ENERGY OF POWER SPECTRAL DENSITY FUNCTION AND...

ENERGY OF POWER SPECTRAL DENSITY FUNCTIONAND WAVELET ANALYSIS OF ABSOLUTE PRESSUREFLUCTUATION MEASUREMENTS IN FLUIDIZED BEDS

M. C. SHOU and L. P. LEU�

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan

The standard deviation of pressure fluctuations is always as a benchmark to characterizefluidized bed, especially in identifying the transition velocities and flow regimes. Inthis study, the energy of power spectral density function (PSDF) was proved to be

a new alternative tool to effectively analyse the pressure fluctuations. Also, the wavelet analy-sis was significant to deal with the signals in multi-resolution decomposition and in identifi-cation of transition velocities and flow regimes. The results from the normalized energy ofPSDF and wavelet analysis are in good agreement with the standard deviation analysis.

Keywords: fluidized bed; transition velocity; pressure fluctuations; turbulent fluidized bed;standard deviation; power spectral density function; wavelet analysis.

INTRODUCTION

Gas–solid fluidized bed reactors have been widely appliedin the industrial application. The existence of different flowregimes in a gas–solids fluidized bed has been discussed.By different measurement techniques and analyticalmethods, the transition velocities can be determined to indi-cate the flow regimes. However, due to influencing factorssuch as gas/solid properties, operating temperature andpressure, the existence of transition velocities arises contro-versies and makes it hard to determine the flow regimesexactly. Thus, several measurement techniques includingvisual observation, local capacitance, pressure fluctuationsand local and overall bed expansion have been utilized todetermine the transition from bubbling regime to turbulentregime. This study focuses on the measurement techniqueof the pressure fluctuation signals.The first quantitative study of local voidage, voidage

fluctuations and pierced void measured by capacitanceprobes seems to have been performed by Lanneau (1960).The first transition criterion from bubbling to turbulent flui-dization was proposed by Yerushalmi et al. (1978). The gasvelocity, Uc at which the standard deviation of pressurefluctuations reached a maximum was said to markthe beginning of the transition to turbulent fluidization,while Uk, where the standard deviation of the pressurefluctuations levels off, was said to denote the end of thetransition. In recent years, Uc has been widely adopted todefine the transition to turbulent fluidization (Bi et al.,

2000). However, Uk often cannot be found (e.g., Satijaand Fan, 1985; Rhodes and Geldart, 1986) and dependson the system used to return particles captured after beingcarried out of the bed (Bi and Grace, 1995) and measure-ment probe location (Chehbouni et al., 1994). On theother hand, Chehbouni et al. (1994) found that Uk couldbe determined from differential pressure measurement,but not from absolute pressure measurement. Nowadays,the controversy on discussing the hydrodynamics of theflow regimes still exists, even the confusion of the defi-nition of transport velocity (Avidan and Yerushalmi,1982; Bi et al., 1995; Grace, 2000).

Also, in recent studies, wavelet analysis has been provento be a very efficient and effective tool in signal processing(Bruce and Gao, 1996). Fan et al. (1990) were the first toapply the concept of Fractional Brownian Motion (FBM)for analysing pressure fluctuations in a gas–liquid–solidfluidized bed in terms of Hurst rescaled range analysis.He et al. (1997) proposed that the pressure fluctuations ina gas–solid fluidized bed could be decomposed into aFBM and Gaussian White Noise (GWN). Lu and Li(1999) and Guo et al. (2002) also employed wavelet analy-sis to investigate the dynamics of pressure fluctuation in abubbling fluidized bed. Park et al. (2001) proved that thetime series of pressure fluctuation signals were analysedby means of discrete wavelet transform coefficients andmulti-resolution decomposition. Zhao and Yang (2003)used wavelet transform, Hurst analysis, multi-scale resol-ution and time-delay embedding to deal with pressure fluc-tuations for characterizing the fluidized bed.

In this study, we analysed absolute pressure fluctuationsmeasured above and below the distributor whereas much ofthe literature measured the pressure fluctuations only above

�Correspondence to: Professor L. P. Leu, Department of Chemical Engi-neering, National Taiwan University, Taipei 106-17, Taiwan.E-mail: [email protected]

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0263–8762/05/$30.00+0.00# 2005 Institution of Chemical Engineers

www.icheme.org/journals Trans IChemE, Part A, May 2005doi: 10.1205/cherd.04123 Chemical Engineering Research and Design, 83(A5): 478–491

the distributor. The aim of the study was to characterize theflow regimes and transition velocities by means of the mul-tiple analyses: standard deviation analysis, PSDF, theenergy of PSDF and wavelet analysis. Among these ana-lyses, effective tools, the energy of PSDF and waveletanalysis, were put to proof.

ANALYTICAL METHODS

Time Domain Analysis

The standard deviation of the measured pressure fluctua-tion signals, pi is defined mathematically as follows:

S.D. ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xni¼1

(pi � �p)2

s; i ¼ 1; 2; 3; . . . ; n (1)

with the average pressure,

�p ¼ 1

n

Xni¼1

pi (2)

Frequency Domain Analysis

The PSDF Pxx(v) of signal x(t) has been derived fromthe autocorrelation function as (Bendat and Piersol, 1980)

Pxx(v) ¼ 2

p

ð10

Rxx(t) cosvt dt (3)

where Rxx(t) is the autocorrelation function of x(t), whichcan be expressed as

Rxx(t) ¼ limT!1

1

T

ðT0

x(t)x(t þ t) dt (4)

Wavelet Analysis

In wavelet analysis, we can use linear combination ofDaubechies (db3) wavelet function to represent pressurefluctuation signals.First, the wavelet transform of a continuous signal x(t)

can be defined as

Wf (a; b) ¼ 1ffiffiffiffiffiffijajpðþ1

�1x(t)w�

a;b

t � b

a

� �dt (5)

where Wf (a; b) is the wavelet coefficient, w�a;b(t) is a basic

wavelet function, a and b are the continuous dilation andtranslation parameters, respectively, they take values inthe range of the amplitude function �1 , a; b , 1 witha = 0.Wavelet functions form a family of functions with high

frequency and small duration that are all normalizeddilations and translation of a prototype ‘wavelet base’ func-tion. Thus

wa;b(t) ¼1ffiffiffiffiffiffijajp w

t � b

a

� �(6)

An original signal is decomposed into its approximationsand details with different frequency bands by means of

wavelet transform. The decomposition process is repeateduntil the desired decomposition level, J, is reached. Theorthogonal wavelet series approximate a continuoussignal x(ti) expressed as

x(ti) � Aj(ti)þ Dj(ti)þ Dj�1(ti)þ � � � þ D1(ti) (7)

where D1(ti);D2(ti); . . . ;Dj(ti) represent detail signals ofmulti-resolution decomposition at different resolution 2j,and Aj(ti) the approximation signal of multi-resolutiondecomposition at resolution 2j.

As we know, the quality of signal decomposition andreconstruction mainly depend on the choice of the motherwavelet (Bruce and Gao, 1996). Wavelets separate asignal into multi-resolution components. The componentsat fine and coarse resolutions indicate the fine- andcoarse-scale features of the signal. In this study, we useDaubechies (db3) wavelet to calculate pressure fluctuationsignals just as Guo et al. (2003) did.

EXPERIMENTS

The schematic diagram of the experimental setup isshown in Figure 1. The apparatus consists of a riser, anexpanded-top section and a cyclone separator. The risercolumn made of Plexiglas is 10.8 cm i.d. and 5.75 m inheight; the expanded top section made of stainless steel is1.5 m in height and is five-times the cross section area ofthe riser bed area. According to Avidan and Yerushalmi(1985), the expanded bed section at high gas velocity canreduce the solid entrainments. The entrained particlesfrom the expanded top section are collected by the cycloneseparator and are recycled to the riser bed by use of theloop-seal type non-mechanical valve. A perforated platewith fine screen was used as the gas distributor. Therewere 250 holes of 1.5 mm in diameter uniformly distributedon the plate, giving an open-area ratio of 4.8%. The bedmaterial used is fluid catalytic cracking (FCC) particles,with an average particle size 71 mm, and a particle densityof 1800 kg m23, i.e., Geldart group A particles (Geldart,1973), the minimum fluidization velocity for FCC particles,0.3 mm s21.

The pressure probe used is made of 3 mm i.d. coppertube. One end of the probe is covered with a fine screento prevent solid particles flowing into the probe, and theother end of the probe is connected to a pressure transducer.The continuous pressure signal from the pressure transdu-cer is amplified and sent to a personal computer for record-ing and further analysis. Data acquisition is performed 100readings per second, and the total number of readings isequal to 10 000. The pressure fluctuation signals are ana-lysed by means of the standard deviation, power spectraldensity function and wavelet analysis. The absolutepressure tap is mounted flush with the wall of the columnat four pressures tap positions 24, 14, 30, 50 cm fromthe distributor.

To obtain sufficient experimental evidence, we measuredpressure fluctuation signals from different static bed heightsvarying from 0.5 m to 0.7 m.

Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A5): 478–491

PSDF AND WAVELET ANALYSIS IN FLUIDIZED BEDS 479

RESULTS AND DISCUSSION

Various kinds of measurements in literature have beenused to predict the transition of flow regimes. In literaturestudies (Lee and Kim, 1988; Schnitzlein and Weinstein,1988; Mori et al., 1988; Sun and Chen, 1989), the absol-ute pressure fluctuation measurements above the distribu-tor are usually applied to obtain the pressure fluctuationsignals. In this study, an absolute pressure probe was usedto measure the pressure fluctuation signals at the mea-suring position not only above but also below thedistributor. We used the standard deviation, PSDF, thenormalized PSDF of pressure fluctuation signals, andwavelet analysis to analyse the pressure fluctuation sig-nals, as a function of the superficial gas velocity forFCC particles.

Standard Deviation of Pressure Fluctuations

Figure 2 shows the measured standard deviation of thepressure fluctuation signals, which change with the super-ficial gas velocity at axial positions (24, 14, 30, and50 cm) above the distributor by an absolute pressureprobe. When the pressure fluctuation signals were measuredbelow the distributor, at 24 cm by an absolute pressureprobe, both Uc ¼ 1.0 m s21 and Uk ¼ 1.65 m s21 werefound. The result is different from those of Chehbouniet al. (1994) who proposed that the transition velocitiescould not be obtained from absolute pressure measurement.Besides, for probe above the distributor, Uc ¼ 1.15 m s21

was obtained, close to the value obtained below the distri-butor, whereas the leveling-off of the standard deviation ofthe pressure fluctuations was not obtained by measuringabove the distributor, at 30 and 50 cm except for an ambig-uous Uk obtained at 14 cm.

In summary, Uc and Uk were found by absolute pressureprobe locating below the distributor. Meanwhile, Uc wasfound and Uk was ambiguous by an absolute pressureprobe locating above the distributor. As seen in the figure,the variation of the transition velocities could describe theflow regimes. With increasing superficial gas velocity, thebubbles became larger and larger, and then sluggingphenomenon appeared over coalescence. The more violentpressure fluctuations became, the more serious sluggingthe bubbles had. When bubbles reached a maximum stablesize, the standard deviation of the pressure fluctuationsreached the maximum and the corresponding superficialgas velocity was denoted as Uc. Uc clarified a change frombubbling to turbulent fluidization. Therefore, Uc was con-sidered as the onset of turbulent flow regime. As the super-ficial gas velocity increased slowly, non-uniformity oftwo-phase (dense-dilute) system gradually vanished andtended to form a homogeneous state. The transition pointwhere the standard deviation of pressure fluctuations leveledoff was the transition velocity, Uk. Thus,Uk was denoted theend of the turbulent bed and the beginning of the fast bed.

Figure 1. Experimental setup of expanded-top fluidized bed: (1) blower;(2) safety valve; (3) gate valve; (4) orifice meter; (5) distributor;(6) riser bed; (7) expanded top section; (8) cyclone separator;(9) bag filter; (10), (11) gate valve; (12) loop-seal; (13) pressuretransducer; (14) amplifier; (15) computer & AD/DA card; P1, P2;pressure tap.

Figure 2. Standard deviation of pressure fluctuations at the static bed,50 cm, in height, measured by an absolute pressure probe at the differentmeasurement positions (24, 14, 30, 50 cm above the distributor).


480 SHOU and LEU

To make sure that the ambiguous Uk did exist, the followinganalytic methods were studied.

Power Spectral Density Function (PSDF)

PSDF is applied to analyse and characterize the hydro-dynamics of fluidized beds. The method is utilized to

describe the dominant frequency. A wide band spectrumis considered to signify an increase in the number ofbubbles, while a narrow band with sharp peaks either sig-nifies a single bubble or slugging bed behaviour.

Figure 3 shows the PSDF of the measured pressure fluctu-ations below the distributor, at24 cm. It shows the PSDF fora wide range of gas velocities covering different fluidization

Figure 3. The power spectral density function (Pa2/Hz) at different superficial gas velocity obtained below the distributor, at 24 cm, and the static bed50 cm, in height, by an absolute pressure probe.



regimes. The band spectrum andmagnitude vary with the gasvelocity. When Ug , 0.93 m s21, a narrow-banded spec-trum was observed [see Figures 3(a-1)–(a-5)]. It indicatedthat the flow regime was from bubbling to turbulent fluidiza-tion.Uc, the maximum point, marked the end of the bubblingfluidized bed and the onset of the turbulent fluidized bed. Thedominant frequency was approximately 0.5–0.75 Hz and its

peak energy increased with the increase of the superficial gasvelocity. When 1.09 m s21 , Ug , 1.93 m s21, the powerspectra became broad and two dominant peak frequencieswere observed [see Figures 3(a-6)–3(a-11)]. With theincrease of the gas velocity, one dominant peak frequency,3.5 � 4.5 Hz increased and its peak energy decreased.When Ug . 2.08 m s21, the power spectra became broader

Figure 4. The power spectral density function (Pa2/Hz) at different superficial gas velocities obtained above the distributor, at 14 cm, and the static bed50 cm, in height, by an absolute pressure probe.


482 SHOU and LEU

and multi-peaks appeared along with the decrease of theenergy of peaks.

Figure 4 shows the PSDF of the measured pressure fluc-tuations at 14 cm above the distributor. At the low gas vel-ocity, when Ug , 0.93 m s21, a narrow-banded spectrumwas observed [see Figures 4(a-1)–(a-5)]. At higher gas vel-ocity, when Ug . 1.09 m s21, the power spectra becamebroad with a moderate spectral magnitude and multi-peaks were noticed [see Figures 4(a-6)–(a-15)], as Mak-kawi and Wright (2002) discussed. The dominant peak fre-quency showed little change and the power spectralmagnitude was considerably lower.

Namely, two trends were seen as a function of Ug. WhenUg , Uc, the amplitude of the pressure fluctuations and theenergy level of the spectral densities increased with thesuperficial gas velocity. Because of the formation andcoalescence of bubbles and slugs, fluctuations became moreand more regular and formed a smaller range of domi-nant frequency. Then, above Uc, in the turbulent regime,fluctuations became more and more irregular with theincrease of the superficial gas velocity. When the superficial

Figure 5. The normalized PSDF, at static bed 50 cm, in height, measuredby four absolute pressure probes, at 24, 14, 30, and 50 cm above thedistributor.

Figure 6. Eight decomposed resolution detail scales of the pressure fluctuation signals.



gas velocity increased, the amplitude of the pressure fluctu-ations decreased and the energy of spectral densities alsodecreased. No dominant frequency still existed. Flow irregu-larity appeared and all fluctuations vanished.From the experimental result, it was found that a precise

estimate of a transition velocity could not be extracted from

the power spectral analysis. This was mainly due to thesimilarity of various spectral shapes at the broad spectraldistribution. Though an obvious peak appeared at everysingle frequency dominant, it was hard to clarify the domi-nant frequency peak in broad-banded power spectra.

The above result agreed with those drawn by Chehbouniet al. (1994) who also utilized PSDF as an analyticalmethod to analyse the differential pressure fluctuationsignals. Thus, we tried to utilize the energy of PSDF toanalyse the pressure fluctuation signals and made sure theexistence of the transition velocities in characterizing thefluidized bed.

The Energy of Power Spectral DensityFunctions (PSDF)

Chehbouni et al. (1994) focused on the peak variation ofthe superficial velocity instead of exploring the energy ofpower spectral density function. However, this study putstress on the energy variation of superficial gas velocity.Some procedures were described as follows:

(1) We try to gain the energy of PSDF.(2) The maximum energy of peak dominant was found and

recorded as (emax. i).(3) The steps from (1) to (2) were repeated and formed

a vector, ~E.

~E ¼ (emax: 1; emax: 2; emax: 3; . . . ; emax: n):

(4) From the vector ~E, we found an element of maximumvalue represented eMAX.

(5) Moreover, we normalized the vector of ~E by eMAX.

Enorm:��! ¼ (emax: 1=eMAX; emax: 2=eMAX; . . . ; emax: n=eMAX)

Figure 5 shows the result of normalized energy of PSDF.The hydrodynamics of the fluidized bed was characterizedsuccessfully similar to the result obtained by the standarddeviation of the pressure fluctuations. We clarified theexistence of Uc, when Uc was found to be 0.9 m s21 atthe different positions (24, 14, 30, 50 cm) by use of anabsolute pressure probe. Also, we obtained a distinctlevel off, Uk ¼ 1.6 m s21, by use of an absolute pressureprobe at 24, þ14 cm above the distributor. At other pos-itions (30, 50 cm above the distributor), there was anambiguous Uk. When superficial gas velocity increased toUc ¼ 0.9 m s21, bubbles reached maximum stable size.And then the pressure fluctuations agitated violently, gradu-ally decayed and finally leveled off to Uk ¼ 1.6 m s21.When massive bubbles began to break up, the flowregime transformed from bubbling to turbulent fluidization.Meanwhile, bubble splitting became dominant. In thisanalysis, it was found that the existence of Uc and Uk

depended on the magnitude of the energy. The energy ofPSDF would be a new alternative tool to effectively analysethe pressure fluctuations. The result was also in agreementwith the standard deviation analysis.

Wavelet Analysis

A peak or peaks in the power spectral density functiondescribed the major periodic component or components in

Figure 7. Wavelet energy (%) profiles for pressure fluctuation signals atdifferent gas velocities.


484 SHOU and LEU

the random variable (Mori et al., 1988). When the bubblesize became bigger, the peak got lower. The peaks weredefined as bubbles. The major frequency was obtained frompower spectral density function but did not reveal thebubbling movement over time. The reason is that Fouriertransform is localized only in frequency but not in time.Figure 6 shows raw pressure data and their decomposed

signals. The pressure fluctuation signals in fluidized bedswere decomposed into eight resolution detail scales withdifferent frequency bands. The original signals wereresolved by order on the basis of sample frequency. The

original signal was plotted in the top row, and the recon-structed wavelets were in the remaining rows. The originalsignal S can be separated into the reconstructed detail, D1

and approximation, A1. The approximation, A1, can bedecomposed into A2 and D2 and the like. In this way, D1

is the detail component of the original signal S, while D2

is the detail component of A1 and D3 is the detail com-ponent of A2 and so forth. The reconstructed waveletsspread from fine scale D1 in the bottom row to coarsescale D8 in the third rows and A8, in the second row. Thefine scale detail was almost dominated by the noise

Figure 8. Wavelet multi-resolution decomposition of fluctuating pressures at different superficial gas velocities (a) Ug ¼ 0.77 m s, (b) Ug ¼ 1.52 m s21,(c) Ug ¼ 2.09 m s21.



(higher frequency). The coarse scale described the low fre-quency oscillations in the pressure signal, large bubbles.Namely, the fine scale corresponded to information aboutthe high frequency of the original signal, and the coarsescale corresponded to information about the low frequencyof the original signal.The energy of a digitized signal is defined as the squared

sum of amplitude as follows:

W ¼Xni¼1

jx(ti)j2 (8)

Let WDj , WA

J be defined as level energy, which is thedecomposed cumulative energy of different level j detailsignals and level J approximation signal.

WDj ¼

Xni¼1

jDj(ti)j2 i ¼ 1;2; . . . ;n; j ¼ 1;2; . . . ; J

(9)

WAJ ¼

Xni¼1

jAJ(ti)j2 (10)

Figure 8. Continued.


486 SHOU and LEU

with the total energy given by

WT ¼XJj¼1

WDj þWA

J (11)

To be more precise, the ratio of energy at different levels tothe total energy is more important in showing how theenergy is displayed at different levels. Let

RDj ¼WD

j

WT

� 100%; j ¼ 1;2; . . . ; J (12)

RAJ ¼WA

J

WT

� 100% (13)

In this manner, RDj and RA

J give a relative wavelet energydistribution in each level. According to equations (12)

and (13), the wavelet energy distribution, which is the per-centage of the energy of the analysed signal over scalesobtained from resolution of discrete wavelet transform, isshown in Figure 7. It shows the variation of energy profilesfrom coarse (approximation) scale to fine (detail) scale withincreasing superficial gas velocities.

Figure 8 illustrates plots of multi-resolution decompo-sition and the reconstruction. In theory, the wavelettransform separates a signal into different scales or multi-resolution levels. The fine scale detail components (i.e.,the wavelet transform) D1 and D2 describe small-scalefluctuations with high frequency oscillations. The coarsescale components (the approximation, the remainder ofthe original signals through wavelet filter) D8 and A8

describe lower frequency oscillations. In Figure 8(a) oflower superficial gas velocity (Ug ¼ 0.77 m s21), the domi-nant scale over the experiment time was the D7 scale,




which was coarser than D6 in Figure 8(b) of higher super-ficial gas velocity (Ug ¼ 1.52 m s21). Also, in Figure 8(b)of lower superficial gas velocity (Ug ¼ 1.52 m s21), thedominant scale over the experiment time was the D6 scale,which was coarser than D5 in Figure 8(c) of higher super-ficial gas velocity (Ug ¼ 2.09 m s21). At the lower super-ficial gas velocity, there was a bigger bubble than the oneat the higher superficial gas velocity. Thus, it indicatedthat an increase in the superficial gas velocity made bubblesin the bed finer.Figure 9 shows the PSDF of multi-resolution decompo-

sition of the pressure fluctuation signals. Figure 9(a)shows the plots of the fine scales and Figure 9(b) showsthe approximation scales. From the fine scale, it wasfound that D7 had the same peak of frequency domain asthe original signal. But the amplitude of the D7 wasabout ratio 0.3 of the amplitude of the original signal.Through filter bank properties of wavelet transform, a

signal was decomposed into sub-signals. Also, the energyand frequency were the main characteristics to describe awave. From Figure 9(a), D7 only presented the frequencyinstead of the energy. To distinctly describe the propertiesof the pressure fluctuation signals, the energy was obtainedfrom the extracted approximation scale. Figure 9(b) isthe PSDF of the approximation scales. The frequency andthe energy from A1 to A6 stayed the same with thoseof the original signal. As for A6, its frequency was alsothe same with the original signal but its energy becamesmaller. Instead of the unstable A6, we chose the approxi-mation scales from A1 to A5 for analysis. We gained themaximum of the approximation scales by PSDF at differentsuperficial gas velocities. Finally, we normalized PSDF.

Figure 10 illustrates the normalized PSDF vs. superficialgas velocity. The absolute pressure probe at 14 cm abovethe distributor was used to measure signals, with the bedheight of 60 cm. From this figure, we noted that there

Figure 9. The PSDF (Pa2/Hz) of multi-resolution decomposition of the pressure fluctuation signals at Ug ¼ 1.52 m s21; (a) the detail scales, D1–D8;(b) the approximation scales, A1–A8.


488 SHOU and LEU

was no much change from A1 to A5 and it representedthe retaining characteristics of original signal. The changebeginning with A6 presented the change toward anotherlow frequency band. From A1 to A5, it was obviouslyfound that Uc value was 1.0 m s21 and Uk value was1.7 m s21.The result in terms of transition velocity agreed with the

analysis of standard deviation of pressure fluctuations andthe energy of PSDF. Thus, the result stated that the flowregimes could also be characterized by the wavelet analy-sis. The signals of low frequency domain characterizedthe bed traits.

CONCLUSION

In this paper, the hydrodynamics of the fluidized bedswas studied on the basis of analysis of standard deviation,

PSDF and wavelet transform of pressure fluctuationsignals. The results follow.

Through the analysis of standard deviation and PSDF ofpressure fluctuations, Uc ¼ 0.9 m s21 was obtained byabsolute pressure measurement above the distributor andbelow the distributor. Through the analysis of standarddeviation, Uk was ambiguous or could not be found.Through the energy of PSDF, Uk ¼ 1.6 m s21 was obtainedfor the absolute pressure measurement above the distributorand below the distributor, though an ambiguous Uk

appeared above the distributor, at 30 cm and 50 cm.By wavelet analysis, the energy of the maximum of the

approximations indicated the transition velocities,Uc ¼ 1.0 m s21 and Uk ¼ 1.7 m s21.

In conclusion, this study provided several analyticalmethods to discuss and identify the existence of the tran-sition velocities, especially the new tools, the energy of




PSDF and wavelet analysis. The result indicated that Uc

and Uk could be found above the distributor and belowthe distributor by absolute pressure measurement.

NOMENCLATURE

a scale parameterAj(ti) approximation of multi-resolution decomposition at level jAJ(ti) approximation of multi-resolution decomposition at level Jb transition parameterdp particle diameter, mmDj(ti) detail of multi-resolution decomposition at level jemax. i maximum of energy of the PSDF at the different superficial

gas velocities, Pa2/HzeMAX an element maximum value of the vector (~E), eMAX ¼

(emax.1, emax. 2, . . . , emax.n), Pa2/Hz

~E the vector of maximum energy of PSDF at differentsuperficial gas velocities, ~E ¼ (emax.1, emax.2, . . . ,emax. n), Pa

2/Hz

Enorm:��!

normalized the vector of ~E by eMAX. Enorm:��! ¼

(emax.1/eMAX, emax.2/eMAX, . . . , emax. n/eMAX)f frequency, HzHs static bed height, mJ maximum multi-resolution leveln integer numberN number of sampling point�p average of time series of pressure fluctuations, k Papi time series of pressure fluctuations, k PaPSDF power spectral density function, Pa2/HzPxx(v) power spectral density function (PSDF), Pa2/HzRAJ relative wavelet energy distribution in level J

(approximation scale), [–]RDj relative wavelet energy distribution in each level j

(detail scale), [–]Rxx autocorrelation functionS.D. standard deviation of pressure fluctuations, k Pat time, sti the discrete time, sT interval time, sUc transition velocity at which standard deviation of

pressure fluctuations reaches a maximum, m s21

Figure 10. The normalized PSDF of the maximum energy vs. superficial gas velocity, at static bed 60 cm, in height, measured above the distributor at14 cm by an absolute pressure probe.


490 SHOU and LEU

Ug superficial gas velocity, m s21

Uk superficial gas velocity corresponding to levelling outof standard deviation of pressure fluctuations as Ug

is increased, m s21

x(t) time series of pressure fluctuations, k Pax(ti) the orthogonal wavelet series approximates

to a continuous signal, k PaW wavelet energy, Pa2

Wf (a, b) the wavelet coefficientWT total wavelet energy, Pa2

WAJ approximation scale energy in level J (J ¼ log2 N)

WDj detail scale energy in each level j, Pa2

Greek letterst time lag, srp particle density, kg m23

v angular frequency, Hz (v ¼ 2pf )wa;b(t) wavelet base (mother wavelet)w�a;b(t) complex conjugation of wa;b(t)

Subscriptj multi-resolution level

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The manuscript was received 19 April 2004 and accepted forpublication after revision 9 February 2005.



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