Evaluation of flow regimes in a semi-cylindrical spouted bed through statistical, mutual information, spectral and Hurst's analysis.

Oliveira, W.P. ; Souza, C.R.F. ; Lim, C.J. 等

INTRODUCTION

Spouted beds were introduced in 1954 by Gishler and Mathur (1957) for drying of grains, especially wheat. Since then, they have emerged as efficient solid-fluid contactors in several industrial operations such as coal combustion, biochemical reactions, drying of solids, drying of solutions and suspensions, granulation, blending, grinding, and particle coating (Mathur and Epstein, 1974; Ormos and Brickle, 1980; Kucharski and Kmiec, 1989; Markowski, 1993; Souza and Oliveira, 2005).

To optimize and control spouted bed operation, the system hydrodynamics need to be well characterized and defined (Xu et al., 2004b). Many spouted bed hydrodynamic studies are performed in half-columns to facilitate visual observations through the transparent walls. However, one must then avoid the corners where results may differ significantly from those in the corresponding fully three-dimensional column. Furthermore, vessels used in industrial operations are not transparent, making visual observations practically impossible. Thus, new methods are required for system monitoring.

Several researchers have investigated fluctuations of pressure (both absolute and differential pressure) and voidage signals to characterize the quality of fluidized beds. Different analytical methods, including statistical or correlation methods, powerspectral density, deterministic chaos, and the Wigner distribution have been used (e.g. Fan et al., 1990; Kage et al., 1991; Bai et al., 1997; He et al., 1997; Briens et al., 1997a,b; Ohara et al., 1999; Briens and Briens, 2002; Chaplin et al., 2004, 2005). However, the application of these methods in spouted beds is scarce, with only a few attempts reported in the literature (Wang et al., 2001; Freitas et al., 2004; Xu et al., 2004a,b,c; Piskova and Morl, 2007).

The work presented in this paper was conducted to compare the suitability of statistical, mutual information function, spectral and Hurst's Rescaled Range analysis for discrimination of flow regime transitions in a semi-cylindrical gas-solid spouted bed. The results may be useful in developing methods for identifying instabilities during industrial spouted bed operations.

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EXPERIMENTAL

The experiments were carried out in a plexiglass semi-cylindrical spouted bed column of diameter 150 mm and height 1000 mm shown schematically in Figure 1. The conical base had an internal angle of G0[degrees]. The inlet orifice diameter was 25 mm with a fine wire mesh to support solids particles. The spouting gas flow rate, supplied by an air compressor (1), was adjusted by valves (2), and measured by an orifice plate (3) connected to pressure transducers ([P.sub.1] and [P.sub.2]). The gas temperature during the experiments was measured by a thermometer. Nearly spherical glass beads with two narrow size distributions (mean diameters 1.2 and 2.4 mm) were used in the hydrodynamic studies. Table 1 presents the main properties of these particles.

The maximum spouting pressure drop, [DELTA][P.sub.m], pressure drop for stable spouting, [DELTA][P.sub.s], and minimum spouting velocity, [U.sub.ms], were determined as recommended by Mathur and Epstein (1974). The pressure drop corresponding to minimum spouting was also recorded, and visual observations were made to verify the findings. Digital videos through the transparent walls were acquired at frequency of 15 frames per second with a digital camera (Logitech Quickcam for Notebooks) connected to a PC computer. Photographs extracted from the videos assisted in characterizing the spouting flow regimes.

Rapid response pressure transducers (Omega Engineering--[P.sub.0]; [P.sub.1]: PX142-030G5V; [P.sub.D]; [P.sub.3]; [P.sub.4]: PX142-00ID5V; [P.sub.5]; [P.sub.6]: PX164-01OD5V; [P.sub.T]: PX142-005D5V) at several axial positions determined pressure fluctuation time series at a frequency of 200 Hz for 2 min intervals. The transducers were connected to a CIO-DAS08 data acquisition card (Measurement Computering Corp., Norton, MA, U.S.A.) installed in a personal computer running Labtech Labview version 12.0 software.

Dimensionless bed heights, [H.sub.0]/[d.sub.c], were varied from 1 to 4 and U/[U.sub.ms] ratios from 0.3 to 1.6 to determine the limits of the spouted bed flow regimes (fixed bed, incipient spouting, stable spouting, pulsating spouting, slugging, bubble spouting, and fluidization).

ANALYSIS OF PRESSURE FLUCTUATION TIME SERIES

Statistical Methods

Pressure fluctuation time series recorded at several axial positions and operating conditions were analyzed by statistical, mutual information theory, spectral and Hurst's Rescaled Range methods in an effort to detect signal properties useful for discrimination of flow regimes transitions in gas-solid spouted beds.

Statistical methods in the time domain that have been used for time series analysis include:

Standard deviation ([sigma]) = [square root of (1/N - 1 [N.summation over (i = 1)] [([Y.sub.i] - Y).sup.2]] (1)

a measurement of the variation of the data around the mean:

Skewness (SK) = 1/(N - 1)[[sigma].sup.3] [N.summation over (i = 1)] [([Y.sub.i] - Y).sup.3] (2)

is a measure of lack of symmetry. A skewness of zero denotes a symmetric data distribution. Positive values signify data skewed to the right side of the mean, whereas negative values denote the opposite.

Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. Data sets with high kurtosis tend to have a distinct peak near the mean, and decline rather rapidly, whereas low-kurtosis sets tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case (NIST, 2004). The kurtosis for a standard normal distribution is 3. Hence, excess kurtosis is defined as:

[K.sub.R] = [N.summation over (i=1)] [([Y.sub.i] - Y).sup.4]/(N - 1)[[sigma].sup.4] - 3 (3)

These statistical parameters have been applied by several authors to identify regime transitions in fluidized and, more recently, spouted beds (Lee and Kim, 1988; Bi and Fan, 1992; Bi and Grace, 1995; Xu et al., 2004a).

Spectral Analysis

Frequency analysis is a common tool for evaluation of pressure fluctuation signals in fluidized beds for characterizing flow regimes. In these systems, changes in frequency distribution in power spectra are used to determine flow regimes transitions, as well as for system monitoring and dynamic scale-up (Johnsson et al., 2000; Trnka et al., 2000; Briongos et al., 2006). However, spectral analysis may be subjective (Johnsson et al., 2000). The main drawbacks refer to the identification of the dominant frequency, which may differ depending on the observer. Although widely used in fluidization, spectral analysis has rarely been applied to spouted beds (Wang et al., 2001; Freitas et al., 2004; Xu et al., 2004a; Piskova and Morl, 2007). The results reported so far are inconclusive and specific for the system configuration, inert material and operating conditions. The number of samples, sampling frequency and number of spectra averaged are also important factors affecting the results.

In this paper, the power spectra density was determined for pressure fluctuation data by Fast Fourier transformation, based on the Welch averaged periodogram method (Johnsson et al., 2000). In order to make reliable comparisons among power spectra obtained for several operating conditions, the experimental pressure time series were first normalized to have a mean of zero and a standard deviation, [sigma], of unity.

Mutual Information Theory

The mutual information (MI) function gives a measure of the dependence between two variables. If the variables are independent, the mutual information between them should be zero, whereas variables having strong interdependency have large mutual information values. Deterministic chaotic signals fall between these two limiting types of behaviour. They are deterministic and therefore fully predictable in principle. However, mainly due to exponential sensitivity to initial conditions, only shortterm predictions are possible. For truly random signals, the system predictability is very low, showing a complete loss of mutual information between the measurements. For completely periodic deterministic signals, there is a direct and fully predictable relationship between successive measurements, permitting the prediction of future behaviour for all future times.

The original works of Shannon and Weaver (1949) and Mansuripur (1987) describe the concepts underlying mutual information. Brief reviews have been presented by Karamavruc et al. (1995) and Xu et al. (2004b), who applied mutual information theory in analyzing experimental data from fluidized and spouted beds, respectively.

In order to define the mutual information, consider a typical experimental data set Y(t) _ {Y([t.sub.1]), Y([t.sub.2]), Y([t.sub.3])...... Y([t.sub.n])}. The Y data may be divided into N bins, denoted by [Y.sub.1], [Y.sub.2]......... [Y.sub.N]. For any data set, the probability of a Y-value falling into a specific bin is P ([Y.sub.i]) . Thus, a set of probabilities, P([Y.sub.1]), P([Y.sub.2])...... P([Y.sub.N]) can be created from the original data set. The average amount of information gained from a measurement that specifies Y, is defined to be the entropy of the system H(X)

H (Y) = -[N.summation over (i=1)] P([Y.sub.i])[log.sub.2]P([Y.sub.i]) (4)

The logarithm is usually taken to the base two, so that the unit of H is the bit (binary digit). Entropy measures the system uncertainty. For a completely deterministic system, the occurrence of an event is practically certain, with probability 1. Therefore, the entropy is zero. However, if all probabilities are equally distributed as P ([Y.sub.i]) = 1/N, the entropy, H (Y) = [log.sub.2] (N), is maximized.

Consider a long time series Y(t) = {Y([t.sub.1]), Y([t.sub.2]), Y([t.sub.3])...... Y([t.sub.n])}, and its time ([tau]) delayed version Y*(t) = {Y([t.sub.1] + [tau]), Y([t.sub.2] + [tau]), Y([t.sub.3] + [tau])...... Y([t.sub.n] + [tau])} with the set of probabilities P([Y*.sub.1]), P([Y*.sub.2])_._ P([Y.sub.N]). The mutual information function can be described as:

I(Y, Y + [tau]) = H(Y) + H(Y + [tau]) - H(Y, Y + [tau]) (5)

where H(Y + [tau]) and H(Y, Y + [tau]) are expressed (Shannon and Weaver, 1949; Karamavruc et al., 1995; Xu et al., 2004b) as:

H (Y + [tau]) = -[N.summation over (j=1)]P([Y*.sub.j]) * [log.sub.2][P([Y*.sub.j])] (6)

H (Y, Y + [tau]) = -[N.summation over (i=1)] [N.summation over (j=1)]P([Y.sub.i], [Y*.sub.j]) * [log.sub.2][P([Y.sub.i], [Y*.sub.j])] (7)

The quantity P([Y.sub.i], [Y*.sub.j]) is the joint probability that measurement Y(t) falls into bin [Y.sub.i] while its delay version Y(t + [tau]) falls into bin [Y*.sub.j]. The mutual information function I (Y, Y + [tau]) can be seen as the average number of bits that can be predicted correctly for a subsequent measurement [tau] times step later, based on a previous value, Y. Mutual information can be used to identify and quantitatively characterize relationships between data sets not detected by commonly used linear measures of correlation. The mutual information function depends on the number of bins used to evaluate the probabilities. In this paper, N was fixed at 50 for all calculations.

Several applications of mutual information in fluidized bed system have been reported. For example, Fraser and Swinney (1986) used mutual information to define the time delay, [tau], for reconstruction of phase space by embedding time series data and to identify periodicity and predictability of the signals. Daw and Halow (1993) reported mutual information from pressure fluctuation signals for various fluidization regimes (slugging, bubbling, and turbulent fluidization). The mutual information function for the various fluidization regimes appeared to fall into bands between purely periodic and purely random. The authors concluded that it is possible to estimate average propagation times and velocities for hydrodynamic wave fronts (i.e., slugs, bubbles, and strands) moving between measurement locations.

Karamavruc et al. (1995), applied mutual information to analyze temperature and differential pressure data from a horizontal heat-transfer tube in a cold bubbling bed. They established the relative signal predictability, showing that local differential pressure signals are less predictable than temperature ones. Karamavruc and Clark (1997) measured the local instantaneous pressure signals for two airflow rates and reported that the levels of all mutual information functions decreased with increasing gas velocity, indicating a rapid loss of information as time progressed. The peaks in mutual information were representative of strong periodic motion.

Dahikar and Sonolikar (2006) utilized mutual information to identify the periodicity and predictability of local instantaneous pressure signals in a magneto-fluidized bed. The mutual information function showed more persistence over time than in the absence of a magnetic field. The level of the mutual information function in the absence of a magnetic field was lower than when a magnetic field was present.

Mutual information was used to analyze differential pressure fluctuation signals by Xu et al. (2004b) for a gas-spouted bed of diameter 0.12 m at different axial and radial positions, with glass beads and silica gel as inert particles. The mutual information function was found to change with the spouting conditions and measurement position. A packed bed had a low level and fast decay of mutual information, suggesting a random signal. Unstable spouting led to a higher level of mutual information and strong periodic motion. For stable spouting, the level of mutual information and the decay rate around time zero were between those for the packed bed and unstable regimes.

Hurst's Rescaled Range Analysis (R/S Analysis)

R/S analysis, introduced by Hurst (1951) to analyze Nile River overflows, has been applied by a number of groups to characterize the hydrodynamics of fluidized and spouted bed systems (e.g. Fan et al., 1990; Briens et al., 1997a,b; Bai et al., 1999, Ellis et al., 2003; Xu et al., 2004c; Briens and Ellis, 2005). It is quicker, simpler and more robust than fractal analysis, quantifying both the persistence and cycle time of time series (Briens et al., 1997a).

The analysis begins by dividing a time series, Y, into N subintervals of length [[tau].sub.H]. For each sub-period, k = 1, ..., n, the signal average, [Y.sub.k], and its standard deviation, [S.sub.k], are calculated. The accumulated departure for each sub-period, [X.sub.k], is then determined by subtracting the mean. For each sub-period, the minimum, [X.sub.k,min], and Maximum, [X.sub.k,max], of the accumulated departure values are calculated next. The range over each subperiod, Rk, is determined as the difference, [X.sub.k,max] - [X.sub.k,min], The rescaled range over the sub-period k is given by: [(R/S).sub.k] = [R.sub.k]/[S.sub.k]. Finally, the average value of the rescaled range for the N subN periods is determined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. After conducting the analysis for several sub-period lengths, the variation of the rescaled range [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] with [[tau].sub.H] can be obtained. The Hurst exponent can then be estimated (Briens et al., 1997b) by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

If the process is white noise (random), then the plot gives roughly a straight line of slope 0.5. If the process is persistent, the trend in the time series will probably continue, and the slope is >0:5. Exponents < 0.5 indicate anti-persistence where the trend likely reverses itself (Feder, 1988; Peters, 1994; Briens et al., 1997a,b, Ellis et al., 2003; Xu et al., 2004c). Signals belonging to the same hydrodynamic structure exhibit the same trend, and therefore the same Hurst exponent.

In analyzing time series for fluidized beds, the slope of the log-log plot of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] against [[tau].sub.H] generally gives different Hurst exponents for small and large sub-periods, with a smooth transition between these two regions. Since the regions of interest are those with H constant, two different values can be obtained, that is, a high-frequency and a low-frequency Hurst exponent (Briens et al., 1997a,b; Ellis et al., 2003, Xu et al., 2004c). The intercept of these two regions corresponds to the dominant signal cycle time. However, it is sometimes difficult to identify the signal cycle time in the log-log plot of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] against [[tau].sub.H]. In some cases, fluctuations due to harmonics for periodic or nearly periodic time series may arise. In other situations, the Hurst exponent changes very gradually. These factors may introduce significant errors in determining the signal cycle time.

The V-statistic (Peters, 1994) or P-statistic (Briens, 2000) may facilitate cycle time identification. The V-statistic, introduced by Peters (1994) to detect cyclic behaviour in stock markets, can be calculated by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

The V-statistic makes the break in the log-log plot of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] against [[tau].sub.H] easily detectable if the break occurs from a region with H > 0.5 to one with H < 0.5.

Briens (2000) generalized the V-statistic to facilitate identification of a break in the log-log plot of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. The P-statistic was defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where [gamma] is a exponent between 0 and 1. If [gamma] = 0.5, the P-statistic is equivalent to the V-statistic. Briens (2000) suggested that [gamma] be taken as the value which maximizes the ratio:

maximum [P.sub.statistic] - maximum of [P.sub.statistic] at the minimum and maximum [[tau].sub.H] Maximum of [P.sub.statistic] at the minimum and maximum [[tau].sub.H] (11)

RESULTS AND DISCUSSION

Measurements of the total pressure drop in the spouted bed system as a function of gas velocity were obtained for several static bed heights for both particle diameters. Figure 2 shows typical results for 2.4 mm particles for stable ([H.sub.0] =250 mm), and unstable ([H.sub.0] = 400 mm) spouting conditions. Graphs similar to those reported in the literature were obtained for all operating conditions studied.

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Identification of Spouted Bed Flow Regimes by Means of Visual Observation

The spouted bed flow regime was significantly affected by the bed depth, [H.sub.0], and gas velocity, U. For U> [U.sub.ms], an increase in bed height caused noticeable alterations in the spouting flow patterns, such as the initiation of a turbulent internal spout, development and rise of bubbles, slugging, and fluidization.

For the particles of mean diameter 2.4 mm, instabilities began at [H.sub.0] -300 mm (for U [greater than or equal to] 1.2 x [U.sub.ms]). Initially, the spout flow pattern changed from stable spouting to a pulsatile state. The frequency of pulsations increased with increasing gas velocity. Another transition in the flow regime occurred at [H.sub.0] [congruent to] 350 mm, with gas bubbles emerging in the upper part of the spout. For these conditions, the system oscillated between pulsatile spouting and "bubbling spouting". A further increment in bed depth to ~400 mm, moved the system towards turbulent spouting. The spout no longer had enough strength to penetrate through the centre of the bed of particles, instead oscillating vigorously from side to side. In this state, the system was very unstable, and the flow regime was defined as "turbulent spouting". Figure 3, shows pictures of the changes in the spouted bed states with increasing static bed height for a particle diameter of 2.4 mm at U/[U.sub.ms] = 1.4. The observed transitions in spouted bed flow regime indicate that the maximum spoutable bed depth should be ~350 mm for 2.4 mm particles, and 350 to 400 mm for 1.2 mm particles.

Similar instabilities occurred for the bed of 1.2 mm glass beads. Figure 4 shows typical flow regimes as a function of bed height and gas velocity. Figure 4 indicates interesting behaviour, mainly at U/[U.sub.ms] = 1.2 and 1.4. Increasing [H.sub.0] moved the system from stable spouting towards less stable flow patterns, for example, pulsatile, bubbling spouting, and turbulent spouting.

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Independent of the static bed depth, the transitions among flow regimes with increasing U/[U.sub.ms] were similar until incipient spouting (U/[U.sub.ms] = 1.0) was reached. The system started as a fixed bed. Increasing the gas velocity caused an internal cavity to form (near U/[U.sub.ms] = 0.3); this extended in length with increasing gas velocity, until incipient spouting was reached. Remarkable transitions in the spouting flow regimes occurred with variations in [H.sub.0] for U> [U.sub.ms] (Figures 4b and c). From these figures it is possible to identify four spouting regions:

(1) Internal spouting;

(2) Stable spouting;

(3) Bubble generation and rise in upper internal spout;

(4) Bubble bursting with or without turbulence (Figures 4b and c at U/[U.sub.ms] =1.2 and 1.4). The frequency of these events increased with increasing gas velocity.

Observations of spouted bed instability indicated that the maximum spoutable bed height, Hmax, for this particle diameter was ~400 mm for this particle diameter.

Curiously, even with these drastic changes in the spouted bed states, the graph of time-mean pressure drop across the bed as a function of the gas velocity showed typical behaviour, independent of [H.sub.0], with small variations at high gas velocities, as shown in Figure 2. This makes it difficult to select the operating conditions based only on the most common information available.

Analysis of Pressure Fluctuation Time Series

The easiest way to detect instabilities is to compare pressure fluctuation time series for the different spouting flow regimes. In this work, pressure fluctuation signals obtained for stable conditions for both particle diameters at [H.sub.0] from 150 to 300 mm were compared with unstable ones in an effort to detect differences, which could be used to monitor the hydrodynamic regimes. Figure 5 presents typical pressure fluctuation signals in the semi-cylindrical spouted bed for several intervals for the 1.2 mm particles at U/[U.sub.ms] = 1.4. Measurements are presented for both stable and unstable conditions in order to identify differences, which may be useful for further analyzes. These traces indicate differences in pressure fluctuation signals for stable and unstable conditions. These differences depend also on the measurement level, and are more pronounced at the cone-cylinder interface and upper levels ([P.sub.4] and [P.sub.5] positions), and for the total pressure drop, [P.sub.T], across the system. Similar results were obtained for the 2.4 mm particles, but the variations were more pronounced (see Figures Ga and b). Thus, the [P.sub.4], [P.sub.5], and [P.sub.T] pressure fluctuation data were subjected to the analysis methods outlined in Analysis of Pressure Fluctuation Time Series Section to evaluate their usefulness for identifying instabilities and flow regimes.

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Statistical Methods

The standard deviation of the pressure fluctuation signals, [[sigma].sub.T], skewness, SK, and excess kurtosis, [K.sub.R], were determined for different spouting velocities, particle diameters and bed depths.

For the 2.4 mm glass particles, independent of the measurement position, the standard deviation of the pressure fluctuation signals increased with increasing U/[U.sub.ms] and [H.sub.0]. Unstable systems resulted in higher standard deviations than stable ones. The effects of U/[U.sub.ms] and [H.sub.0] on [[sigma].sub.T], for the 2.4 mm particles are presented in Figures 7a and b, respectively. The increase of standard deviation with increasing U/[U.sub.ms] is due to the increased complexity of the gas-solid dynamics. However, for the 1.2 mm particles, no clear trend of standard deviation of pressure fluctuation signals with increasing U/[U.sub.ms] ratio was evident; 6T initially increased for U < [U.sub.ms], followed by a substantial drop. For U > [U.sub.ms], [[sigma].sub.t] initially increased with increasing U/[U.sub.ms], especially for deep beds, as observed for the 2.4 mm particles.

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Figures 8a and b plot kurtosis results for the total pressure fluctuation signals as a function of the static bed depth and gas velocity for both particle diameters. These show that the excess kurtosis of the pressure fluctuation signals depended on the operating conditions. For the 2.4 mm particles, the system became platykurtic when it reached unstable spouting. This behaviour was more detectable at higher gas velocities, indicating broadening of the distribution about the mean. It would therefore appear that the kurtosis of the pressure fluctuation signals may be used to detect flow regime transitions in spouted beds of larger particles. However, a complex trend appeared for the 1.2 mm particles. For a lower gas velocity (U=0.615 m/s), the kurtosis tended to decrease with increasing static bed depth, [H.sub.0], whereas, for a higher velocity, the kurtosis first declined when [H.sub.0] increased from 150 to 250 mm, and then tended to increase until the spouting became unstable. However, a further increase in bed depth caused a decrease in kurtosis. This behaviour of the kurtosis can make the use of this statistical parameter difficult as a tool for detection of system instabilities/flow regimes, especially for relatively small particles. However, it may be useful for monitoring existing systems.

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Figure 9 shows the skewness of the total pressure fluctuation signals for the 2.4 mm particles. It is seen that the skewness tended to increase when the system moved from stable to unstable spouting. However, increasing the static bed depth beyond [H.sub.max] caused a decrease in SK for U= 1.34 and 2.24 m/s. Detection of differences in spouted bed hydrodynamics based on the SK values for 1.2 mm particles appears to not be feasible, since the experimental results show no clear tendency.

It can be concluded, however, that for larger particles, statistical analysis gives satisfactory results, and is useful for detecting system instabilities caused by changes in operating conditions in the semi-cylindrical spouted bed. Although, the statistical analysis of the pressure fluctuation signals does not show clear dependence on operating conditions for smaller glass particles, it may be useful for monitoring and control purposes.

Spectral Analysis

The power spectral density of the pressure fluctuation signals at various levels was obtained by FFT based on the Welch method. The results were normalized through the time-integral squared pressure amplitude power spectral density (PSDTISA). The objective was to identify dominant frequencies in the pressure fluctuation signals for application as an index to characterize bed dynamics. This index could then be applied to monitor spouted bed-processing operations, like drying of pastes, solutions and suspensions, granulation and particle coating. Figure 10 presents typical [PSD.sub.TISA] results from the total pressure fluctuation signals as a function of U/[U.sub.ms] for 1.2 mm particles, for stable and unstable systems. It can be observed that PSDTISA for U < [U.sub.ms] are similar for stable and unstable systems. The finding, which also applied to the 2.4 mm particles, is expected, given that for U< [U.sub.ms] the system remains a packed bed or a packed bed with an internal spout.

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Differences between stable and unstable systems became more perceptible for U > [U.sub.ms]. Figure 10 reveals two or three peaks of dominant frequency for the unstable system (at 9, 5, and 2 Hz). The frequency near 9 Hz also occurred for the stable system, and can be considered a characteristic frequency, as reported by Xu et al. (2004b). The dominant frequencies and intensities of the [PSD.sub.TISA] are related to the flow regimes for the system. These features also occur for the 2.4 mm particles but at lower frequencies, for unstable conditions, and may be masked by energetic bubbles bursting at the bed surface. Also, instability observed in the core zone, where the spouting swayed back and forth to the sides of the column, may have affected the PSD results. Figure 11 presents photographs taken at intervals of 0.1 s over a period of 2 s for the unstable system ([d.sub.p] =1.2 mm, U/[U.sub.ms] = 1.4, [H.sub.0] = 500 mm). These photographs indicate that the frequency of the transient events decreased in the order: Bubble generation > Bubble growth and rise > Bubble bursting.

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The hydrodynamic events for the 1.2 mm particles also occurred for the 2.4 mm beads. Nevertheless, due to the strong turbulence observed under unstable conditions for the 2.4 mm particles, dominant frequencies were less than for the 1.2 mm beads. Some peaks may have been masked by energetic bubbles bursting at the bed surface. Moreover, the instability observed in the core zone, where the spout swayed back and forth, may have contributed.

The effect of static bed depth, [H.sub.0], on [PSD.sub.TISA] for [d.sub.p] = 2.4 mm and two gas velocities (1.34 and 1.68 m/s) is presented in Figure 12. For shallow beds, the results do not indicate dominant frequencies. In this case, the transverse solids cross-flow from annulus to spout near the bottom of the conical base strongly influenced the pressure signals. This solids cross-flow fluctuated at high frequency. Figure 12 shows a reduction in the dominant frequency with increasing [H.sub.0] for [H.sub.0] > [H.sub.max] ([approximately equal to] 350 mm).

Mutual Information (MI)

Figures 13a and b present the mutual information of the total pressure fluctuation signals as a function of [H.sub.0] for both particle diameters and U/[U.sub.ms] = 1.2 and 1.4. MI is seen to be a function of the operating conditions, reflecting the distinct hydrodynamics of the system. The rapid decrease of MI for [H.sub.0] [less than or equal to] [H.sub.max] indicates a quick loss of information over time, that is, the predictability between successive signals was short-lived. The MI was lower for the 2.4 mm particles than for the 1.2 mm ones, suggesting faster loss of information. The latter is probably due to the higher gas velocity needed to spout the 2.4 mm beads. A slight oscillation on the MI curves is observed for [H.sub.0] [greater than or equal to] [H.sub.max] (~350 mm), mainly for the smaller particles. This indicates the persistence in the pressure fluctuation signals, in conformity with the flow patterns for the system. For U/[U.sub.ms] = 1.6, the mutual information clearly changed with increasing [H.sub.0].

For the 2.4 mm particles, the system remained practically stable until [H.sub.0] [approximately equal to] [H.sub.max] (~350 mm). For these conditions, MI decayed quickly, corresponding to a practically random response, as expected. With a further increase in [H.sub.0] to ~400 and 450 mm, there were small oscillations in the MI signals, indicating slight periodicity in the pressure fluctuation signals. The correspondence between the MI results and the system hydrodynamics was clearer for the 1.2 mm particles. For shallow beds ([H.sub.0] = 150 and 250 mm), there was a strong periodic component in the signals, consistent with the pulsating flow regime described above. Further increases in [H.sub.0], to 300 and 350 mm led to an increase in the MI oscillation period, perhaps due to augmentation of the layer of particles to be overcome by the pressure waves, thereby reducing the pulsation frequency. The frequency of peaks in the MI graphs continued to decrease with additional increments in [H.sub.0], reflecting transitions in the flow regimes. For [H.sub.0] = 450 mm, the system approached fluidization, and the MI indicates small frequency periodicities, reflecting intermittent appearance and bursting of bubbles at the bed surface.

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Hurst's Rescaled Range Analysis (RIS Analysis)

Figures 14a and b plot log-log [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] against [[tau].sub.H] (POX diagram) for the 1.2 mm particles for different static bed depths with U= 0.615 and 0.758 m/s. These graphs indicate two distinct linear zones (bi-fractal behaviour). From the regions where the slopes of log-log (RIS) [[tau].sub.H] versus [[tau].sub.H] are constant, it is possible to determine two distinct Hurst exponent. The graphs for higher gas velocities (Figure 14b) show greater sensitivity to variations in [H.sub.0]. The oscillations, especially apparent for [H.sub.0] = 250 and 300 mm, are related to the pulsating spout regime observed for these operating conditions.

The low-frequency Hurst exponents were estimated from Figure 14b, to investigate their relationship with the various spouting flow regimes. The calculated HLF values (corresponding to frequencies of 1 to 5 Hz), depended on the operating conditions. Initially for [H.sub.0] = 150 mm, HLF was found to be 0.446, indicating a nearly random system, probably due to the low bed depth. Increasing [H.sub.0] first produced a significant drop in HLF (to ~0.18), indicating an anti-persistent system. Subsequent increases in [H.sub.0] led to a rise in [H.sub.LF] (0.24 for [H.sub.0] = 300 mm; 0.30 for [H.sub.0] = 350 mm; 0.36 for [H.sub.0] = 400 mm; ~0.50 for [H.sub.0] = 450 mm). On the other hand, the high-frequency Hurst exponent, [H.sub.HF], estimated from the pressure fluctuation signals, was ~1.0, relatively insensitive to changes in [H.sub.0]. These results are in agreement with Xu et al. (2004b) and suggest an unstable system with a periodic component.

Figures 15a and b show the POX diagram for the 2.4 mm particles as a function of bed height for U = 1.34 and 1.68 m/s, respectively. The low- and high-frequency Hurst exponents can also be determined and related to the system conditions, confirming the utility of R/S analysis for characterizing the spouting flow regimes and/or for monitoring methods. Note also that the [[tau].sub.H] values corresponding to the change in the slope in the POX diagrams tended to increase with [H.sub.0], indicating higher cycle times, tc, especially for [H.sub.0] [greater than or equal to] [H.sub.max]. A higher cycle time corresponds to a lower dominant frequency, corroborating the results of the spectral analysis.

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

Cycle Times (P-Statistics)

Cycle times of time series of total pressure fluctuations were determined for several operating conditions to investigate their feasibility for characterizing the spouted bed hydrodynamic regimes. The cycle times were detected using the V- and P-statistics.

Figures 16a and b plot the P-statistics of the total pressure fluctuation time series for 1.2 mm particles at different static bed heights for U= 0.615 and 0.758 m/s. The graphs for U= 0.615 m/s show a peak, corresponding to the process cycle time. However, due to the small gas velocity, the cycle time remained practically constant and equal to 95 ms.

Visual examination of the spouting flow regime corresponding to this condition indicates a well-organized system with a fine channel through the centre of the bed. This behaviour remained almost constant and independent of [H.sub.0]. The existence of a periodic component is evident for U= 0.748 m/s, in agreement with the pulsating spouting regime detected mainly for 150 [less than or equal to] [H.sub.0] [less than or equal to] 300 mm. For [H.sub.0] = 150 mm, a "jet spouting" flow regime was observed. Increasing [H.sub.0] beyond 350 mm led to a transition, with the formation of bubbles and their bursting at the bed surface. Figure 16b shows that the system cycle time and P-statistic changed as the flow regime varied.

[FIGURE 17 OMITTED]

[FIGURE 18 OMITTED]

Figures 17a and b for the 2.4 mm glass beads show clear changes in the cycle time resulting from variation in system hydrodynamics, confirming the usefulness of this parameter as a tool for process monitoring. It is noteworthy that these results are in good agreement with those from spectral analysis, mutual information and Hurst analysis. The cycle times from Figures 16 and 17 are summarized in Figure 18.

CONCLUSIONS

Spouted bed flow regimes were characterized through mathematical analysis of pressure fluctuation signals measured at several axial positions and by visual observation. Several analysis methods were utilized: statistical, spectral analysis, mutual information, Hurst rescaled range analysis, and the P statistic, in order to evaluate their usefulness for characterization of flow regimes. There were significant changes in flow regime with varying U/[U.sub.ms] and static bed height, [H.sub.0], for glass beads of diameter both 1.2 and 2.4 mm. Instabilities were identified as the static bed depth exceeded [H.sub.max] (internal spouting zone; stable spouting; bubble growth, ascent and bursting). For the larger particles, the statistical analyses could be used to detect system instabilities caused by changes in operating conditions. The spectral analysis, mutual information, R/S analysis, and P-statistic all showed dependence on the hydrodynamic spout flow regimes. Almost all of the methods evaluated in this work were reasonably effective in identifying spouted bed flow regimes, furnishing complementary results. Integration of these methods into a software package to monitor or control fluidized and spouted beds hydrodynamics would be a worthwhile exercise for future research.

NOTATION

[d.sub.p] particle diameter (m)

[d.sub.c] column diameter (m)

[H.sub.HF] high frequency Hurst exponent

[H.sub.LF] low frequency Hurst exponent

[H.sub.0] static bed depth (m)

[H.sub.max] maximum static bed depth (m)

H(Y) system entropy (bits)

I(Y, Y + [tau]) mutual information (bits)

[K.sub.R] excess kurtosis, defined by Equation (3)

MI mutual information

N number of information arrays

[P.sub.D] distributor pressure drop (kPa)

[P.sub.T] total pressure drop (kPa)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] P-statistic, defined by Equation (10)

P([Y.sub.i]) probability of occurrence of event i

[PSD.sub.TISA] time-integral squared pressure amplitude power spectral density

SK skewness

[t.sub.c] cycle time (s)

[t.sub.1],...,[t.sub.n] time periods (s)

U superficial gas velocity (m/s)

[U.sub.ms] minimum spouting velocity (m/s)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] V-statistic, defined by Equation (9)

[X.sub.k,min] minimum accumulated departure for each subperiod k (kPa)

[X.sub.k,max] maximum accumulated departure for each subperiod k (kPa)

Y mean value (various)

Y Y(t) time series (various)

[DELTA][P.sub.m] maximum pressure drop (kPa)

[DELTA][P.sub.s] spouting pressure drop (kPa)

[DELTA]Z axial position (m)

[DELTA]Z particle sphericity

[gamma] exponent in Equation (10)

[sigma] standard deviation, same as Y

[[tau].sub.H] time sub-period (s)

ACKNOWLEDGEMENTS

The authors are grateful to the Brazilian Council for Research and Development (CNPq), for granting a Post-Doctoral fellowship to the first author (Grant 200832/2005-0). Research funding from the Natural Sciences and Engineering Research Council of Canada in support of the research is also acknowledged with gratitude.

Manuscript received December 24, 2007; revised ruannscript received February 15, 2008; accepted for publication February 21, 2008.

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W.P. Oliveira, (1,2) C.R.F. Souza, C.J. Lim (3) * and J.R. Grace (3)

(1.) Universidade de Sao Paulo, Faculdade de Ciencias Formoceuticas de Ribeirao Preto, Av. do Cafe SIN., BL. Q, 14040-903, Ribeirao Preto, SP, Brazil

(2.) Post-doctoral fellow of the Brazilian Council for Scientific and Technological Development (CNPQ) at the University of the British Columbia, Vancouver, BC, Canada

(3.) Department of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, BC, Canada V6T 1Z3

* Author to whom correspondence may be addressed.

E-mail address: cjlim@chml.ubc.ca Table 1. Properties of inert materials (Wang, 2006) Material [d.sub.p] [[rho].sub.s] [phi] [[epsilon].sub.0] (mm) (kg/[m.sup.3]) Glass beads 1.2 2500 1 0.39 Glass beads 2.4 2500 1 0.39