Time-series analysis of pressure fluctuations in gas–solid fluidized beds – A review

International Journal of Multiphase Flow 37 (2011) 403–428

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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w

Review

Time-series analysis of pressure fluctuations in gas–solid fluidized beds – A review J. Ruud van Ommen a,⇑, Srdjan Sasic b, John van der Schaaf c, Stefan Gheorghiu d, Filip Johnsson e, Marc-Olivier Coppens f a

Delft University of Technology, Dept. of Chemical Engineering, Product & Process Engineering, Julianalaan 136, 2628 BL Delft, The Netherlands Chalmers Univ. of Technology, Dept. of Applied Mechanics, SE-41296 Gothenburg, Sweden c Eindhoven Univ. of Technology, Lab. of Chemical Reactor Engineering, P.O. Box 513, 5600 MB Eindhoren, The Netherlands d Center for Complexity Studies, Aleea Parva 5, Bucharest 061942, Romania e Chalmers Univ. of Technology, Dept. of Energy and Environment, SE-41296 Gothenburg, Sweden f Rensselaer Polytechnic Institute, Isermann Department of Chemical and Biological Engineering, 110 8th Street, Troy, NY 12180, USA b

a r t i c l e

i n f o

Article history: Received 17 July 2010 Received in revised form 8 December 2010 Accepted 20 December 2010 Available online 30 December 2010 Keywords: Fluidization Pressure measurements Signal analysis Statistics Spectral analysis Chaos analysis

a b s t r a c t This work reviews methods for time-series analysis for characterization of the dynamics of gas–solid fluidized beds from in-bed pressure measurements for different fluidization regimes. The paper covers analysis in time domain, frequency domain, and in state space. It is a follow-up and an update of a similar review paper written a decade ago. We use the same pressure time-series as used by Johnsson et al. (2000). The paper updates the previous review and includes additional methods for time-series analysis, which have been proposed to investigate dynamics of gas–solid fluidized beds. Results and underlying assumptions of the methods are discussed. Analysis in the time domain is often the simplest approach. The standard deviation of pressure fluctuations is widely used to identify regimes in fluidized beds, but its disadvantage is that it is an indirect measure of the dynamics of the flow. The so-called average cycle time provides information about the relevant time scales of the system, making it an easy-to-calculate alternative to frequency analysis. Autoregressive methods can be used to show an analogy between a fluidized bed and a single or a set of simple mechanical systems acting in parallel. The most common frequency domain method is the power spectrum. We show that – as an alternative to the often used non-parametric methods to estimate the power spectrum – parametric methods can be useful. To capture transient effects on a longer time scale (>1 s), either the transient power spectral density or wavelet analysis can be applied. For the state space analysis, the information given by the Kolmogorov entropy is equivalent to that of the average frequency, obtained in the frequency domain. However, an advantage of certain state space methods, such as attractor comparison, is that they are more sensitive to small changes than frequency domain methods; this feature can be used for, e.g., on-line monitoring. In general, we conclude that, over the past decade, progress has been made in understanding fluidized-bed dynamics by extracting the relevant information from pressure fluctuation data, but the picture is still incomplete. Ó 2010 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Scope and outline of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Standard deviation and higher-order moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Probability distribution of pressure increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Cycle time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Rescaled range analysis and the associated V-statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Autoregressive (AR) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. Summary of time domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⇑ Corresponding author. E-mail address: [email protected] (J.R. van Ommen). 0301-9322/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmultiphaseflow.2010.12.007

404 404 404 405 405 406 407 408 409 410

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4.

5.

6.

7. 8.

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Frequency domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Autoregressive (AR) models to estimate the power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Power spectrum fall-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Transient power spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Wavelet analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1. Summary of frequency domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State space analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Attractor reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Correlation dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Attractor comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1. Summary of state space analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . User guideline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Proper measurement procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Choice of analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A wider scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Supplementary data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Pressure is often chosen to characterize fluid dynamics of gas– solid fluidized beds. The advantage of using pressure is that it is easily measured, even under harsh, industrial conditions. In addition, a pressure measurement system, including pressure sensor and pressure tap, is robust, relatively cheap and virtually nonintrusive, thus avoiding distortion of the flow around the point of measurement. Yet, the interpretation of pressure signals is far from straightforward. It is known that in-bed pressure fluctuations are mainly related to bubble motion within the bed, but the exact origin of the fluctuations has already been debated for a long time (Verloop and Heertjes, 1974; Kage et al., 1991; Musmarra et al., 1995; M’Chirgui et al., 1997; van der Schaaf et al., 1998; Bi, 2007; Sasic et al., 2007). The main limitation in understanding the nature of the pressure signal lies in its intrinsically non-local nature. In that manner, the interpretation of pressure measurements is far more complicated than of a more local measurement such as local solids concentration measurements using optical probes. In recent times, there have been a few comprehensive review studies on pressure fluctuations in a fluidized bed. Bi (2007) reviewed attempts to elucidate the underlying mechanisms of the fluctuations. Sasic et al. (2007) investigated various models for the description of fluidized-bed hydrodynamics, and their use in understanding the involved phenomena. Van Ommen and Mudde (2008) recently reviewed methods to determine the gas– solids distribution in fluidized beds. Since pressure measurements are probably the most common measurement technique for verifying models as well as to determine gas–solids distribution, it is of interest to review available analysis techniques with respect to their ability to describe the dynamics of the bed. Therefore, in this paper, we will apply and critically evaluate time-series analysis techniques, employed in recent papers dealing with pressure signals in fluidized beds. The aim is to investigate which methods provide the possibility to link the characteristics of pressure timeseries to physical phenomena observed in a fluidized bed. This work is a follow-up of the earlier review by Johnsson et al. (2000). Since 2000, a number of additional analysis techniques have been applied on pressure signals in fluidized beds and these recent techniques are in focus here. We will also revisit some of the analysis methods already discussed by Johnsson et al. for those cases where important new developments have been reported over the past decade. This paper is organized in a similar way as the Johnsson et al. paper and applies the same pressure time-series

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as used by Johnsson et al. Thus, by applying a different set of techniques compared to the original work, we will be able to deliver further insight into the relationship between the characteristics of in-bed pressure fluctuations and the dynamics of the bed. In addition, we will present discussion on some of the important open questions in gas–solid fluidization, such as the existence of turbulence and the nature of bed dynamics (chaotic versus stochastic). Finally, we will provide recommendations on the use of the methods examined. Given the considerable amount of work published in the field, we here adopt a somewhat subjective approach and restrict ourselves to methods that, in our opinion, represent a clear advance in understanding or otherwise received considerable attention. 1.1. Scope and outline of the study Literature presents a large number of methods for the analysis of time-series of pressure signals recorded in fluidized beds. Such methods can generally be grouped into three categories: (1) time domain methods, (2) frequency domain methods, and (3) state space methods. The discussion of the various methods in this paper is organized correspondingly. In Section 3 on time domain methods, we first consider simple measures, such as standard deviation and probability density function. Then, we discuss more advanced methods, such as autoregressive models and rescaled range analysis. In Section 4, dealing with frequency domain methods, we discuss power spectral techniques and wavelets. The latter is actually a time–frequency representation of the signal, and therefore it does not belong strictly to either of the domains, but we have nevertheless chosen Section 4 to discuss this method. Section 5 discusses the state space methods. This is done by first reconstructing an attractor from a pressure signal. Then, methods to characterize the attractor, such as the entropy, are evaluated. Sections 3–5 are organized so that they first provide a brief introduction to the theory of the analysis method, followed by the application to pressure data from the four fluidization regimes, and, finally, conclusions and recommendations for the use of each particular method. 2. Experimental The data sets applied here are the same as those used in the previous review paper (Johnsson et al., 2000), in which details on

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600

Pressure [Pa]

400 200 0 -200 -400 -600 0

0.1

0.2

0.3

0.4

0.5

Time [s] Fig. 1. Short time-series of the pressure fluctuations measured in the multiplebubble regime.

were stored on a PC. The sampling frequency was 400 Hz in all cases; 786,432 samples were taken, corresponding to 33 min of total sampling time. In this way, the pressure signals are obtained with a very high precision, and with virtually no random noise present (see Fig. 1). The CFB rig was operated with a constant total inventory of solids and with gas velocities ranging from 0.6 to 5 m/s. In the Johnsson et al. paper, as well as in this paper, we use 17 pressure time-series obtained at various gas velocities to apply the various analysis methods to. Special focus is on four time-series selected from this set of 17. These four are representing four fluidization regimes: the multiple-bubble regime, the single-bubble regime, the exploding bubble regime and the transport regime (see Fig. 2). To obtain the multiple-bubble regime, a distributor with a higher pressure drop was used (Johnsson et al., 2000). The main conditions are given in Table 1. All pressure time-series can be downloaded from the web-site of this journal (see Appendix A). 3. Time domain analysis

pressure measurements are given. The experiments were carried out in a CFB unit operated under ambient conditions. The riser has a cross-section of 0.12  0.7 m and a total height of 8.5 m. The bed material was silica sand with an average particle size of 0.32 mm and a particle density of 2600 kg/m3, i.e., Group B particles. In the riser, pressure fluctuations were measured at 0.2 m above the air distributor through a 50 mm long and 4 mm ID steel tube with a fine mesh net at the side facing the fluidized bed; these probe dimensions in combination with the transducer minimize the distortion of the pressure signal (van Ommen et al., 1999b). The pressure is measured ‘‘single ended’’: the fluctuations are recorded, while the average is set to zero. The transducer (Kistler type 7261) has a response frequency greater than 1 kHz and a pressure resolution of 1.5 Pa. It is connected to a charge amplifier (Kistler type 5011A10) with an adjustable range, facilitating a high response for each condition. The measurement range is typically some hundreds Pa up to a few kPa, depending on the gas velocity. The charge amplifier acts as a high-pass filter with a filter frequency of 0.1 Hz. The pressure transducer was connected to a 16-bit data acquisition board (Difa ABP 200). The signals were low-pass filtered at the Nyquist frequency, and the recorded data

The simplest method of analysis in the time domain is plotting a sequence of data points of the measured signal. It is advisable to always inspect the signal in this way before further processing in order to identify possible abnormalities, for example due to deviating bed behaviour or problems with the data acquisition. The obtained visualisation of the pressure signal gives a qualitative indication of the relevant time scales and of the complexity of the flow. In most fluidized-bed systems, the dominant frequencies are of the order of 1–5 Hz, meaning that a sequence of 10 s is suitable for this purpose (see Fig. 2). In the remainder of this section, we will discuss various methods to characterize pressure data directly in the time domain, i.e. without first carrying out a transformation as in frequency domain analysis and state space analysis. 3.1. Standard deviation and higher-order moments A widely applied time domain analysis method is to determine the amplitude of the pressure signals, expressed in the form of standard deviation or variance (viz., second order statistical moment), or as the average absolute deviation (sometimes referred

multiple bubble 2000 0 -2000 0

2

4

6 single bubble

8

10

2

4 6 exploding bubble

8

10

2

4 6 transport conditions

8

10

8

10

Pressure [Pa]

2000 0 -2000 0 5000 0 -5000 0 500 0 -500 0

2

4

6

Time [s] Fig. 2. Time-series of the pressure fluctuations measured in the four regimes (note the different scale on the vertical axis). The measurements were taken at 0.2 m above the air distributor. Operating conditions according to Table 1.

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Table 1 Operating conditions for the four pressure time-series used in this paper (as well as in the paper by Johnsson et al., 2000).

a

Regime condition

Multiple bubble

Single bubble

Exploding bubble

Transport condition

Gas velocity [m/s] Solids mass flux [kg m2 s1] Bottom bed height [m] Bottom bed voidage [–] Bottom bed pressure drop [Pa] Distributor pressure drop [Pa]

0.6 0 0.40 0.51 4960 4200

0.6 0 0.37 0.50 4730 660

2.2 1 0.30 0.58 3310 3090

4.1 25 – 0.80a 1120a 13,700

No bottom bed present, values given over the lower 20 cm of the columns.

to as the absolute deviation). The latter measure is about 0.8 times the standard deviation for a Gaussian distribution. Its advantage over the standard deviation is that it is less sensitive to outliers. However, typically the two measures show a similar trend (see Fig. 3). The change in the standard deviation (or another measure of the amplitude) has often been used to identify a regime change. For example, it can be used to determine the minimum fluidization velocity (Wilkinson, 1995; Puncˇochárˇ and Drahoš, 2005; Felipe and Rocha, 2007; Sobrino et al., 2008) or as an on-line tool to detect defluidization in industrial fluidized-bed reactors (van Ommen et al., 2004a). The standard deviation is also often used to determine the regime transition from bubbling to turbulent fluidization. A maximum in the standard deviation as a function of the superficial gas velocity indicates the transition velocity, which is commonly denoted as Uc (Bi et al., 2000). Bi et al. showed that Uc is lower for absolute pressure fluctuations than for differential pressure fluctuations. Moreover, Uc from the latter varies with the measurement interval, while Uc from absolute pressure fluctuations is relatively insensitive to the axial location. For our time-series, the maximum in the standard deviation is found around 1.1 m/s, which – with some reservation (Johnsson et al., 2000) – can be interpreted as the change from bubbling to turbulent fluidization. Andreux et al. (2005) showed that the maximum in the pressure fluctuations might well overpredict the gas velocity at which the transition takes place. The standard deviation has also been used for on-line monitoring of fluidized-bed hydrodynamics, for example to determine the particle size (Davies et al., 2008) or the ‘quality of fluidization’ (Chong et al., 1987; Kai and Furusaki, 1987). However, the strong dependence of the standard deviation on the gas velocity – also within a regime – makes it questionable whether these methods are practically applicable in industrial

Stand. dev. [Pa]; Avg. abs. dev. [Pa]

2500 stand. dev.

avg. abs. dev.

2000

1500

1000

500

0

0

1

2

3

4

5

6

Superficial gas velocity [m/s] Fig. 3. Standard deviation and the average absolute deviation as a function of gas velocity for the 16 pressure time-series.

installations, in which the gas velocity is seldom completely constant. Johnsson et al. (2000) concluded that a disadvantage of characterizing the regime by amplitude is that it gives no information on the time scale. Moreover, the amplitude is influenced by the dynamics of the flow, by the distribution of bed material in the system, and by changes in the average suspension density. Since the link between these three effects is unknown, this measure of fluidized bed hydrodynamics should be applied with great care. Higher-order moments, i.e., skewness (normalized third-order statistical moment) and flatness (normalized fourth-order statistical moment, also called kurtosis), which express lack of symmetry (S = 0 for a Gaussian distribution) and sharpness in a probability distribution (F = 3 for a Gaussian distribution) have been applied by only a few researchers in fluidization (Saxena et al., 1993; Johnsson et al., 2000). Instead of using just one or a few measures characterizing the probability density function, one can also use the complete probability density functions (i.e., a histogram of the pressure fluctuations). However, just like using the moments of the distribution, this has the disadvantage that all information on the time scale is lost.

3.2. Probability distribution of pressure increments A suitable alternative to the probability density function of pressure values is to consider the probability density function of pressure increments. Gheorghiu et al. (2003) proposed a statistical analysis of pressure increments Dp = p(t + Dt)  p(t), over a variable time delay Dt. This study was inspired by signal processing methods in turbulence research, where the relevant variable is typically the velocity increment Dv. The advantage of using pressure increments is that, implicitly, the time scale and dynamics of the pressure are included. Moreover, the method is more robust, since it removes artefacts due to fluctuations in the gas flow and any long-term trends in the data. Fig. 4 shows the probability density functions (pdf) of pressure increments Dp, for time delays Dt = 5 ms, 10 ms, 50 ms and 500 ms. The first two values are below the typical cycle time, while the other two are above. The representation is semi-logarithmic, to emphasize the shape of the pdf’s (in this representation, the Gaussian distribution is a parabola). For short time delays the shape of the four pdf’s is very similar, while they differ significantly at large delays Dt. This is in agreement with the picture than can be obtained from power spectra for high frequencies versus low frequencies (see Section 4). Fig. 5 shows a log–log representation of the pdf’s for the four regimes, at short time delays (Dt = 5 ms). Two of the graphs (multiple- and exploding-bubble regimes) show reasonably good power-law tails, while the other two pdf’s feature more complicated shapes. The pdf’s are non-Gaussian for all signals and all selected time delays. In only two regimes, the multiple-bubble and exploding-bubble regime, the pdf’s resemble Gaussian distributions at large Dt. If the short-time signal increments were identically distributed independent variables (‘‘noise’’), this would

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Fig. 4. Probability density functions of pressure increments, Dp = p(t + Dt)  p(t), corresponding to the four selected signals. In every graph, the four lines correspond to Dt = 5 ms (circles), 10 ms (stars), 50 ms (triangles), and 500 ms (crosses), from bottom to top, respectively. The bottom line (circles) is drawn to scale; the others are shifted upwards for clarity.

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bottom zone with high local velocities, thus generating perturbations unevenly distributed in time and space. Although it is possible that such observations are reminiscent of intermittency in fully developed single-phase turbulence, Gheorghiu et al. (2003) proposed an alternative explanation for the non-Gaussian pdf’s based on the polydispersity of the passing voids that generate the pressure fluctuations. We believe that the non-Gaussian statistics are a consequence of the bubble component of the pressure signal rather than a proof for the existence of turbulence. For the three bubbling regimes treated here, the gas–particle mixture is too dense and viscous for turbulence to develop. The situation may be somewhat different for much higher gas flow rates, as in transport conditions. Finally, the very magnitude of the pressure fluctuations analyzed in this paper makes it unlikely that they are produced by the turbulence of the gas phase within individual bubbles. Another important application of this method can be in the timely detection of changes in the fluidization behaviour. Gheorghiu et al. (2004) have shown that the shape of the pdf of pressure increments is rather sensitive to agglomeration during gasification of straw in a fluidized bed, and as such, it may be used for monitoring purposes. We can conclude that probability distributions of pressure increments have only received limited attention so far. It seems to be a promising method, but further research is needed to show whether the method really has significant added value to existing approaches. 3.3. Cycle time

Fig. 5. Log–log representation of the probability density functions of pressure increments for the four selected signals, with time delay Dt = 5 ms. Pressure increments are normalized to their standard deviation r, to emphasize the shape of the distribution. The bottom line (circles) is drawn to scale; the other lines are shifted upwards for clarity.

mean that the long-time increments are Gaussian, by virtue of the fact that they are sums of short-time increments. This behaviour is based on the central limit theorem that states that the sum of a large number of independent observations from the same distribution has, under certain general conditions, an approximate normal distribution. These conditions are that there exist no or only weak correlations between the observations, and that the observations have a finite mean and a finite standard deviation. However, the strongly non-Gaussian nature of the single-bubble signal proves that there are significant correlations on the time scale of 500 ms (the signal of the single-bubble regime is the ‘‘least random’’ of the four regimes). At this point, it is interesting to discuss the existence of turbulence in gas–solid fluidized beds. In the turbulence literature, a non-Gaussian pdf is seen as a signature of intermittency (Mandelbrot, 1982; Frisch, 1995). In bubbling beds or in a dense bottom bed of a circulating bed, gas passes through bubbles and leaves the

In the literature on fluidized beds, power spectral analysis is often applied to analyse pressure fluctuations. This analysis method belongs to the frequency domain methods, and will be discussed in detail in that section. However, applying power spectral analysis to profoundly non-periodic signals, such as those recorded in fluidized beds, may not always be beneficial. Therefore, it is useful to look for alternatives in the time domain. An easy-to-calculate characteristic is the average cycle time, obtained from the number of times a pressure signal crosses its average value. Briens and Briens (2002) state that this measure gives good results for model data, but often fails with experimental time-series since high-frequency fluctuations cause the signal to repeatedly cross the average. It is indeed true that the cycle time is more sensitive to noise in the data than most other analysis methods. Our experience is, however, that with applying proper measurement equipment and low-pass filtering the signal with a cut-off frequency of half or one-third of the sample frequency, the average cycle time normally yields useful values. Bartels et al. (2009b) showed that within the bubbling regime, the average cycle time is virtually independent of gas velocity, bed mass, and particle size, while it strongly changes when particle agglomeration takes place in the bed. Fig. 6 shows the average cycle time as a function of the gas velocity. A change in the trend of the average cycle time typically indicates a regime change. For the time-series applied in this work (cf. Table 1), this would mean that a regime change, most likely from bubbling to turbulent fluidization, takes place around a gas velocity of 0.87 m/s. Note that this value is considerably lower than the 1.1 m/s found from the standard deviation (cf. Fig. 3). Instead of only calculating the average cycle time, one can also plot the cycle-time distribution. Since fluidized-bed pressure signals are typically non-periodic signals containing information at multiple time scale, the presentation of the cycle-time distribution gives a more complete picture. Van Ommen et al. (1999a) showed that the average cycle-time distribution can be used to detect changes in the particle-size distribution. An alternative, but also more complex, way of determining the cycle time of a signal is via the rescaled range analysis, which will be discussed below.

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and momentum transfer (Bai et al., 1997b). Unfortunately, most studies fail to clarify which additional insights the results obtained from the Hurst analysis add to the information obtained by more conventional analysis techniques. The V-statistic is derived from R/S analysis and quantifies the departure of the measured signal from an uncorrelated Brownian process. Briens et al. (1999) successfully used the method to detect local defluidization using triboelectric current signals in a precise manner. To our knowledge, the V-statistic has not yet been applied to pressure signals from gas–solid fluidized beds. Hurst analysis of a signal involves determining the scaling properties of the ratio of the range of the signal to the standard deviation of the signal increments in time-windows of variable size. If x(t) denotes the signal, then let d(t) = x(t + dt)  x(t) denote the signal increments, where dt is the sampling interval. For a chosen time-window of length s, one computes the cumulative departure y(t) of the increments d(t) from the average increment hdis in that time window,

0.8 Circ. conditions with dense bottom bed

Average cycle time [-]

0.7 0.6

Transport conditions

0.5 0.4 0.3 0.2

Non-circulating conditions

0.1 0 0

1

2

3

4

5

6

Superficial gas velocity [m/s] Fig. 6. The average cycle time as a function of the superficial gas velocity. The error bars give the standard deviation as obtained from dividing each time-series in 10 parts, and calculating the average cycle time for each part. The dashed vertical lines give the boundaries between the different regimes with regard to the circulating conditions, as reported by Johnsson et al. (2000). The large squares indicate, from left to right, the values for the selected data sets for the single-bubble, the exploding-bubble, and the transport regime, respectively. The average cycle time for the multiple-bubble regime, 0.178 s, is not shown in this graph.

yðt; sÞ ¼

tþs X

Some of the methods for time-series analysis have focused on characterizing the degree of stochasticity present in fluidized-bed measurements. The so-called rescaled range analysis, also known as R/S or Hurst analysis (Hurst, 1951; Mandelbrot and Wallis, 1969), is one of the cornerstones of fractal analysis and has been used across disciplines, from surface science to physiology and the stock market. Starting with Fan et al. (1990), Hurst analysis has been applied to characterize hydrodynamic regimes and assessment of the quality of flow behaviour in various multiphase systems: bubble columns (Drahoš et al., 1992), dilute flow in horizontal pipes (Cabrejos and Klinzing, 1995), three-phase fluidized beds (Briens et al., 1997), and circulating fluidized beds (Kikuchi and Tsutsumi, 2001). Signals originating from different types of probes were analyzed in this manner: differential pressure in the majority of studies, but also temperature (Karamavruc and Clark, 1997), voidage signals from optical probes (Kikuchi et al., 1997),

Rðt; sÞ ¼ max yði; sÞ  min yði; sÞ and t
s

RðsÞ / sH SðsÞ

ð3:4:3Þ

where 0 < H < 1 denotes the Hurst exponent (H = 0.5 for an ordinary Brownian walk). Practically, H is determined from the slope of a log–log graph of R/S versus s.

single bubble 10

2

0.18 ± 0.05 10

10

0.9 ± 0.05

10

10

0 -2

10

10

-1

10

0

10

1

10

1

0.15 ± 0.1

0.85 ± 0.1

0

10

exploding bubble

2

-2

10

-1

R/S [-]

10

10

10

0

10

-1

10 10 window size τ [s]

0

0

10

1

0.2 ± 0.05

0.85 ± 0.05

-2

10

transport conditions

0.2 ± 0.1 1

ð3:4:2Þ

The ratio of the time-averaged range to time-averaged standard deviation is then studied as a function of window size s. If the signal has self-similar scaling properties (e.g. for a Brownian random walk), the ratio R/S is a power-law of the window size:

multiple bubble

1

ð3:4:1Þ

The new signal y(t, s) is a de-trended version of the original signal x(t), in which the linear trend on scale s is removed. Next, the range of signal y in the given window and the standard deviation of the signal increments in the same window are computed:

3.4. Rescaled range analysis and the associated V-statistic

R/S [-]

ðdðiÞ  hdis Þ

i¼tþ1

10

1

10

1

0.85 ± 0.05

0

10

-2

-1

0

10 10 window size τ [s]

10

1

Fig. 7. Rescaled range analysis of the signals corresponding to the four fluidization regimes, in log–log representation. Two regions of the graph are fitted with straight lines; the Hurst exponent (with its standard deviation) calculated from the slopes is shown next to each region. The dashed line marks the average cycle time for each signal.

J.R. van Ommen et al. / International Journal of Multiphase Flow 37 (2011) 403–428

The V-statistic associated with the signal at a particular scale s is given by:

VðsÞ ¼

RðsÞ=SðsÞ

s1=2

ð3:4:4Þ

It intuitively gives the departure of the signal from the s1/2 scaling, which is characteristic of a Brownian random walk. The R/S analysis results for our four selected pressure timeseries are shown in Fig. 7. All four signals show similar features. Instead of a unique scaling behaviour for R/S, the plots clearly show two distinct regimes, due to the (pseudo)periodic nature of the signals. The shoulder of the R/S curve approximately coincides with the average cycle time of each signal. Within time windows s smaller than the average cycle time, all the signals are strongly correlated (H  1.0 is the mark of a noiseless, smooth signal). This reflects the fact that fluidized-bed pressure signals are composed of individual ‘‘oscillations’’ generated by various events (bubble passage, bed-mass oscillations, bubble formation at the distributor, etc.). On the time scale of a single event, the signal is strongly correlated. In the range of s larger than the average cycle time, the R/S graphs of all four time-series show fairly straight lines with slope around 0.15–0.20. Time windows larger than the average cycle time typically contain more than one pressure ‘‘oscillation’’, so scaling in this range may be due to inter-event correlations and/ or to the existence of a polydisperse distribution of voids. We favour the latter explanation: the distribution of bubble sizes, at least in the multiple-bubble regime, is likely to be a power-law with exponential cut-off for small bubble sizes (Bai et al., 2005), which may explain the scaling of the R/S curve beyond the average cycle time. The V-statistic was computed for all four time-series, for s values of 10 ms (below the average cycle time), 50 ms (roughly equal to the average cycle time), and 500 ms (above the average cycle time). Fig. 8 shows that the V-statistic does not vary significantly among the different flow regimes, and therefore cannot be used to discriminate between them. This is consistent with the previous observation that the R/S graphs corresponding to the four flow regimes are very similar (see Fig. 7). We would like here to caution the reader against any hasty interpretation of H values. The original interpretation of Mandelbrot and Wallis (1969) was in the framework of fractional Brownian motion. In this context, signals are fractal and long-range correlated, with H > 0.5 associated to persistence (conservation of current trends), while H < 0.5 may be interpreted as antipersistence

(fluctuation, frequent reversal of trends). Since then, many authors have somewhat abusively extended the interpretation to other classes of signals. We believe that the four CFB signals that are the object of this paper are examples of signals for which the classical interpretation of rescaled range analysis, in terms of fractional correlated noise, is rather meaningless. There is no fractality or long-range fractal-like correlation present in the CFB signals; the observed scaling properties of the R/S curve can be explained entirely based on short-term correlations of individual events, coupled to the polydispersity of event sizes. Mandelbrot (1982) warns about this distinction of polydispersity from long-range correlation to interpret two distinct classes of phenomena with power-law tails in their pdf (see also Gheorghiu and Coppens, 2004). 3.5. Autoregressive (AR) models Apart from the Hurst Analysis, autoregressive models (AR) can be also used to characterize the stochastic nature of the pressure signal. Brown and Brue (2001) reported that the dynamics of a gas–solid fluidized bed are more or less analogous to those of a mechanical system of a certain order. In such a case, the pressure signal recorded in the bed is assumed to be an output of a linear time-invariant system driven by a forcing function. The forcing function represents a number of apparently random events (e.g., formation of bubbles at the distributor, bubble eruptions at the surface of the bed, etc.) and can thus be approximated as white noise. The former assumption is not without flaws, since the stochastic nature of the input forces creating the pressure signal is not fully understood, and has also been represented as coloured random noise or deterministic chaos. Autoregressive models belong to the class of parametric methods, in which a time-series x(n) is first created by the following difference function:

xðnÞ ¼ 

p X

ak xðn  kÞ þ

k¼1

q X

bk uðn  kÞ

ð3:5:1Þ

k¼0

where u(n) defines the input to the system and ak and bk are coefficients. The process generated by the zero-pole model, Eq. (3.5.1), is called an autoregressive-moving average (ARMA) process of order (p, q). If q = 0 and b0 = 1, the output is called an autoregressive (AR) process of order p. The term ‘‘autoregressive’’ reflects the fact that the signal has a regression on its own past. On the other hand, if p = 0 in Eq. (3.5.1), a moving average (MA) process of order q is defined. So far, only the AR process has been applied to pressure signals in fluidized beds (Zhong and Zhang, 2005) because of its suitability of representing spectra with narrow peaks, such as those encountered in flows governed by large structures (bubbles in fluidized beds), in contrast to the MA process that requires significantly more coefficients to describe a narrow spectrum (Proakis and Manolakis, 1989). Also, Drahoš et al. (1988) have proposed a firstorder AR model to describe regime transitions in a circulating fluidized bed. For the latter purpose, the authors recommended the model for the cases when very short time-series of the pressure signal were available. The AR model of a certain order and at time t is represented by:

AðqÞxðtÞ ¼ eðtÞ

Fig. 8. V-statistic computed for the signals corresponding to the four flow regimes, for window sizes s = 10 ms (circles), 50 ms (triangles) and 500 ms (squares). The error bars give the standard deviation.

409

ð3:5:2Þ

where e(t) is a white noise process with zero mean and variance r2, while A(q) represents the shift operator with the following property: qnf(t) = f(t  n). Here, an autoregressive model (AR) of the pressure time-series is proposed, and it will be shown that the order of the model identifies a mechanical equivalent of certain fluidization behaviour. In addition, parametric methods are applied in this paper

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er-order models are necessary if the lower frequency content is not dominant in the signal. The order of the model identifies a mechanical equivalent of the fluidization behaviour. For example, a model of order 2 or 4 indicates that a certain case can be conceived as a single oscillator (with or without damping) or as a set of coupled oscillators acting in parallel. On the other hand, if the analysis results in a higher-order model, the series oscillates rapidly, and, in such a case, the pressure signal is of fairly stochastic nature. The analysis also shows that the part of the entire signal from which the data set is taken does not play a significant role in determining the model order. The latter is, however, somewhat affected by the length of the time-series and the sampling frequency (see Table 2). The same results obtained for the single and the exploding bubble regimes point out that their dynamical behaviour can be approximated by a second order differential equation. The conclusions presented in Table 2 are further supported by using the relative Akaike criterion, defined by:

for estimation of power spectra of pressure fluctuations. In contrast to more commonly applied methods based on the Fast Fourier Transform (FFT), the parametric methods do not necessitate the availability of long data records to yield the necessary frequency resolution required in many applications (Proakis and Manolakis, 1989). Moreover, these methods do not suffer from spectral leakage effects, originating from windowing as the inherent feature of the FFT methods when using the finite-length data records. In general, the parametric methods are recommended whenever long data records are not available or when the quality of the recorded signal is poor (Proakis and Manolakis, 1989). Also, when evaluating results from CFD simulations of fluidized beds, time-series of sufficient length are often difficult to produce (due to considerable computational time, especially in 3D simulations). The key points in the modelling procedure are the selection of the order (p) and the validation of a model. If the order is too low, excessively smoothed spectra will be obtained. On the other hand, if p is selected too high, false low-energy peaks may occur in the spectrum. The most common criteria for this purpose used in literature are Akaike’s Final Prediction Error (FPE) and Akaike’s Information Criterion (AIC), closely related to the FPE (Proakis and Manolakis, 1989). The FPE criterion is defined by:

FPE ¼ r2p

1 þ p=N 1  p=N

aK ¼

ð3:5:5Þ

Here, the 10th order model is selected as an optimal order of the model in all regimes, since no significant difference is found between (AIC)10 and (AIC)opt. The length of the signals is 60 s with a sampling frequency of 40 Hz (resampled from our original data). Again, the results show (Fig. 9) that the single-bubble and the exploding-bubble regimes can be reproduced even by a second order AR model (relatively low values of a2), whereas higher orders are evidently needed to describe the other two regimes.

ð3:5:3Þ

where r2p is the estimated variance of the prediction error of the order p and N is the length of the data record used in calculations. Note that r2p also represents the variance of the input white noise signal. In the same manner, the prediction coefficients in a linear predictor of the order p are equal to the parameters of an AR process of the same order. The predictor error is defined as the difference between the signal recorded and a linear predictor of the order p. The AIC criterion (Akaike, 1974) is given by:

AIC ¼ ln r2p þ 2p=N

ðAICÞk  ðAICÞopt ðAICÞ1  ðAICÞopt

3.5.1. Summary of time domain analysis In summary, we have discussed in this section a number of both frequently used methods (e.g. the standard deviation or the Hurst analysis) and recently proposed procedures (e.g. studying probability distribution of pressure increments) for investigating pressure signals from fluidized beds in the time domain. We have demonstrated that great caution has to be applied when using the standard deviation for identifying a regime change or studying the quality of fluidization. Using the standard deviation together with pressure drop data will make the picture clearer (Johnsson et al., 2000). For identifying regime changes, the average cycle time can be a good additional method to use. Moreover, the average cycle time can be effectively used for on-line monitoring purposes. We recommend using the average cycle time over the rescaled range analysis (Hurst analysis) for monitoring. The problem with the latter is that it is very difficult to obtain an unambiguous interpretation of the Hurst exponent (i.e. there is no long-range, fractal correlation present in the signals). The probability distribution of pressure increments could give more insight in the polydisperse nature of fluidized beds, but further work is needed at this point. Using autoregressive (AR) models shows that there is a clear analogy between simple mechanical systems (mechanical oscillators) and fluidized beds. It has been shown that the order of the AR model is closely related to the appearance of the pressure signal in the time domain.

ð3:5:4Þ

As the order of the model (p) is increased, the first term in Eq. (3.5.4) decreases. At the same time, the second term increases with the increase of p, and, therefore, a desired, minimal value of AIC is obtained for some optimal value of p. Here, we will demonstrate how the AR models can be used to establish an analogy between fluidized-bed dynamics and a set of well-defined mechanical systems. Since parametric methods are in general advocated when the length of the data record is short, only fractions of the entire signals (20, 60, 120 and 300 s long), taken from different parts of the overall signal, are employed here. The analysis is carried out with resampled signals, with sampling frequencies of 100, 40 and 20 Hz. The optimal order of the AR model applied to the four fluidization cases is determined according to criteria (3.5.3) and (3.5.4). The results, presented in Table 2, clearly indicate that the order of the model is closely related to the appearance of the signal in the time domain and its representation in the frequency domain. Signals with marked trends and whose lower frequency component is dominant in the power spectrum, are generally described by lower order processes. Conversely, high-

Table 2 Optimal orders of the AR model for the discussed cases. Length of the signal (s)

20

Sampling frequency (Hz)

20

40

100

60 20

40

100

120 20

40

100

300 20

40

100

Order of the AR model (–) Single-bubble regime Multiple-bubble regime Exploding-bubble regime Transport conditions

3 7 3 8

4 7 4 9

5 7 5 9

3 8 3 9

4 8 4 10

5 8 5 10

3 8 3 10

4 8 4 10

4 8 4 10

3 8 3 10

3 8 3 10

4 8 4 10

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J.R. van Ommen et al. / International Journal of Multiphase Flow 37 (2011) 403–428

single bubble

0.5

0

0

2

4

6

multiple bubble

1

αk

αk

1

8

0.5

0

10

0

2

4

p exploding bubble

0.5

0

0

2

4

8

10

6

8

10

transport conditions

1

αk

αk

1

6

p

8

10

0.5

0

0

2

4

p

6

p

Fig. 9. Relative Akaike criterion as a function of the order of the autoregressive model (p).

4. Frequency domain analysis Analysis in the frequency domain is a common tool for investigating pressure signals recorded in fluidized beds. This type of analysis is typically carried out using the Fourier transform. Because of the univariate nature of the time-series we study in this paper, we will not discuss methods based on spectral analysis of multiple signals, such as spectral decomposition to determine the average bubble size (van der Schaaf et al., 2002; Zhang et al., 2010). In this section, we will first use a method that treats the ‘regular’ power spectrum in a way somewhat different from what is common practice in fluidization literature in relation to pressure fluctuations. In this method, we will consider the fluctuations as an output of a linear system driven by a forcing function. After that, the power spectra will be evaluated by a method in which the pressure data have to be first generated by a model. Subsequently, we will study dynamics of fluidized beds as inferred from the falloff of a power spectrum. Finally, we will apply wavelet analysis to provide the representation of a signal simultaneously in the time and frequency domains. As a preamble to wavelets, the concept of the so-called transient power spectral density is introduced and discussed. 4.1. Power spectrum Spectral analysis often aims at obtaining dominant frequencies present in the time-series and assigning them to various physical phenomena (Kage et al., 1991). Additionally, it has been used to validate scale-up strategies of fluidized beds by comparing spectra from a model with those from a full-scale unit (Nicastro and Glicksman, 1984). Parise et al. (2009) fitted power spectra with a Gaussian curve in order to detect defluidization of the bed from changes in the average and the standard deviation. The aim of spectral analysis governs the requirements imposed on data sampling. To determine dominant frequencies, sampling with 20 Hz is considered sufficient, since the major fluidized-bed frequency content is typically below 10 Hz. Extension of the range of frequencies beyond those corresponding to bubble dynamics, however, obviously requires higher sampling frequencies.

Procedures for calculating or estimating a power spectrum belong to two categories: parametric and non-parametric methods. The former will be treated in detail in Section 4.2. In the latter, the spectrum is estimated directly from the signal and there is a need for long data records to produce a necessary frequency resolution of the spectrum (Proakis and Manolakis, 1989). Usually, the spectrum is estimated as an average of a number of sub-spectra in order to decrease variance. The number of sub-spectra is chosen to obtain a satisfactory trade-off between frequency resolution and variance. The procedure was already explained in detail in Johnsson et al. (2000). However, a somewhat different approach to this group of methods is provided by Brown and Brue (2001). The idea is to consider pressure fluctuations as an output of a linear time-invariant system driven by a forcing function. The forcing function is represented by a number of random events (forming of bubbles at the air distributor, bubble eruptions at the surface of the bed, etc.) and is therefore approximated as white noise. Pressure spectra of the four fluidization cases are presented as Bode plots and the results indicate that fluidized beds behave like single or multiple second-order mechanical systems acting in parallel. By definition, the frequency response of a linear dynamical model describes how the model reacts to sinusoidal inputs. The response is presented by a Bode plot, which consists of two diagrams: one that illustrates the dependence of the amplitude of the output on the frequency of the input, and one that presents the phase shift, again as a function of the input frequency (Schwarzenbach et al., 1982). The dynamics of the system are characterized by a transfer function, which is related to power spectral density functions of the input and output:

Pxx ðjxÞ ¼ jGðjxÞj2 Puu ðjxÞ

ð4:1:1Þ

If the input signal is assumed to be an uncorrelated random process (white noise) with a zero mean and a variance r2, Eq. (4.1.1) takes the following form:

10 log P xx ðjxÞ ¼ 20 log jGðjxÞj  20 log r

ð4:1:2Þ

The Bode plot is created by plotting 10 log Pxx(jx) (in decibels) versus log x (in rad/s). The second-order transfer function (with the damping n and the natural frequency xn) takes the following form:

J.R. van Ommen et al. / International Journal of Multiphase Flow 37 (2011) 403–428

100

100

80

80

Magnitude (dB)

Magnitude (dB)

412

60 40 20 single bubble

0 -20 -2 10

0

10

2

10

60 40 20 0 -20 -2 10

4

10

80

80

Magnitude (dB)

Magnitude (dB)

100

60 40 20 exploding bubble

-20 -2 10

0

10

2

10

0

10

2

4

10

10

Angular frequency (rad/s)

Angular frequency (rad/s) 100

0

multiple bubble

4

10

Angular frequency (rad/s)

60 40 20 0

transport conditions

-20 -2 10

0

10

2

4

10

10

Angular frequency (rad/s)

Fig. 10. Bode plots for the four fluidization regimes.

GðjxÞ ¼ 1=½1  ðx2 =x2n Þ þ jð2nx=xn Þ or; in Laplace space : GðsÞ ¼ x

2 2 n =ðs

þ 2nxn s þ x

2 nÞ

The magnitude, which we are primarily interested in, will be: 10 logf½1  ðx2 =x2n Þ2 þ ½2nðx=xn Þ2 g dB. It is straightforward to see that: for x=xn  1; jGðjxÞj  10 log 1 ¼ 0 dB, whereas, for x=xn  1jGðjxÞj  20 logðx2 =x2n Þ ¼ 40 log ðx=xn ÞdB: Therefore, the straight line approximation for magnitude is a line at 0 dB for low frequencies, changing to a line with a slope of 40 dB/decade at the cornering frequency (defined by x ¼ xn ). Different curves may be obtained for different values of the damping constant. Brue and Brown (2001) suggested that the dynamics of fluidized beds are better modelled as two second order systems acting in parallel (with their damping constants and natural frequencies, respectively):

Fig. 11. Bode plot of an idealized second-order dynamical system.

1 1   þ    GðjxÞ ¼  2 2 x x x 1  x2 þ j 2n1 xn1 1  x2 þ j 2n2 xxn2 n1

ð4:1:3Þ

n2

Fig. 10 presents Bode plots of pressure fluctuations for the four cases treated in this work. By comparing Fig. 8 and a Bode plot of an idealized second-order dynamical system (Fig. 11), it seems reasonable to represent pressure fluctuations in fluidized beds either as a single, second-order oscillator or as a set of second-order oscillators acting in parallel, with Eq. (4.1.3) as the transfer function. The order of the system defines the number of independent oscillatory phenomena present. Characteristic features of those processes are therefore obtained by fitting the appropriate transfer functions to the Bode plots (Fig. 10). The results are presented in Table 3. From the above, it seems that the systems that exhibit a dominant frequency in the power spectrum of pressure fluctuations may successfully be modelled by single, second-order transfer functions. On the other hand, when the character of a system resembles the narrow-band random noise, higher orders are necessarily used, modelled as multiple systems acting in parallel. Note that the type of behaviour is more significant than the actual numbers obtained from the fitting process. In that manner, we see that the single- and the exploding-bubble regime behave like pronounced, under-damped oscillators. On the other hand, the multiple-bubble regime is still an under-damped system, but with higher damping constants. The regime called ‘‘transport conditions’’ seems to be closest to the ‘‘straight line approximation’’, which means that the Bode plot can be approximated by using a horizontal and a vertical asymptote for low and high frequencies respectively. The conclusions obtained in this section agree well with those obtained in Section 3.5 on autoregressive models. This is not surprising, since the assumptions behind the fluidized-bed dynamics in the two cases are rather similar. Moreover, AR models and Bode plots are a good example of analysis methods in which it is beneficial to have prior information about the system of interest before applying a particular procedure.

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J.R. van Ommen et al. / International Journal of Multiphase Flow 37 (2011) 403–428 Table 3 Characteristic features (best fit) of the transfer functions for the four discussed cases.

Type of dynamical system Damping (n) Natural frequency (x (rad/s))

Single bubble

Multiple bubble

Exploding bubble

Transport conditions

Single oscillator or two oscillators acting in parallel 0.1 and 0.2 4

Multiple oscillators acting in parallel Range: 0.32–0.4 Range: 20–40

Single oscillator

Two oscillators acting in parallel 0.5 and 0.65 14 and 19

4.2. Autoregressive (AR) models to estimate the power spectrum In the fluidization literature, the power spectra of pressure fluctuations are almost always calculated using non-parametric methods, which make no assumption on how the data were generated. In contrast, when using parametric methods, the data are first generated by a model and only then the corresponding spectral estimates are made. In the area of multiphase flows, the use of parametric methods is recommended whenever long data records are not available or when the quality of the recorded signal is poor. Also, when evaluating results from CFD simulations of fluidized beds, time-series of sufficient length are often difficult to produce, due to considerable computational time, especially in 3D simulations. In this work, parametric methods are applied for estimation of power spectra of pressure fluctuations for the four fluidization cases. It will be demonstrated that the order of the model (p, already mentioned in Section 3.5) plays a crucial role in the estimation of the spectra. If the order is too low, excessively smoothed spectra will be obtained. On the other hand, if p is selected too high, false low-energy peaks may occur in the spectrum. An estimation of power spectra by an AR process consists of the following steps: estimation of the autocorrelation function (rxx) from the data, finding the relationship between rxx and the model parameters, computation of those parameters and the output of an AR process by filtering the white noise signal through the AR filter, and, finally, estimation of the spectra. The estimate of the spectrum for the model chosen in this work (the Yule–Walker method) is:

2  p   X 2  j2pfk  PYW ðf Þ ¼ r = 1 þ a ðkÞe   p p xx   k¼1

ð4:2:1Þ

where ap(k) are estimates of the AR model parameters, and x = 2pf. The estimated variance of the pth order prediction error is:

r2p ¼ rxx ð0Þ

p Y

ð1  jap ðkÞj2 Þ

ð4:2:2Þ

k¼1

Apart from the Yule–Walker method, other models, reported in literature, can be used, such as the Burg or the covariance method. A number of parameters influence the performance of a model: length of the data period, stability of the model (it can happen that the model is stable, while the actual time-series is not), signal-to-noise ratio, etc. Consequently, the selection of a model can often be a matter of performing multiple tests and comparisons, and, eventually, an optimization. Here, the results are again presented as Bode plots to accentuate the behaviour of fluidized beds as linear dynamical systems. There are three parameters that determine the quality of a spectral estimate by a certain type of parametric model. These are: the length of a signal (number of samples N), the sampling frequency (fs), and the order of the model (p). Here, the aim is to illustrate the ability of simple parametric models to describe the dynamics of pressure fluctuations, and therefore, the order of the model (p) is kept as shown in Table 2 or Fig. 9. In other words, the effect on the quality of the spectral estimate of increasing the order of the model is not of interest in the present work. Consequently, the focus is on the effect of the other two parameters mentioned above.

0.14 7

In that manner, Fig. 12 compares Bode plots (only magnitude is relevant in this work, i.e., the phase shift is not shown) of pressure fluctuations in the cases treated, obtained by the non-parametric method applied on the entire length of the signals (thin solid lines in Fig. 12), and from the reduced signals estimated by the parametric method (thick solid lines in Fig. 12) described above (Eqs. (4.2.1) and (4.2.2)). The reduced signals are 60 s long and the sampling frequency is 20 Hz. The length and sampling frequency of the latter are chosen based on the appearance of the time scale of the time series recorded. There is no optimal choice, but the selection should be done so that the AR model fully reproduces the dynamics of the system, in our case both with respect to large scale fluctuations present in the single and the exploding-bubble regimes, and to the shape of the fall-off. A reduction in the sampling frequency, for example to 10 Hz, while keeping the same length of the signals, would provide information related only to the dynamics of the large scale fluctuations, i.e. the bubble flow (low-frequency fluctuations). Alternatively, a doubling of the number of samples taken (the length of the signals), while keeping the same sampling frequency (20 Hz), does not provide any additional information compared to the one already presented in Fig. 12. If no attention is paid to the appearance of the signal, there is a risk of oversampling, which will result in that the large scale fluctuations are not resolved in the spectrum. The thick dashed lines in Fig. 12 show such an example, where the length of the signals is kept to 60 s while increasing the sampling frequency to 100 Hz. As can be seen, the large scale fluctuations (single-bubble and exploding-bubble regimes) are not resolved, and, thus, only for the cases where there are no such structures, the spectra are correctly represented (the multiple-bubble and the transport regimes). The only way to capture the low-frequency peaks for an oversampled signal would be to considerably increase the order of the model. But, as indicated above, the goal is to keep the order of the model as low as possible.

4.3. Power spectrum fall-off Unpredictable behaviour in physical systems may be present in two entirely different types of systems: deterministic ones, even when the number of degrees of freedom is relatively small (e.g., Lorenz model), and in systems dominated by noise. In mathematical terms, the question is whether a certain, apparently random behaviour can be explained by a small number of deterministic equations or by a stochastic process. The fall-off in power spectra at high frequencies represents a possible tool for distinguishing those two types of systems. Furthermore, it should be used prior to applying some other methods that already predefine the type of the system to be investigated, such as calculating dimensions and Lyapunov exponents. to characterize chaotic behaviour in dynamical systems. It has been hypothesized that power spectra from systems exhibiting deterministic chaos decay exponentially at high frequencies, whereas the spectra from systems dominated by noise decay via a power-law (Sigeti, 1995). Considerable effort has been taken in the past to mathematically prove these statements. For example, Ruelle (1986) investigated deterministic systems, while others (Brey et al., 1984; Caroli et al., 1982; Sigeti and Horsthemke, 1987) looked into stochastic systems.

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Fig. 12. Bode plots of pressure fluctuations for the four regimes. Thick solid lines: parametric model with order of the model given in Table 2, based on 60 s signals with 20 Hz sampling frequency. Thick dashed lines: parametric model based on 60 s signals with 100 Hz sampling frequency. Thin solid lines: non-parametric method, the spectral estimate based on the entire signal length.

Sigeti (1995) also concluded that the exponential decay constant for systems that are chaotic in nature represents an invariant of the system. A brief outline of the theory of stochastic processes helps to understand this. For a single-variable, stationary stochastic process, the increment over small time steps is given by a Langevin equation:

dX ¼ AðXÞdt þ BðXÞdW

ð4:3:1Þ

where W is a set of independent Wiener processes (whose statistical moments are worked out from the properties of white noise), A(X) is the drift vector for X and B(X) is the diffusion matrix for X. Equations of this type appear naturally in engineering applications and statistical mechanical systems, where noise is to be introduced into differential equations that describe the evolution of a physical system (Arnold, 1974). The stochastic differential equation corresponding to Eq. (4.3.1) describes the trajectory of a stochastic process:

dX ¼ AðXÞ þ BðXÞn dt

ð4:3:2Þ

where n is the vector of independent Gaussian white noises. Regarding the behaviour of stochastic processes at high frequencies, Brey et al. (1984) established that the high-frequency spectra of systems described by a first-order Langevin equation have a x2 behaviour, irrespective of the possible nonlinearities present in the system. On the other hand, Caroli et al. (1982) showed that, for systems described by second-order Langevin equations, the high-frequency spectra have an x4 asymptotic behaviour. Sigeti and Horsthemke (1987) derived the asymptotic series describing the high-frequency fall-off applicable to both classes of systems (for those with a finite number of degrees of freedom and for the ones with an infinite

number of degrees of freedom). Furthermore, they studied effects of adding coloured noise to the system, and showed that there might be a crossover in power spectra, indicating the correlation time of the noise. Power spectra of pressure fluctuations for the four cases treated here can be divided into three regions: a region corresponding to larger structures of the flow field (related to bubbles and clusters) and, at higher frequencies, two regions representing finer structures (regions 2 and 3 in what follows). Fig. 13 zooms in on regions 2 and 3, and we will discuss whether the nature of the dynamical system of interest can be deduced from the type of fall-off (exponential or power-law) in those two regions. Both logarithmic and semi-logarithmic plots are given, since a power-law fall-off gives a straight line in the logarithmic, whereas an exponential fall-off provides a straight line in the semi-logarithmic plot. In most cases, it is not an easy task to obtain firm conclusions for measured time-series by studying the fall-off of a power spectrum. The reason is that there is always some noise present in the signals, even if it is not directly observable. If we start with region 2 (roughly between 4 and 10 Hz), there are two possibilities in general: an exponential and a power-law fall-off. If the former is present, it implies that region 2 is the region where deterministic dynamics are expected to be still preserved. The actual size of this region depends on the level of noise, assumed to have the tendency to flatten the spectrum (i.e., to turn it into the power-law fall-off, characteristic for noise). Generally, a system with a low level of noise will experience a longer region of exponential decay, and eventually power-law decay at high frequencies. Therefore, the longer the region of exponential decay, the more relevant the deterministic dynamics are, up to longer time scales and larger amplitudes. In other words, the exponential decay preserves the

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Fig. 13. Regions 2 (below 10 Hz) and 3 (above 10 Hz) of the power spectra of pressure fluctuations for the four regimes in (a) a logarithmic plot and (b) a semi-logarithmic plot.

deterministic dynamics at larger scales at the expense of noise that is, in real systems, always dominant at smaller scales. Also, there is another interesting aspect of this issue in real systems: deterministic dynamics imply that all characteristic features of the system, including the decay constant, originate from the solutions of deterministic equations of motion, which are, by definition, analytic functions of time t, for real t. On the other hand, it has been proven that the presence of noise violates the analytic character of the solution. This would mean that noisy (i.e. real) systems cannot have exponential decay in the power spectrum at all! This difficulty is overcome by introducing the concept of so-called shadowing orbits, which preserves the region of the exponential decay in the presence of noise. For further details, the reader should consult the work of Sigeti (1995) and the references therein. Also, another question is of relevance when concerning the presence of exponential decay in the power spectra in the cases treated: from Figs. 13a and b it is seen that region 2 is limited only to the frequency range of 4–10 Hz (in the multiple-bubble case, it is not even straightforward how to clearly separate regions 1 and 2) and is significantly smaller than region 3. Keeping in mind the rather rapid decline of the amplitude with the increase of the frequency in region 2, and additionally, its limited size, it is possible that this region is seriously affected by the so-called end effects (Abarbanel et al., 1993). The authors recommend a method to check for the presence of the end effects (for details, see the reference mentioned above). In summary, great caution should be exercised when a certain

region in the power spectrum is to be modelled using exponential decay. If, on the other hand, the power-law fall-off is suggested for region 2, it means that, in this case, we deal with two regions with this type of decay, with, however, two different slopes (Johnsson et al., 2000). Sigeti and Horsthemke (1987) derived the asymptotic series for the power spectra and applied the formula to both white and coloured noise. They showed that the white noise case produces the x2 fall-off, whereas the coloured noise yields the x4 fall-off. In our case, it is evident that constants have the value of approximately 2 for region 2 (except for the exploding-bubble case) and the value close to 4 (with exceptions present in low velocity cases) in region 3. The point where those two asymptotes (x2 and x4) intersect is especially interesting. It defines the correlation time of noise, and if the characteristic frequency of the system is smaller than the frequency value at the intersection, the noise in region 2 will be white. For high frequencies the power spectrum will be a straight line with a slope of 4 (coloured noise). In our case, it seems that the correlation time of noise present in the system is of the order of 1/10 s. The presence of the power-law fall-off at high frequencies again provokes a hypothesis of the existence of gas phase turbulence in fluidized beds. The latter is motivated by stating that pressure fluctuations are predominantly caused by oscillations of the kinetic energy in the gas phase. In such a scenario, the gas flow of relatively high velocity, passing through the bubbles in the bottom bed, is

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cedure is as follows: the signal treated is divided into a number of segments and the power spectral density is calculated for each of them. The process results in a set of spectral densities as a function of time covering the entire time sequence. The outcome of the analysis is a surface plot with different colour intensities indicating the intensity of the power. Fig. 14 illustrates the transient power spectral densities calculated for the four pressure signals treated here. The calculations are performed for a variety of different cases with the following parameters being systematically changed in the time-series: (a) sampling frequency (400 Hz as the original sampling frequency, 40 Hz and 20 Hz). The 20 Hz sampling frequency is still acceptable for capturing the major frequency content in fluidized beds (Johnsson et al., 2000) and (b) the length of each segment for which the average power spectral density is calculated (1 min and 2 min), which implies that either 32 or 16 segments are used. Fig. 14 is representative for all the cases investigated, which is the reason why only one figure is presented. It can be observed that, for the single-bubble regime, only a single dominant frequency exists. For the multiple-bubble case, no dominant frequency exists. The same can be concluded from the power spectrum of the entire signal, which contains a broad band of frequencies and with a maximum at about 3–5 Hz. As for the exploding-bubble regime, the power spectrum exhibits a pronounced peak at 1.2 Hz and no other significant modes are found. Finally, for the transport conditions, the dominant frequency of bed-mass oscillations has no physical meaning. Nevertheless, maximum energy is within the frequency range of 0–3 Hz, which is also seen in Fig. 14. In summary, the presence of multiple modes of bed mass oscillations may be expected in beds with a relatively large diameter. In such cases, recording pressure signals from different sides of the bed (at the same height) can produce different dominant frequencies in the power spectra. However for the cases investigated in this work, and having in mind the geometry of the bed, this is not the case. The technique described here cannot provide accurate and quantitative information on time localization of particular frequency components: it does not have the accuracy in time localization in the order of seconds, like wavelet analysis has (see next section). It is not straightforward how to select a suitable number

assumed to be the main source of turbulence (Sternéus et al., 1999). However, we believe that turbulence is not a likely explanation for the observed shape of the fall-off. We have already discussed this issue in detail in Section 3.2. In support of the conclusions obtained, Bai et al. (2005) have shown that, for bubbling fluidized beds, the bubble size distribution in the bed has a power-law tail, which explains the power-law tail of the distribution of pressure fluctuations. Finally, region 3 is unambiguously modelled using a power-law fall-off, representative of a stochastic system. In conclusion, we have shown that all the methods treated in Sections 4.1–4.3 yield valuable information on the dynamics of fluidized beds. However, when a transformation from the time to the frequency domain is carried out, it is not possible any more to know when a certain phenomenon has taken place. An attempt to introduce time localization of the already identified frequency components in the signals will be introduced in Sections 4.4 and 4.5. 4.4. Transient power spectral density Gas–solid fluidized beds are often characterized by a single, dominant (also called ‘‘natural’’) frequency of bed-mass oscillations (Davidson, 1968; Baskakov et al., 1986; Roy et al., 1990). This frequency is dependent on the fluidization conditions, and the properties of the corresponding expressions have been outlined in detail in literature (van der Schaaf et al., 1999). The presence of different modes of bed-mass oscillations at constant measurement conditions has also been reported in literature (van der Schaaf et al., 1999). The phenomenon is explained by the existence of a sloshing motion of the bed surface. The oscillations generated by this phenomenon are not presented as the natural frequency of a fluidized bed. It is concluded that this mechanism of bed-mass oscillations can be important only for beds with a relatively large bed diameter. The presence of multiple modes can be investigated by determining the so-called transient power spectral density. It is a simple technique, using standard Fourier methods, to observe whether the frequency composition of a certain signal changes in time. The pro-

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Fig. 14. The transient spectral density for the four selected cases, applied to normalized pressure signals. The sampling frequency is 20 Hz and the entire time-series is divided into 32 segments of 1 min each. Red colour indicates the maximum intensity of the power spectral density for the corresponding case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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of segments into which the signal should be divided. Instead, the goal of such a method is to give a first indication whether there exists a transition between different frequency components (i.e. modes of oscillation) in a pressure signal. In summary, the procedure may point to the existence of temporal changes in the oscillation mode of a certain fluidization regime, and thus, advocate the usefulness of employing more advanced methods, such as wavelet analysis, described in the next section.

4.5. Wavelet analysis Wavelets provide an approach to signal processing which allows for the representation of a signal simultaneously in time and in frequency. In this respect, wavelet analysis is similar to the short time (or windowed) Fourier transform. Since wavelets have only rather recently been introduced into analysis of dynamics of fluidized beds, we have chosen here to present a somewhat more detailed theoretical background of the technique. Wavelet analysis exists in two forms: a continuous and a discrete wavelet transform. The main applications of the continuous version are as a discontinuity detector in signal and image processing, and as means to extract self-affine or fractal-like features of signals. The discrete wavelet transform is mostly used for de-noising and data compression applications. Several authors have recognized the advantages of applying wavelet analysis tools to fluidized-bed signals, in characterizing the heterogeneous nature of fluidization, and for the study of short-time or transient phenomena. Since fluidization is a multiscale phenomenon, signals measured in fluidized beds typically contain components on at least three frequency scales: high frequency ‘‘noise’’ associated to particle motion, medium frequency structures from particle clusters, and low frequency components quantifying the formation and motion of voids. The capability of wavelets to naturally separate these three frequency bands, without erasing the position information, was shown by several researchers (e.g., Li and Kwauk, 1994; Li, 2000; Ren et al., 2001; Ellis et al., 2004). Yang and Leu (2008) advocated using the ratio of energies contained in the high and low frequency wavelet bands to characterize fluidization regimes. Several researchers proposed that the information about bubbles is concentrated in certain scales of the wavelet transform (Lu and Li, 1999; Guo et al., 2003; Yurong et al., 2004; Sasic et al., 2006). The wavelet transform has been used by Wu et al. (2007) to compute the average cycle time, and show differences in fluidization behaviour between different particle systems. Signal de-noising algorithms based on wavelets have been used to enhance commonly used methods of determining the void-size and particle-size distribution (Seleghim and Milioli, 2001; Chen and Chen, 2008). Briens et al. (2003) used the ratio between small and large pressure fluctuations obtained using wavelets to assess the bed fluidity. In recent years, the wavelet transform modulus maxima (WTMM) method has become the standard tool for extracting fractal features from signals and images. Mallat and Hwang (1992) and Muzy et al. (1994) showed that maxima of the continuous wavelet transform at different scales contain information about Lipschitz– Hölder exponents of the signal (local equivalents of the Hurst exponent). From this perspective, WTMM can be seen as a more sophisticated version of the R/S analysis. As such, it has been applied by Chen et al. (2004) to a fluidized-bed pressure signal. Zhao and Yang (2003) combined continuous and discrete wavelet transforms for Hurst exponent determination. The localization property of wavelets has been used by Ren et al. (2001) to detect contours of voids and clusters in images of fluidized beds, while a study by Ellis et al. (2004) attempted a wavelet approach to the shape of optical probe signals.

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The continuous wavelet transform (CWT) of a signal f at time t and scale s (to be explained in detail below) is defined as:

1 W s f ðtÞ ¼ pffiffi s

Z

  ts ds f ðtÞw s

ð4:5:1Þ

i.e., a convolution of the signal with a dilated version of an analyzing function w called ‘‘wavelet’’. To qualify as a wavelet, function w must be well localized both in time and in frequency. Its tails must decay fast at ±1; compactly supported functions are the most desirable (Daubechies, 1992). Optionally, it can have a number of vanishing moments, which are important for edge or singularity pffiffi detection. The normalization factor s ensures that the energy (power) of the signal is maintained upon transformation. The discrete version of the wavelet transform (DWT) is based on a pair of digital filters, which decompose the signal into a low frequency component A1 called the ‘‘approximation’’, and a high frequency component D1 called the ‘‘detail’’. The operation is then repeated using the approximation A1 as the input signal. By doing this operation recursively up to a desired level N, one obtains a hierarchical multiresolution representation of a signal f (Mallat, 1989):

f ¼ D1 þ D2 þ þ DN þ AN

ð4:5:2Þ

such that each detail Dk contains frequency information in a range around fs/2k, where fs is the sampling frequency, and k is an integer. Each of those frequency bands defines a scale. At scale index 1, there are N/2 wavelet coefficients (N is the total number of data points in the signal), at index 2 there are N/4 and so on, to the single coefficient at the final scale m determined as m = log 2(N). In summary, the scale is inversely proportional to the frequency of the classical Fourier analysis, which means that every scale is spanned by a single period wave. This operation is fast and yields a very compact representation of the signal. Additionally, the inverse wavelet transform allows for reconstruction of a signal without loss of information. The Daubechies wavelets are the most frequently used, but several other families of such filters exist. Low-pass wavelet filtering is implemented by the inverse transform of the approximation coefficients AN. High-pass filtering involves a similar reconstruction, using only the detail coefficients D1, . . . , DN up to the desired scale. A drawback of wavelet analysis is that the frequency scale is logarithmic, and therefore not very accurate, especially at high frequencies. If the studied phenomenon does not fall exactly into one of the frequency bands of the wavelet decomposition, its energy will spill over into the neighbouring bands, which may make interpretation more difficult. This inconvenience can be overcome by using an extension of wavelet theory, wavelet packets (Coifman and Wickerhauser, 1992), which can be designed with a discrete, but linear frequency scale. One can even use continuous wavelet transforms to obtain a smooth scale of frequencies, although for continuous transforms no fast computation algorithm is available. An alternative multi-resolution method based on the Hilbert– Huang transform method has been proposed by Villa Briongos et al. (2006, 2007), of which the authors claim that it is better able to deal with non-linear and non-stationary data. Recently, de Martín et al. (2010) showed that the Hilbert transform can be used to extract low-frequency information from acoustic signals, similar to the information obtained from pressure measurements. Since the discrete wavelet transform has been used most in fluidization research, we will also apply this method to analyse our set of pressure signals. The signals are decomposed up to the 9th level, using the discrete version of the Meyer wavelet, implemented in the MatlabÒ Wavelet Toolbox. The choice of a particular wavelet family is not essential for the results; the Meyer wavelet is selected because it has higher regularity than Daubechies wavelets. For every level k of the decomposition, a reconstruction was

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10-6 A9 D9 D8 D7 D6 D5 D4 D3 D2 D1 Fig. 16. Log–log representation of the wavelet power spectrum of the four selected signals, computed from a decomposition up to level 9, using the Meyer wavelet. The horizontal axis corresponds to frequency, increasing from left to right. A stands for approximation (low frequency) and D for detail (high frequency) at different scales.

computed using only the detail coefficients Dk (see Fig. 15). This is somewhat equivalent to band-pass filtering the original signal with the frequency window [fs/2k+1, fs/2k]. The variance of this reconstruction is proportional to the power of the signal in that particular frequency window, and is used as the equivalent of the power spectral density. The resulting wavelet spectrum is shown in Fig. 16. The main features of the power spectral density analysis of the four signals are nicely recovered by wavelet analysis (in general, peaks at low frequencies and power-law tails at high frequencies). The multiple-bubble signal shows a pronounced peak in energy corresponding to intermediate frequency (details D5–D7), which roughly corresponds to pressure fluctuations due to bubble passage. One can also see a broadening of the high-frequency tail of the wavelet spectrum with increasing gas flow rate, as more and more power moves from lower to higher frequencies. Another natural application of discrete wavelet analysis is in the study of turbulence already discussed in Sections 3.2 and

4.3 by other methods. As shown by Johnsson et al. (2000), highpass filtering with an increasing cut-off frequency, combined with a computation of the flatness (kurtosis, or fourth statistical moment) of the filtered signals, can reveal the degree of intermittency present in the analyzed signal. Here we repeat the calculation, using wavelet high-pass filtering instead of usual Fourier filtering. To achieve that, reconstructions Rk of the signal f(t) are computed using only detail coefficients up to a varying level k: Rk = D1 + D2 + + Dk. (Fig. 17). The ‘‘bursting’’ nature of signals R2–R5 can be interpreted as intermittency. A quantitative assessment of this phenomenon is obtained by computing the flatness (F) of the reconstructed signals. A plot of F against the level of the reconstruction is given in Fig. 18. For all hydrodynamic regimes with the exception of transport conditions, flatness increases dramatically with increasing cut-off frequency, signalling intermittency. The fact that wavelets classify the multiple-bubble regime signal also as ‘‘intermittent’’ is in contradiction with the original calculation of Johnsson et al. (2000), which used regular Fourier filtering. This inconsistency may be due to the lower capacity of wavelets to discern high frequencies with good enough resolution. In summary and along with the discussion presented in Sections 3.2 and 4.3, we believe that the observed ‘‘intermittent’’ behaviour of all the signals cannot be attributed to the presence of turbulence in fluidized beds. As indicated above, wavelet packets can be used in order to overcome the problem of interpretation when the studied phenomenon does not reside precisely into one of the energy bands of the wavelet decomposition. With wavelet packets, instead of decomposing only the approximation Ai at stage i, both Ai and Di are passed through the low- and high-pass filters, producing four components: an approximation of the approximation, a detail of the approximation (so far, it is just the wavelet decomposition), an approximation of the detail, and a detail of the detail (these two are new). For a three-level decomposition, the tree is presented in Fig. 19. As a consequence, a large number of wavelet packet representations are possible. In the example above, we can have S = AAA + AAD + ADA + ADD + D, or S = AA + ADA + ADD + D, etc. The various combinations are useful when compressing and de-noising signals, having in mind that the ‘‘essence’’ of the signal may reside in any of these nodes or combinations of nodes, so that

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the freedom to choose the ‘‘best tree’’ allows for obtaining the optimal effect. We will now have a look at decomposition, comprising only terminal nodes: S = AAA + AAD + ADA + ADD + DAA + DAD + DDA + DDD, using the notation of Fig. 19. In this representation, the frequency scale is much more uniformly divided, being ‘‘almost’’ uniform, and one can compute ‘‘pseudo-frequencies’’ of the centre of the band corresponding to each node. We take exemplary slices of 211 points = 2048 points = 5 s of data from all four signals, starting at position 10,000 (roughly 25 s into each data set). We use the Meyer wavelet and run the WP decomposition up to level 7. The choice of wavelet is not essential in any of these studies, although ideally one would want a shape that is closest to the phenomenon of interest. The Meyer wavelet looks more like

Fig. 18. Intermittency analysis based on calculation of the flatness of wavelet-based high-pass filtering of the pressure signals. The horizontal axis is logarithmic, and corresponds to increasing cut-off frequency. Rk denote signals reconstructed from details up to scale k.

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Fig. 20. Logarithm of the wavelet packets coefficients for the exploding bubble regime.

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4.5.1. Summary of frequency domain analysis We have chosen in this section to carry out the frequency domain analysis using methods less commonly employed in the literature related to fluidized beds. Our motivation for this choice is that the use of the classical, non-parametric methods on the same set of signals has already been discussed in detail in the previous study by Johnsson et al. (2000). Therefore, as an alternative modelling approach, we have considered here the fluctuations as an output of a linear time-invariant system driven by a forcing function. Comparing the Bode plots of the signals treated here with the plots of idealized mechanical systems, it is concluded that fluidized beds behave like single or multiple second-order mechanical systems (oscillators) acting in parallel. At the same time, one has to keep in mind that it is somewhat controversial to accept white noise

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a ‘‘bubble’’: it is fairly symmetric, wiggly, and smooth, as opposed to Daubechies which are asymmetric and somewhat sharp-edged. We have also tested the Coiffman wavelets and obtained the same picture. The first step is to plot the wavelet packet (WP) coefficients of the terminal nodes in order to get the WP equivalent of the Fourier spectrum. The result is that WP coefficients decrease rapidly with increasing frequency, so that the figure only contains the approximation coefficients (not shown here). However, if the logarithm of the coefficients is plotted, the results are equivalent to the spectrum on a semi-log scale. As an example, Fig. 20 shows the results for the exploding-bubble regime. It is difficult to claim that there is a clear separation in frequency between different components of the signal, and the situation is similar for all the cases. The frequencies seem to be mixed together, and the conclusion is that the broader bands of an ordinary wavelet analysis do a better job of separating them. If, alternatively, the reconstructions of the signals from the coefficients plotted above are presented, we get back the ‘‘total’’ time resolution (Fig. 21, the exploding-bubble regime). In the procedure, all coefficients of the terminal nodes are set to zero, except for one terminal node, and the signal is reconstructed from just the coefficients of that terminal node. It is now possible to observe the various components as ‘‘shadows’’. There is a different situation for the single-bubble regime, Fig. 22, in which the ‘‘sea of blue’’ is clearly dominant in the figure.

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time [s] Fig. 22. Reconstruction of the original signal from the wavelet packets coefficients for the single-bubble regime.

as input or that the obtained similarity is actually a proof that a fluidized bed is a linear system. Also, we have demonstrated in this section the use of parametric, model-based methods for estimation of power spectrum of pressure fluctuations. It is argued that those methods should be used whenever long data records and/or highquality data are not available for some reason. The shape of the fall-off of pressure spectra is proposed to deduce the dynamical nature (i.e. deterministic versus stochastic) of the system. To study different modes of bed-mass oscillation, a simple method called the transient power spectral density is suggested here. For the signals we have used, the method is not useful for proving a simultaneous presence of different frequency components in the signals. Possibly, the method can be useful in beds with a relatively large cross-sectional area. To provide accurate and quantitative information on time localization of particular frequency components in a pressure signal, we have analyzed the signals using wavelets and wavelet packets. It is demonstrated that wavelets can be a useful tool when studying intermittency in fluidized beds, and, moreover, that the results can be somewhat different from those obtained when using regular Fourier filtering. In addition, it is shown that the main features of the spectral density analysis can be adequately reproduced by wavelets analysis. Wavelet packets are used to show a recognizable separation in frequency between different components of the signals. The latter is achieved with varying success for the four regimes treated here, and it is not clear whether that result is a consequence of the method itself or of the particular signals used in this work. Having in mind that the wavelets are at present used in fluidization literature mostly to confirm results obtained by simpler methods, it appears that their full potential is yet to be demonstrated.

5. State space analysis Complementing analysis in the time and frequency domain, the fluidized-bed pressure signal can be studied in the state space or phase space. This approach is typical for non-linear analysis, and

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exploding bubble

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of a single characteristic variable of the system. Using the so-called time-delay coordinates, it is possible to convert a normalized pressure time-series (p1, p2, . . . , pNp) consisting of Np values into a set of Np–m delay vectors Pk with m elements, where Pk = (pk, pk+1, . . . , pk+m1)T. For example, if we use an embedding dimension (m) of 4 then we obtain P1 = (p1, p2, p3, p4), P2 = (p2, p3, p4, p5), etc. The value of 4 is just chosen for the sake of illustration; typically higher values are used. The subsequent delay vectors P1, P2, etc. can be regarded as points in an m-dimensional state space yielding a reconstructed attractor. Fig. 23 shows the attractors for the four timeseries we study in this paper. To make visualisation possible, we have just given two-dimensional attractors. In practice, typically embedding dimensions in the order of 10–100 are used. The attractor plots themselves do not give too much information: we will need to derive characteristic numbers from them, as will be described in the next sections.

single bubble

4

2

-4 -4

p (t+τ)

multiple bubble

-2

0

2

4

transport conditions

-4 -4

-2

p(t) [-]

0

2

5.2. Entropy

4

p(t) [-]

Fig. 23. Two-dimensional attractors obtained by plotting p(t + s) versus p(t) with a s of 100 ms for the four selected data sets. The signals are normalized before attractor reconstruction.

became popular after Stringer raised the question in 1989 whether a fluidized bed can be regarded as a chaotic system, in the mathematical sense of the word (Stringer, 1989). Daw et al. (1990, 1995) and Schouten and van den Bleek (1991; van den Bleek and Schouten, 1993) demonstrated the application of techniques from chaos analysis to pressure data from a fluidized bed. State space analysis or chaos analysis of fluidized-bed pressure data has been extensively applied since the second half of the 1990s. In this section, we will treat the commonly applied state space methods, with emphasis on the question whether it can be proven if a fluidized bed exhibit chaotic data, and whether the method can also be applied for on-line monitoring purposes. 5.1. Attractor reconstruction In theory, the state of a system (such as a fluidized bed) can be determined by projecting all variables governing the system into a multidimensional space, the state space, for a certain moment in time. The collection of the successive states of the system during its evolution in time is called the ‘attractor’. This attractor can be regarded as a ‘finger print’ of the dynamics of the system. In practice, it is impossible to know all governing variables of a complex multiphase system. However, Takens (1981) proved that the attractor of a system can be reconstructed from the time-series

6

(a)

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Superficial gas velocity [m/s]

Avg. cycle freq. [s]

Entropy [bits/s]

35

In the past 20 years, various characteristics have been used to characterize the chaotic behaviour of a fluidized bed, such as the correlation dimension, the Lyapunov exponents, and the Kolmogorov entropy (also known as correlation or Shannon entropy). These characteristic numbers are typically calculated via attractor reconstruction. Especially the Kolmogorov entropy has often been used to characterize fluidized-bed hydrodynamics. A convenient procedure to calculate a maximum likelihood estimate of the Kolmogorov entropy from experimental time-series is given by Schouten et al. (1994b). The basic idea is that two points on the attractor that are closer than a certain (small) length scale are followed in time, until their distance has grown larger then this chosen length scale. By using a large number of pairs, the entropy can be determined with a rather high precision; see the small standard deviation given by the error bars in Fig. 24a. The shorter the time two initially nearby points need to diverge, the higher the entropy. Linear systems have a Kolmogorov entropy of zero and are predictable at infinitum, whereas random systems have an infinite correlation entropy and are thus unpredictable. For a chaotic system, the Kolmogorov entropy should be independent of the length scale at which it is calculated, if this length scale is chosen small enough. However, the Kolmogorov entropy of fluidized beds is a function of the length scale (van der Schaaf et al., 2004; Zarghami et al., 2008). For high-dimensional or highentropy systems, independence of length scale – a so-called scaling region – is only found at extremely small values of the length scale (Kantz and Olbrich, 1997). For experimental systems with a high dimension and/or entropy, it is hard or practically impossible to attain the scaling region, making it difficult or impossible to distinguish them from random oscillating systems (Gaspard and

40

Entropy [bits/s]

p (t+τ)

4

(b)

30 20 10 0 0

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6

Avg. cycle frequency [s]

Fig. 24. (a) The maximum likelihood entropy and the average cycle frequency as functions of the gas velocity. The error bars give the standard deviation as obtained from dividing each time-series in 10 parts, and calculating the entropy for each part. (b) The maximum likelihood entropy versus the average cycle frequency. The large squares indicate the four datasets; the other markers represent the additional measurements at intermediate gas velocities.

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Wang, 1993). In practice, it has not been proven possible to capture the low-dimensional dynamics by applying proper embedding of filtering techniques, as proposed by Daw et al. (1995). Even for a near-periodical slugging fluidized bed, a scaling region has not been found (van der Schaaf et al., 2004). This means that the entropy analysis does not prove that fluidized beds exhibit low-dimensional chaotic behaviour. This absence of a proper scaling region is often the case for experimental systems where chaotic behaviour is suspected: also for climatological, economical, and physiological data uncontested evidence of chaos is often lacking (Kantz and Olbrich, 1997). Zhao et al. (2001) showed that data from fluidized-bed experiments and a data-based model gave basically the same entropy (and also similar correlation dimensions and Lyapunov exponents), indicating that the system has a certain degree of determinism. Villa Briongos et al. (2006, 2007) showed that using a multi-resolution analysis based on the Hilbert–Huang Transfer Method, the pressure signal can be divided in several components. This is a way to obtain a component that just contains information about the bubble dynamics, which might exhibit low-dimensional chaotic behaviour; further research is needed at this point. Assuming that the fluidized-bed pressure time-series are randomly generated, van der Schaaf et al. (2004) demonstrated how the power spectral density is linked to the value of the Kolmogorov entropy at the length scale at which it is evaluated. Furthermore, this result led to a formula that predicted how the maximum likelihood dimension or correlation dimension depends on the length scale at which it is evaluated and on the embedding dimension, another calculation parameter. The results agreed well with the experimentally determined Kolmogorov entropy and the maximum likelihood dimension. With this result, van der Schaaf et al. (2004) additionally showed that the Kolmogorov entropy calculated at a specific length scale is directly proportional to the average frequency of the fluidized bed. The proportionality was confirmed for a large number of pressure time-series in bubbling, slugging, and circulating fluidized beds. If we make the same comparison for the data set studied in this paper, we also observe a strong correlation between the Kolmogorov entropy and the average cycle frequency; see Fig. 24a. If we plot the entropy versus the average cycle frequency, it appears that they are practically linearly proportional (Fig. 24b). The proportionality constant was shown to be directly related to the shape of the PSD (van der Schaaf et al., 2004). This proportionality between the Kolmogorov entropy and the average cycle frequency explains why it is possible to derive correlations that describe the dependency between the Kolmogorov entropy and number of bubbles erupting from the bed per unit of time (van den Bleek et al., 2002): it is more direct to relate bubble eruption frequency and average cycle frequency of the pressure signal. The strong relationship between the Kolmogorov entropy and the average cycle frequency was already reported earlier (Johnsson et al., 2000; Manyele et al., 2002), but van der Schaaf et al. (2004) were the first to explain the mechanism behind it. Because the average frequency and the PSD are more easily correlated to physical phenomena, these characteristics are preferred over the correlation entropy and correlation dimension. Moreover, the average frequency is not dependent on calculation parameters whereas the Kolmogorov entropy and correlation dimension are. 5.3. Correlation dimension The correlation dimension expresses the amount of space that is occupied by the attractor, characterizing the complexity of the time-series. It is a fractal dimension, and, therefore, does not have to be an integer number. Schouten et al. (1994a) proposed a method to estimate the correlation dimension of an attractor recon-

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Superficial gas velocity [m/s] Fig. 25. The maximum likelihood correlation dimension as a function of the gas velocity for the 17 pressure time-series. The blue diamonds indicate the data obtained for the regular distributor; the red square indicates the value obtained with the high-pressure drop distributor at a low gas velocity (the multiple-bubble regime). The error bars give the standard deviation as obtained from dividing each time-series in 10 parts, and calculating the correlation dimension for each part.

structed from a noisy time-series. Hay et al. (1995) have indicated that the correlation dimension can remain constant over a wide range of operating conditions: this makes it a less useful property to characterize the hydrodynamics. Fig. 25 shows that the correlation dimension strongly increases with increasing gas velocity for the lower gas velocities. Above 1.5 m/s, the variations are much smaller but still significant. The red marker shows that moving from the single-bubble to the multiple-bubble regime – an increase in complexity of the hydrodynamics – indeed leads to a higher correlation dimension. Bai et al. (1997a) showed that there is also a distinct difference between the correlation dimension (and other parameters) derived from absolute pressure signals (as shown in Fig. 25) and the correlation dimension from differential pressure signals.

5.4. Lyapunov exponents The Lyapunov exponents express the local rate of convergence or divergence of two neighbouring points on the attractor. For each direction, a Lyapunov exponent can be determined, meaning that their number is equal to the embedding dimension. For a chaotic system, at least one of the exponents is positive, quantifying the ‘sensitivity to initial conditions’. The Kolmogorov entropy equals the sum of all positive Lyapunov exponents (Eckmann and Ruelle, 1985). A major problem in the determination of the Lyapunov spectrum from experimental data is the appearance of spurious exponents when the embedding dimension (i.e., the dimension of the reconstructed state space) is larger than the dimension of the true state space. Therefore, most researchers decided do leave aside the Lyapunov exponents, also because they do not give much added value compared to the Kolmogorov entropy (van der Stappen, 1996). Researchers that have been applying this analysis method to fluidized-bed pressure date typically just calculate the largest Lyapunov exponent, characteristic for the chaotic behaviour of the system. Most publications only report a very limited study of the largest Lyapunov exponent. They show that for specific cases the value decreases (Guo et al., 2003) or slightly increases (Hay et al., 1995) with increasing gas velocity. Bai et al. (1997a) showed that for some cases the Lyapunov exponents from absolute pressure signals, differential pressure signals, and optical signals were in reasonable agreement, while in other cases there was a large difference. In general, calculating the Lyapunov exponents of fluidized-bed pressure data has shown little added value.

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5.5. Attractor comparison

10

pffiffiffiffi Q ¼ ð2d pÞm

Z

½qx ðRÞ  qy ðRÞ2 dR

ð5:5:1Þ

Diks et al. (1996) have given a procedure to calculate an unbiased ^ of the squared distance for two given time-series. The estimator Q larger the difference between two attractors, the larger their squared distance Q will be. However, we will need to know whether ^ indicates a significant difor not a certain value of the estimator Q ference between the two delay vector sets, and thus between the two hydrodynamic states (reference and evaluation) of the fluidized bed from which they originate. Therefore, we also need an estimate ^ . Diks et al. (1996) also derived an expression for the variance V of Q to calculate this variance for two given time-series. We can define a dimensionless squared distance S as

^ Q S ¼ qffiffiffiffiffiffiffiffiffiffiffi ^Þ VðQ

ð5:5:2Þ

Subsequently, the null hypothesis is tested that the two sets of vectors are drawn from the same probability density distribution, viz., that the two sets are generated by the same dynamic mechanism. Diks et al. (1996) did not prove if S has a normal distribution, but Pukelsheim (1994) has shown that for any unimodal probability density, the probability for values larger than three times the standard deviation is smaller than 5%, leading to a threshold value of +3 for S at a confidence level of 95%. This is a conservative choice, since the test is only one-sided: differences between attractors always lead to large positive S-value, not to large negative S-values. Diks et al. (1996) showed by numerical simulations that the confidence level is about 98–99%, well above the 95% that was strictly proven. The Diks et al. test does not make a priori assumptions about the system under study: it can be applied to chaotic, regular, and noisy systems. The method involves the choice of a number of parameters, such as embedding dimension, band width (describing the smoothing of points on the attractor), and the segment length

(a) Single bubble Transport conditions

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As stated before, no convincing evidence has been delivered that fluidized beds exhibit low-dimensional chaotic behaviour. Moreover, Drahoš and co-workers have rightly stated that the application of chaos analysis is useful only if it brings information that cannot be more easily discovered by a simpler linear analysis (Drahoš, 2003; Drahoš and Ruzicka, 2004). The detection of small changes in the hydrodynamics is however a typical example where state space analysis can outperform analysis in the time domain or frequency domain. Daw and Hallow (1993) were the first to propose the monitoring of fluidization hydrodynamics through nonlinear time-series. They suggested that monitoring certain timeaveraged properties of the reconstructed attractor could be used in practice for on-line detection of changes in the fluidization dynamics. As long as such a property would remain within certain limits, the fluidized bed exhibits stationary behaviour. If the property would exceed the limits, certain action, for example by a process operator, would be needed to return to the desired fluidizedbed behaviour. However, Daw and Hallow (1993) neither indicated how these limits had to be determined, nor how sensitive such a method would be. Instead of focussing on a single attractor property, van Ommen et al. (2000) proposed to determine possible changes in the attractor as a whole by comparing attractors for two different points in time using a statistical test developed by Diks et al. (1996). This test considers two sets of delay vectors (i.e., two attractors, qx and qy) as multidimensional probability distributions. These two sets can, depending on the situation, be considered as a reference situation and a situation to be evaluated. The extent to which two attractors differ can be expressed by the squared distance Q between them:

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Superficial gas velocity [m/s] Fig. 26. (a) S-value as a function of time for the four selected data sets (i.e., constant gas velocity) and (b) time-average S-value as a function of the gas velocity with three different references. For both plots, the reference time-series is taken from t = 15 min to t = 18 min. Every time-series to be evaluated is divided in 10 parts of 3 min long. The embedding dimension is 40, the band width is 0.5, and the segment length is 1.5 s.

(related to avoiding dependence among delay vectors). These parameters are discussed in more detail in van Ommen et al. (2000). Fig. 26a shows the average S-value as a function of time for the four selected data sets. S stays below zero, indicating that the timeseries are stationary (i.e., no significant change is taking place over time). As the reference, the data from 15 to 18 min is taken, which leads to more negative S-values at that point in time; this has no physical meaning. Fig. 26b shows the average S-value (average of the 10 values obtained per time-series) as a function of the gas velocity, with three different references. The graph shows that the S-value is only below 3 when the gas velocity for reference time-series and evaluation time-series are the same. This means that every change in gas velocity is detected as a significant difference, and demonstrates the sensitivity of the method. However, it should be kept in mind that the method is not intended to detect changes in gas velocity, but to detect agglomeration or other phenomena leading to a change in particle-size distribution or other particle properties. The method has, for example, been used to detect agglomeration during biomass combustion in bubbling fluidized bed (Nijenhuis et al., 2007; Bartels et al., 2009a) and circulating fluidized beds (Bartels et al., 2010a), and to monitor drying of pharmaceutical powders (Chaplin et al., 2005a,b). An extensive comparison of various methods suited for monitoring purposes is made by Bartels et al. (2008, 2010b).

5.5.1. Summary of state space analysis Chaos analysis has widely been applied to data from fluidized beds during the past two decades. An attractive feature of this approach is that it handles fluidized-bed dynamics in a way that is different from many conventional methods: rather than a

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reductionist approach (e.g., trying to capture the movement of all individual particles) it considers the spatio-temporal patterns encountered in a fluidized bed as a whole. However, as for many real-life systems (e.g., climatologic, economical and physiological systems) it is extremely difficult to prove that a fluidized bed is a chaotic system or that it shows certain chaotic features. Commonly applied characteristics, such as entropy, the correlation dimension, and the Lyapunov exponents, should be used with great care: the validity of calculated results is questionable when conditions, such as independence of length scale, are not fulfilled. We also showed for our data sets that the Kolmogorov entropy is linearly proportional to the average frequency, obtained in the frequency domain. Since the latter is easier to calculate, it is a more obvious choice for characterization of time-series, e.g. for detecting regime transitions. However, methods closely associated to chaos theory, such as attractor comparison, are useful to detect small changes in fluidized-bed dynamics (e.g., the onset of agglomeration) with a higher sensitivity to small differences than other methods. While the current data sets are not very well suited to prove this, since they only differ in gas velocity, it has been shown in other papers that this method is very sensitive to changes in the particle properties. 6. User guideline In this paper, we have discussed a number of techniques that can be used to analyse univariate pressure signals originating from a fluidized bed. Although specific requirements to carry out proper measurements and analysis differ from case to case, this section presents some general guidelines. First, we will discuss the conditions for properly measuring pressure time-series. Second, we will briefly recap the different methods discussed before with their specific pros and cons. 6.1. Proper measurement procedure In order to be able to analyse pressure signals using the various analysis methods, the pressure has to be sampled with sufficiently high frequency. This requires suitable sensors with a high enough response frequency. In addition, the following point should be taken into account (van Ommen and Mudde, 2008): 1. Dimensioning of the probe-transducer system. In most cases, the pressure transducer is not in direct contact with the bed, but connected to it by a probe or tube. In order to minimize resonance effects, it is recommended to keep the probe length as short as possible. The inner diameter of the probe is preferably between 2 and 5 mm (van Ommen et al., 1999b). Smaller diameters lead to dampening of the signal. Larger diameters increase resonance effects and increasingly disturb the local hydrodynamics, especially when a purge flow is applied. 2. Free entrance of the probe. One solution to avoid blocking of the probe entrance is to cover it with a wire mesh. As long as the mesh is open enough, it has no significant influence on the pressure fluctuations (van Ommen et al., 1999b). An alternative is to apply a constant purge flow. This flow should be high enough to prevent particles from entering the probe, but low enough to avoid (excessive) bubble formation at the probe tip. Recommended values are 0.5–1.0 m/s for Geldart A particles (Geldart and Xie, 1992; van Ommen et al., 2004a) and 1.0–2.0 m/s for Geldart B particles (Nijenhuis et al., 2007). 3. Proper placement of the probe(s). Pressure probes are often placed flush with the wall, as this minimizes the disturbances of the bed hydrodynamics. Croxford et al. (2005) reported that for a small-scale fluidized bed, in principle, one probe is sufficient to characterize the hydrodynamics. However, to obtain

the bubble size as a function of height, several probes are needed. Van Ommen et al. (2004b) showed that a pressure probe in a large-scale fluidized bed can detect local pressure waves (due to, e.g., bubbles) up to a distance of about 0.5 m from their origin. 4. Sampling frequency. In this paper, we used a sampling frequency of 400 Hz, which is sufficient for most combinations of analysis technique and fluidized-bed system. A few methods discussed before, mainly those based on the standard deviation and the average cycle time, can work well with a sample frequency that is an order of magnitude smaller. For systems with very small time scales (e.g., narrow, high-velocity risers), one can consider using a higher sampling frequency (e.g., 1000 Hz). 5. Filtering the signal. As with any measured signal, proper lowpass filtering should be applied at half the sample frequency or lower (satisfying the Nyquist criterion) to avoid signal distortion due to aliasing. The low-pass filtering should be carried out during the A/D conversion of the sampling process, in which the continuous signal is converted into a numeric sequence. It cannot be applied afterwards. In addition, it is recommendable to apply a high-pass filter to remove slow trends from the signal, especially for absolute pressure measurements. A typical cutoff frequency that can be used is 0.1 Hz. This prevents the signal from slowly moving out of the measurement range. In addition, it puts the average of the signal at zero, which gives an optimal use of the measurement range (i.e., the best resolution). Highpass filtering can be applied both during A/D conversion and in a later stage. 6. Length of the time-series. To be able to derive statistically sound results from the measurements, the data series should be sufficiently long. The proper length strongly depends on the type of analysis. For example, a length of the time-series of up to 30 min is recommended for spectral and non-linear analysis (Johnsson et al., 2000). For time domain analysis, shorter time-series (e.g., 5 min) could be used. One should, however, keep in mind that fluidized beds might exhibit slow dynamics (e.g., changing bubble or mixing patterns), which have a time scale of several minutes or longer.

6.2. Choice of analysis method Table 4 gives an overview of the methods treated in this paper. It summarizes the kind of information given by each method, and mentions briefly the restrictions one should keep in mind when using the method. The table suggests that every method has its limitations, which is indeed true: there is no single method that yields the complete picture about the fluidized-bed dynamics. Using a combination of methods will thus lead to the most comprehensive view.

7. A wider scope In the introduction of this paper, we discussed the advantages of pressure measurements in gas–solid fluidized beds. However, there can be good reasons to choose other measurement techniques instead of, or in addition to, pressure measurements. For example, optical probes, capacitance probes, acoustic sensors, or gamma-ray densitometry can be used (see, e.g., van Ommen and Mudde, 2008). The analysis methods discussed in this paper can typically also be applied to those other signals, provided that they are sampled with high enough frequency. Some of the measurements, such as those with optical probes and capacitance probes, are intrinsically limited to more local information, compared to pressure measurements. Therefore, the outcomes of time-series

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Table 4 Overview of the signal analysis methods discussed in this paper, with some hints on proper use and interpretation. Method Time domain methods Standard deviation Probability density function of pressure increments Cycle time Hurst analysis Autoregressive (AR) models Frequency domain methods Power spectrum via non-parametric methods Power spectrum via AR models Power spectrum fall-off Transient power spectral density (TPSD) Wavelets State space methods Attractor reconstruction Entropy Correlation dimension Lyapunov exponents Attractor comparison

Information about

Comments

Regime transitions Intermittency, presence or absence of long-term correlations Main time scale of the signal Nature of the signal, deviation from Brownian motion Representing a fluidized bed in analogy with a simple mechanical system

Easy-to-use, but info on time-scale lost Still under debate Easy-to-use, but caution needed with noisy data Often unreliable, due to limited scaling range Assumptions about model input might be oversimplified

Relevant time scales

Widely used; rather long time-series needed to get reliable estimate

Representing a fluidized bed in analogy with a simple mechanical system Origin of unpredictability (chaos versus stochasticity) Multiple modes of operation

Assumptions about model input might be oversimplified. Useful with noisy data or short time-series Still under debate

Time localization of frequency components

Still to reach full potential in studying fluidized beds

Complexity of signal Predictability Complexity of signal Sensitivity to initial conditions Small changes in signals over time

Useful as a basis for further analysis Gives average frequency instead of true predictability for fluidized beds Not very sensitive for fluidized bed pressure signals Calculation prone to errors; little added value Useful for non-stationary signals, e.g., in case of agglomeration

analysis of both types of signals can show significant differences (see, e.g., Bi et al., 2000). The discussed analysis techniques are not limited to gas–solid fluidized beds. These techniques can also be applied to signals – either pressure signals or other signal types – measured in related systems such as gas–liquid fluidized beds, bubble columns, and slurry bubble columns. We will only briefly state a few examples here. Drahoš et al. (1991) used various methods to characterize pressure fluctuations measured in bubble columns. Villa et al. (2003) applied attractor comparison to pressure fluctuations from bubble columns to detect foaming. Kwon et al. (1994) analyzed pressure fluctuations measured in a three-phase fluidized bed, and tried to correlate the Hurst exponent, obtained from rescaled range analysis, to the bubble size. Mehrabi et al. (1997) proposed alternative methods to obtain the Hurst exponent from pressure signals measured in three-phase fluidized bed. Ruthiya et al. (2005) applied spectral analysis to pressure signals to determine multiple regime transitions in slurry bubble columns. Also for other multiphase systems than fluidized beds and (slurry) bubble columns, time-series analysis of measured signals can be very useful. Horowitz et al. (1997) compared several analysis methods to determine flow transitions in trickle beds from pressure fluctuations. Paglianti et al. (2000) analyzed conductance measurements with multiple methods to determine the flooding/loading transition in gas–liquid stirred vessels. Albion et al. (2008) determined regime transitions in liquid–solid transport systems by analysing acoustic signals, using a variety of methods. These examples show that the toolkit of signal analysis methods has a much wider range of applicability than pressure fluctuations measured in gas–solid fluidized beds, the topic of this review. When properly applied, time-series analysis can reveal a wealth of information about the complex dynamic nature of various manifestations of multiphase flow. 8. Conclusions Pressure measurements are cheap and relatively easy to perform, and therefore widely applied in fluidized beds. In industrial conditions, typically only measurements of the average pressure

Mainly useful for larger beds

drop are carried out. However, when the pressure is sampled at a sufficiently high frequency, the obtained fluctuation signal can yield a lot of information about the fluidized-bed dynamics. These time-resolved or dynamic pressure measurements are nowadays broadly applied in fluidized-bed research. The sample frequency used for pressure fluctuation measurements depends on the system under investigation and on the way the pressure signals are analyzed, but 20 Hz can be considered a lower limit. Typically, a sample frequency in the order of 200 Hz is applied. Although pressure signals are relatively easy to measure, their interpretation is not always straightforward. Care should be taken when determining which analysis method should be applied to reach a certain goal. In this paper, we have reviewed the analysis methods that have been applied to fluidized-bed pressure signals, with a focus on the last decade. We divided the analysis methods into three groups: time domain methods, frequency domain methods, and state space methods. It is important to note here that the pressure signals used in this work do not exhibit all possible modes of behaviour and phenomena observed in fluidized beds. Therefore, the full potential of some of the methods used exceeds what could be demonstrated with those signals. Analysis in the time domain is often the simplest approach. The standard deviation of pressure fluctuations is widely used, e.g. to identify the regime change from bubbling to turbulent fluidization. The method should be applied with caution, since a change in standard deviation can be caused by a number of phenomena, not all of them indicating a regime change. Higher-order moments have scarcely been applied. The probability distribution of pressure increments is used to provide information about relevant time scales of a process. The method has only recently been applied to pressure fluctuations in fluidized beds and needs further research. The cycle time and its distribution also provide information about the relevant time scales of the system. It is a useful and easy-tocalculate alternative to frequency analysis. Some researchers have applied the rescaled range analysis (Hurst analysis) to fluidizedbed pressure fluctuations. For the four regimes studies in this paper, this characteristic was not discriminative. The autoregressive (AR) methods can be used to show an analogy between a fluidized bed and a single or a set of simple mechanical systems acting in

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parallel. From this equivalence one can deduce the values of parameters describing such a system (natural frequency and damping). The most common frequency domain method is the power spectrum. It can be calculated or estimated by non-parametric and parametric methods. Nowadays, the former clearly dominate the literature, since long data records and suitable analysis methods are readily available. In a similar way to AR models, the parametric methods are, however, useful to describe fluidized beds in analogy with well-defined mechanical systems (e.g., a single or a set of multiple oscillators acting in parallel). The slope at higher frequencies (either in a log-normal or a log–log representation) represents an alternative way for characterization. To capture transient effects on a longer time scale (>1 s), either the transient power spectral density or wavelet analysis can be applied. The true potential of the latter is to provide an accurate time localization of different phenomena present in the pressure signal and observed in the frequency domain. Analysis in state space is a relatively new development that became popular in the nineties of the last century. Interpretation is often based on the assumption that a fluidized bed is a chaotic system, but there is currently no conclusive evidence that either fluidized beds truly exhibit low-dimensional chaotic behaviour, or that it is essential to assume the latter to use some attractor-based methods. It has been shown that the information given by the Kolmogorov entropy – one of the most widely used characteristics in state space analysis – is equivalent to that of the average frequency, obtained in the frequency domain. For non-linear analysis it is often more difficult to make a direct connection between the physical phenomena taking place in the bed and the results from the analysis. Moreover, non-linear analysis methods typically require a more complex calculation procedure. Therefore, this type of analysis should be applied with care. An advantage of certain state space methods, such as attractor comparison, is, however, that they are more sensitive to small changes than frequency domain methods. This makes them especially suited to determine whether or not a fluidized bed is showing stationary behaviour. From a number of analysis methods presented in the paper, we conclude that there is no conclusive evidence of the existence of turbulence in a bubbling bed or a in a dense bottom bed of a circulating fluidized bed. We claim that the observations reminiscent of those seen in single-phase turbulence are a consequence of the bubble component (e.g., coalescence and breakup of bubbles) of the pressure signal. A strong point of high-frequency pressure measurements is that their use is not limited to academic research: industrial application is certainly feasible, since the required sensors can be implemented in a robust and cost-effective way. However, this requires reliable analysis methods that can translate the industrial pressure data into relevant information about the fluidized-bed hydrodynamics. While progress has been made during the past decade, further research will still be needed in the coming years. From a scientific point of view it is still challenging to fully explain the complex nature of a fluidized bed, as reflected by the pressure waves and other experimental data. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.ijmultiphaseflow.2010.12.007. References Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., Tsimring, L.S., 1993. The analysis of observed chaotic data in physical systems. Rev. Mod. Phys. 65, 1331–1392. Akaike, H., 1974. New look at the statistical model identification. IEEE Trans. Autom. Control AC-19, 716–723.

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