Characterization of Regime Transition in Fluidized Beds at High

Article pubs.acs.org/IECR

Characterization of Regime Transition in Fluidized Beds at High Velocities by Analysis of Vibration Signals Hedayat Azizpour, Rahmat Sotudeh-Gharebagh,* Navid Mostoufi, and Reza Zarghami Oil and Gas Processing Centre of Excellence, School of Chemical Engineering, College of Engineering, University of Tehran, P.O. Box 11155/4563, Tehran, Iran ABSTRACT: Vibration measurement, as a reliable nonintrusive technique, was used to investigate the hydrodynamics of a gas− solid fluidized bed. The extensive experiments were carried out on a laboratory scale fluidized bed, operated under ambient conditions and various particle sizes of sand, aspect ratios, and superficial gas velocities. Pressure fluctuations and vibration signals were recorded at sample frequencies of 400 Hz and 65.6 kHz, respectively. The signals were processed using various techniques such as the standard deviation, Kolmogorov entropy, average cycle frequency, and power spectral density function. The results showed that the vibration technique is able to predict the regime transition from bubbling to turbulent fluidization conditions. It was shown that the transition from bubbling to turbulent can be determined from the variation of the standard deviation, Kolmogorov entropy, and average cycle frequency of vibration signals. However, the velocity can be determined only from the standard deviation of the pressure fluctuation. Other methods of analysis, applied to pressure fluctuations, predicted the transition between the macrostructure and the finer structure of the fluidized bed system, but not the transition velocity from bubbling to turbulent. investigating the fluidized bed hydrodynamics.6−8 This method is simple and sensitive to changes in the hydrodynamics. The formation and rising of gas bubbles and clusters and the coalescence of bubbles and their eruption are the source of vibration fluctuations. Abbasi et al.6 used vibration techniques to determine the minimum fluidization velocity (Umf). A low frequency accelerometer has been used for monitoring fluidized beds instead of conventional pressure measurements.8 The potential of acoustic signals for determining bubble properties has been investigated, and this method has been compared with pressure signals;9 also, Albion et al.10 used acoustic signals to determine regime transitions in liquid−solid transport systems. The time series of the measured signal is characterized in three domains including time, frequency, and phase. Information about the hydrodynamics of the bed can be obtained by means of many statistical methods such as standard deviation, skewness, and kurtosis in the time domain.2,5,11,12 Most analyses in the time domain have been utilized to determine regime transitions.13−16 This research indicated that the maximum point of the standard deviation versus superficial gas velocity graph corresponds to the transition from the bubbling to the turbulent regime. In frequency domain analyses, fast Fourier transform (FFT) and wavelet transform16−18 are the mathematical tools which express the behavior of a time series in the frequency domain. The dominant frequency in a power spectrum in bubbling and slugging beds corresponds to the pass of bubbles through the bed.16 Many researchers have utilized the average cycle frequency19−21 because it can be obtained easily. Nevertheless,

1. INTRODUCTION Fluidization is a process in which solid particles become suspended and fluidized at a high enough gas or liquid velocity (higher than minimum fluidization velocity) and the bed adopts a fluidlike property. Today, due to many advantages such as high rates of heat and mass transfer, fluidized bed reactors are extensively used in various industries (oil, petrochemical, chemical, biochemical, pharmaceutical, food, etc.), in comparison with the conventional packed beds. In spite of these advantages, fluidized beds have disadvantages which limit their industrial applications. Possible defluidization due to changes in the hydrodynamics of the fluidized bed which can occur by particle agglomeration and gas velocity reduction, sudden change in the hydrodynamics due to unwanted change in the excess gas velocity, and blocking parts of the gas distributor are some of these problems. In addition, fluidization is one of the most complex systems in practical application. Therefore, permanent monitoring of hydrodynamic conditions and accurate and reliable methods to determine hydrodynamic properties of the fluidized bed is necessary and further investigations are still needed on this topic to understand the complex flow structures in these reactors. Many intrusive and nonintrusive measuring methods have been used to study the hydrodynamics of fluidized beds. One accurate interpretation of the hydrodynamics of fluidized beds can be reached through the analysis of the time series of measurable signals, such as pressure fluctuations. Since measuring the pressure fluctuations is rather easy and they can represent many different phenomena happening in the bed, many investigators have utilized this method to study the sizes and velocities of the bubbles1 and regime transitions.2 Fiber optic probes have also been used for measuring cluster characteristics3 and local velocity in fluidized beds.4,5 Analyzing the vibration signals is a new nonintrusive technique which is a reliable method for © 2012 American Chemical Society

Received: Revised: Accepted: Published: 2855

April 23, 2011 January 24, 2012 January 24, 2012 January 24, 2012 dx.doi.org/10.1021/ie200863y | Ind. Eng. Chem. Res. 2012, 51, 2855−2863

Industrial & Engineering Chemistry Research

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Nyquist criterion which is greater than the maximum frequency component within the frequency spectrum.28 All measurements were repeated three times to ensure reproducibility of signals.

this analysis has not been applied to determine the transition point and conditional monitoring of the fluidized bed. Evaluation of the correlation dimension and Kolmogorov entropy16,22−24 in the phase domain has been investigated by Grassberger and Procaccia.25 Autocorrelation and mutual information functions were also used in nonlinear state space analysis to determine time delay of reconstructed attractors.26,27 In the present work, vibration signals of the shell of a fluidized bed were used to identify its hydrodynamic status and determine the transition point from bubbling to turbulent fluidization. For the sake of comparison, pressure fluctuations were recorded, and standard deviation, Kolmogorov entropy, average cycle frequency, and fast Fourier transform (FFT) were used to analyze both the transient vibration data and pressure fluctuations measured in the fluidized bed.

3. METHOD OF ANALYSIS 3.1. Standard Deviation. Standard deviation is a common measure of analysis in the time domain that investigates the amplitude of the signal and is obtained by measuring the data set divergence from its mean. The deviation will be higher the more spread apart the data are. Standard deviation is defined as 1 n−1

σ=

n

∑ (xi − x ̅ )2 i=1

(1)

The mean value (x)̅ is calculated from

2. EXPERIMENTS The setup was a Plexiglas gas−solid fluidized bed of 0.15 m inner diameter shown in Figure 1. The gas distributor was a

x̅ =

1 n

n

∑ xi (2)

i=1

3.2. Discrete Fourier Transform. Discrete Fourier transforms (DFTs) and fast Fourier transform (FFT) are helpful tools for analyzing data in the frequency domain. This analysis disintegrates a signal into constituent sinusoids of different frequencies. The estimated Fourier transform, F(f), of the measured time series x(n) which consists of N points is16,19 N

F (f ) =

∑

x(n)e−2j πnf (3)

n=1

in which f and j are the frequency and a complex number, respectively. This relation becomes the fast Fourier algorithm if N is a power of 2 and is an efficient algorithm for computing the DFT of a time series. For analysis of the signal in the frequency domain, the power spectral density function (PSDF) can be calculated which represents the contribution of every spectrum frequency to the power of the overall signal. The PSDF is calculated from

Figure 1. Schematic of the experimental fluidized bed setup.

⎡ N ⎤2 i ⎢ ∑ x (n) w(n)e−2j πnf ⎥ Pxx (f ) = N i ⎥ ∑n = 1 w 2(n) ⎢⎣ n = 1 ⎦

perforated plate containing 435 holes with a 7 mm triangular pitch. Air was supplied by a compressor, and its flow rate was measured by an orifice meter. A cyclone was placed at the column exit to return the entrained solids back to the bed. Sand particles with mean sizes of 226, 470, and 700 μm and particle density of 2600 kg/m3 were used in the experiments. The system was electrically grounded to decrease electrostatic effects. The experiments were carried out with static bed heights of 7.5, 15, and 22.5 cm. Two identical DJB Model A/120/V accelerometers with a cutoff frequency of 25 kHz, resonant frequency of 53 kHz, and sensitivity of 100 mV/m s−2 were used to measure vibration signals. The maximum frequency analyzable without amplification effects is around 1/3 resonant frequency. These measuring probes were mounted on the column at 5, 10, and 15 cm above the distributor by means of a magnet to minimize sudden fluctuations. Absolute pressure fluctuation signals were measured at various axial locations along the bed using a piezoresistive pressure transducer type SEN-3248(B075). To avoid blockage by fine particles, the tips of the pressure probes were covered with glass wool and were located in the holes drilled in the wall of the column. To prevent wave interference and losing information, the sampling frequency (fs) of vibration and pressure fluctuation signals were set to 65 kHz and 400 Hz, respectively, based on the Shannon−

1

(4)

The averaged power spectrum becomes Pxx(f ) =

1 L

L

∑ n=1

i Pxx (f ) (5)

3.3. Average Cycle Frequency. The average cycle frequency is calculated from fc =

Nc N Δt

(6)

in which N is the number of data points, Δt is the time step between the data points, and Nc is half the times that the data set crosses its average value. In fact, the average cycle frequency is the number of cycles per total time of the signal. The number of cycles is equal to half the times that the data set crosses its average value. 3.4. Entropy. Entropy is a common method for measuring the amount of disturbance in a system. It is also related to the amount of information which is lost along the attractor. The attractor of a system can be reconstructed using only a smoothed single measured variable of the system. This can be 2856

dx.doi.org/10.1021/ie200863y | Ind. Eng. Chem. Res. 2012, 51, 2855−2863


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achieved by applying time delay embedding theory to measured values. Time delay and the embedding dimension are most important parameters selected for the Kolmogorov entropy estimation and for the attractor reconstruction. Selecting an appropriate embedding dimension is very important due to the fact that an incorrect embedding dimension would result in a reconstructed attractor that does not represent the real attractor of the system. In this work, the method of time delay has been used for the state space reconstruction. The normalized cutoff length is set to 1.0; this means that the maximum cutoff length is set equal to the average absolute deviation of the point in the time series. The embedding dimension is set at 50. In order to have a low error in the Kolmogorov entropy estimation, the attractor should have around 100 points per cycle and around 1000 cycles, and to achieve correct data, it is provided by the mentioned sampling frequency. In other words, two points on different trajectories of the attractor (i.e., different initial conditions), which are very close in the phase space, are evolving into two different trajectories; thus, they represent two different states due to the divergence between the nearby trajectories.16 Consequently, the initial information would be missed after a time and is expressed in bits per second (the amount of information which is lost in the time unit).27,29 According to information theory, Grassberger and Procaccia25 declared that the necessary information for predicting a system during the time interval [t1, t2] is I[t1, t2] = I[t1] + K (t2 − t1)

Figure 2. Correlation integral versus log ε for a vibration signal.

dimension of the system which is equal to the slope of the first section of this graph is about 12.485. According to the correlation suggested by Ruelle,30 the number of data points should be more than 1 747 800. Since the sampling frequency for vibration signal was about 65 kHz (65 536 Hz), the required time for attractor reconstruction is close to 26 s, and to be on the safe side, is considered to be 30 (s). Schouten et al.19 suggested another method to estimate the entropy of the fluidized beds. The advantage of this method is its simple algorithm and easiness to apply for pressure fluctuations and vibration signals. This entropy is called by Schouten et al.19 the Kolmogorov entropy, which can be calculated by using the maximum likelihood method. It has been shown that the close trajectories diverge exponentially on the reconstructed attractors.19,21 This exponential divergence can be defined by the following accumulative distribution:

(7)

where I[t1] is the information at time t1 and K is the entropy. For a predictable system (e.g., fully periodic phenomenon), K is zero, while it is infinity for a stochastic system and a positive finite value for a chaotic system. Considering that the dominant frequency spectrum of vibration fluctuations in a fluidized bed is less than 10 kHz (Figure 5), the sampling frequency for vibration fluctuation signals was about 65 kHz, which is much more than 10 kHz, satisfying the Nyquist criterion. This sampling frequency is also in accordance with a criterion of 50−100 times the average cycle frequency (maximum 10 kHz) which is required for nonlinear evaluation in bubbling fluidized beds.19,20 In addition, the sufficient data points for attractor reconstruction are related to the correlation dimension of the system, and a high value of the correlation dimension implies a high sampling frequency. Ruelle30 showed that the correlation dimension of the system should be less than 2 log N, where N is the number of data points. The correlation dimension can be calculated from the powerlaw relation between the correlation integral, C, of an attractor and a neighborhood radius, ε. Generally, the correlation dimension, which is the slope of the log plot of C versus ε, initially increases with the embedding dimension and reaches a limiting value in which further increase in the embedding dimension would not increase the correlation dimension; the correlation dimension reaches its saturation state.16,25,27 This power-law relation can be used to provide an estimate of the correlation dimension:16,29 ∂ ln C(ε, M ) ∂ ln ε M →∞ ε→ 0

DC = lim

lim

⎛ Kb ⎞ C(b) = exp⎜⎜ − ⎟⎟ ⎝ fs ⎠

(9)

where b is the escape time and is defined as the number of time steps that the trajectories begin from an arbitrary pair of initial points on the attractor and remain close within a specified cutoff length ε0. The probability to see a pair of trajectories of the same escape time is equal to C(b−1) − C(b). For this probability distribution, the maximum likelihood, estimation for the entropy, K, corresponding to M observations of a random sequence value of b, {b1, b2, ..., bM}, is derived as16 ⎛ 1⎞ K = −fs ln⎜1 − ⎟ ⎝ b̅ ⎠

(10)

where

1 b̅ = M

M

∑ bi i=1

(11)

4. RESULTS AND DISCUSSION 4.1. Standard Deviation. The standard deviation of pressure and vibration signals versus superficial gas velocity in a bed with the initial aspect ratio L/D = 1.5 are shown in parts a and b, respectively, of Figure 3. The measuring probes were located 10 cm above the distributor. For both vibration and pressure signals, the standard deviation initially rises with increasing gas velocity and then declines with further increasing gas velocity. This change in the trend of the standard deviation can be related to the regime transition of the fluidized bed. It was shown that the gas velocity at the maximum standard

(8)

Figure 2 shows the correlation integral versus log ε at its saturation state (other states are not shown) for a sample vibration signal of about 65 kHz; as can be seen, the correlation 2857



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Table 1. Calculated Transition Velocities (m/s) from Bubbling to Turbulent Fluidization by Different Methods, and Minimum Fluidization Velocity (Umf) from σ7 dp (μm)

from ref 32

press. fluctuation

vibration

from fc (vibration)

from Kml (vibration)

Umf7

226 470 700

0.93 1.22 1.41

0.89 1.15 1.34

0.91 1.25 1.38

1.04 1.18 1.49

0.87 1.26 1.47

0.043 0.15 0.34

regime as well as pressure fluctuations. These values are in close agreement with values predicted by the correlation of Bi and Grace.32 Variations of the standard deviations of pressure fluctuations and vibration signals versus superficial gas velocity for sand with a mean particle size of 700 μm at various static bed heights are shown in parts a and b, respectively, of Figure 4. It can be seen

Figure 3. Standard deviations obtained at 10 cm above the distributor with L/D = 1.5: (a) pressure fluctuation; (b) vibration.

deviation for pressure fluctuation denotes the onset of the turbulent fluidization regime.2,10,16,31 Increase in the standard deviation of the pressure fluctuations can be related to the growth of the bubble size due to the gas velocity increase in the bubbling regime. However, further increase in the gas velocity in the turbulent regime results in decreasing the standard deviation of pressure fluctuations because of the breakdown of large bubbles to voids and small bubbles. A similar trend can be seen in Figure 3b for the standard deviation of vibration signals, which shows that the vibration signals can also be used to determine the onset of the turbulent fluidizing regime. Figure 3b also indicates that the value of the standard deviation is higher for larger particles since larger particles produce larger and more stable bubbles. Figure 3 also reveals that the small particles show higher standard deviation than the larger particles at low gas velocities. This effect can be explained by the fact that small particles produce larger bubbles at low velocities compared to large particles due to the immobility of large particles under this condition. However, at high gas velocities, the mobility of large particles increases and causes more fluctuations, and larger bubbles are formed. Consequently, the value of the standard deviation rises considerably more sharply in a bed of large particles than in a bed of small particles. Transition gas velocities from bubbling to turbulent regime extracted from Figure 3 are summarized in Table 1. As seen in Table 1, the standard deviation of vibration signals can predict the onset of turbulent

Figure 4. Variations of standard deviations for sand with mean particle size of 700 μm at different aspect ratios of bed: (a) pressure fluctuation; (b) vibration.

in Figure 4a that the standard deviation increases with the growth of the aspect ratio. In fact, increasing the aspect ratio results in the formation of larger bubbles since they have enough time to grow up when they rise along the bed. In addition, the velocity at the onset of the turbulent fluidization regime (maximum point) shifts to lower velocities by increasing the aspect ratio. This trend can be attributed to the amount of energy transfer from the gas phase to solid particles. 2858



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By increasing the bed height, more energy is transferred from the gas phase to solid particles. Therefore, it causes the transition from the bubbling to the turbulent regime to occur at a lower velocity. Figure 4b shows the same trend for the dependency of the standard deviation of vibration signals and the relation between transition velocities on the aspect ratio. However, the dependency of standard deviation and transition velocity on the aspect ratio in the case of pressure measurement is more severe than for the vibration signal. 4.2. Power Spectral Density Function Analysis. Figure 5 illustrates the power spectral density resulting from the mea-

Figure 6. Kolmogorov entropy versus superficial gas velocity for different sizes of particles: (a) pressure fluctuation; (b) vibration.

bubbles increases, the system disorder increases and the entropy increases. Thus, it can be concluded that when there is a minimum deviation from the periodicity of the bed, the entropy is minimum. In addition, as can be seen in Figure 6a, the Kolmogorov entropy is higher for smaller particle sizes, which indicates that finer structures become more important in a bed of smaller particles. In fact, interaction among single particles and between particles and fluid corresponds to very high frequencies of microstructures. Macrostructure represents the behavior of large pressure fluctuations, and large bubbles (a size comparable to the physical dimension of the bed diameter such as large bubble eruptions and movement of large bubbles).16 Figure 6b shows the Kolmogorov entropy of the vibration signals versus the superficial gas velocity for various particle sizes. It can be seen in Figure 6b that the entropy decreases by increasing the particle size. This can be related to increasing the bubble size with increasing the particle sizes because the larger particles produce larger bubbles and the behavior of the system approaches that of a periodic system by the growth of the bubbles. The same trend was seen for the standard deviation (Figure 3) for which the size of bubbles is related to the value of the standard deviation. Figure 6b also shows that there are two minimums in each curve. The first minimum can be related to the structure transition at different velocities. The second minimum matches the transition velocity from the bubbling to the turbulent regime. Because of the coalescence of bubbles and the formation of larger bubbles by increasing the gas velocity in the bubbling regime, the system behavior approaches periodic status and the entropy of the system decreases and reaches a minimum (second minimum). Increasing the gas velocity after the transition velocity from the bubbling to the turbulent regime leads to the breakdown of larger bubbles to small ones and increasing disorder; consequently, the entropy of the system increases. The Kolmogorov entropies of pressure fluctuations and vibration versus superficial gas velocity for sand with a particle size of 700 μm at various bed heights are presented in parts a

Figure 5. Power spectra of the measured vibration fluctuations at height 10 cm above the distributor with different superficial gas velocities, particle size 700 μm, and L/D = 1.5.

surement of vibration signals for the bed of 700 μm sand at various gas velocities. Figure 5 demonstrates that the peak corresponding to the dominant frequency is located around 2500 Hz, which is related to the main phenomena occurring in the fluidized bed. In fact, this frequency of vibration of the wall of the bed is related to the continuous passage of bubbles through the bed. Before the transition point from the bubbling to the turbulent regime as shown in Figure 5a,b, the value of the power spectrum rises as the gas velocity increases because of the formation and coalescence of bubbles; thus, a maximum is observed in the power spectrum diagram (Figure 5c). As shown in Figure 5d, two distinct peaks with less energy than the transition velocity peak energy are formed by further increasing the gas velocity beyond the transition velocity, which can be related to the breakdown of large bubbles to small ones and the formation of a multiple-bubble structure in the bed. This trend has been observed for other sands, but Figure 5 is shown for sand with a mean particle size of 700 μm. 4.3. Kolmogorov Entropy. The maximum likelihood estimation of the Kolmogorov entropy of pressure fluctuations for beds of various particle sizes and an initial bed height of 22.5 cm is shown in Figure 6a. As shown in Figure 6a, the maximum likelihood estimation of the Kolmogorov entropy initially decreases for all three particle sizes and then increases with increasing gas velocity. Fluidized beds with larger bubbles behave the same as a periodic system, when the number of small 2859



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and b, respectively, of Figure 7. As shown in Figure 7a, the entropy reaches a minimum which is related to the structure

4.4. Average Cycle Frequency. Figure 8a shows the average cycle frequency of the pressure fluctuations versus gas

Figure 7. Kolmogorov entropy versus superficial gas velocity for different heights of bed for 700 μm sand: (a) pressure fluctuation; (b) vibration.

Figure 8. Average cycle frequency versus superficial gas velocity for different particle sizes of sand: (a) pressure fluctuation; (b) vibration.

transition in the bed. Also, the Kolmogorov entropy decreases by increasing the aspect ratio. Increasing the aspect ratio results in the formation of larger bubbles which correspond to macrostructures of the bed and show more periodic behavior. The entropy or the amount of disorder in the system decreases as a result of greater periodicity of the system. The same behavior can be seen in Figure 4a, in which the value of the standard deviation of pressure fluctuations increased by increasing the aspect ratio of the bed, which indicates the formation of large bubbles. Figure 7b shows the trend of the Kolmogorov entropy of vibration signals versus gas velocity at various aspect ratios. The same dependency of entropy on the aspect ratio that was seen in the case of pressure fluctuations (Figure 7a) is also seen for vibration signals. Similarly, as the aspect ratio becomes higher, the system becomes more periodic and the Kolmogorov entropy becomes lower. Also, the other trend (the second minimum in the curve) that is seen in Figure 7b can be related to the regime transition from bubbling to turbulent. These values are listed in Table 1 and are in good agreement with the values obtained by the standard deviation method. In the bubbling regime, the entropy of the system increases because of the existence of the bubbles of different sizes. However, after the coalescence of bubbles and the formation of larger bubbles, entropy decreases until the regime transition is reached; at this point the minimum value of entropy can be seen. By further increasing the gas velocity after the regime transition, the large bubbles break into small ones and the entropy increases again. Although we see three minimums in Figures 6b and 7b instead of two minimums, overall two sharp minimums are distinguished in these figures. Also, Figures 6b and 7b are much less clear than Figure 3a when it comes to determining the transition from bubbling to turbulent fluidization.

velocity at various particle sizes and initial bed height L/D = 1.5. Figure 8a demonstrates that the average cycle frequency decreases initially and approaches a minimum for all particle sizes. This minimum is related to the dominant frequency of the bed and then increases with increasing gas velocity. The dominant frequency in the bed corresponds to the main phenomena occurring in the bed at the frequency at which the power spectrum density reaches a maximum. The average cycle frequency of a perfect periodic time series is identical to the dominant frequency. A difference between the average cycle frequency and the dominant frequency of the bed indicates deviation from a completely periodic system. The dominant frequencies of a bubbling fluidized bed (between 1.5 and 2.5 Hz for pressure fluctuation) correspond to the bubbles of the macrostructures of the bed.6,13,16 As the difference between the average cycle frequency and the main frequency of the bed is increased, the system deviates further from the larger structures in the bed and the finer structures become more dominant. Thus, a decrease in the average cycle frequency of the pressure fluctuation signals indicates less deviation from periodicity. This minimum average cycle frequency occurs at gas velocities of 0.3, 0.7, and 0.8 m/s for 226, 470, and 700 μm sands, respectively. This trend can be seen in Figure 5a, in which minimum values of the Kolmogorov entropy of pressure fluctuation occur at 0.4, 0.8, and 0.9 m/s for the mentioned sands. Moreover, as can be seen in Figure 8a, the average cycle frequency is larger for small particles, which shows that finer structures become more important in a bed of small particles. Since the periodicity of the bed corresponds mostly to the existence of the larger structures in the bed, a minimum in the Kolmogorov entropy and average cycle frequency of the pressure fluctuations indicates a minimum deviation from the macrostructures of the bed. This minimum can be considered as the transition 2860



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ratio is similar for both pressure fluctuations and vibration signals. As the aspect ratio becomes higher, the system becomes more periodic and the average cycle frequency decreases. Figure 9b also demonstrates that the minimum of the average cycle frequency at various aspect ratios occurs at the transition velocity from bubbling to turbulent. Comparison of Figures 6 and 7 with Figures 8 and 9 shows that the trends of the average cycle frequency and the Kolmogorov entropy versus the gas velocity are similar, which shows a close relationship between these parameters. This similarity between the behavior of the average cycle frequency and the Kolmogorov entropy and using other analyses helps to obtain a clear description of the hydrodynamics of the fluidized bed system at high velocities. These results are in agreement with obtained results reported in the literature. Many investigators indicated that the information which is given by the Kolmogorov entropy is similar to that of the average cycle frequency of the fluidized bed;16,17,33 however van der Schaaf et al.17 were the first investigators to clearly explain the mechanism behind it and describe this similarity completely.33

point between macrostructures and finer structures in the bed. This shows that the contribution of finer structures of the bed decreases initially by increasing the gas velocity. After passing the transition velocity, which corresponds to the minimum entropy, their contribution is enhanced by increasing the gas velocity. Figure 8b shows the average cycle frequency of the vibration signal versus the gas velocity at various particle sizes and initial bed height L/D = 1.5. As can be seen in Figure 8b, the minimum average cycle frequency occurs at the velocity in which the regime transition from bubbling to turbulent occurs for all particle sizes. Since vibration signals are not truly local phenomena of the bed and are caused by vibration at different points of the wall, they would be combined at the tip of the probe. The same trend is not expected for pressure fluctuations because pressure fluctuations are local phenomena and can detect a structure transition instead of a regime transition. The average cycle frequency of pressure fluctuations and vibration signals versus the superficial gas velocity at various aspect ratios is presented in Figure 9. As shown in Figure 9a,

5. CONCLUSION The hydrodynamic features of gas−solid fluidized beds were determined by analysis of vibration signals and pressure fluctuations. In both cases, the gas velocity at the maximum standard deviation denotes the onset of the turbulent fluidization regime. Also, it was shown that, by increasing the aspect ratio, the standard deviation increases because bubbles have enough time to grow up at higher bed height. Small particles showed a higher standard deviation than the larger particles at low gas velocity due to the generation of more bubbles. However, at higher gas velocity, larger particles produced larger bubbles and consequently a higher standard deviation. The average cycle frequency and Kolmogorov entropy of vibration signals and pressure fluctuations were calculated. A minimum in the average cycle frequency indicated a minimum deviation from periodicity or a minimum deviation from the larger structures of the bed. The average cycle frequency showed a similar trend of the Kolmogorov entropy against the gas velocity, although the resemblance is not very strong. It was found that the minimum in the average cycle frequency and the Kolmogorov entropy of vibration signals corresponds to the transition velocity from the bubbling to the turbulent regime. The values obtained by this method are in good agreement with those obtained from the standard deviation and the power spectral density function. It is found that the Kolmogorov entropy and the average cycle frequency are higher for smaller particle sizes, which shows that the importance of the contribution of the finer structures is higher for the bed of smaller particles. Also, the higher aspect ratio of the bed caused the behavior of the fluidized bed system to tend to the periodic system and decrease in the average cycle frequency and the Kolmogorov entropy. It was shown that the regime transition from bubbling to turbulent cannot be determined from the average cycle frequency and the Kolmogrov entropy of the pressure fluctuations. However, the average cycle frequency and the Kolmogrov entropy of pressure fluctuations can detect the structure transition between macrostructures and finer structures. Results of the present work demonstrate that the vibration signals are highly feasible to determine the hydrodynamics of gas−solid fluidized beds, especially at high gas velocities.

Figure 9. Average cycle frequency versus superficial gas velocity for different bed heights of 700 μm sand: (a) pressure fluctuation; (b) vibration.

the cycle frequency reaches a minimum which can be related to the structure transition in the bed. Also, it can be seen in Figure 9a that the average cycle frequency decreases by increasing the aspect ratio. Increasing the aspect ratio results in the formation of larger bubbles in the bed. As mentioned before, larger bubbles are macrostructures of the bed which show more periodic behavior. Therefore, the average cycle frequency of the system becomes close to the dominant frequency of the bed. The same behavior is seen in Figure 4a, in which the value of the standard deviation of pressure fluctuations increased by increasing the aspect ratio, which indicates the formation of larger bubbles. Figure 9b shows the average cycle frequency of vibration signals versus the gas superficial velocity at various aspect ratios. Comparing Figure 9a and Figure 9b reveals that the dependency of the average cycle frequency on the aspect 2861



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AUTHOR INFORMATION

Corresponding Author

*Fax: (0098) 21-6646-1024. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE b = escape time C = correlation integral D = bed diameter, m DC = correlation dimension DFT = discrete Fourier transform dp = particle size, μm f = frequency, Hz F( f) = estimated Fourier transform of the measured pressure time series fc = average cycle frequency, Hz FFT = fast Fourier transform fs = sampling frequency, Hz H = bed height, m I[t1] = information in bits at time t1 K = maximum likelihood estimation of the Kolmogorov entropy, bits/s L = number of windows M = number of points on attractor N = number of data points Nc = number of cycles PSDF = power spectral density function Pxx = averaged power spectrum i Pxx = power spectrum estimate of each segment STFT = short-time Fourier transform t = time, s Δt = total time of measured signal, s U = superficial gas velocity, m/s Uc = transition velocity from bubbling to turbulent, m/s Umf = minimum fluidization velocity, m/s w = window function x(n) = time-series signal, m/s2

Greek Symbols

σ = standard deviation ε = neighborhood radius around a point on attractor

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REFERENCES

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Characterization of Regime Transition in Fluidized Beds at High

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