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Simultaneous measurement of hold-up profiles and interfacial area using LDA in bubble columns: predictions by multiresolution analysis and comparison with experiments
Chemical Engineering Science 56 (2001) 6437–6445
www.elsevier.com/locate/ces
Simultaneous measurement of hold-up pro%les and interfacial area using LDA in bubble columns: predictions by multiresolution analysis and comparison with experiments Amol A. Kulkarnia , Jyeshtharaj B. Joshia; ∗ , V. Ravi Kumarb , Bhaskar D. Kulkarnib a Department
of Chemical Technology, University of Mumbai, Matunga, Mumbai 400 019, India Engineering Division, National Chemical Laboratory, Pune 411 008, India
b Chemical
Abstract A wavelet-based technique was used for the identi%cation of bubble events from the velocity-time data obtained using LDA in bubble column reactors. Bubble rise velocity distribution was also measured from the velocity-time series, which enabled the estimation of bubble size and shape distributions. From this information, the values of local fractional gas hold-up (G ) and e8ective interfacial area (a) were calculated. The estimated values of a have been compared with those obtained from the chemical method of air-oxidation of sodium sul%te. For this purpose, LDA measurements were carried out during the reaction of sul%te oxidation. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Bubble size distribution; Bubble column; Gas hold-up; Interfacial area; LDA; Wavelets
1. Introduction E8ective interfacial area (a) is an important design parameter for a variety of mass transfer operations in bubble column reactors. For its estimation, a number of empirical correlations have been reported in the published literature (Deckwer, 1993) on the basis of experimental measurements. Two categories of experimental techniques have been used: physical and chemical. In the physical methods, bubble size distribution and fractional gas hold-up are measured for the determination of e8ective interfacial area. These methods have been reviewed by Joshi, Patil, Ranade, and Shah (1990). Chemical methods have been described by Danckwerts (1970) and Doraiswamy and Sharma (1984). The gas– liquid reaction systems are selected in such a way that the absorption is accompanied by fast chemical reaction. As a result, the overall rate of absorption (Ra) is proportional to interfacial area and is given by the following
∗ Corresponding author. Tel.: +91-22-414-5616; fax: +91-22414-5614. E-mail address: [email protected] (J. B. Joshi).
equation: 1=2 2 n ∗ m+1 Ra = a DA kmn [A ] [B0 ] m+1
(1)
where, [A∗ ] is the saturation concentration of solute gas (A) in liquid. [B0 ] is the concentration of liquid phase reactive species (B), m and n are the orders with respect to A and B, respectively. kmn is the rate constant and DA is the di8usivity of A in liquid. A variety of chemical systems have been reported in the published literature, which include: CO2 -aqueous NaOH, CO2 -aqueous diethanolamine, H2 -hydrazine, O2 -sodium sul%te. In all the cases, the bracketed term ((2=m + 1)DA kmn [A∗ ]m+1 [B0 ]n ) ∼ (F) is measured using a model contactor (where a is known apriori), over a wide range of [A∗ ]; [B0 ] and temperature. These characterized systems (with known F) are then used in bubble columns,√where the measurement of Ra directly gives a (a = Ra= F). The values of a are expected to depend upon D, HD , sparger design, VG , the regime of operation and the nature of gas–liquid system. However when HD =D ¿ 5, the %rst three parameters are relatively unimportant. In the present work, the chemical method has been used for the measurement of a. In addition, a new
0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 2 2 6 - 3
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physical method has been developed using velocity-time data obtained from laser Doppler anemometry (LDA). In the second section, we have discussed the chemical method (i) for the measurement of F in a model stirred cell and (ii) for the measurement of a in bubble column. Section 3 describes the LDA setup and the analysis of velocity time data for the measurement of fractional gas hold-up and bubble rise velocity. Section 4 brings out the variation in the bubble size and shape distribution, rise velocity distribution. A comparison has been presented between the e8ective interfacial area (a) obtained from the chemical method and the LDA measurements. A novel methodology has been developed for the estimation of a. 2. Experimental: chemical method 2.1. Experiments with model contactor The details pertaining to chemical method have been given in Doraiswamy and Sharma (1984). Stirred cell with a Jat interface was used as a model contactor. The gas–liquid system consisted of air oxidation of aqueous solutions of sodium sul%te in the presence of CoSO4 as a catalyst. The rate constant (kmn ) is a function of catalyst l . Further, concentration and may be written as kmn = k Ccat in a stirred cell, mass transfer resistance may contribute to the overall rate of reaction (Ra). Under these conditions, the Danckwert’s formulation can be used for Ra and Eq. (1) takes the following form: 1=2 2 n ∗ ∗ m−1 l 2 DA k [A ] Ccat [B0 ] + kL : Ra = a[A ] m+1 (2) In a stirred cell, the mass transfer resistance can be eliminated by manipulating the impeller speed. Under these conditions, the second term inside the bracket becomes negligible. In order to %nd the values of l and n, the catalyst [Ccat ] and Na2 SO3 [B0 ] concentrations were varied in the range of 0.25 –1:2 × 10−3 M and 0.2–0:8 M, respectively. The decrease in Na2 SO3 concentration was followed using iodometry. The values of l and n were found to be 0.5 and 1.5, respectively. These are in agreement with those reported by Linek and Vacek (1981) and Linek, Benes and Sinkule (1990). Further, these authors reported the value of m to be zero and was accepted in this work. With this information, Eq. (2) takes the following form: 0:5 [B0 ]1:5 + kL2 }1=2 : Ra = a[A∗ ]{2DA k [A∗ ]−1 Ccat
(3) 1:5
0:5 [B0 ] gives the The Danckwert’s plot of Ra2 versus Ccat ∗ −1 value of 2DA k [A ] (henceforth called P) from slope and was found to be 0:00065 m2 =s2 . This value of P represents the physico-chemical properties of the system under consideration.
2.2. Measurements of e7ective interfacial area in bubble columns Two columns of 100 and 150 mm i.d. were used with a height to diameter ratio of six in both the columns. An oil free diaphragm type compressor was used to sparge air through a sieve plate sparger with the hole diameter of 0:8 mm and free area of 0.347%. A precalibrated rotameter was used to measure the volumetric gas Jow rate. The experiments were carried out at three super%cial gas velocities (VG ); 24; 29 and 39 mm=s, respectively. For each VG , the catalyst and Na2 SO3 concentrations were varied in a similar range as that of a stirred cell. The Danckwert’s plot was constructed at each VG; [B0 ] and D combination (typical plots are shown in Fig. 1, VG = 24; 29 mm=s for [B0 ] 0:2 M for di8erent [Ccat ] mentioned in Table 1). The knowledge of P from the stirred cell enabled the estimation of e8ective interfacial area. Again the rates were obtained using iodometry. In this case, in Eq. (1), only the values of a and kL were unknown. The slope of plot (Fig. 1) gave the values of a. For various experimental conditions it was observed that, a increased with VG . Further, for the same VG , a was found to increase with concentration of the electrolyte. 3. LDA and data processing 3.1. LDA and data acquisition The LDA setup comprised of a Dantec 55X modular series along with electronic instrumentation and a personal computer (80586). A 5 W Argon-ion laser and optics were mounted on a bench having a traversing mechanism. The traversing mechanism could focus the beams at 5 m increments by displacing the front lens (±300 mm) with the help of a linearly encoded stepping motor that could be operated either by computer or manually. To identify the Jow reversals correctly, a frequency shift was given to one of the beams by means of a Bragg cell with electronic downmixing. Data validation and signal processing were carried out with the help of a Dantec 57N21 Burst Spectrum Analyzer (BSA). The personal computer functioned also as the central data acquisition and preprocessing unit. Since the forward scatter optical arrangements provide 102 –103 times better signal to noise ratio as compared to the back-scatter mode, all the measurements were made in the forward scatter mode. Additional details pertaining to the LDA measurements in two-phase Jows are given in Deshpande, Prasad, Kulkarni, and Joshi (2000) and Kulkarni, Joshi, Ravi Kumar, and Kulkarni (2001). LDA measurements of the instantaneous axial velocity were carried out in both the columns at 5 di8erent axial levels and 11 radial locations (Table 1). Each cylindrical bubble column was enclosed in an another column of
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Fig. 1. Ra2 vs: P[cat]0:5 [B0 ]1:5 for bubble column. ( ; VG = 24 mm=s; ♦; VG = 29 mm=s).
Table 1 Experimental conditions Column speci%cations Catalyst concentrations Sparger Axial levels Radial locations (mm from center) Super%cial gas velocities
D = 100; 150 mm; H = 800 and 1020 mm, respectively 2.5e-4, 5.0e-4, 7.5e-4, 8.0e-4, 9.0e-4 and 1.2e-3 M CoSO4 Sieve plate (0:8 mm home diameter, triangular pitch), 0.347% free area 100 mm i.d.: 200, 300, 400, 500 and 600 mm 150 mm i.d.: 300, 405, 510, 615 and 720 away from the sparger 100 mm i.d.: −45; −40; −30; −20; −10; 0; 10; 20; 30; 40; 45 150 mm i.d.: −70; −60; −45; −30; −15; 0; 15; 30; 45; 60; 70 24, 29 and 39 mm=s
Liquids (International Critical Tables, 1929)
Water
Density (kg=m3 ) Viscosity (mPa s) Surface tension (mN=m)
998 0.8 72
Aqueous Na2 SO3 solution kmol=m3 (M) 0.2 1018 1.04 54
square cross section and the space between the two was %lled with water to minimize the refraction e8ects. At every measurement location, the data (10 000 data points) was collected at a sampling frequency of 200 ± 5 Hz in each run. 3.2. Data processing When the velocity-time data from a two-phase Jow is compared with a single-phase Jow, the former is characterized by time gaps during the acquisition of velocity
0.4 1046 1.186 69
0.8 1080 1.374 73
information as against practically uniform data arrival in the latter case. The time gap in a two-phase system is basically because of the interception of laser beams by the bubbles. As mentioned earlier, the bubbles may intercept the beam at the point of or prior to the beam intersection (measurement volume) and it was thought desirable to clarify the time gaps accordingly. When a bubble passes through the measurement volume (the bubble is also accompanied by its wake) a discontinuity in the velocity is observed corresponding to bubble rise velocity. If, the time gap is not followed by any burst, it represents that the measurement volume was in liquid for the time when
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a bubble had interrupted beam path. This interruption will not result into any measurement and hence no question of any burst. Thus, we can consider this hypothetical bubble gap as the time for which the measurement volume is in liquid. From the foregoing discussion, it is clear that the measurement volume occupied by the bubble is represented by a time gap accompanied by a burst in local energy. At all other times, the measurement volume is in the liquid phase. Therefore, the local fractional gas hold-up was estimated as the ratio of time spent by the measurement volume in gas to the total measurement time phase (discussed later in detail). This result was further con%rmed at four data rates of 148; 185; 359 and 442 Hz and the di8erence between maximum and minimum hold-up values was less than 0.04%. At this stage, we would like to point out that, for all the measurements reported in this paper, the data rate was maintained constant from the center to wall by manipulating the voltage across PM tube. A care was also taken that this variation in voltage will not a8ect the % validation to a great extent. There is a possibility that, when a bubble crosses the incident beam before it forms the measurement volume, another bubble may pass through position where the measurement volume would have formed and it may be missed. However, this happens only when both the events overlap completely. If, there is a small time difference, then the bubbles at the measurement volume will get detected. Bubbles will not get detected if there is a wide bubble size distribution and a small bubble passes through the measurement volume while a large bubble passes through the incident beam path. However, in the present case, we did not have a wide bubble size distribution. It is also likely that all the bubbles may not move perfectly in the vertical direction. The real path may be at an angle. However, it must be emphasized that the vertical component of the bubble velocity consists of two parts. One is the slip velocity (as a result of drag, gravity and buoyancy forces) and the other is the vertical component of the local liquid velocity. Further, the radial component of bubble velocity usually equals the radial component of local liquid velocity. Therefore, the LDA measurement of axial velocity captures the bubble rise velocity irrespective of its angle of movement. A typical velocity-time data x(i) in bubble column is shown in Fig. 2. It has been shown by many investigators that, the wavelet transform (WT) can be successfully applied for the characterization of a multiphase system (Bakshi, Zhong, Jiang, & Fan, 1995) and thus does not need a detailed explanation. In WT, the window size analysis may be carried out for every single spectral component without any information loss. In mathematical terms WT is de%ned as (see, Daubechies, 1992) 1 i−b ; (4) a; b (i) = a |a|
where a; b (i) is the chosen wavelet basis with two parameters, namely, the scaling parameter a and the translation parameter b. The scaling parameter a, compresses or expands the basis without changing the number of oscillations while, b shifts the wavelet to a di8erent local region for analysis. In this study, we have used orthogonal Daubachie’s wavelets (Db4) as the wavelet basis function. The N number of data points x(i); i = 1; 2; : : : ; N; sampled in time intervals t when subjected to a discrete wavelet transform (DWT) (Farge, 1992) yields wavelet coeRcients Wj; k at dyadic scales in j and displacements k, de%ned as 1 j=2 x(i) j; k (i); (2j i − k) Wj; k = √ j; k (i) = 2 N (5) for j = 0; : : : ; p; k = 1; : : : ; 2p , where = log N=log 2 corresponding to the a dyadic scale. A low value of j implies a %ne scale analysis while high j brings out the data features at a coarse scale. The velocity-time data was subjected to the DWT and an algorithm by Roy et al. (1999) was applied for the denoising procedure. Further, the denoised data was used for its further scale-wise decomposition where, the events happening at various frequencies would get captured at di8erent scales. In turbulent two-phase Jows like in a bubble column, the passage of the bubbles at a particular measurement location is a discontinuous process and thus happens intermittently. The signi%cantly large gradients in velocity-time data can be used to locate these intermittent events. For the case of a bubble column, it may be visualized that the passage of bubbles through the measurement volume results in bursts of energy. It is possible to identify these abnormalities and the associated time information from the properties of the scalewise wavelet transformed reconstruction [xj (i)], by evaluating a quantity termed as the local intermittency measure (LIM). For bubble columns, the interest is to identify the sharp gradients in velocity, which arise due to passage of bubbles by calculating the LIMj (i) over all wavelet scales (j = 0; 2; : : : ; p − 1) given by: LIMj (i) =
|xˆj (i)|2 ; |xˆj (i)|2
i = 1; 2; 3; : : : ; N
(6)
representing the deviation in local energy distribution for a scale j and time index i from the average (¡ ¿) in time for the particular scale j. Thus, a graphical representation of LIMj (i) is a good indicator of the local intermittency in time (Farge, 1992) and thus the local energy changes that occur in each scale. In the present study, the analysis of the LIMj (i) was used e8ectively to detect bubble passage events from the time variant velocity measurements in a bubble column. It was observed that, although the LIM for %ner scales (j = 1; : : : ; 4) showed higher values of local intermittency with dense distributions, these scales need not be considered for further analysis because
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Fig. 2. Typical velocity-time data obtained from bubble column.
they contained a negligible fraction of the total energy content (Pˆ j ) in the entire time-series. The bursts in energy were detected by applying a threshold to the LIM data. The threshold value was chosen based on the root mean square (RMS) value of local intermittency measure (LIM) data for each scale. It was varied from 40% of RMS to 120% RMS and a sensitivity analysis were carried out. It was found that, for low values of threshold, the number of dominant peaks (indicating the bubbles) detected was very high while for higher values of threshold, the number of bubbles detected were too small. The selection of proper threshold was done with reference to the predicted values of average fractional gas hold-up, which is discussed in the next section. It was observed that, these peaks in LIM can be revealed for the plots of the LIM within j = 6; : : : ; 11 and thus, a proper choice of threshold value by an elegant procedure helped to identify the bubble events. 3.3. Bubble dynamics and the prediction of interfacial area It is well known that, the bubble size, size distribution and the shape distribution are the important parameters for the understanding of bubble dynamics. In view of this, it was thought desirable to estimate the bubble parameters using LDA data and further apply it for the prediction of interfacial area through a stepwise procedure. (a) It is known that, while ascending, a bubble carries a wake having the same velocity as that of bubble. Thus, the velocity of wake (identi%ed from the LIM analysis) was considered as the bubble slip velocity. For all the bubbles, the variation in the rise velocity was found to be within 0.15 –0:60 m=s. At this moment, it can be noted that, bubbles smaller than 1:5 mm could not be detected using the time series analysis of LDA velocity-time data. (b) From the information on the rise velocity, equivalent bubble diameter was estimated using the following
correlations. It was assumed that the bubble motion in the system occurs at its terminal rise velocity and this velocity is attained in very small time even if it may or may not have experienced a break-up or coalescence before passing through the measurement point. (i) Karamanev (1994) has used a Tadaki number (Ta) for taking into account the contribution of bubble shapes in correlating the bubble size and rise velocity. ub (i)1−b 'b C(i)0:5 (7) db (i) = 0:862 0:5 b 0:23b g ) Mo where, C(i) = 24(1 + 0:173Re(i)0:65 )Re(i) + 0:143=(1 + 16300Re(i)−1:09 );
(8)
b is a constant the values of which have been reported by Karamanev (1994) as a function of Ta, and Mo is Morton number. (ii) Nguyen (1998) has proposed the following correlation in terms of dimensionless numbers: ub (i) db (i) = : 2−2b 2b +1 2−2b g'L 0:33 4a2 Mo0:46b )2L g 6−6b )L 1 '2 2:85 2 − 2b (9) Nguyen (1998) also has reported the values of a and b as a function of Ta. The physical properties of the liquid phase (reported in Table 1) and the measured bubble rise velocity were used for the estimation of the equivalent bubble diameter (from Eqs. (7) and (9)) by an iterative procedure, where values of a ; b and C(i) needs to be satis%ed to be in a certain range. Fig. 3 shows that the bubble number variation with equivalent bubble diameters obtained from Eqs. (7) and (9) and are fairly in close agreement. Since the Nguyen has reported the correlation developed speci%cally for contaminated liquids and since the present system can be considered to be contaminated liquid, Eq. (9) has been used for further calculations in
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Fig. 3. Bubble size distribution from the correlation by (♦) Karamanev (1994) and (O) Nguyen (1998).
this work. In order to obtain reliable and reproducible information, the e8ect of number of data points on the results was analyzed for 4096; 8192; 16384 and 32768 data points. Reproducibility was obtained at 8192 data points. (c) Since bubble shape also has an e8ect on the e8ecT was estitive interfacial area, the mean aspect ratio (E) mated using the Eotvos number (Eo ): 1 : (10) ET = 1 + 0:163Eo0:757 It was observed that, for 90% of the identi%ed bubbles, the aspect ratio was found to be greater than 1. (d) As explained earlier, if the LIM values for the energetic scales (scales containing noticeable part of the total energy in signal), were above a certain threshold value, the events were identi%ed as passage of gas bubbles. The arrival time gap associated with this event (when no data was obtained) indicates the bubble passage through the measurement volume. The sum of all such time gaps is the total time for which the measurement volume is occupied by the gas phase. Thus, for a suRciently longer data set, the ratio of total time occupied by the gas phase to that of the total length of the time series gave the local fractional gas hold-up (G ). This procedure was carried out for a typical threshold value and the estimated gas hold-up at eleven radial locations was used for getting the volume average gas hold-up
(TG ) for the entire column. In a similar way, the volume average hold-ups were estimated at various levels of threshold. The appropriate threshold value was selected in such a way that the estimated hold-up agreed with the experimentally measured average hold-up (calculated from the bed expansion). Fig. 4 shows hold-up (G ) pro%les at %ve di8erent axial locations (at 200; 300; 400; 500 and 600 mm away from the sparger) for three di8erent gas velocities, 24; 29 and 39 mm=s; respectively, in the 100 mm column containing aqueous Na2 SO3 solutions. (e) From the knowledge of the bubble shape, the actual interfacial area (A) and the bubble volume (v) were estimated for all the identi%ed bubbles. The overall effective interfacial area was estimated using the following equation: N N b b A(i) v(i) ; (11) a(r; z) = G (r; z) i=1
i=1
where, a(r; z) is the local interfacial area at co-ordinate position (r; z) and Nb is the total number of bubbles encountered during the measurements at (r; z): A(i) and v(i) are the surface area and volume of a ith bubble. (f) The area average a(z) was estimated (at every axial location) using the following equation: R 1 20ra(r; z) dr: (12) a(z) = 2 0R −R
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Fig. 5. Bubble shape map for di8erent experimental conditions. Re versus Eo .
(g) As mentioned earlier, the LDA measurements were made at %ve axial locations, and the corresponding values of a(z) were estimated. The axial variation was found to be within 4.06%. Therefore, the arithmetic average of %ve values of a(z) was taken for estimating the overall volume averaged value of a. 4. Results and discussions
Fig. 4. Radial variation of fractional gas hold-up (G ) , VG = 24 mm=s; U; VG = 29 mm=s; ; VG = 39 mm=s at %ve axial levels. (L1 = 200 mm, L2 = 300 mm, L3 = 400 mm, L4 = 500 mm and L5 = 600 mm away from the sparger).
The hold-up pro%les from Fig. 4 showed that for the 0:2 M aqueous solution of sodium sul%te, the hold-up is more in the center and less near wall region indicating an upward liquid Jow in the central region and downward in the wall region. This observation remained consistent for all the three gas velocities at all %ve axial positions mentioned in Table 1. Clift, Grace, and Weber (1978) have reported the region map for bubble shapes in the form of a plot of Re versus Eo with Mo as parameter. The results of the present work have also been shown in Fig. 5. It can be seen that, the size and shape distribution obtained in this work follows the trend reported by Clift et al. (1978). The parity plot for a predicted from the velocity-time data and those obtained by the chemical method is shown in Fig. 6. Excellent agreement can be seen over a range of VG , D and electrolyte concentrations covered in this work. This analysis can perhaps be applied for the industrial columns where the time series of a characteris-
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[BO ] Ccat C db (i) D DA Eo ET g HD i j kmn
Fig. 6. Parity plot for the e8ective interfacial area in both the bubble columns. 100 mm; •; 150 mm; ◦. (Each point corresponds to one [B0 ] and one value of VG .)
k LIM m
tic parameter of the system (pressure, conductivity, etc.) can be obtained and a similar methodology can be developed and used as a diagnostic tool during the operation of bubble columns.
Mo Nb N p Pˆ j
5. Conclusion
r r=R
Analysis of a LDA velocity-time series from bubble columns has been carried out using wavelet transforms. A method has been developed for the measurement of bubble size and shape distribution, which favorably agrees with the bubble-shape map (Clift et al., 1978). The velocity data were also used for the estimation of local gas hold-up (G ) and the local e8ective interfacial area (a). The resulting hold-up pro%les obtained from the multiresolution analysis method showed consistency in trend for the range of VG in these studies at %ve axial positions. Further, an excellent agreement has also been shown between a estimated by time series analysis and those a obtained by the chemical method. The agreement holds irrespective of the regime of operation, which is a signi%cant result. Notation a and b a and b a (r; z) a A [A∗ ]
scaling and translation parameter constants in Eqs. (7) and (9) e8ective interfacial area of bubbles at co-ordinate (r; z); 1=m overall e8ective interfacial area, 1=m actual interfacial area of a gas bubble, m2 concentration of gas phase at the interface, kmol=m3
R Re Ta ub v VG W x z
concentration of liquid reactant, kmol=m3 catalyst concentration, kmol=m3 drag coeRcient of the bubble, dimensionless equivalent bubble diameter, mm column diameter, mm di8usivity of gas in liquid, m=s g()L − )G )d2b ; dimensionEotvos number, Eo = 2 less mean aspect ratio, dimensionless acceleration due to gravity, m=s2 total height of dispersion, mm bubble number and the time index for instantaneous velocity wavelet scales second order rate constant, kmn = k [Ccat ]; m3 =s=kmol translation parameter, dimensionless local intermittency measure the %rst energetic scale available for LIM analysis Morton number Mo = g'4 =)23 , dimensionless number of bubbles identi%ed at (r; z) number of data points largest resolvable wavelet scale total content in the denoised time-series, N energy 2 2 2 x(i) ˆ ; m =s 1 radial co-ordinate, mm normalized radial distance from centre, dimensionless speci%c rate of absorption, kmol=m2 s d b ub ) L Reynolds number, Re = ; dimensionless 'L 0:23 Tadaki number, Ta = ReMo , dimensionless rise velocity of bubble, m=s volume of individual bubble identi%ed at, r; z; m3 super%cial gas velocity, m=s wavelet coeRcient time series data, m=s axial co-ordinate, mm
Greek letters TG G ' ) 2
average fractional gas gold-up, dimensionless local fractional gas hold-up, dimensionless viscosity of liquid, kg=m=s liquid density, kg=m3 surface tension of liquid, kg=s2 wavelet function
Subscripts b G
bubble gas phase
A. A. Kulkarni et al. / Chemical Engineering Science 56 (2001) 6437–6445
j k L r t
wavelet scale translation parameter in Eq. (2) liquid phase refers to reconstructed velocity-time data time, s
Superscripts ˆ l; m&n
refers to the denoised sets orders with respect to catalyst, gas and liquid concentrations, respectively
Acknowledgements The %nancial assistance of the “Professor M. M. Sharma Endowment” (AAK) and Department of Science and Technology, New Delhi, India, (VRK and BDK) under the program [III.5(19)=98-ET] in carrying out this work is gratefully acknowledged. References Bakshi, B. R., Zhong, H., Jiang, P., & Fan, L. S. (1995). Analysis of Jow in gas–liquid bubble columns using multiresolution methods. Transactions of Institution of Chemical Engineers, 73(A), 608. Clift, R., Grace, J. R., & Weber, M. E. (1978). Bubbles, drops, and particles. New York: Academic Press. Danckwerts, P. V. (1970). Gas–liquid reactions. New York: McGraw Hill. Daubechies, I. (1992). Ten lectures on wavelets. Society for Industrial and Applied Mathematics.
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Deckwer, W. D. (1993). Bubble column reactors. New York: Wiley. Deshpande, N. S., Prasad, Ch. V., Kulkarni, A. A., & Joshi, J. B. (2000). Hydrodynamic characterization of dispersed two-phase Jows by laser doppler velocimeter. Chemical Engineering Research & Design, 78(A), 903. Doraiswamy, L. K., & Sharma, M. M. (1984). Heterogeneous reactions. Vol. II. New York: Wiley. Farge, M. (1992). Wavelet transforms and their applications to turbulence. Annual Reviews in Fluid Mechanics, 24, 395. International Critical Tables, 1929. Vol. 5, New York: McGraw-Hill Book Company. Joshi, J. B., Patil, T. A., Ranade, V. V., & Shah, Y. T. (1990). Measurement of hydrodynamic parameters in multiphase sparged reactors. Reviews in Chemical Engineering, 6, 74–227. Karamanev, D. G. (1994). Rise of gas bubbles in quiescent liquids. American Institute of Chemical Engineers Journal, 40, 1418. Kulkarni, A. A., Joshi, J. B., V Ravi Kumar, & Kulkarni, B. D. (2001). Application of multiresolution analysis for simultaneous measurement of gas and liquid velocities and fractional gas hold-up in bubble column using LDA. Chemical Engineering Science, in press. Linek, V., & Vacek, V. (1981). Chemical engineering use of catalyzed sul%te oxidation kinetics for the determination of mass transfer characteristics of gas–liquid contactrors. Chemical Engineering Science, 36(11), 1747–1768. Linek, V., Benes, P., & Sinkule, J. (1990). Critical assessment of the steady-state Na2 SO3 feeding method for kL a measurement in fermentors. Biotechnology and Bioengineering, 35, 766–770. Nguyen, A. V. (1998). Prediction of bubble terminal velocities in contaminated water. American Institute of Chemical Engineers Journal, 44(1), 226–230. Roy, M., Ravi Kumar, V., Kulkarni, B. D., Sanderson, J., Rhodes, M., & Van der Stappen, M. (1999). Simple denoising algorithm using wavelet transform. American Institute of Chemical Engineers Journal, 45(11), 2461.
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Simultaneous measurement of hold-up pro%les and interfacial area using LDA in bubble columns: predictions by multiresolution analysis and comparison with experiments Amol A. Kulkarnia , Jyeshtharaj B. Joshia; ∗ , V. Ravi Kumarb , Bhaskar D. Kulkarnib a Department
of Chemical Technology, University of Mumbai, Matunga, Mumbai 400 019, India Engineering Division, National Chemical Laboratory, Pune 411 008, India
b Chemical
Abstract A wavelet-based technique was used for the identi%cation of bubble events from the velocity-time data obtained using LDA in bubble column reactors. Bubble rise velocity distribution was also measured from the velocity-time series, which enabled the estimation of bubble size and shape distributions. From this information, the values of local fractional gas hold-up (G ) and e8ective interfacial area (a) were calculated. The estimated values of a have been compared with those obtained from the chemical method of air-oxidation of sodium sul%te. For this purpose, LDA measurements were carried out during the reaction of sul%te oxidation. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Bubble size distribution; Bubble column; Gas hold-up; Interfacial area; LDA; Wavelets
1. Introduction E8ective interfacial area (a) is an important design parameter for a variety of mass transfer operations in bubble column reactors. For its estimation, a number of empirical correlations have been reported in the published literature (Deckwer, 1993) on the basis of experimental measurements. Two categories of experimental techniques have been used: physical and chemical. In the physical methods, bubble size distribution and fractional gas hold-up are measured for the determination of e8ective interfacial area. These methods have been reviewed by Joshi, Patil, Ranade, and Shah (1990). Chemical methods have been described by Danckwerts (1970) and Doraiswamy and Sharma (1984). The gas– liquid reaction systems are selected in such a way that the absorption is accompanied by fast chemical reaction. As a result, the overall rate of absorption (Ra) is proportional to interfacial area and is given by the following
∗ Corresponding author. Tel.: +91-22-414-5616; fax: +91-22414-5614. E-mail address: [email protected] (J. B. Joshi).
equation: 1=2 2 n ∗ m+1 Ra = a DA kmn [A ] [B0 ] m+1
(1)
where, [A∗ ] is the saturation concentration of solute gas (A) in liquid. [B0 ] is the concentration of liquid phase reactive species (B), m and n are the orders with respect to A and B, respectively. kmn is the rate constant and DA is the di8usivity of A in liquid. A variety of chemical systems have been reported in the published literature, which include: CO2 -aqueous NaOH, CO2 -aqueous diethanolamine, H2 -hydrazine, O2 -sodium sul%te. In all the cases, the bracketed term ((2=m + 1)DA kmn [A∗ ]m+1 [B0 ]n ) ∼ (F) is measured using a model contactor (where a is known apriori), over a wide range of [A∗ ]; [B0 ] and temperature. These characterized systems (with known F) are then used in bubble columns,√where the measurement of Ra directly gives a (a = Ra= F). The values of a are expected to depend upon D, HD , sparger design, VG , the regime of operation and the nature of gas–liquid system. However when HD =D ¿ 5, the %rst three parameters are relatively unimportant. In the present work, the chemical method has been used for the measurement of a. In addition, a new
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physical method has been developed using velocity-time data obtained from laser Doppler anemometry (LDA). In the second section, we have discussed the chemical method (i) for the measurement of F in a model stirred cell and (ii) for the measurement of a in bubble column. Section 3 describes the LDA setup and the analysis of velocity time data for the measurement of fractional gas hold-up and bubble rise velocity. Section 4 brings out the variation in the bubble size and shape distribution, rise velocity distribution. A comparison has been presented between the e8ective interfacial area (a) obtained from the chemical method and the LDA measurements. A novel methodology has been developed for the estimation of a. 2. Experimental: chemical method 2.1. Experiments with model contactor The details pertaining to chemical method have been given in Doraiswamy and Sharma (1984). Stirred cell with a Jat interface was used as a model contactor. The gas–liquid system consisted of air oxidation of aqueous solutions of sodium sul%te in the presence of CoSO4 as a catalyst. The rate constant (kmn ) is a function of catalyst l . Further, concentration and may be written as kmn = k Ccat in a stirred cell, mass transfer resistance may contribute to the overall rate of reaction (Ra). Under these conditions, the Danckwert’s formulation can be used for Ra and Eq. (1) takes the following form: 1=2 2 n ∗ ∗ m−1 l 2 DA k [A ] Ccat [B0 ] + kL : Ra = a[A ] m+1 (2) In a stirred cell, the mass transfer resistance can be eliminated by manipulating the impeller speed. Under these conditions, the second term inside the bracket becomes negligible. In order to %nd the values of l and n, the catalyst [Ccat ] and Na2 SO3 [B0 ] concentrations were varied in the range of 0.25 –1:2 × 10−3 M and 0.2–0:8 M, respectively. The decrease in Na2 SO3 concentration was followed using iodometry. The values of l and n were found to be 0.5 and 1.5, respectively. These are in agreement with those reported by Linek and Vacek (1981) and Linek, Benes and Sinkule (1990). Further, these authors reported the value of m to be zero and was accepted in this work. With this information, Eq. (2) takes the following form: 0:5 [B0 ]1:5 + kL2 }1=2 : Ra = a[A∗ ]{2DA k [A∗ ]−1 Ccat
(3) 1:5
0:5 [B0 ] gives the The Danckwert’s plot of Ra2 versus Ccat ∗ −1 value of 2DA k [A ] (henceforth called P) from slope and was found to be 0:00065 m2 =s2 . This value of P represents the physico-chemical properties of the system under consideration.
2.2. Measurements of e7ective interfacial area in bubble columns Two columns of 100 and 150 mm i.d. were used with a height to diameter ratio of six in both the columns. An oil free diaphragm type compressor was used to sparge air through a sieve plate sparger with the hole diameter of 0:8 mm and free area of 0.347%. A precalibrated rotameter was used to measure the volumetric gas Jow rate. The experiments were carried out at three super%cial gas velocities (VG ); 24; 29 and 39 mm=s, respectively. For each VG , the catalyst and Na2 SO3 concentrations were varied in a similar range as that of a stirred cell. The Danckwert’s plot was constructed at each VG; [B0 ] and D combination (typical plots are shown in Fig. 1, VG = 24; 29 mm=s for [B0 ] 0:2 M for di8erent [Ccat ] mentioned in Table 1). The knowledge of P from the stirred cell enabled the estimation of e8ective interfacial area. Again the rates were obtained using iodometry. In this case, in Eq. (1), only the values of a and kL were unknown. The slope of plot (Fig. 1) gave the values of a. For various experimental conditions it was observed that, a increased with VG . Further, for the same VG , a was found to increase with concentration of the electrolyte. 3. LDA and data processing 3.1. LDA and data acquisition The LDA setup comprised of a Dantec 55X modular series along with electronic instrumentation and a personal computer (80586). A 5 W Argon-ion laser and optics were mounted on a bench having a traversing mechanism. The traversing mechanism could focus the beams at 5 m increments by displacing the front lens (±300 mm) with the help of a linearly encoded stepping motor that could be operated either by computer or manually. To identify the Jow reversals correctly, a frequency shift was given to one of the beams by means of a Bragg cell with electronic downmixing. Data validation and signal processing were carried out with the help of a Dantec 57N21 Burst Spectrum Analyzer (BSA). The personal computer functioned also as the central data acquisition and preprocessing unit. Since the forward scatter optical arrangements provide 102 –103 times better signal to noise ratio as compared to the back-scatter mode, all the measurements were made in the forward scatter mode. Additional details pertaining to the LDA measurements in two-phase Jows are given in Deshpande, Prasad, Kulkarni, and Joshi (2000) and Kulkarni, Joshi, Ravi Kumar, and Kulkarni (2001). LDA measurements of the instantaneous axial velocity were carried out in both the columns at 5 di8erent axial levels and 11 radial locations (Table 1). Each cylindrical bubble column was enclosed in an another column of
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Fig. 1. Ra2 vs: P[cat]0:5 [B0 ]1:5 for bubble column. ( ; VG = 24 mm=s; ♦; VG = 29 mm=s).
Table 1 Experimental conditions Column speci%cations Catalyst concentrations Sparger Axial levels Radial locations (mm from center) Super%cial gas velocities
D = 100; 150 mm; H = 800 and 1020 mm, respectively 2.5e-4, 5.0e-4, 7.5e-4, 8.0e-4, 9.0e-4 and 1.2e-3 M CoSO4 Sieve plate (0:8 mm home diameter, triangular pitch), 0.347% free area 100 mm i.d.: 200, 300, 400, 500 and 600 mm 150 mm i.d.: 300, 405, 510, 615 and 720 away from the sparger 100 mm i.d.: −45; −40; −30; −20; −10; 0; 10; 20; 30; 40; 45 150 mm i.d.: −70; −60; −45; −30; −15; 0; 15; 30; 45; 60; 70 24, 29 and 39 mm=s
Liquids (International Critical Tables, 1929)
Water
Density (kg=m3 ) Viscosity (mPa s) Surface tension (mN=m)
998 0.8 72
Aqueous Na2 SO3 solution kmol=m3 (M) 0.2 1018 1.04 54
square cross section and the space between the two was %lled with water to minimize the refraction e8ects. At every measurement location, the data (10 000 data points) was collected at a sampling frequency of 200 ± 5 Hz in each run. 3.2. Data processing When the velocity-time data from a two-phase Jow is compared with a single-phase Jow, the former is characterized by time gaps during the acquisition of velocity
0.4 1046 1.186 69
0.8 1080 1.374 73
information as against practically uniform data arrival in the latter case. The time gap in a two-phase system is basically because of the interception of laser beams by the bubbles. As mentioned earlier, the bubbles may intercept the beam at the point of or prior to the beam intersection (measurement volume) and it was thought desirable to clarify the time gaps accordingly. When a bubble passes through the measurement volume (the bubble is also accompanied by its wake) a discontinuity in the velocity is observed corresponding to bubble rise velocity. If, the time gap is not followed by any burst, it represents that the measurement volume was in liquid for the time when
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a bubble had interrupted beam path. This interruption will not result into any measurement and hence no question of any burst. Thus, we can consider this hypothetical bubble gap as the time for which the measurement volume is in liquid. From the foregoing discussion, it is clear that the measurement volume occupied by the bubble is represented by a time gap accompanied by a burst in local energy. At all other times, the measurement volume is in the liquid phase. Therefore, the local fractional gas hold-up was estimated as the ratio of time spent by the measurement volume in gas to the total measurement time phase (discussed later in detail). This result was further con%rmed at four data rates of 148; 185; 359 and 442 Hz and the di8erence between maximum and minimum hold-up values was less than 0.04%. At this stage, we would like to point out that, for all the measurements reported in this paper, the data rate was maintained constant from the center to wall by manipulating the voltage across PM tube. A care was also taken that this variation in voltage will not a8ect the % validation to a great extent. There is a possibility that, when a bubble crosses the incident beam before it forms the measurement volume, another bubble may pass through position where the measurement volume would have formed and it may be missed. However, this happens only when both the events overlap completely. If, there is a small time difference, then the bubbles at the measurement volume will get detected. Bubbles will not get detected if there is a wide bubble size distribution and a small bubble passes through the measurement volume while a large bubble passes through the incident beam path. However, in the present case, we did not have a wide bubble size distribution. It is also likely that all the bubbles may not move perfectly in the vertical direction. The real path may be at an angle. However, it must be emphasized that the vertical component of the bubble velocity consists of two parts. One is the slip velocity (as a result of drag, gravity and buoyancy forces) and the other is the vertical component of the local liquid velocity. Further, the radial component of bubble velocity usually equals the radial component of local liquid velocity. Therefore, the LDA measurement of axial velocity captures the bubble rise velocity irrespective of its angle of movement. A typical velocity-time data x(i) in bubble column is shown in Fig. 2. It has been shown by many investigators that, the wavelet transform (WT) can be successfully applied for the characterization of a multiphase system (Bakshi, Zhong, Jiang, & Fan, 1995) and thus does not need a detailed explanation. In WT, the window size analysis may be carried out for every single spectral component without any information loss. In mathematical terms WT is de%ned as (see, Daubechies, 1992) 1 i−b ; (4) a; b (i) = a |a|
where a; b (i) is the chosen wavelet basis with two parameters, namely, the scaling parameter a and the translation parameter b. The scaling parameter a, compresses or expands the basis without changing the number of oscillations while, b shifts the wavelet to a di8erent local region for analysis. In this study, we have used orthogonal Daubachie’s wavelets (Db4) as the wavelet basis function. The N number of data points x(i); i = 1; 2; : : : ; N; sampled in time intervals t when subjected to a discrete wavelet transform (DWT) (Farge, 1992) yields wavelet coeRcients Wj; k at dyadic scales in j and displacements k, de%ned as 1 j=2 x(i) j; k (i); (2j i − k) Wj; k = √ j; k (i) = 2 N (5) for j = 0; : : : ; p; k = 1; : : : ; 2p , where = log N=log 2 corresponding to the a dyadic scale. A low value of j implies a %ne scale analysis while high j brings out the data features at a coarse scale. The velocity-time data was subjected to the DWT and an algorithm by Roy et al. (1999) was applied for the denoising procedure. Further, the denoised data was used for its further scale-wise decomposition where, the events happening at various frequencies would get captured at di8erent scales. In turbulent two-phase Jows like in a bubble column, the passage of the bubbles at a particular measurement location is a discontinuous process and thus happens intermittently. The signi%cantly large gradients in velocity-time data can be used to locate these intermittent events. For the case of a bubble column, it may be visualized that the passage of bubbles through the measurement volume results in bursts of energy. It is possible to identify these abnormalities and the associated time information from the properties of the scalewise wavelet transformed reconstruction [xj (i)], by evaluating a quantity termed as the local intermittency measure (LIM). For bubble columns, the interest is to identify the sharp gradients in velocity, which arise due to passage of bubbles by calculating the LIMj (i) over all wavelet scales (j = 0; 2; : : : ; p − 1) given by: LIMj (i) =
|xˆj (i)|2 ; |xˆj (i)|2
i = 1; 2; 3; : : : ; N
(6)
representing the deviation in local energy distribution for a scale j and time index i from the average (¡ ¿) in time for the particular scale j. Thus, a graphical representation of LIMj (i) is a good indicator of the local intermittency in time (Farge, 1992) and thus the local energy changes that occur in each scale. In the present study, the analysis of the LIMj (i) was used e8ectively to detect bubble passage events from the time variant velocity measurements in a bubble column. It was observed that, although the LIM for %ner scales (j = 1; : : : ; 4) showed higher values of local intermittency with dense distributions, these scales need not be considered for further analysis because
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Fig. 2. Typical velocity-time data obtained from bubble column.
they contained a negligible fraction of the total energy content (Pˆ j ) in the entire time-series. The bursts in energy were detected by applying a threshold to the LIM data. The threshold value was chosen based on the root mean square (RMS) value of local intermittency measure (LIM) data for each scale. It was varied from 40% of RMS to 120% RMS and a sensitivity analysis were carried out. It was found that, for low values of threshold, the number of dominant peaks (indicating the bubbles) detected was very high while for higher values of threshold, the number of bubbles detected were too small. The selection of proper threshold was done with reference to the predicted values of average fractional gas hold-up, which is discussed in the next section. It was observed that, these peaks in LIM can be revealed for the plots of the LIM within j = 6; : : : ; 11 and thus, a proper choice of threshold value by an elegant procedure helped to identify the bubble events. 3.3. Bubble dynamics and the prediction of interfacial area It is well known that, the bubble size, size distribution and the shape distribution are the important parameters for the understanding of bubble dynamics. In view of this, it was thought desirable to estimate the bubble parameters using LDA data and further apply it for the prediction of interfacial area through a stepwise procedure. (a) It is known that, while ascending, a bubble carries a wake having the same velocity as that of bubble. Thus, the velocity of wake (identi%ed from the LIM analysis) was considered as the bubble slip velocity. For all the bubbles, the variation in the rise velocity was found to be within 0.15 –0:60 m=s. At this moment, it can be noted that, bubbles smaller than 1:5 mm could not be detected using the time series analysis of LDA velocity-time data. (b) From the information on the rise velocity, equivalent bubble diameter was estimated using the following
correlations. It was assumed that the bubble motion in the system occurs at its terminal rise velocity and this velocity is attained in very small time even if it may or may not have experienced a break-up or coalescence before passing through the measurement point. (i) Karamanev (1994) has used a Tadaki number (Ta) for taking into account the contribution of bubble shapes in correlating the bubble size and rise velocity. ub (i)1−b 'b C(i)0:5 (7) db (i) = 0:862 0:5 b 0:23b g ) Mo where, C(i) = 24(1 + 0:173Re(i)0:65 )Re(i) + 0:143=(1 + 16300Re(i)−1:09 );
(8)
b is a constant the values of which have been reported by Karamanev (1994) as a function of Ta, and Mo is Morton number. (ii) Nguyen (1998) has proposed the following correlation in terms of dimensionless numbers: ub (i) db (i) = : 2−2b 2b +1 2−2b g'L 0:33 4a2 Mo0:46b )2L g 6−6b )L 1 '2 2:85 2 − 2b (9) Nguyen (1998) also has reported the values of a and b as a function of Ta. The physical properties of the liquid phase (reported in Table 1) and the measured bubble rise velocity were used for the estimation of the equivalent bubble diameter (from Eqs. (7) and (9)) by an iterative procedure, where values of a ; b and C(i) needs to be satis%ed to be in a certain range. Fig. 3 shows that the bubble number variation with equivalent bubble diameters obtained from Eqs. (7) and (9) and are fairly in close agreement. Since the Nguyen has reported the correlation developed speci%cally for contaminated liquids and since the present system can be considered to be contaminated liquid, Eq. (9) has been used for further calculations in
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Fig. 3. Bubble size distribution from the correlation by (♦) Karamanev (1994) and (O) Nguyen (1998).
this work. In order to obtain reliable and reproducible information, the e8ect of number of data points on the results was analyzed for 4096; 8192; 16384 and 32768 data points. Reproducibility was obtained at 8192 data points. (c) Since bubble shape also has an e8ect on the e8ecT was estitive interfacial area, the mean aspect ratio (E) mated using the Eotvos number (Eo ): 1 : (10) ET = 1 + 0:163Eo0:757 It was observed that, for 90% of the identi%ed bubbles, the aspect ratio was found to be greater than 1. (d) As explained earlier, if the LIM values for the energetic scales (scales containing noticeable part of the total energy in signal), were above a certain threshold value, the events were identi%ed as passage of gas bubbles. The arrival time gap associated with this event (when no data was obtained) indicates the bubble passage through the measurement volume. The sum of all such time gaps is the total time for which the measurement volume is occupied by the gas phase. Thus, for a suRciently longer data set, the ratio of total time occupied by the gas phase to that of the total length of the time series gave the local fractional gas hold-up (G ). This procedure was carried out for a typical threshold value and the estimated gas hold-up at eleven radial locations was used for getting the volume average gas hold-up
(TG ) for the entire column. In a similar way, the volume average hold-ups were estimated at various levels of threshold. The appropriate threshold value was selected in such a way that the estimated hold-up agreed with the experimentally measured average hold-up (calculated from the bed expansion). Fig. 4 shows hold-up (G ) pro%les at %ve di8erent axial locations (at 200; 300; 400; 500 and 600 mm away from the sparger) for three di8erent gas velocities, 24; 29 and 39 mm=s; respectively, in the 100 mm column containing aqueous Na2 SO3 solutions. (e) From the knowledge of the bubble shape, the actual interfacial area (A) and the bubble volume (v) were estimated for all the identi%ed bubbles. The overall effective interfacial area was estimated using the following equation: N N b b A(i) v(i) ; (11) a(r; z) = G (r; z) i=1
i=1
where, a(r; z) is the local interfacial area at co-ordinate position (r; z) and Nb is the total number of bubbles encountered during the measurements at (r; z): A(i) and v(i) are the surface area and volume of a ith bubble. (f) The area average a(z) was estimated (at every axial location) using the following equation: R 1 20ra(r; z) dr: (12) a(z) = 2 0R −R
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Fig. 5. Bubble shape map for di8erent experimental conditions. Re versus Eo .
(g) As mentioned earlier, the LDA measurements were made at %ve axial locations, and the corresponding values of a(z) were estimated. The axial variation was found to be within 4.06%. Therefore, the arithmetic average of %ve values of a(z) was taken for estimating the overall volume averaged value of a. 4. Results and discussions
Fig. 4. Radial variation of fractional gas hold-up (G ) , VG = 24 mm=s; U; VG = 29 mm=s; ; VG = 39 mm=s at %ve axial levels. (L1 = 200 mm, L2 = 300 mm, L3 = 400 mm, L4 = 500 mm and L5 = 600 mm away from the sparger).
The hold-up pro%les from Fig. 4 showed that for the 0:2 M aqueous solution of sodium sul%te, the hold-up is more in the center and less near wall region indicating an upward liquid Jow in the central region and downward in the wall region. This observation remained consistent for all the three gas velocities at all %ve axial positions mentioned in Table 1. Clift, Grace, and Weber (1978) have reported the region map for bubble shapes in the form of a plot of Re versus Eo with Mo as parameter. The results of the present work have also been shown in Fig. 5. It can be seen that, the size and shape distribution obtained in this work follows the trend reported by Clift et al. (1978). The parity plot for a predicted from the velocity-time data and those obtained by the chemical method is shown in Fig. 6. Excellent agreement can be seen over a range of VG , D and electrolyte concentrations covered in this work. This analysis can perhaps be applied for the industrial columns where the time series of a characteris-
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[BO ] Ccat C db (i) D DA Eo ET g HD i j kmn
Fig. 6. Parity plot for the e8ective interfacial area in both the bubble columns. 100 mm; •; 150 mm; ◦. (Each point corresponds to one [B0 ] and one value of VG .)
k LIM m
tic parameter of the system (pressure, conductivity, etc.) can be obtained and a similar methodology can be developed and used as a diagnostic tool during the operation of bubble columns.
Mo Nb N p Pˆ j
5. Conclusion
r r=R
Analysis of a LDA velocity-time series from bubble columns has been carried out using wavelet transforms. A method has been developed for the measurement of bubble size and shape distribution, which favorably agrees with the bubble-shape map (Clift et al., 1978). The velocity data were also used for the estimation of local gas hold-up (G ) and the local e8ective interfacial area (a). The resulting hold-up pro%les obtained from the multiresolution analysis method showed consistency in trend for the range of VG in these studies at %ve axial positions. Further, an excellent agreement has also been shown between a estimated by time series analysis and those a obtained by the chemical method. The agreement holds irrespective of the regime of operation, which is a signi%cant result. Notation a and b a and b a (r; z) a A [A∗ ]
scaling and translation parameter constants in Eqs. (7) and (9) e8ective interfacial area of bubbles at co-ordinate (r; z); 1=m overall e8ective interfacial area, 1=m actual interfacial area of a gas bubble, m2 concentration of gas phase at the interface, kmol=m3
R Re Ta ub v VG W x z
concentration of liquid reactant, kmol=m3 catalyst concentration, kmol=m3 drag coeRcient of the bubble, dimensionless equivalent bubble diameter, mm column diameter, mm di8usivity of gas in liquid, m=s g()L − )G )d2b ; dimensionEotvos number, Eo = 2 less mean aspect ratio, dimensionless acceleration due to gravity, m=s2 total height of dispersion, mm bubble number and the time index for instantaneous velocity wavelet scales second order rate constant, kmn = k [Ccat ]; m3 =s=kmol translation parameter, dimensionless local intermittency measure the %rst energetic scale available for LIM analysis Morton number Mo = g'4 =)23 , dimensionless number of bubbles identi%ed at (r; z) number of data points largest resolvable wavelet scale total content in the denoised time-series, N energy 2 2 2 x(i) ˆ ; m =s 1 radial co-ordinate, mm normalized radial distance from centre, dimensionless speci%c rate of absorption, kmol=m2 s d b ub ) L Reynolds number, Re = ; dimensionless 'L 0:23 Tadaki number, Ta = ReMo , dimensionless rise velocity of bubble, m=s volume of individual bubble identi%ed at, r; z; m3 super%cial gas velocity, m=s wavelet coeRcient time series data, m=s axial co-ordinate, mm
Greek letters TG G ' ) 2
average fractional gas gold-up, dimensionless local fractional gas hold-up, dimensionless viscosity of liquid, kg=m=s liquid density, kg=m3 surface tension of liquid, kg=s2 wavelet function
Subscripts b G
bubble gas phase
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j k L r t
wavelet scale translation parameter in Eq. (2) liquid phase refers to reconstructed velocity-time data time, s
Superscripts ˆ l; m&n
refers to the denoised sets orders with respect to catalyst, gas and liquid concentrations, respectively
Acknowledgements The %nancial assistance of the “Professor M. M. Sharma Endowment” (AAK) and Department of Science and Technology, New Delhi, India, (VRK and BDK) under the program [III.5(19)=98-ET] in carrying out this work is gratefully acknowledged. References Bakshi, B. R., Zhong, H., Jiang, P., & Fan, L. S. (1995). Analysis of Jow in gas–liquid bubble columns using multiresolution methods. Transactions of Institution of Chemical Engineers, 73(A), 608. Clift, R., Grace, J. R., & Weber, M. E. (1978). Bubbles, drops, and particles. New York: Academic Press. Danckwerts, P. V. (1970). Gas–liquid reactions. New York: McGraw Hill. Daubechies, I. (1992). Ten lectures on wavelets. Society for Industrial and Applied Mathematics.
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