Flow Regime Diagnosis in Bubble Columns via Pressure Fluctuations

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Ind. Eng. Chem. Res. 2009, 48, 1072–1080

Flow Regime Diagnosis in Bubble Columns via Pressure Fluctuations and Computer-Assisted Radioactive Particle Tracking Measurements Marıá Sol Fraguıó,† Miryan C. Cassanello,*,† Sujatha Degaleesan,‡,§ and Milorad Dudukovic‡ PINMATE, Dep. de Industrias, FCEyN, UniVersidad de Buenos Aires, Int. Guïraldes 2620, Buenos Aires, Argentina, C1428BGA, CREL, and Department of Energy, EnVironmental and Chemical Engineering, Washington UniVersity in St. Louis, 1 Brookings Dr., St. Louis, Missouri, 63130

The importance of diagnosing the flow regime in bubble columns by noninvasive and easy-to-implement methods is well-known. Hence, the aim of this work is to diagnose the flow regime in a pilot scale bubble column by comparing the attractor that gives the fingerprint of a tested underlying hydrodynamic condition against the attractor of a reference condition, using the statistical S test, developed by Diks et al.1 The attractors are reconstructed from the time series of two characteristic variables: the trajectory of a liquid flow follower, determined by CARPT (computer-assisted particle tracking), and pressure fluctuations. Since CARPT fully maps the hydrodynamics in multiphase systems in a Lagrangian sense, the tracer particle trajectory time series is used to establish the optimal set of parameters required for the S test when analyzing pressure fluctuations. This work demonstrates that the same set of optimal parameters determined when applying the S test to CARPT experimental time series leads to successful flow regime identification when applying the global S test to pressure fluctuation signals detected at various locations. This validates the use of pressure fluctuation signals in industrial settings as an economic way to detect flow regimes. Introduction

conditions,3-11 while Cassanello et al.,12 Larachi et al.,13 and Limtrakul et al.14 described the behavior of three-phase fluidized beds. Fraguıó et al.15 recently demonstrated that Diks’ test is a powerful tool in diagnosing the flow regime transition in threephase fluidized beds when applied to compare attractors reconstructed from experimental time series of one of the coordinates of the solid tracer particle trajectories recovered by CARPT. It was shown that, by using the test and a proper reference, it was possible to classify the flow regime for all studied conditions, regardless whether the experiments were performed in different CARPT facilities, with different liquid velocities or tracer particles of different sizes and mixtures or within columns of different sizes.

Bubble columns are gas-liquid contactors widely used for petrochemical, biochemical, and chemical applications. Flow regime largely affects their performance particularly when the desired operation is close to the transition, which is usually the situation in many applications. Hence, it is important to diagnose the flow regime by objective noninvasive methods that could be readily implemented. Diks et al.1 proposed a statistical test that allows comparison of the strange attractors describing systems dynamics. These attractors, which give a fingerprint of the hydrodynamic conditions in many multiphase systems, can be reconstructed in an embedding phase space from experimental time series of a characteristic variable using the delay method originally proposed by Takens.2 The test defines a statistics (actually a number), S, from the distance between the two compared attractors. Therefore, it is a quantitative, mathematically founded tool to judge when differences are significant between two hydrodynamic conditions. The computer-aided radioactive particle tracking (CARPT) technique provides trajectories of a tracer particle representing the liquid or the solid phase in multiphase systems and allows different ways of extracting information of the underlying dynamics. The motion of the tracer particle provides the full Lagrangian description of the hydrodynamics of the system. The ability of the CARPT technique to characterize different multiphase systems has been thoroughly demonstrated. Bubble columns and slurry bubble columns hydrodynamics have been exhaustively examined by CARPT under different flow

Therefore, the objective of this work is to extend the application of the S test to compare attractors reconstructed from experimental time series obtained by CARPT in bubble columns. The additional goal is to compare the output of the test when the time series of different characteristic variables are observed. Many reported applications of the S test to compare attractors reconstructed from experimental time series measured in large scale multiphase systems were carried out using pressure fluctuations measurements.16-19 Although good results have been reported using pressure fluctuations, it is important to compare the results arising from the application of the test to different variables. Moreover, in using pressure sensors flushed to the column wall, uncertainties related to the optimal position of the sensor arise when applying this kind of measurements, especially in large scale equipment.

* To whom correspondence should be addressed. Phone: (+5411)45763383. Fax: (54-11)45763366. E-mail: [email protected]. † Universidad de Buenos Aires. ‡ Washington University in St. Louis. § Present address: Shell Global Solutions (US) Inc., Westhollow Technology Center, 3333 Highway 6 South, Houston, TX 77082-3101, USA.

In this context, we decided to apply the S test to compare attractors of a reference and a test condition, both reconstructed from two different characteristic variables measured in the same system: the displacements of a tracer that represents the motion of the liquid obtained by CARPT and the pressure fluctuations measured by four sensors located flushed to the wall. Thus,

10.1021/ie800549d CCC: $40.75  2009 American Chemical Society Published on Web 12/12/2008

Ind. Eng. Chem. Res., Vol. 48, No. 3, 2009 1073

CARPT experiments are used to validate a more economic and easy to implement technique of pressure fluctuation measurements. Diks Test Diks et al.1 proposed a statistical test based on computing the distances between two reconstructed attractors obtained from the time series of an observed characteristic variable. From the chaos theory viewpoint, the attractor reconstructed from an experimental time series measured in a multiphase system represents a fingerprint of the hydrodynamics of the system under study for given conditions. The main idea of comparing attractors is based on the fact that, if we were able to determine all the governing variables of a system and their evolution in time and project them in the corresponding multidimensional space, we would obtain an attractor that is unique for each system. Certainly, it is impossible to know and measure all the variables governing the system. Takens2 showed that the dynamic states of a system can be reconstructed from the evolution in time of one characteristic variable. In this work, attractors will be reconstructed, for each examined operating condition, using the time series provided either by the solid tracer trajectories obtained from CARPT experiments or by pressure fluctuations measured with conveniently located pressure transducers. From the time series of the considered characteristic variable, consisting of N values, time delay coordinates of dimension m are extracted. In this way, a set of (N - m) delay vectors is built. The time evolution of these vectors in the m multidimensional space constitutes the reconstructed attractor. Takens2 proved that this reconstructed attractor has the same dynamic characteristics as the one arising from all the variables governing the system. The power of Diks’ test is that it allows the estimation of a statistic number, S, which quantifies the differences between two attractors. If S e 3 the null hypothesis that there are no differences between two compared attractors is valid with 95% confidence. If S > 3, there are significant differences between the attractors describing the dynamics of the systems under comparison. As the S value increases, the probability that the two compared attractors belong to different underlying flow regimes is larger. The statistics S is defined as

Figure 1. Picture of the four pressure sensors located at 18, 33, 67, and 82 cm above the gas distributor in the 0.19 m i.d. bubble column. Table 1. Experimental Conditions under which CARPT Experiments and Pressure Fluctuation Measurements Were Carried Out column diameter (m)

gas velocity, ug (m/s)

flow regime

0.19

0.020 0.050 0.120

homogeneous transition heterogeneous

where Q is an estimator of the distance between the two attractors and Vc its conditional variance. The test involves the selection of key parameters for each studied system: the embedding dimension, m, the bandwidth, d, the number of segments, L, and the length of the time series. For a detailed explanation of the test and the meaning of these parameters, the readers are referred to the work of Diks et al.1

is reconstructed by inverse reconstruction algorithms. The radioactive source, a minimum amount of 46Sc, was covered by a hollowed plastic to build a neutrally buoyant tracer; i.e., with the same density as the fluid that was followed. For details on the CARPT experiments, readers are referred to Degaleesan.7 Pressure fluctuations were measured using four piezoelectric pressure sensors (PCB Piezotronics, model 106B) arranged along the column at 18, 33, 67, and 82 cm above the gas distributor. The sensors were connected to a four channel signal conditioner (PCB Piezotronics, model 482A16). The conditioner output was recorded by a PC through an acquisition system (Power DAQ 12bit A/D ( 5 V). Pressure fluctuations were acquired at 400 Hz with a gain of 100. Figure 1 shows a picture illustrating the location of the four pressure transducers along the column. Both kind of experimental time series used, CARPT and pressure signals, were standardized before applying the test in order to reduce the sensitivity of the method to small changes in the superficial gas velocity, as pointed out by van Ommen et al.16 Table 1 shows the operating conditions under which the CARPT experiments were performed. Pressure fluctuations were first measured at the same conditions and then extended to a wider range of superficial gas velocities, between 0.02m/s and 0.22m/s.

Experimental Section

Results

The experiments were performed in a bubble column of 0.19 m internal diameter (i.d.) operated in batch for the liquid phase (water) and varying the gas (air) velocity within a wide range to span the homogeneous and heterogeneous flow regimes. The flow transition occurred between 0.05 and 0.07 m/s. The CARPT facility consisted of an array of 20 NaI(Tl) scintillation detectors that continuously recorded the γ rays emitted by a radioactive liquid flow follower. With the combined response of the detectors, the trajectory of the radioactive tracer

Figure 2 shows the test output when it is applied to the x coordinate of the liquid flow follower trajectories obtained by CARPT. Results of the S statistics for the three examined operating conditions are represented vs the gas velocity. Optimal parameters found by Fraguıó et al.15 for three-phase fluidized beds were chosen to perform the test (m ) 20; d ) 1; L ) 33; series length, 72000 points). The experimental condition in the homogeneous flow regime (0.02m/s) is taken as the reference time series for this figure and all the following ones. For the

S)

Q

√Vc(Q)

(1)

1074 Ind. Eng. Chem. Res., Vol. 48, No. 3, 2009

Figure 2. Flow regime classification in a 0.19 m i.d. bubble column from the reconstructed attractors comparison performed by the S test using the optimal parameters found by Fraguıó et al.15 for three-phase fluidized beds. Operating conditions: ug ) 0.02 m/s (homogeneous regime), ug ) 0.05 m/s (transition), ug ) 0.12m/s (heterogeneous regime). (a) complete scale; (b) enlargement.

Figure 3. Reconstructed attractors from CARPT data obtained in a 0.19 m i.d. bubble column under different operation conditions projected onto the first two principal components obtained by SVD. (a) ug ) 0.02 m/s (homogeneous regime), (b) ug ) 0.05 m/s (transition), (c) ug ) 0.12 m/s (heterogeneous regime).

condition in the homogeneous flow regime (0.02 m/s), Figure 2 shows that the computed values of S statistics remain below three and close to zero, as expected when computing the statistics S from the time series belonging to the same hydrodynamic state. When ug ) 0.12 m/s, the flow regime is heterogeneous, and the test, using the above specified parameters and reference, indicates a significant probability that the two compared attractors (reference and test) belong to different hydrodynamic states. An enlargement of the ordinate, to highlight the results obtained from the time series measured under conditions that lead to a regime transition, is illustrated in Figure 2b. For this intermediate situation (transition) the test output shows the same trend found for the transition in three-phase fluidized beds.15 Values of the statistics S are close to three, alternating below and above, depending on the region of the column visited by

the tracer particle during the time interval corresponding to the portion of the time series used to calculate the S statistics for each case. It is worthwhile mentioning that CARPT experiments usually render time series of several hours from which excerpts of 72000 points are extracted. Further examination of CARPT data is instructive. Figure 3 shows examples of the compared reconstructed attractors obtained from CARPT data for the three examined conditions, projected onto the first two principal components obtained by singular value decomposition (SVD).20 It can be observed that the second component variability increases as the gas velocity is increased. Since the S statistics is related (through Q, see eq 1) to the distances between two different attractors (i.e., hydrodynamic states), Figure 3 explains the good results obtained in Figure 2.


Figure 4. Attractor reconstructed from pressure fluctuations determined by the sensor located at H ) 82 cm above the gas distributor in the 0.19 m i.d. bubble column, projected onto the first two principal components obtained by SVD. (a) ug ) 0.02 m/s (homogeneous regime), (b) ug ) 0.05 m/s (transition), (c) ug ) 0.12 m/s (heterogeneous regime).

Figure 5. Effect of the parameter L, the number of segments, on the output of the S test for: (a) CARPT data; (b) global S statistics computed from the joint response of the four pressure sensors; (c) and (d) local S statistics computed from the response of the sensors located at H ) 82 cm and H ) 33 cm, respectively. Bubble column of 0.19 m i.d. operating under different gas velocities: ug ) 0.02 m/s (homogeneous regime), ug ) 0.05 m/s (transition), ug ) 0.12 m/s (heterogeneous regime).


Figure 6. Effect of the parameter d, the bandwidth, on the output of the test for: (a) CARPT data; (b) global S statistics computed from the joint response of the four pressure sensors; (c) and (d) local S statistics computed from the response of the sensors located, respectively, at H ) 82 cm and H ) 33 cm above the gas distributor. Bubble column of 0.19 m i.d. operating under different gas velocities: ug ) 0.02 m/s (homogeneous regime), ug ) 0.05 m/s (transition), ug ) 0.12 m/s (heterogeneous regime).

Since the classification was successful while applying the S statistic test to compare attractors reconstructed from CARPT data in bubble columns, the next objective was to examine the results obtained when using a different characteristic variable in the same system. Pressure fluctuations were examined as the second characteristic variable considering that this variable has been used in the majority of the studies reported in the literature that applied the S statistic test to quantify differences between two hydrodynamic states by comparing the underlying reconstructed attractors. Additionally, pressure fluctuations measurements are relatively simple and fast to implement in industry. Figure 4 shows the projections of the reconstructed attractors calculated from the time series of pressure fluctuations measured by the sensor located 82 cm above the gas distributor for the same operating conditions as in Figure 3. Once more, the size of the attractor grows as the gas velocity is increased. In this case, the attractor size increases in both directions. These results lead us to expect that the S statistics applied to compare attractors reconstructed from pressure fluctuations time series will also discriminate properly between the hydrodynamic states in bubble columns. To establish the optimal parameters for applying the S test with either of the characteristic variables used, the trend in the S test output, while progressively modifying the selected parameters, was studied for both variables at the same operating conditions. For pilot scale columns it is advisable to follow the response of more than one pressure sensor to monitor the column hydrodynamics. The sensors response can be analyzed independently or jointly to get an overall panorama of the underlying dynamics. Therefore, local values of the S statistics have been

computed from the response of each sensor and a global S statistics for the whole column was calculated from the joint response. For the last, the global S statistics was determined either as the average of the individual local S values or by considering the joint response of the four sensors following the data treatment proposed by van Ommen et al.16 for multisignals. The length of the time series used for the analysis was fixed in 72000 data points considering that, with this time series length, good results were obtained for CARPT in three-phase fluidized beds15 and for pressure fluctuations measured with the same sample frequency by van Ommen et al.16 and Villa et al.17 The embedding dimension was set to m ) 20 also based on similar considerations. Experimental time series measured in the homogeneous flow regime (0.02 m/s) are taken as reference. Figure 5 shows the effect of parameter L, the number of segments, on the output of the S test applied to compare attractors reconstructed from both variables: CARPT data and pressure fluctuations. The effect of the parameter L on the S statistics computed from either characteristic variable is similar at the three studied operating conditions (homogeneous, transition, and heterogeneous flow regimes). The value of parameter L does not seem to have an effect on the S test output while the system is within the homogeneous flow regime or at transition conditions; the S test produces almost the same result for all the studied range of L values. However, when the column is operated in the heterogeneous flow regime, the computed S statistics decreases as the used value of L increases and tends to a constant when L is large. The only difference between the results obtained from both studied variables and between local and global determinations of S values is the sensitivity of the test while comparing attractors reconstructed from time series


Figure 7. Flow regime classification in a 0.19 m i.d. bubble column from the reconstructed attractors comparison performed by the S test using optimal parameters. Operating conditions: ug ) 0.02 m/s (homogeneous regime), ug ) 0.05 m/s (transition), ug ) 0.12 m/s (heterogeneous regime). (a) S statistics calculated from CARPT data, (b) global S statistics computed from the joint response of the four sensors, (c) local S statistics calculated from time series of pressure fluctuations measured by each of the four sensors located along the column, (d) enlargement of part c.

measured under the heterogeneous flow regime to the reference taken in the homogeneous regime. While computing a local S for each sensor, the sensitivity with respect to the L parameter also depends on the location of the sensor. The maximum sensitivity is obtained for the sensor located farthest up from the gas distributor. However, the test satisfactorily indicates significant differences between the compared attractors from the individual analysis of any of the four sensors. The global outputs obtained by the joint analysis or from the analysis of the tracer trajectories from CARPT experiments are also able to properly classify the flow regimes using any value of the L parameter within the examined range. Figure 6 shows the performance of the test when the parameter d, the bandwidth, is varied for the same operating conditions as those shown in Figure 5. It is evident that this parameter has a larger impact on the test output than parameter L. Values of parameter d in certain range affect the S test output because this parameter sets the scale at which the differences between attractors are observed.1 If parameter d is too small, the S test starts to pick up local differences, while if d is set too large the S test may not be able to distinguish between two different hydrodynamic states. Figure 6 highlights that the S test is very sensitive to this parameter particularly for the conditions in the heterogeneous regime but also at regime transition. In most of the cases studied in the literature using the S statistic, the main objective was to distinguish between two different hydrodynamic states. Therefore, the criterion for selecting the optimal parameters relied in obtaining the S values below three and close to zero, when computing the time series

belonging to the same hydrodynamic state, and generating the largest possible values when computing the time series from two different hydrodynamic states. In this work, the criterion has to be stricter, since the objective is to classify flow regimes and the transition based on the computed S values. Hence, it is important to study the effect of the selected parameter on S values at transition conditions using a characteristic variable that gives full information about the phenomena taking place in the system. Figure 6 shows that, at transition conditions, the S values are more sensitive to parameter d than to parameter L. While applying the test to CARPT data at transition conditions S values close to 3 are obtained, with some of them slightly higher and some slightly lower than three. As already argued,15 this behavior arises from the fact that the flow transition is not an instantaneous phenomena and that it may not be established in the whole column. Then, a tracer trajectory arising from different locations at different periods of time will lead to values below and above three, depending on the region of the column mostly visited by the tracer particle during the period for which the time series for the S test was collected. This situation is encountered when analyzing CARPT data at transition gas velocity of ug ) 0.05 m/s. As Figure 6a illustrates for values of d larger than 1, the transition is well identified with S values scattered around the value of three. The differences arise due to the local heterogeneity of regimes in the column as explained above. When the chosen value of d is lower than one, the S test picks up local differences1 leading to the conclusion that the column is operating under the heterogeneous regime. Hence, values of d around 1 are recommended. Parts c and d of Figure 6 show the influence of parameter d on the S test output while using the separate responses of two


Figure 8. Flow regime classification within a wide range of gas velocities by comparing the attractors reconstructed from the local response of each of the four sensors located along the 0.19 m i.d. bubble column, using the S test with optimal parameters. Probe distance from the distributor: (a) H ) 18 cm, (b) H ) 33 cm, (c) H ) 67 cm, (d) H ) 82 cm.

sensors located at different axial positions in the column. Once again, the general trend is the same as the one obtained from CARPT data but less evident, and the sensitivity of the S value to parameter d is affected by the location of the sensor in the column. The correct expected output of the test is obtained with any of the four sensors for d ) 1. When a global S value is determined from the joint response of the four pressure sensors, the influence of parameter d is mitigated when compared to the results arising from CARPT experiments or from the separate analysis of each pressure sensor. Values of d around one lead again to the best output of the test. It is worthwhile to remark that Figures 5 and 6 indicate that the S test output is not extremely sensitive to the selected parameters. From these figures, one observes that many sets of parameters lead to a correct output of the test and provide for correct classification of the flow regimes. This result is in agreement with observations already pointed out by van Ommen el al.16 while implementing the test to get an early warning on agglomeration in gas-solid fluidized beds. The analysis of the parameters’ effect on the test output for the studied characteristic variables leads to the conclusion that in this case the same set of “optimal” parameters can be used for both variables (i.e., CARPT data and pressure fluctuations). However, this may not be a general conclusion that holds for any system and/or characteristic variable. Therefore, it will always be advisable to adjust the optimal parameters for known conditions, for instance by applying the test to CARPT data, and then adjust the parameters for any other measured characteristic variable until they show the same trend found for the CARPT data. In that sense “calibrating” the parameters for the

S test based on data like CARPT that truly captures the dynamics of flow before applying the test to other more readily measure variables, like pressure drop, seems advisable. In Figure 7, the results of the S test using the optimal parameters (m ) 20, d ) 1, L ) 33) are presented for both studied variables and for the three different examined operating conditions corresponding to the homogeneous, transition, and heterogeneous flow regimes. Figure 7a illustrates the results of the S statistics computed from the tracer coordinate time series determined by CARPT and shows that tested hydrodynamic conditions are properly classified when the S test is used to compare the underlying attractors to a reference one. Figure 7b shows the global S statistics computed from the joint response of the four pressure sensors, as suggested by van Ommen et al.16 for multiple signals. Again, the flow regime conditions are classified as expected by comparing the underlying attractors to a proper reference. The S statistics evaluated from the pressure fluctuations time series measured by each of the four sensors located along the column are represented in Figure 7c; an enlargement of the ordinate scale is illustrated in Figure 7d. As observed, the underlying flow regime is properly diagnosed by comparing the tested attractors reconstructed from any of the four sensors for the homogeneous and heterogeneous conditions, although the sensitivity of the test is different at different sensor locations. The transition is best diagnosed by analyzing the time series measured by the sensors located farthest from the distributor. Once the optimal parameter set for analyzing pressure fluctuation time series was established by comparison with the results obtained from the CARPT data, we examined more


Figure 9. Flow regime classification in the 0.19 m i.d. bubble column within a wide range of operating conditions by comparing the attractors reconstructed from pressure fluctuations time series computing global S statistics using the established optimal parameters. (a) Global S statistics calculated from the joint response of the four pressure sensors distributed along the column with the methodology proposed by van Ommen et al.16 for multiple signals. (b) Global S statistics determined as the average of the local values computed from the individual response of each of the four sensors. (c) Enlargement of part a. (d) Enlargement of part b.

exhaustively the capability of the method to classify the underlying flow regime. Particularly, conditions close to transition and high gas velocities were explored. Figure 8 shows the test output within a wide range of gas velocities, evaluated from time series of pressure fluctuations measured by each of the four pressure sensors located along the column. Figure 8 highlights the ability of the method to classify the flow regime in the bubble column under the examined operating conditions. The S test yields values below 3 for conditions in the homogeneous flow regime and around 3 for conditions close to the transition (ug ≈ 0.05 m/s). Then, at higher gas superficial velocities, the S statistic becomes rapidly larger than 3 and increases sharply with the increase in superficial gas velocity (up to ug of about 0.12 m/s) due to an increase in the distance Q (see eq 1) between attractors of the test and reference (homogeneous regime) hydrodynamic conditions. Simultaneously, there is a decrease in the conditional variance of Q, leading to a larger certainty in the result of the comparison. For sufficiently large gas velocities in highly churn-turbulent flow, the S statistics apparently tends to a constant value or even decreases slightly with gas velocity. Although the analysis of each of the four sensor pressure fluctuations provided satisfactory results, those sensors closer to the column ends (H ) 18 cm and H ) 82 cm) are most subject to misclassification errors when the column operates in the homogeneous flow regime. This is understandable since at the inlet the sparger creates heterogeneities of flow while at the top of the column disengagement of gas changes the flow regime.

Figure 9 shows the output of the global S statistics evaluated by two different means, as the average of the individual results computed from the response of each pressure sensor and from the analysis of the joint response. Scale enlargements are provided around the transition region. It is observed that comparing attractors computing the global S statistics leads to a good classification of the flow regimes at all the examined experimental conditions, mitigating the occasional misclassifications found from the analysis of the individual signals. These results emphasize the importance of monitoring a pilot or industrial column with several sensors located along the column to avoid local effects. The results illustrated in Figure 9 show that a similar output is obtained by either computing the global S statistics from the joint response of the four pressure sensors, as proposed by van Ommen et al.16 for multiple signals or by considering the average of the S statistics determined from the individual four responses. However, the way of computing the global S statistics proposed by van Ommen et al.16 leads to a better estimator of the distance between attractors showing less variability among different portions of the time series and properly mitigating the misclassification errors found from the analysis of the local response. Therefore, this methodology of analysis is apparently more robust and is recommended. Conclusions This work demonstrates that the S statistic test is a powerful tool for comparing the attractors that describe the underlying


hydrodynamics in a bubble column. Hence, the comparison can be used to diagnose the flow regime from the analysis of experimental time series of either the trajectory of a liquid flow follower, determined by CARPT or pressure fluctuations. The S test provided adequate results from both variables using the same set of optimal parameters. This validates the use of pressure fluctuation signals in industrial settings as an economic way to detect flow regimes. While studying the behavior of a pilot or industrial column, it is important to monitor the column at different locations to improve the robustness and reliability of the test output. By comparing results to those obtained from the analysis of CARPT experiments, the optimal set of parameters, the number of pressure sensors required, and the best localization of the sensors for getting proper classification of the flow regime in a bubble column can be established. Acknowledgment Authors from Argentina acknowledge the financial support from UBA and CONICET. Authors from Argentina and the US are grateful to the International Atomic Energy Agency and to the Fulbright Commission for providing the avenues for this cooperative research. The CREL authors appreciate the funding form CREL industrial sponsors, which made CARPT studies possible. We also acknowledge the unconditional assistance of Prof. Rud van Ommen (Delft University of Technology, The Netherlands) for implementing the S statistic test. Nomenclature d ) bandwidth for smoothing of points in the state space L ) segment length m ) embedding dimension N ) number of values in experimental time series Q ) estimator for the squared distance between two attractors S ) estimator for the normalized squared distance between two attractors ug ) superficial gas velocity, m/s V ) estimator for the conditional variance

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ReceiVed for reView April 06, 2008 ReVised manuscript receiVed October 17, 2008 Accepted October 20, 2008 IE800549D

Flow Regime Diagnosis in Bubble Columns via Pressure Fluctuations

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