3 dead volume, m3 relative dead volume (=Vdead /Vtot ), dimensionless withdrawn gas volume, m3 total gas–liquid dispersion volume, m3

Greek letters     

parameter of gas holdup homogeneity model, dimensionless specific interfacial area exponent in Eq. (15), dimensionless gas holdup, dimensionless standard deviation, m arithmetic average of all intervals of class, m

Acknowledgments The authors acknowledge the financial support of the Instituto Mexicano del Petróleo (IMP) through a FIES project (FIES-98-107-VI) and CONACyT (México). The authors also thank Engs. J. Sánchez Labrada and V. Vital Martínez for their

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technical assistance and L. Dendooven and L.B. Flores-Cotera for their critical reading of the manuscript. S. García-Salas was holder of a grant from COFAA-I.P.N. and SUPERA and gratefully acknowledges the COTEPABE-I.P.N., México. References Adler, I., Schügerl, K., 1983. Cultivation of E. coli in single and ten stage tower loop reactors. Biotechnology and Bioengineering 25, 417–436. Akita, K., Yoshida, F., 1974. Bubble size, interfacial area, and liquid-phase mass transfer coefficient in bubble columns. Industrial and Engineering Chemistry, Process Design and Development 13 (1), 84–91. Alves, S.S., Maia, J.M., Vasconcelos, J.M.T., 2002. Experimental and modelling study of gas dispersion in a double turbine stirred tank. Chemical Engineering Science 57, 487–496. Bartsch, H.J., 1974. Handbook of Mathematical Formulas. Academic Press, New York. pp. 238–240. Bin, A., Duczmal, B., Machniewski, P., 2001. Hydrodynamics and ozone mass transfer in a tall bubble column. Chemical Engineering Science 56, 6233–6240. Boyd, J.W.R., Varley, J., 1998. Sound measurement as a means of gas-bubble sizing in aerated agitated tanks. A.I.Ch.E. Journal 44, 1731–1739. Buchholz, R., Shügerl, K., 1979. Bubble column bioreactors. I. Methods for measuring the bubble size. European Journal of Applied Microbiology and Biotechnology 6, 301–313. Buchholz, R., Adler, I., Shügerl, K., 1979. Investigations of the structure of two phase flow model media in bubble column bioreactors. I. Transverse variations of local properties when coalescence-promoting media are used. European Journal of Applied Microbiology and Biotechnology 6, 135–145. Buchholz, R., Zakrzewski, W., Schügerl, K., 1981. Techniques for determining the properties of bubbles in bubble columns. International Chemical Engineering 21, 180–187. Calderbank, P.H., 1958. Physical rate processes in industrial fermentation. Part I: the interfacial area in gas–liquid contacting with mechanical agitation. Transactions of the Institution of Chemical Engineers 36, 443–463. Cents, A.H.G., Brilman, D.W.F., Versteeg, G.F., 2001. Gas absorption in an agitated gas–liquid system. Chemical Engineering Science 56, 1075–1083. Deckwer, W.D., 1992. Bubble Column Reactors. Wiley, New York. pp. 157–254. García-Salas, S., Orozco-Álvarez, C., Porter, R.M., Thalasso, F., 2005. Measurement of local gas holdup in bubble columns via a non-isokinetic withdrawal method. Chemical Engineering Science 60, 6929–6938. Godbole, S.P., Schumpe, A., Shah, Y.T., Carr, N.L., 1984. Hydrodynamics and mass transfer in non-Newtonian solutions in a bubble column. A.I.Ch.E. Journal 30, 213–220. Greaves, M., Kobbacy, K., 1984. Measurement of bubble size distribution in turbulent gas–liquid dispersions. Chemical Engineering Research and Design 62, 3–12. Hofmeester, J.J.M., 1988. Gas holdup measurements in bioreactors. Trends in Biotechnology 6, 19–22.

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