Evaluating the Probabilities of Fluidization Regimes - Industrial

ARTICLE pubs.acs.org/IECR

Evaluating the Probabilities of Fluidization Regimes Milad Aghabararnejad, Navid Mostoufi, Rahmat Sotudeh-Gharebagh,* and Reza Zarghami Multiphase Systems Research Laboratory, Oil and Gas Processing Centre of Excellence, School of Chemical Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran ABSTRACT: Probabilities of fluidization regimes were determined experimentally using frequency domain analysis of pressure fluctuation data. It was assumed that regime transition does not occur at a distinct gas velocity and fluidization regimes coexist at each gas velocity. Three different flow structures (macro-, meso-, and microstructures) were recognized in a fluidized bed. Macrostructure is represented by bubbles, mesostructures are represented by clusters, and microstructure is represented by particles. These structures are the characteristics of bubbling, turbulent, and fast fluidization regimes. Regime probabilities were determined using power spectrum energies of these structures. Experiments were carried out in a 1 m high and 5 cm inner diameter glass column. Pressure fluctuations were measured using a piezoresistive pressure sensor. Sand particles with various diameters (150, 280, and 490 μm) and fluid catalytic cracking (FCC) particles (73 μm) were used as solids. Probabilities of fluidization regimes were calculated at various gas velocities. A new procedure was proposed for evaluating the regime probabilities versus gas velocity. It was shown that the proposed correlation is in good agreement with the experimental data and is noticeably better than the method proposed in the literature.

1. INTRODUCTION Fluidization is an operation that involves direct contact between freely moving solids and the flow of a fluid. Having excellent mixing between the phases and good controllability and good handling of fluidized bed reactors has led to a wide variety of applications including gasification, adsorption, polymerization, and catalytic cracking.1,2 Fluidized beds cover wide range of operating conditions, including bubbling, turbulent, and fast fluidization regimes. Most of hydrodynamic models have been developed for a specific fluidization regime. Thus, effective and exact discrimination of flow regimes is very important and essential prior to design of any practical application. There are plenty of works related to determination of the flow regime transition velocity. Bi and Grace3 reported a diagram for identifying the hydrodynamic regime based on gas velocity. In other work, Bi and Grace4 proposed a correlation for calculating the transition velocity from bubbling to turbulent. Bi et al.5 proposed a correlation for calculating transition velocity from turbulent to fast fluidization. However, considerable uncertainty exists in flow regime transition criteria and existing models mostly predict discontinuity at the boundaries. There were few attempts to evaluate reactor model performance using probabilities of different regimes. Thompson et al.6 proposed that transition from bubbling to turbulent regime does not occur at a distinct gas velocity. Based on their work, a twophase model for bubbling regime and an axial dispersed plug flow model for turbulent regime were merged by allowing key parameters to vary continuously with the superficial gas velocity. Abba et al.7,8 extended this approach to fast fluidization, and Constantineau et al.9 developed the same model for bubblingslugging regime. It was shown by Jafari et al.10 that such a generalized modeling of fluidized beds approach can predict the performance of fluidized bed reactors considerably better than previous ones. r 2011 American Chemical Society

In the present work, probabilities of fluidization regimes are determined experimentally using frequency domain analysis of pressure fluctuation data. Three groups of phenomena are characterized based on their frequencies and energies. The ratio of energy of these groups to the total energy of the signal is defined as the regime probability. A generalized model is presented to predict each fluidization regime probability. This model can be applied to all fluidization regimes.

2. EXPERIMENTS Experiments were carried out in a fluidized bed column shown in Figure 1. The column was made of glass with 5 cm inner diameter and 1 m height. Air at room temperature was supplied by a compressor and entered the column through a porous plate distributor with 5 μm holes and free area of 3.02%. The air flow rate was measured by an orifice-manometer system. Two cyclones, placed at the column exit, returned the entrained solids back to the bed. Solids used in these experiments were sand with mean diameters of 150, 280, and 490 μm and fluid catalytic cracking (FCC) particles with an average size of 73 μm. Properties of the solids are given in Table 1. Absolute pressure fluctuations were measured using a piezoresistive pressure transducer, type SEN-3248B075. To avoid blockage by fine particles, the tip of the pressure probe was covered with glass wool. In each test, the pressure fluctuations were recorded by an analog-to-digital convertor card connected to a Kistler charge amplifier, type 5011, and then a data acquisition board, type Advantech (PCI1712), to save the signals on a personal computer. The sampling frequency was 400 Hz, and 65 536 data were gathered in each test in approximately Received: March 16, 2010 Accepted: February 18, 2011 Revised: December 15, 2010 Published: March 09, 2011 4245

dx.doi.org/10.1021/ie100642b | Ind. Eng. Chem. Res. 2011, 50, 4245–4251

Industrial & Engineering Chemistry Research

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Figure 2. Pressure fluctuation signal. Sand; dp = 150 μm; U0 = 0.6 m/s.

Figure 1. Schematics of the fluidized bed.

Table 1. Properties of Solid Particles

a

particle

dp (μm)

Fp (kg/m3)

Umf (m/s)

Uc (m/s)

Use (m/s)

sand sand

150 280

2640 2640

0.033 0.052

0.85 1.04

2.94 3.53a

sand

490

2640

0.150

1.29

4.04a

FCC

90

1580

0.005b

0.52

1.65

Calculated from Bi et al.4 b The value is related to Umb.

164 s. To ensure the reproducibility of the sampled signals, the measurements were repeated three times at each measuring position. Superficial air velocity varied in the range 0-3.4 m/s to cover different regimes of fluidization including bubbling, turbulent, and fast fluidization. Pressure fluctuations were measured at H = 10, 15, and 20 cm above distributor. The aspect ratios L/D = 0.5, 1, and 1.5 were used in the experiments. Results showed that there is no significant difference in the energy distribution for different probe heights and initial bed heights. Therefore, in the following only the results for H = 20 cm and L/D = 1 are presented.

3. RESULTS AND DISCUSSION A quantitative description of flow regimes is obtained from a time series analysis of fluctuating signals.11,12 The key to such quantification is an appropriate measurement method, as well as appropriate methods of time series analysis of the measured fluctuating signals. Analysis of experimental data was done in the time domain, in the frequency domain, and in state-space; the latter was used in nonlinear time series analysis.13 Figure 2 shows a sample time series signal of the absolute pressure fluctuations. The complex shape of Figure 2 illustrates the complexity of different phenomena in a fluidized bed. In order to simplify the pressure fluctuation signal, frequency domain analysis was used. 3.1. Frequency Domain Analysis. Fourier transform was used to analyze the time series signal in the frequency domain. In a bubbling bed, the dominant frequency in the power spectrum corresponds to the bubbles passing through the bed.14 Regime

transitions were identified by a change in the frequency distribution in the power spectra when the intensity of higher frequencies overcomes that of lower frequencies. The shape of the spectrum depends on the number of samples, the sampling frequency, and the number of spectra averaged.15 In most of the previous works related to frequency domain analysis, only dominant frequencies were identified and little information was extracted on the distribution of energy over a wider range of frequencies.1,14 The discrete Fourier transform of time series signal x(n) is defined as N

xðf Þ ¼

∑ xðnÞ expð- j2πnf Þ n¼1

ð1Þ

where x(f) is the frequency series signal. In the present study, Welch’s method was used to improve the frequency resolution.15 In this method, time series are divided into L equal parts of length NL. The power spectrum of each part is calculated using 2 N 1 L xðnÞ wðnÞ expð- j2πnf Þ ð2Þ Pxx ðf Þ ¼ NL U n ¼ 1

∑

where w(n) is the window function, Pxx is the power spectral density function,and the normalization coefficient U is defined as N

U ¼

∑

1 L 2 w ðnÞ NL n ¼ 1

ð3Þ

The Hamming window, used in Welch’s method, is defined as16 2πn wðnÞ ¼ 0:54 - 0:46 cos ð4Þ NL - 1 To normalize the frequency data series, the following equations were used: NL

PNxx ðnÞ ¼

∑ Pxx ðnÞΔf n¼1 Δf

ð5Þ

NL

∑ PNxx ðnÞ ¼ 1

n¼1

ð6Þ

Here, PN is the normalized power spectral density function (PSDF) of pressure fluctuations. To calculate PSDF, the 4246

dx.doi.org/10.1021/ie100642b |Ind. Eng. Chem. Res. 2011, 50, 4245–4251


ARTICLE

Figure 5. Energy of different structures. Sand; dp = 150 μm. Figure 3. Normalized PSDF of pressure fluctuation signal. Sand; dp = 150 μm.

Figure 6. Normalized energy of different structures. Sand; dp = 150 μm. Figure 4. Logarithmic curve of PSDF. Sand; dp = 150 μm; U0 = 0.6 m/s.

Hamming window was used and the original time series signal was divided into eight equal parts with 8192 data.15 Figure 3 shows the normalized PSDF at various gas velocities. Figure 3 demonstrates that the major frequencies of pressure fluctuation in a fluidized bed occur below 10 Hz and the dominant frequency is about 2 Hz in all cases. By increasing the gas velocity, the intensity related to the higher frequencies increases because of changing the flow structure in the fluidized bed. Three different structures, i.e., macrostructure, mesostructure, and microstructure, have been identified in the literature.15,17 On the basis of frequency domain analysis, Johnsson et al.15 showed that the low frequency region (up to 4 Hz) in the power spectrum diagram is related to the macrostructures, the frequency range of 4-10 Hz corresponds to the mesostructures, and an area of high frequency (20-200 Hz) is related to the microstructures of the bed. The same ranges were used, up to 4 Hz for macroscale, 4-20 Hz for mesoscale, and 20-200 Hz for microscale, in Zarghami's work.17 In order to analyze the PSDF more thoroughly, a sample PSDF is illustrated in Figure 4 in logarithmic scale. Three different regions are recognized in Figure 4.17 The first region from 0 to 4 Hz corresponds to the macrostructure of the flow in which the dominant frequencies are present. The distribution and amount of energy in this region differ significantly between

the fluidization regimes studied. The region ranges typically up to about 4 Hz and is best presented on a linear scale. The second region from 4 to 20 Hz has a falloff in frequency, which is fitted as a power-law falloff with an approximate R = 1.73. The third region from about 20 to 200 Hz (the Nyquist frequency) has a power-law falloff of R = 5.14. The energies of these structures are defined as Z 3 Pxx df ð7Þ Ema ¼ 0

Z

20

Emz ¼

Pxx df

ð8Þ

Pxx df

ð9Þ

3

Z Emi ¼

200

20

ET ¼ Ema þ Emz þ Emi

ð10Þ

where ET is the total energy of the signal; Ema, Emz, and Emi are the energy of the macro-, meso-, and microstructures, respectively. Figure 5 shows the value of these energies against the gas velocity for 150 μm sand particles. Figure 5 illustrates that the energies of macro- and mesostructures reach a maximum against the gas velocity. The energy of the microstructure appears only at high velocities and grows monotonically. This trend can be 4247



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Figure 7. Regime probability diagrams: (a) 150 μm sand; (b) 280 μm sand; (c) 490 μm sand; (d) FCC.

explained by considering macro-, meso-, and microstructures as bubbles, clusters, and particles, respectively. At low gas velocity, the bubbles (macrostructure) are the dominant flow characteristics in the bed. Increasing the gas velocity results in more bubbles; thus, the energy of the macrostructure increases initially. However, a further increase in the gas velocity leads to the formation of clusters (mesostructure). At the onset of turbulent fluidization, the bubbles reach their maximum size and number. As a result, the energy of the macrostructure decreases while the energy of the mesostructure increases in this region. Further increase in the gas velocity results in breakage of the clusters into particles (microstructure) which can be related to increase in the energy of microstructure and decrease in the energy of the mesostructure. Energies of different structures were normalized in order to be compared with each other according to the following formula: ENx ¼

Ex ET

ð11Þ

in which ENx indicates the normalized energy of structure x. Figure 6 shows relative normalized energies of different structures for 150 μm sand particles. The energy of macrostructures is at maximum at superficial gas velocity U0 =1.6. Beyond this velocity, the energies of the meso- and microstructures tend to

increase. Figure 6 suggests that the macrostructures are significant in the bubbling regime while meso- and microstructures correspond to turbulent and fast fluidization regimes, respectively.18 Therefore, quantification of the probabilities of fluidization regimes was done by identifying the flow structures of the bed.19 3.2. Regime Probabilities. In order to calculate the probability of the bubbling regime, frequencies with behaviors similar to that of the macrostructures were identified and the probability of the bubbling regime was defined as s

PB ðU0 Þ ¼

∑ Ei ðU0 Þ i¼1 ET ðU0 Þ

ð12Þ

where PB(U0) is the bubbling regime probability at velocity U0, Ei(U0) is the energy of frequency i, and s is the total number of frequencies with behavior similar to frequency i. Similarly, the probability of being in the fast fluidization regime was calculated from the energies of the frequencies of microstructures as follows: t

PF ðU0 Þ ¼ 4248

∑ Ej ðU0Þ

j¼1

ET ðU0 Þ

ð13Þ



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Table 2. Comparison of Uc and Use Values from This Work with Literature Values Uc (m/s)

Use (m/s) 4

this work

Bi et al.4

2.94

2.83

particle

this work

Bi et al.

sand (150 μm)

0.85

0.84

sand (280 μm)

1.04

1.06

-

3.53

sand (490 μm) FCC (90 μm)

1.29 0.52

1.31 0.54

1.65

4.04 1.72

where PF(U0) is the probability of the fast fluidization regime at velocity U0, Ej(U0) is the energy of frequency j, and t is the total number of frequencies with behavior similar to frequency j. After evaluating probabilities of bubbling and fast fluidization regimes from eqs 12 and 13, the probability of being in turbulent regime of fluidization was obtained from PT ðU0 Þ ¼ 1 - PB ðU0 Þ - PF ðU0 Þ

turbulent regimes of fluidization. In such a situation, an increase in superficial velocity results in a change in the hydrodynamic status of the bed such that the probability of being in the bubbling regime decreases and the probability of being in the turbulent regime increases. Likewise, a decrease in superficial velocity results in a change in the hydrodynamic status of the bed such that the probability of being in the turbulent regime decreases and the probability of being in the bubbling regime increases. Therefore, it can be imagined that bubbling and turbulent probabilities are in equilibrium with the superficial velocity as a reagent for changing between these two regimes. Forgetting the numerical values of these parameters for a short time and treating those as chemical species in equilibrium, the above concept can be shown as the following: PB þ nU0 T PT

where the equilibrium constant for such reaction is given as

ð14Þ

It should be mentioned that calculating PT from energy of the mesostructures is difficult because the behavior of the mesostructures is complicated compared to other structures. Figure 7 illustrates the probability of different fluidization regimes against gas velocities. At low gas velocity, PB is dominant because bubbles are the major phenomenon in the bed. By increasing the gas velocity, PB decreases due to bubble breakage20,21 while PT starts to increase. The value of PB becomes equal to PT at U0 = Uc, where Uc is the transition velocity from the bubbling to the turbulent regime. It should be noted that Uc values obtained by the probability method (Figure 7 ) are equal to Uc obtained experimentally from standard deviation of pressure fluctuations. Increasing the gas velocity beyond Uc results in decreasing PB until it eventually vanishes. However, PT reaches a maximum and then decreases when the gas velocity increases because the energy of structures like clusters, which are characteristic of the turbulent regime, decreases. The value of PT reaches PF at U0 = Use, where Use is the transition velocity from the turbulent to the fast fluidization. This velocity is very close to the value of Use obtained from Bi et al.3 At very high gas velocity, PF becomes dominant and other regime probabilities are negligible. Table 2 compares the values obtained for Uc and Use with those obtained from the literature.

4. CORRELATION FOR PROBABILITY DISTRIBUTION Looking at Figure 7 , one can recognize the similarity between this figure and a species distribution diagram for the following chemical reaction system: AþX T B

ð15Þ

BþX T C

ð16Þ

Knowing equilibrium constants for reactions 15 and 16, the distribution of concentrations of A, B, and C against the concentration of X shows a diagram very similar to what is seen in Figure 7 for the regime probability distribution against gas velocity. In the following, the same approach was employed to propose a correlation for the regime probability distribution. Consider a bed operating at a relatively low superficial velocity, where the probability of being in fast fluidization regime can be neglected and the bed can be characterized by only bubbling and

ð17Þ

PT PB U 0 n

KBT ¼

ð18Þ

According to Abba,7 PB = PT at U0 = Uc; thus, the equilibrium constant, KBT, is determined by 1 ð19Þ Uc n The same argument can be made at relatively high superficial velocity where the probability of being in the bubbling fluidization regime may be neglected and the bed is characterized by only turbulent and fast fluidization regimes. In this region, the above thought can be expressed as following: KBT ¼

PT þ mU0 T PF

ð20Þ

where the equilibrium constant for such a reaction is given as KTF ¼

PF P T U0 m

ð21Þ

According to Abba,7 PT = PF at U0 = Use; thus the equilibrium constant, KTF, is determined by 1 ð22Þ Use m Based on the above concept, it is possible to have all fluidization regimes in a fluidized bed. In this situation, probabilities of different regimes are given according to the following equations: n PT U0 ¼ ð23Þ PB Uc KTF ¼

PF ¼ PT

U0 Use

m

PB þ PT þ PF ¼ 1

ð24Þ ð25Þ

Solving eqs 23-25 for the probabilities results in PB ¼

4249

1 n n m U0 U0 U0 1þ þ Uc Uc Use

ð26Þ



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Table 3. Fitting parameters m and n particle

dp (μm)

m

n

R2

sand

150

4.53

1.98

0.9865

sand

280

5.72

1.85

0.9924

sand

490

5.92

2.07

0.9876

FCC

73

4.65

1.97

0.9938

n U0 U n c n m PT ¼ U0 U0 U0 1þ þ þ Uc Uc Use

ð27Þ

n m U0 U0 U U nc sen m ð28Þ PF ¼ U0 U0 U0 1þ þ Uc Uc Use Equations 26-28 can be further improved by replacing the superficial velocities with the excess superficial velocities (U Umf) which makes the probability of being in bubbling regime of fluidization equal to 1 at minimum fluidization velocity (Umf): PB ¼

1 n U0 - Umf U0 - Umf n U0 - Umf m 1þ þ Uc - Umf Uc - Umf Use - Umf ð29Þ

U0 - Umf n Uc - Umf PT ¼ U0 - Umf n U0 - Umf n U0 - Umf m 1þ þ Uc - Umf Uc - Umf Use - Umf ð30Þ U0 - Umf n U0 - Umf m Uc - Umf Use - Umf PF ¼ U0 - Umf n U0 - Umf n U0 - Umf m 1þ þ Uc - Umf Uc - Umf Use - Umf ð31Þ Of course, for Geldart A particles, bubbling starts at Umb. Consequently, it is recommended to use Umb instead of Umf for Geldart A particles. However, since both Umf and Umb accept small values, this replacement would not have considerable effect on the numerical values of probabilities obtained from eqs 29-31. In the above equations, the values of Umf, Uc, and Use are known for a powder; thus, the probabilities of being in each fluidization regime can be evaluated. However, there are two constants (i.e., m and n) which have to be determined experimentally. Table 3 shows values of m and n for different sand particles which were calculated by fitting eqs 29-31 to the experimental values of probabilities. Table 3 illustrates that the values of m and n for different particles are almost the same. In fact, the effect of the particle property on the probability distribution has been taken into account in terms of Uc and Use. Thus, the constants m and n are less sensitive to the particle property. Figure 8 illustrates the quality of fitting of the proposed

Figure 8. Regime probability diagram for FCC particles.

correlation to the experimental data for FCC particles. Figure 8 demonstrates that the proposed correlation adequately fits the experimental probabilities. Figure 8 also includes the prediction of Abba8 for regime probabilities, who provided the regime probabilities based on the gamma distribution function. Comparison between the model developed in this work and prediction of Abba8 reveals that the correlation proposed in this work is considerably more accurate than Abba's approach.

5. CONCLUSION In the present study, a method was proposed to evaluate regime probabilities from pressure fluctuations of the fluidized bed. In order to calculate the probability of each fluidization regime, time series data of pressure fluctuations were analyzed by Fourier transform and three ranges of frequencies were characterized: with decreasing energy, increasing-decreasing energy, and increasing energy against gas velocity. These frequencies show behaviors similar to macro-, meso-, and microstructures that correspond to bubbling, turbulent, and fast fluidization, respectively. These structures were considered equivalent to bubbles, clusters, and particles. By evaluating the energy of the power spectrum for each group, probabilities of different regimes were specified at various gas velocities. A new correlation was proposed to predict regime probabilities against gas velocity, and it was shown that it fits the experimental data satisfactorily. ’ AUTHOR INFORMATION Corresponding Author

*Tel.: (þ98-21) 6696-7797. Fax: (þ98-21) 6646-1024. E-mail: [email protected].

’ ACKNOWLEDGMENT The authors gratefully acknowledge financial support (Grant 84094/28) from the Iran National Science Foundation (INSF). ’ NOMENCLATURE D = column diameter (m) dp = particle diameter (m) Ema = energy of macrostructures (Pa2) 4250


Industrial & Engineering Chemistry Research Emi = energy of microstructures (Pa2) Emz = energy of mesostructures (Pa2) ENx = normalized energy ET = total energy of the signal (Pa2) f = frequency (Hz) H = measuring height (m) L = bed height (m) m = model parameter n = model parameter PB = bubbling regime probability PF = fast fluidization regime probability PN = normalized power spectral density function (Pa2/Hz) PT = turbulent regime probability Pxx = power spectral density function (Pa2/Hz) Uc = transition velocity from bubbling to turbulent (m/s) Umb = minimum bubbling velocity (m/s) Umf = minimum fluidization velocity (m/s) Use = transition velocity from turbulent to fast fluidization (m/s) U0 = superficial gas velocity (m/s) Fp = particle density (kg/m3)

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(17) Zarghami, R.; Mostoufi, N.; Sotudeh-Gharebagh, R. Nonlinear characterization of pressure fluctuation in fluidized beds. Ind. Eng. Chem. Res. 2008, 47, 9497. (18) Kim, S. W.; Kirbas, G.; Bi, H. T.; Jim Lim, C.; Grace, J. R. Flow behavior and regime transition in a high-density circulating fluidized bed riser. Chem. Eng. Sci. 2004, 59, 3955. (19) Falkowski, D.; Brown, R. C. Analysis of pressure fluctuation in fluidized beds. Ind. Eng. Chem. Res. 2004, 43, 5721. (20) Sasic, S.; Leckner, B.; Johnson, F. Time-frequency investigation of different modes of bubble flow in a gas-solid fluidized bed. Chem. Eng. J. 2006, 121, 27. (21) Yang, T. U.; Leu, L. P. Study of transition velocities from bubbling to turbulent fluidization by statistic and wavelet multi-resolution analysis on absolute pressure fluctuations. Chem. Eng. Sci. 2008, 63, 1950.

’ REFERENCES (1) Yang, W. C. Fluidization, Solid Handling and Processing; Siemens Westinghouse Power Corp.: Pittsburgh, PA, USA, 1998. (2) Kunii, D.; Levenspiel, O. Fluidization Engineering; ButterworthHeinemann: Boston, MA, USA, 1991. (3) Bi, H. T.; Grace, J. R. Flow regime diagram for gas solid fluidization and upward transport. Int. J. Multiphase Flow 1995, 21, 1229. (4) Bi, H. T.; Grace, J. R. Transition from bubbling to turbulent fluidization. Presented at the AIChE Annual Meeting, San Francisco, CA, USA, 1994. (5) Bi, H. T.; Grace, J. R.; Zhu, J. Regime Transitions Affecting GasSolids Suspensions and Fluidized Beds. Trans. Inst. Chem. Eng. 1995, 73, 154. (6) Thompson, M. L.; Bi, H. T.; Grace, J. R. A generalized bubbling/ turbulent fluidized-bed reactor model. Chem. Eng. Sci. 1999, 54, 2175. (7) Abba, I.; Grace, J. R.; Bi, H. T. Variable-gas density fluidized bed reactor model for catalytic processes. Chem. Eng. Sci. 2002, 57, 4797. (8) Abba, I. A generalized fluidized bed reactor model across the flow regimes. Ph.D. Dissertation, University of British Columbia, Vancouver, Canada, 2001. (9) Constantineaua, J. P.; Grace, J. R.; Lima, C. J.; Richards, G. G. Generalized bubbling-slugging fluidized bed reactor model. Chem. Eng. Sci. 2007, 62, 70. (10) Jafari, R.; Sotudeh-Garebagh, R.; Mostoufi, N. Performance of the wide-ranging models for fluidized bed reactors. Adv. Powder Technol. 2004, 15 (5), 533. (11) Dhodapkar, S. V.; Klinzing, G. E. Pressure fluctuation analysis for a fluidized bed. AIChE Symp. Ser. 1993, 89, 170. (12) Felipe, C. A. S.; Rocha, S. C. S. Time series analysis of pressure fluctuation in gas-solid fluidized beds. Braz. J. Chem. Eng. 2004, 21, 497. (13) Piskova, E.; Moral, L. Characterization of spouted bed regimes using pressure fluctuation signals. Chem. Eng. Sci. 2008, 63, 2307. (14) Sobrino, C.; Sanchez-Delgado, S.; Garcia-Hernando, N.; De Vega, M. Standard deviation of absolute and differential pressure fluctuations in fluidized beds of group B particles. Chem. Eng. Res. Des. 2008, 5, 1236. (15) Johnson, F.; Zijerveld, R. C.; Schouten, J. C.; Van Den Bleek, C. M.; Leckner, B. Characterization of fluidization regime by time series analysis of pressure fluctuation. Int. J. Multiphase Flow 2000, 26, 663. (16) Enochson, L. D.; Otnes, R. K. Programming and Analysis for Digital Time Series Data; Shock and Vibration Monograph Series; Department of Defense, Shock and Vibration Information Center: Washington, DC, USA, 1968. 4251


Evaluating the Probabilities of Fluidization Regimes - Industrial

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