7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Spectral analysis of dynamic behaviour of the continuous phase in
liquid fluidized bed
Kechroud Nassima, Brahimi Malek and Djati A.Halim
Department of process engineering, Faculty of Technology, University A.MIRA of Bejaia,
Bejaia, Bejaia, Algeria
Kechroud.nassima(vahoo.fr, mk brahimi(vahoo.fr, diatihalim@(vahoo.fr
Keywords: Liquid fluidization; laser anemometry, spectral analysis; multiresolution analysis,
Coherent structures
Abstract
The spectral analysis of dynamic behaviour of the continuous phase in liquid solid fluidized bed is
characterized through velocity measurements by laser anemometry at the top of the bed. The experiments
were conducted using glass particles of 2, 4 and 8 mm diameter fluidized by water.
The spectra analysis of the velocity time series has revealed a specific spectral dynamic of liquid fluidized
bed for the high frequencies range which does neither follow strictly the kolmogorov law nor a Brownian
process power law. The continuous phase does not reach a fully developed turbulence as is the case for
single phase high Reynolds number flow. A time frequencyscale decomposition combined to an
autocorrelation analysis of velocity signal was pertinent to capture the impact of porosity waves and
cooperative movements of particles on the liquid phase dynamic, and to characterize these coherent
structures by low frequencyscales (below 1 Hz). The results compare well with the available data obtained
directly from void propagation studies by light transmission techniques (elkaissi and Homsy (1976), Ham
et al. (1990), Poletto et al. (1995)).
1. Introduction
Fluidization is an operation which
involves the suspension of solids in contact with
gas, liquid or both. A large variety of processes of
interest to chemical industry and biotechnology is
concerned with fluidized bed operations.
Examples include crystallisation, ion exchange,
adsorption as well as chemical reaction. It is of
great interest to understand the dynamics of
fluidized beds for a good control and optimal
design of the process as well as for its realistic
simulation. To this end, experimental studies have
been conducted to highlight some of the physical
mechanisms of the unsteady behaviour of
fluidized beds and many of them have concerned
the dispersed phase dynamics. A detailed analysis
of the flow regimes and transition in liquid
fluidized beds has been conducted by Didwania
and Homsy (1981). Using an optical technique,
they characterized four distinct regimes, in terms
of the time and length scales of the particle
motion. The regimes include wavy flow, wavy
flow with transverse structure, fine scale turbulent
flow and bubbling states in the order of decreasing
solid fraction. Digital video recordings were used
by Poletto and al. (1995) to obtain voidage
distribution in a narrow rectangular fluidized bed.
From these recordings, the authors determined
temporal and spatial autocorrelations, the
corresponding spectra, and found that the porosity
waves are characterized by frequencyscales lower
than 1 Hz, and the dynamic becomes more chaotic
when the porosity increases. Zenit and Hunt
(2000) analysed the nonsteady component of the
volume fraction signal obtained from an
impedance volume fraction meter. The root mean
square (RMS) of the void fluctuations and their
spectra have been analysed and show also that the
wavelike bands of low concentration are
confined in low frequency scales while high
frequency fluctuations are more random in nature.
To realise a finer analysis and to further
exhibit some other aspects related to the unsteady
behaviour of fluidized beds, researchers have
introduced multiresolution analysis (Percival and
Walden (2000) and quantitative methods of
dynamical systems (Abarbanel ( 1996), Kantz and
Schreiber (1997)).
Till now, these tools have been used to
study gas fluidized beds dynamics from
essentially pressure fluctuation recordings (Ellis
and al. (2003), Sasic and al. (2006), Kulkami and
al., (2001); Lu and al. (1999); Guo and al. (2002))
and rarely used for liquid fluidized beds. Despite
their valuable contribution to improve our
understanding of the bubbles dynamic and to give
a finer characterization of the fluidization
regimes, there is a consensus about the need to
investigate further the origin and nature of large
and small scale motions in both phases as pointed
out by Johnson and al. (2000) and by Ellis and al.
(2003), among others.
The best experimental approach to have
access to the different significant scales of motion
is to do local measurements capturing, with
enough resolution, either the continuous phase
dynamic or the particles dynamic. We observe a
severe lack of experimental studies on the
continuous phase dynamic probably because of
the practical difficulties to do measurements
inside the bed with a good reliability.
Nevertheless, we can mention the work of
Handley and al. (1966) who made measurements
inside a liquid fluidized bed with a micro Pitot
tube, but recognize the inaccurate estimation of
turbulence characteristics, and the work of
Bernard and al. (1981) who used laser
anemometry in a completely transparent liquid
fluidized bed. These latter authors has considered
two porosities and limited the turbulence study
only to the RMS of the velocity fluctuations of the
liquid phase and did not conduct a spectral
analysis of the dynamic.
This paper reports the results of a spectral
analysis of liquid velocity fluctuations obtained by
laser anemometry. The measurements have been
done just above the top of the fluidized bed
considering, by the frozen velocity field
hypothesis of Taylor (Monin and Yaglom (1975)),
that they represent with a good approximation the
liquid phase dynamic inside the bed. Fourier
spectra are studied in different operating
conditions and the high frequency decaying range
is compared to characteristic power laws of
equilibrium inertial subrange (Kolmogorov) and
of Brownian type process. To precise and
determine the origin and nature of low and high
frequency scales, we have proceeded to multi
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
resolution analysis of the velocity time series.
Coherent and random motions have been
identified and characterized by their frequency
scales. When possible, we have found useful to
reveal the similarity between the liquid and
dispersed phase dynamics.
Nomenclature
Archimedes number
particle diameter
superficial velocity
minimal fluidisation velocity
terminal velocity
Greek letters
particle density
fluide density
porosity or void fraction
2. Experimental setup and analysis methods
2.1 Experiments
A schematic diagram of the experimental
setup of fluidized bed is shown in Fig 1. The
column was made of Plexiglas with an inner
diameter of 9.3 cm and a height of 2m. It was
marked with a graph paper along its length for
measuring the height of the bed, a parameter
which is used for determining the bed void
porosity (s). Particular care was taken to obtain
uniform fluid distribution at the base, for this,
several assembly distributors were tested, and we
found that a fine wire mesh preceded by an empty
calming section provide uniform distribution of
water (Kechroud (2000)).
Water was pumped from a storage tank
through calibrated rotameters and admitted at the
bottom of the column through a distributor
assembly. A pair of rotameters was used to cover
wide range of liquid flow rate. The water exited
from the top of the column and recirculated back
to the storage tank through an overflow weir. All
piping was made in PVC and had 28 / 32 mm
diameter.
Table 1 gives the properties of the beads under
investigation as well as the range of flow rates
covered for each bead size, in addition to
experimentally obtained values of bed voidage
and liquid velocity at minimum fluidization
conditions.
The velocity of the liquid phase has been
measured by Laser Doppler Anemometer (LDA).
The LDA set up include a Dantec 55 X modular
series along with instrumentation such as an
electronic frequency shifter connected to the Brag
cell and a counter which delivers the validated
data. The light source is an argonion laser and the
power of emitted beam can be regulated up to 5W.
Beams from the splitter optics passed the focusing
optics to intersect and form a measuring volume,
just above the top of the fluidized bed, on the axis
of the column. For each superficial velocity, the
measuring volume was displaced upward as the
bed height increases in order to avoid crossing the
volume by the particles. In all the experiments, tap
water was used as working fluid. No additional
tracer particles were added. The natural seeding of
tap water was sufficient to our purposes. Two
modes of sampling frequency allowed by the
LDA technique can be used to generate time
series of the velocity: random and fixed sampling
frequencies. We have investigated the effect of
signal sampling rate and sample size on the
reproducibility of the mean velocity and the root
mean square of the velocity fluctuations values in
different operating conditions (Kechroud (2000)).
The results presented in this paper are obtained
from time series of 10240 data points with a data
rate of 20 Hz for the fixed sampling frequency
mode, the only mode used for the spectral and
multiresolution analysis because time spacing of
the data must be constant as required by the
algorithms (Percival and Walden (2000),
Nievergelt (2001)). Several authors have observed
that the significant frequency content is below 10
Hz and then, a sampling frequency of 20 Hz is
sufficient (Didwania and Homsy (1981); Poletto
and al. (1995), ElKaissy and Homsy (1976);
Johnson (2000)).
2.2 Analysis methods
A power spectral density estimation of the
fluctuations has been conducted to reveal the
energy distribution among the different
frequencies, and the spectral shape of the liquid
phase dynamic. The results of the Fourier spectra
have been obtained using Welch's method, in
order to reduce the variance of power spectral
density estimation (Condy (1988). The estimation
is based on an average of several subspectra.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Therefore, the whole time series is divided into
segments of equal duration and the power
spectrum of each segment is calculated. The
averaged power spectrum is given by
1
P,(f) = Pd(f)
N
where N is the number of segments and P' sdO) is
the power spectrum of segment i.
In the present study, N = 20, then one segment
contains 512 data points of 25.5 s duration, and
the frequency resolution is 0.04 Hz.
In addition to Fourier spectral analysis
presented in this paper, we have also conducted a
multiresolution analysis of the velocity signal.
Our purpose is to quantify the frequency scales of
the coherent structures which appear in the
unsteady behaviour of liquid fluidized beds. This
structures may be the porosity wave like
movements, clusters of particles with cooperative
movements and voids as observed by several
authors with increasing porosity (Didwania and
Homsy (1981), Poletto and al. (1995) ; ElKaissy
and Homsy (1976). To this end, we need a time
frequency scale decomposition of the velocity
time series combined to an autocorrelation
analysis (Percival and Walden (2000), Li (1998)).
The wavelet transform allows localization,
both in the time and frequencyscale domains via
translation and dilation of the wavelet (Mallat
(1999)).
Such a representation gives a multiresolution
framework for analyzing the phenomena present
in a time series. In practice, the discrete wavelet
transform is a pair of digital filters, which
decompose a signal into a low frequency
component A1 (called the approximation) and a
high frequency part Di (called the detail) (Percival
and Walden (2000)). In the next step, the
approximation A1 is used as an input, and by
performing this operation recursively up to a level
k, a hierarchical representation of the signal is
obtained:
k
u(t)= DJ +Ak
The detail Dj contains frequency information in
the band [f,/2'+1,f,/2'] where fs is the
sampling frequency and j is an integer. Each of
these frequency bands defines a frequencyscale.
Finally, the original signal can be reconstructed
from wavelet coefficients by the inverse wavelet
transform without losing information (Percival
and Walden (2000). In this study, we have used
orthogonal Daubechies wavelets of order five (Db5)
as the mother wavelet, following some authors such
as Ellis and al. (2003).
The autocorrelation function, which
expresses the relation of earlier to later values in
the time series, is applied to different levels of
signal decomposition in order to show the
coherence of the corresponding frequencyscale
movements. The correlation between two points
separated by a time lag kAt is defined by:
Nkl
Nk1
DD (k) = Z(D (n) D )(D (n k) Dj )
n=0
This becomes, when normalized with the value at
zero lag, cDD (O)
CDD (k) CDD (k)
CDD(O)
We note that the calculations of the above
quantitative characteristics were made using
Matlab toolboxes.
3. Results and discussion
3.1 Spectral analysis
Power spectra of time fluctuations of local
liquid velocity have been determined for different
flow conditions and the results for three particle
diameters are reported on figures 2a)b)c). The
shape of the spectra shows two distinct ranges of
energy distribution; an energy containing range
with no distinguishable peak, and a decaying
energy range. We observe, when the flow rate
grows, that the energy containing range includes
progressively higher frequencies. But in all cases,
the most energetic movement scales of the liquid
phase are confined in low frequencies, less than 2
Hz. We note also a slight change in the slope of
the decaying range, more clear for the 2 mm
particles, with increasing superficial velocity or,
equivalently, increasing particle Reynolds
number. This dynamic change is not noticeable
beyond U/Umf = 2 and the spectra are similar. For
comparison, we have reported on the figures,
some characteristic power laws, 2 for some
stochastic processes and the famous 5/3 power
law of Kolmogorov.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The Kolmogorov slope characterizes the so
called inertial subrange of the spectra in single
phase high Reynolds number. For the inertial sub
range to appear, even in single phase turbulence,
there must be a large separation of scales, several
decades, between the low and high frequencies,
which is obtained at high Reynolds numbers. This
is necessary to claim that the flow has reached a
fully developed turbulent state. The spectra of the
liquid phase fluctuations obtained in the present
study exhibit only one decade of separation
between low and high frequencies because of the
low and moderate values of the Reynolds number.
It is not sure th'i the continuous phase
fluctuations of fluidized bed could reach a
universal equilibrium state in the decaying range
of the spectra.
In the system under investigation in the present
work, the Reynolds numbers based on the
superficial velocity varies from 49 to 250, 175 to
700 and%820 to 1750 for the particles of 2 mm, 4
mm and 8 mm respectively. These values are far
from representing those encountered in developed
turbulent flows and, in the case of the flow around
a single sphere, the flow structure shows different
dynamics at moderate Reynolds numbers (Clift
and al. (1978), Qiang and al. (1994), Tighzert,
(2002), Benabbas and al. (2003), Veldhuis and al.
(2004)). If we increase the flow rate to obtain
higher Reynolds numbers, the fluidized state will
rapidly reach its limit and hydraulic transport of
particles will take place (Hadinoto and
Curtis,(2009)). So, the fluidization state will
almost always be characterized by low and
moderate Reynolds numbers. We believe that the
multiple hydrodynamic interactions between the
particles and the liquid phase with additional
mechanisms of generating fluctuations, other than
the classical interaction between Reynolds
stresses and the mean velocity gradients, and a
specific energy transfer between scales prevent
probably from a rigorous observation of the
Kolmogorov law. Thus, the 5/3 power law does
not apply strictly in the decaying range of the
spectra although it is approached in some
conditions.
The 2 power law is observed for some stochastic
processes as those approached by an Omstein
Uhlenbeck model (Pope (2000)). This type of
process includes, in addition to a deterministic
contribution, a Brownian motion dynamic which
is a completely uncorrelated movement at small
scales. The spectra of the liquid phase dynamic
show that this power law is not strictly followed
too. The particles when observed carefully
suggest effectively a stochastic approach for the
mathematical modelling of their dynamics and, by
coupling, the continuous phase too. This
comparison with two characteristic dynamics at
high frequencies reveals the complex specific
behaviour of fluidized bed. We face a non
equilibrium system in statistical mechanical sense.
Unfortunately, there is a severe lack of
investigations on spectral analysis of liquid phase
fluctuations and we cannot compare directly our
results with others, even obtained in different
operating conditions (kechroud and al. (2000)).
But, for void fluctuations analysis, several works
have been conducted (Didwania and Homsy
(1981), Polleto and al. (1995), Zenit and Hunt,
2000; Ham and al., 1990; ElKaissy and Homsy
(1976)). It is interesting to show the similarities
and differences in the frequency domain between
the liquid velocity and void fluctuations in the
spectral representation.
Zenit and Hunt (2000) have presented spectra in
the same form as those reported on figures 2, but
have not smoothed them by averaging. Their
results show an energy containing range which
becomes wider with the superficial velocity, and a
decaying range, for different sphere diameters and
densities. The low frequency events do not exceed
3 Hz. These results are qualitatively similar to
ours. But the slope of the decaying range where
the high frequency random fluctuations are
confined, as indicated by the authors, seems to be
higher than 2 in practically all the operating
conditions. This power law is quite different from
those we have obtained. Poletto and al. (1995)
who also analysed void fluctuations and spectra,
observe qualitatively the same frequency range
subdivision and confine the coherent dynamic,
which is related to porosity wave composed of
different wave lengths, in the frequency range
below 1 Hz. The decaying range of high
frequencies present a slope rather lower than 2,
different from that obtained by Zenit and Hunt
(2000). We would like to note that the
measurements done by Zenit and Hunt (2000) are
cross sectional averages and seem to produce a
screening effect for high frequencies and so,
observes a more rapid decreasing frequency
range. In contrary, Poletto and al.(1995) have
done the measurements in a column of three
particle diameter depth and so resolve more
accurately the high frequencies.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Didwania and Homsy (1981) performed a detailed
study of the spectral behaviour of void
fluctuations using a light transmission technique.
The range of the significant frequencies does not
exceed 10 Hz. The spectra at low porosities are
dominated by a single peak at a frequency less
than 1.5 Hz, characterizing a porosity wave
propagating through the bed. Peaks tend to flatten
at intermediate and higher voidage indicating a
change in the dynamic and the presence of
different wave scales. The spectra of the liquid
phase fluctuations do not contain a clear peak at
low frequencies but this does not mean that there
is no impact of porosity wave on the liquid phase.
We need a time frequencyscale decomposition of
the fluctuations combined to an autocorrelation
analysis, to exhibit the coherent character of the
dynamic and to specify its frequencyscale (Li,
1998). This is what we present and discuss in the
following section.
3.2 Multiresolution analysis
We can consider the fluid phase dynamic as
the result of more or less coherent low frequency
movements, but deterministic, combined to
random fluctuations of higher frequencies
characterising stochastic processes. There is
actually a growing interest for stochastic
modelling in fluid mechanics (Pope (2000),
BamdorfNielsen and Schmiegel (2005),
Lamorgese and al. (2007), Balachandar (2009))
but we need further experimental guidance to
precise this new approach of modelling for
fluidized beds. Careful visual observations and
reported quantitative video measurements of the
behaviour of fluidized beds (Didwania and Homsy
(1981), Poletto and al. (1995)), suggest that the
coherent part of the dynamic has essentially two
origins: a porosity wave, from planar to complex
one, and a large scale coherent movement of
clusters of particles and voids (called bubbles),
which are observed at high porosities. These
movements may be called cooperative movements
of the particles and are characterized effectively
by low frequencies as reported by several authors
who analysed the fluctuations of the porosity in
different operating conditions (Didwania and
Homsy (1981), Poletto and al. (1995), Ham and
al. (1990), ElKaissy and Homsy (1976)). The
random part of the dynamic is to be related to the
small scale motions of the particles. This section
is intended to give some experimental support to
the above representation of the fluidized bed
dynamic through the multiresolution analysis of
the liquid phase dynamic.
To this end, the original time series of the axial
velocity is decomposed into several time series of
fixed frequencyscales by discrete wavelet
transform (Kechroud and Brahimi, 2005). Figure
3 shows an example of the decomposition
obtained for the particle of 4mm and s=0.7. This
time frequencyscale representation reveals the
presence of structures with high amplitude
regularly separated in time. These structures are
seemingly the wave like coherent structures. As
criteria, the frequencyscale of the coherent
movement is determined when the corresponding
autocorrelation presents a cyclic behaviour with
the largest amplitude. The cyclic feature
corresponds to the repeating occurrence of the
coherent structure. Figure 4 presents a sample of
the autocorrelation observed for the coherent
component of 0.469 Hz frequencyscale in the
operating conditions of figure 3. For higher
frequency (3.75 Hz), in the decaying range, and
same operating conditions, the autocorrelation
decreases rapidly as showed by the sample in
figure 4, indicating the random nature of the
corresponding small scale movement. The whole
multiresolution analysis is summarized in figure
5, were we have reported the frequencyscales of
the coherent components, according to the
retained criteria, as a function of porosity. For
comparison, we have also represented the
valuable data of Ham and al. (1990), ElKaissy
and Homsy (1976) and Poletto and al. (1995)
obtained from void propagation analysis directly.
The results show clearly that the propagating
coherent movement in the liquid phase is
characterized by low frequencies. Almost all the
frequencyscales are below 1 Hz. Thus, this
phenomenon is buried in the energy containing
range of the spectra presented above. We observe
also from the beginning of fluidization, and for
the lowest porosities, a slight increase of the
frequencyscale. Beyond e=0.5, the frequencies
are rather distributed around 0.4 Hz and fall
within the segments representing the results of
Poletto and al. (1995). In our knowledge, only
Poletto and al. (1995) have considered moderate
and high porosities (e=0.59, 0.66, 0.73) in the
investigation of propagation of voidage waves.
Ham and al.(1990) and ElKaissy and Homsy
(1976) have limited their analysis to low
porosities (E < 0.5) for which a clear peak appears
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
in the spectra and consider that there is no wave at
higher porosities for which bubbles appear. Our
results compare well with their data only in a
narrow range of porosities, just after the
fluidization of the particles. When approaching
e=0.5, they observe much higher values of the
frequencyscales. May be this difference is to be
related to the small dimensions of the particles
and the apparatus used by the authors. We cannot
draw a clear conclusion about the effect of particle
diameters on the frequencyscale of the coherent
structures because of the dispersion of the data.
The procedure followed in the present study to
determine the frequencyscale of the coherent
movement of the liquid phase has proved to be
pertinent. We have shown that the voidage wave
has an effect on the liquid phase dynamic and its
trace can be captured by multiresolution analysis.
The absence of peaks in the spectra should not be
automatically associated to the nonexistence of
coherent movements. So, the classical procedure
of Fourier analysis is not sufficient to detect and
evaluate the frequencyscales of coherent
structures. The random part of the dynamic is
confined in the high frequency range which
corresponds to rather small scale movements
induced by the particles. So, a dynamic model for
the two phases which does not take into account
these random small scale movements as those
based on a systematic averaged procedure, will
not reflect the real behaviour of the fluidized beds.
4. Conclusion
We have shown in the present study that the
local liquid velocity time series recorded at the top
of a liquid fluidized bed is representative of the
continuous phase behaviour inside the bed. The
procedure has allowed us to characterize
quantitatively spectral aspects of the dynamics.
The spectra of the liquid velocity fluctuations and
the multiresolution analysis are complementary
and have revealed some new features of liquid
fluidized bed dynamics.
The continuous phase does not reach a fully
developed turbulence as is the case for single
phase high Reynolds number flows and thus does
not exhibit clearly the so called inertial
equilibrium subrange of Kolmogorov. The high
frequencyscales are also found to have a rapid
decreasing autocorrelation function and then
qualified as random fluctuations, but their spectra
do not correspond strictly to the power law of an
OmsteinUhlenbeck type process. So, fluidized
beds appear to follow a specific spectral dynamic
which nevertheless contains coherent structures
identified by cyclic large amplitude auto
correlation function at low frequencies (below 1
Hz). This large scale coherent part has been linked
to cooperative movements of the particles while
the random part is related to the small scale of the
particle Brownian type motion. This spectral
characterization and subdivision of the liquid
phase dynamic constitute an experimental basis
for the development of advanced mathematical
modelling of liquid fluidized beds of coarse
particles such as stochastic processes based
modelling.
We have clearly shown the analogy between the
void dynamic and liquid phase dynamic through
the comparison of the spectra and frequency
scales of the coherent movements of both
dynamics. We have observed a good similarity
which is probably a consequence of an underlying
correlation between particle and continuous phase
fluctuations. It is of great interest to highlight the
details of the fluctuation dynamics of both phases
and their correlation. It is one of the scopes of our
investigations in connection with the present and
future quantitative experimental studies.
Acknowledgements
The authors are grateful for the financial
support of the Ministry of Higher Studies and
Scientific Research of Algeria.
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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
1Fluidized bed
2 Rotameters
3 Pump
4 Low water reservoir
5 Upper water reservoir
6 Laser source
7 Bragg cell
8 Photomultiplier
9 Frequency shift
10 Counter
11 Data acquisition system
Figure 1: Schematic of experimental setup
Table 1: properties of particles used in experiments
dp Umf Ut
Material Pp/Pf U(cm/s) 8 Ar. 10
(mm) (cm/s) (cm/s)
2 2.554 2.45 21.07 0.88 12.25 0.4 0.81 1.22
glass
4 2.564 4.38 30.43 1,59 19,7 0,43 0,81 9.82
8 2.595 7.8 42.18 2.819.5 0.440.7 80.1
1E1
tM5/3
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
b)
1E2
1E3
1E5
0.01 0.10 1.00
Frequency,Hz
10.00
0.01 0.10 1.00 10.00
Frequency,Hz
) 1E
1E1
1 E2
a
S1E3
1E4
s U/Umf Re
0.45 1 624
0.55 1.5 922
0.62 2 1232
0.7 2.5 1560
0.73 2.8 1747
0.10 1.00
Frequency,Hz
10.00
Figure 2: Effect of superficial liquid velocity (or bed voidage s) on power spectral density functions: a):
dp=2mm, b): dp=4mm, c): dp=8mm
1E3
1E4
1E5
1E6
s U/U Re
0.42 1 49
0.48 1.5 74
0.55 2 98
0.6 2.5 123
0.64 3 147
0.81 5 245
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Uecomposition at level 9: u = aUd + db + dt + d b + d d + cB + d4 + d l + d2 + 01 .
0.14
d ,
a 0.13 iii i i i
9 o .oi . ..__ i ___I_ I ___ I ___ __ L
9 0.0
d8 0: 0
42 0.02 . ..
10 0 02
6 00
d 0 A.. 1Z L1.A A.j ^I l
d, .1:LA LJ o^tAj j
d .
d2 0.
d 0 1 11 Il 11
d^ . ^^^t^^^it>^4il~lii~llt*llli*l~l
Frequency
bands (Hz)
[0.04 0.02]
[0.04 0.02]
[0.08 0.04]
[0.156 0.08]
[0.313 0.156]
[0.625 0.313]
[1.25 0.625]
[2.5 1.25]
[5 2.5]
[105]
Frequency scale
(Hz)
0.029
0.029
0.058
0.117
0.234
0.469
0.937
1.87
3.75
7.5
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Figure 3: Discrete wavelet decomposition of liquid velocity fluctuations
for dp =4mm, U/Unf=3 (s = 0.7)
08 0.469 Hz
o 04
02
8 0 2
0 4
0 6
0 8
0
Time delay, s
125
Time delay, s
Figure 4: Multiresolution autocorrelation coefficients of velocity components
dp=4mm, U/Umf=3 (s = 0.7)
3.75 Hz
C rJ
C)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Our experimental results
Sdp 2 mm
Sdp= 4mm
Sdp 8 mm
+ f
06 0.8
bed void fraction, e
Experimental results of
Elkaissy and Homsy,(1976)
dp=0.59 mm
S dp 0.83 mm
A dp 1.1 mm
A dp 1.56 mm
Experimental results
of Ham and al.,(1990)
dp=0.325 mm
Experimental results
of Poletto and al., (1995)
+ dp =6.35 mm
Figure 5: Frequencyscales of the coherent structures versus s
1.5
N
=
0.5
0
A
* A
0.4
0.4
