On coherent structures in gas–solid fluidization

Journal Pre-proof On coherent structures in gas-solid fluidization Musango Lungu, Haotong Wang, Jingyuan Sun, Jingdai Wang, Yongrong Yang, Fengqiu Chen, John Siame

PII:

S0263-8762(19)30452-6

DOI:

https://doi.org/10.1016/j.cherd.2019.09.035

Reference:

CHERD 3825

To appear in:

Chemical Engineering Research and Design

Received Date:

29 April 2019

Revised Date:

25 September 2019

Accepted Date:

26 September 2019

Please cite this article as: Lungu M, Wang H, Sun J, Wang J, Yang Y, Chen F, Siame J, On coherent structures in gas-solid fluidization, Chemical Engineering Research and Design (2019), doi: https://doi.org/10.1016/j.cherd.2019.09.035

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On coherent structures in gas-solid fluidization

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Musango Lungu1,2, Haotong Wang1, Jingyuan Sun1, Jingdai Wang1*, and Yongrong Yang1, Fengqiu Chen1, John Siame2

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State Key Laboratory of Chemical Engineering, College of Chemical and Biological Engineering, Zhejiang University; Hangzhou 310027, P.R.China. 2

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Multiphase Flow Research Group, Chemical Engineering Department, School of Mines and Mineral Sciences, Copperbelt University, Kitwe 10101, Zambia.

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Graphical Abstract

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Highlights

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Hilbert spectral analysis of fluctuating axial particle velocity at x/X = 0.0 for (a) Ug/Umf = 2, (b) Ug/Umf = 3 and (c) Ug/Umf = 4.

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Coherent structure identification and characterization using advanced mathematical tools.

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Spectral analysis of pressure and axial particle velocity fluctuations.

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Multiscale resolution of particle velocity fluctuations using EMD.

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Time-frequency analysis using wavelet coherence analysis and Hilbert spectral analysis.

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Abstract In this work, experimental data from the novel high speed particle image velocimetry (HSPIV) measurements and pressure fluctuations complemented by numerical predictions from TFM simulations are processed using advanced signal processing protocols to identify and characterize coherent structures in a gas-solid fluidized bed with Geldart D particles. The time-frequency properties of the bed dynamics were studied using the wavelet coherent analysis (WCA) and the Hilbert Huang transform. The WCA of numerical pressure drop signals at

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different measurement positions showed the existence of spatial-temporal coherent structures. At close measurement positions phase locked coherent phenomenon due to bubble generation dominate most of the

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frequency and time scales due the proximity to the distributor. At a wider measurement spacing the fast

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traveling waves are attenuated due to gas bubble/void coalescence and acceleration and only pockets of highly coherent oscillations are visible mostly in the frequency range of 0.5-3 Hz at certain times. Multi-resolution of

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the particle fluctuations realized with the Hilbert-Huang transform. The structures were resolved into the micro, meso and macro scales of fluidization based on the energy and frequency distributions. The meso scale

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and ultimately the mixing.

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structures form the main contribution to the normal axial Reynolds stresses and bubble granular temperature

Keywords: Coherence; Hilbert-Huang Transform; Fluidization; Wavelet analysis

1. Introduction

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Fluidization technology offers several advantages compared to other gas-solid contactors including good mixing, small temperature gradients and high heat transfer coefficients (J. R. Grace, 1986). It is therefore not surprising that this technology finds a wide range of applications including roasting of ores in the metallurgical industry, coal and biomass gasification (Kunii and Levenspiel, 1991) and lately chemical looping combustion (Gauthier et al., 2017) among others. Despite its wide industrial usage, systematic design, scale-up and operation of fluidized beds still remains a challenge mostly due to the poorly understood relationship between 3

the fluid mechanics of the bed and the design objectives such as residence time distribution, mixing time, dispersion coefficients and heat and mass transfer coefficients. Typical fluid mechanic parameters include the mean velocity components and turbulent intensities; turbulent kinetic energy and turbulent energy dissipation rate; eddy diffusivity; components of mean and turbulent stress; size, shape, velocity and energy distribution of turbulent structures; energy spectra and so on (Joshi et al., 2017). Therefore current design and scale up procedures of such equipment still heavily relies on empirical correlations and past experience. Fluidized beds like other multiphase reactors are operated under turbulent conditions to enhance contacting

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between the different phases and achieve better performance. The turbulent motion encountered in these

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reactors consists of coherent and incoherent motions. Coherent motions or structures are defined as discernible patterns in the flow which occur with sufficient regularity, in space and/or time, to be recognizable as

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quasiperiodic or near-deterministic (Joshi et al., 2009) whereas incoherent motions are exact the opposite. Gas-

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solid flows are characterized by an abundance of these transient structures with varying degree of vorticity and from a reactor design perspective there is need to qualitatively and quantitatively characterize them and

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determine their effect on reactor performance vis-a-vis mixing, transport phenomena and chemical reaction. Research on coherent structures in multiphase reactors gained traction in the 1990s, a paradigm shift from the

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common steady state treatment but has mostly focused on liquid-gas systems as reviewed by (Joshi et al., 2002) and (Mudde, 2005). The steady-state or uniform flow approach masks the structures which are revealed by analyzing fluctuating component of phase fractions, velocities, pressure and temperature (van den Akker, 1998). Improvement of computational tools, advanced experimental and signal processing techniques now

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makes it possible to study multiscale flow structures in both liquid-gas and gas-solid systems with relative ease. By using laser doppler anemometry, LDA, (Groen et al., 1996) showed that the time averaged profiles of the liquid velocities in bubble columns were as a result of the contribution different velocities from different coherent structures. A year later, (R. F. Mudde et al., 1997) investigated the role of these coherent structures on normal Reynolds stresses in a pseudo 2D bubble column using PIV. It was demonstrated in this work that the 4

low frequency vortical structures were responsible for the characteristic parabolic shape of the normal stresses and that neglecting these structures gave rise to flat profiles of the stresses. (Harteveld et al., 2003) studied the influence of gas distribution on coherent structures using short time frequency transform in liquid-gas bubble column. It was observed that uniform gas aeration generated weak low frequency coherent structures as opposed to non uniform gas aeration which generated strong coherent structures at low frequencies. (Kulkarni et al., 2001) used wavelet analysis for the multiresolution of local liquid velocity in a bubble column to extract the different coherent structures.

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Parrallel studies in solid-gas systems include the work of (Mudde et al., 1999) who studied the dynamics of a

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fluidized bed with Geldart D particles using gamma radiation. Application of probability density functions and cross correlation techniques to the time series data qualitatively showed the existence of coherent motions

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moving up and down the bed . Simultaneous measurement of pressure fluctuations within the bed and in the

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plenum (van der Schaaf et al., 2002) revealed the existence of spatially coherent structures i.e. voids and clusters in gas-solid fluidized beds. Estimates of charactristic length scales of flow structures that is bubbles,

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slugs and solid clusters were also obtained . In her doctoral disseration (Naoko, 2003), applied statistical, spectral and coherence function analyses to pressure and optical probe voidage signals in the characterization of

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coherent motions in turbulent fluidization. TFM simulations by (Sun et al., 2012) yielded promosing results for characterization of coherent motion and turbulence. In their work, the effects of coherent structures on the intermittency of the flow field was investigated using a novel coherent structure extraction procedure and extended self similarty (ESS) scaling law. (Sánchez-Delgado et al., 2018) applied wavelet coherence over the

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bubble void fraction (BVF) time series for different distributor designs and other operation conditions in an attempt to correlate the short and long term bubbling dynamics. Unlike in liquid-gas bubble columns, it was observed that the distributor design in gas-solid bubbling fluidization had minmal effect on the long term bubbling dynamics which are quintessentially low frequency coherent structures. The mechanism of how coherent strutures are formed is not yet fully understood and has continued to stir debate among the fluidization community. According to (Naoko, 2003), turbulence leads to the presence of 5

local coherent structures. Professor Jinghai Li’s research group at the institute of process engineering, Chinese academy of sciences advanced the idea of the so called energy minimization (Li, 2000) in which competing effects of the discrete and continuous phases leads to the creation of non-uniform and non-linear heterogeneous structures as a compromise for stability. Further multi-scale analysis and multi resolution of the heterogeneous structures with respect to the particle, cluster/void and unit scales was proposed to elucidate the undermining mechanisms. The study of multiphase flow at different scales has now been generally accepted as the norm and wavelet based methods have been the choice for multiresolution. An alternative to wavelet based methods for (N E Huang et al., 1998) capable of handling non-linear

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multiresolution is the Hilbert-Huang transform

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and non-sttionary. In this work, wavelet based methods as well as the Hilbert-Huang transform have been used. (Briongos et al., 2006) applied the HHT for multiresolution of gas-solid fluidized bed pressure fluctuation

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signals operated under slugging conditions into particle dynamics, local bubble and bulk components . (Sun and Yan, 2017a) identified and characterized clusters in a CFB test rig by applying the HHT to electrostatic and

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accelerometer signals. (Llop and Gascons, 2018) combined the HHT and recurrence quantification analysis to

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characterize gas-solid fluidized bed dynamics in the bubbling and and slugging regimes using pressure fluctuations. More recently (Zhang et al., 2018) studied the non-linear fluidization behavior of a vibration gas

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solid fluidized bed, VGFB by means of multi scale reolution of pressure fluctuations using HHT. The aforementioned studies using HHT have made use of pressure fluctuations signals which provide global information as opposed to fluctuations measured say using optical or capacitance probes which provide local information. It is for this reason that the fluctuating component of particle velocity is normally used for studying

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turbulence and characterization of coherent structures. To the best of the authors’ knowledge it is the first time that the HHT is applied to particle velocity measurements in this paper for coherent structure idenfication and characterization.

A review of the current experimental techniques for particle velocity measurements by (Bhusarapu et al., 2006) shows that most measuring techniques suffer a drawback of low data sampling rates and poor resolution making it diffcult for complete temporal resolution and charactrization of the different coherent structures and 6

also for the validation of CFD models. The typical frequency range for multiphase flows covering the complete temporal range of flow structures is 10-1 - 103 Hz (Drahoš and Čermák, 1989) . To overcome this challenge the National Energy Technology Laboratory (NETL) developed a novel high speed particle image velocimetry, HSPIV (Gopalan and Shaffer, 2012) capable of higher sampling rates making it more useful for the complete temporal resolution of the flow structures encountered in gas-solid fluidization and also for validating CFD models. It is worth noting that the need for systematic CFD model validation and uncertainty quantification was the motivation for NETL to generate the small scale challenge problem

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https://mfix.netl.doe.gov/experimentation/challenge-problems/ which is also the basis for the current study. The

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HSPIV method has been previously applied successfully to CFB risers operated under dense suspension upflow and annulus regimes (Gopalan and Shaffer, 2013) .

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Our previous works (Lungu et al., 2018, 2016) focused on validation of TFM using this experimental data. In

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this work attention is paid mostly to the identification and quantification of coherent structures based on the NETL experimental measurements. Our motivation for conducting this work stems from the fact that such work

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for gas-solid systems is rather scanty in the open literature in comparison to liquid-gas systems. This might be probably due to the difficulty in designing experiments and obtaining experimental measurements with

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sufficiently high sampling rates from dense particulate systems. Results from our previous TFM simulations have been used to complement experimental measurements where necessary. The rest of the paper is structured as follows; section 2 gives a brief introduction of the experimental setup, the TFM is described in section 3, the methods used to treat the data have been described in section 4 and the results are reported and discussed in

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section 5. The first part of the results and discussion gives the picture of the fluidization pattern as observed from the TFM numerical predictions. Findings of the spectral analysis of particle velocity and pressure flcutuations are then presented and discussed there after. Finally results of multiresolution analysis using the wavelet coherence analysis and the Hilbert-Huang transform are displayed and analyzed . The conclusions drawn from this work are given in section 6.

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2.0 NETL experimental set up The test rig for the small scale challenge problem consists of a transparent thin rectangular column with dimensions of 0.075 m depth, 0.23 m width and 1.22 m length constructed by NETL for the small scale challenge problem. The column is filled with Nylon beads (Sauter mean diameter of 3256 μm and particle density of 1131 kg/m3 ) giving a height of 0.173 m and void fraction of 0.42 at minimum fluidization conditions. The bed was fluidized with air at standard conditions and the minimum fluidization velocity of the system was determined to be 1.05 m/s from the pressure drop versus superficial gas velocity curve. Three

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experimental trials were conducted at superficial gas velocities of 2.19 m/s, 3.28 m/s and 4.38 m/s respectively

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corresponding to so called fluidization numbers, i.e. Ug/Umf of approximately 2, 3 and 4.

Eulerian particle velocity measurements were taken at five lateral positions of 0.02356 m, 0.06928 m , 0.115 m,

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0.16072 m and 0.20644 m and height of 0.0762 m above the gas distributor using HsPIV at sampling rates of

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1, 1.2 and 1.5 kHz and for total sampling times of 21.096 s, 17.88 s and 14.665 s respectively for the three operating conditions. The five lateral positions correspond to dimensionless positions , x/X of -0.795, -0.398,

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0.0, 0.398 and 0.795 respectively. Granular temperatures were computed from the Eulerian particle velocity measurements following (Gopalan and Shaffer, 2012). The pressure fluctuations were measured for each of the

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three operating conditions with a Setra differential pressure transducer between the taps at heights of 0.0413 m and 0.3461 m above the distributor at a frequency of 1 kHz for 300 s. Detailed experimental description and other information can be obtained from the NETL website https://mfix.netl.doe.gov/experimentation/ .

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3.0 Numerical model

Numerical simulations were performed to extract information not readily available from the experimental measurements as a means of complementing the experimental work. For instance the use of the wavelet coherence analysis method requires bivariate time series which the experimental work did not provide but could easily be extracted from the numerical simulations.

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The Euler granular model with closures from the kinetic theory of granular flow available in ANSYS Fluent 15.0 was used for 3D simulations based on the experimental setup. The governing mass and momentum equations together with the full partial differential equation for the conservation of granular energy are solved in the finite volume framework with appropriate initial and boundary conditions. The gas-solid interaction is modeled according to the expression by (Syamlal and O’Brien, 2003) tuned using the experimental minimum fluidization velocity which ensures that the predicted drag force at minimum fluidization conditions matches that from the experimental measurements . A detailed description of the model equations, numerical techniques

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and set up is reported in our previous work (Lungu et al., 2018) and not repeated here for brevity.The

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simulations were run for a total of 40 s and the first 15 s of the simulations were not considered due to the start up transients and instead the last 25 seconds were used for analysis. Pressure and particle velocity time series

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were obtained at numerical probes inserted in the flow field corresponding to the actual measurement positions

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in the experimental set up. 4.0 Data treatment

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As aforementioned in section 1, coherent structures can be identified and characterized using advanced signal

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processing protocols. In our study we have restricted our analyses to frequency and time-frequency domains.

4.1 Frequency domain analysis

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4.1.1 Welch method

The Welch method (Johnsson et al., 2000) is adopted for the Fourier spectral analysis. In this method, several sub spectra are averaged to obtain an average estimate of the power spectrum in order to reduce the variance. The sub spectra are chosen as a compromise between frequency resolution and variance. Firstly the time series of the quantity of interest is divided into L segments of individual length Ns as follows:

xi  n   x  n  iNs  n  1, 2,...N s , i  1, 2,..., L

(1) 9

The power spectrum estimate of each segment is given as

1 P f  N sU i xx

Ns

 x  n  w  n  exp   j 2 fn  n 1

i

2

(2)

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where U is a normalization factor of the power in the window function, w (n) expressed as:

U

1 Ns

Ns

 w  n 2

(3)

n 1

1 L i  Pxx  f  L i 1

(4)

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Pxx  f  

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Finally the averaged power spectrum for a time series signal x is given as:

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An important property of spectral analysis is the conservation of energy of the signal in the frequency domain

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otherwise known as Parseval’s theorem expressed as: 

 2   Pxx  f  df 0

(5)

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where  2 is the variance of the fluctuating component of the signal in the time domain. Thus the variance can be estimated directly from PSD curves simply by integration of the area under the curve over the frequency

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range of interest and taking the square root of the result yields the standard deviation.

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4.2 Time-frequency domain analysis

Fluidized bed dynamics are inherently non-linear and non-stationary implying that their frequency content changes over time and therefore the use of frequency based methods results in the loss of temporal information. The need for capturing the temporal changing properties of signals has long been recognised by researchers in various fields leading to the development of time-frequency methods including short time Fourier time, wavelet

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analysis and the Hilbert Huang Transform. Several researchers have demonstrated the usefulness of timefrequency methods for investigating multiphase reactor dynamics including fluidized beds.

4.2.1 Wavelet coherence analysis The WCA expresses the joint common properties of a bivariate time series in the time-frequency domain. For a reference time series X and any other time series Y at a vertical position upstream , WCA is expressed as the cross wavelet spectrum squared normalized by the auto wavelet spectra (Grinsted et al., 2004) :

2

 

 S s 1 WnY  s 

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

S s 1 WnX  s 

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(6)

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Rn2 

S  s 1WnXY  s  

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with WnXY  s   WnX  s WnY   s  where  indicates complex conjugate. S is a smoothing function in both time and scale. It can be seen the definition of WCA is similar to the conventional Fourier based magnitude squared

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coherence function. It is advantageous to compute Rn2 rather than the wavelet cross spectra because

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normalization of the wavelet cross spectra removes the dependency of the individual wavelet power spectra which might otherwise have an ambiguous interpretation.

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The phase difference,  xy in radians of the bivariate time series is (Dashtian and Sahimi, 2013):

 I Wn  s     xy  s   arctan   XY   XY

  R Wn

(7)

 s  

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where I and R are the imaginary and real components of the complex function respectively with  xy    ,   . 4.2.2 Hilbert-Huang Transform The Hilbert-Huang transform (HHT) was developed by researchers at NASA led by Norden Huang to handle nonlinear and non-stationary data and has been successfully applied in several fields including geophysical studies, mechanical systems analysis and financial markets. It is adaptive and highly efficient consisting of two

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parts; the empirical mode of decomposition (EMD) and Hilbert spectral analysis (HSA). Table below summarizes the differences between HHT and other commonly used analyses.

Application of the EMD to non-linear and non-stationary time series data yields a set of so called intrinsic mode functions (IMFs) which represent the oscillation modes embedded in the data. IMFs are required to satisfy two conditions , firstly the number of extrema and the number of zero crossings must either equal or differ at most

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by one in the whole data set and secondly at any point the mean value defined by the local maxima and the envelope defined by the local minima must be zero (Norden E Huang et al., 1998). EMD uses an iterative

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sifting process to obtain the IMFs as follows:

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1. Lower and upper envelopes are defined for a uniformly sampled time series, x (t) by connecting identified local maxima and minima using the cubic spline fit.

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2. The mean of the two envelopes for the ith iteration mk ,i  t  is computed.

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3. For the first iteration, ck  t   x  t  , the mean envelope is subtracted from the residual signal to obtain an

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IMF, ck  t   ck  t   mk ,i  t  subject to fulfilling the requirements for an IMF otherwise steps 4 and 5 are skipped and the procedure is iterated again at step 1 with the new value of ck  t  . The iterative process is terminated when the standard deviation from two consecutive iterations is less than 0.3 so as to guarantee that the resulting IMF contains information with physical meaning.

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4. If ck  t  satisfies the conditions for an IMF, a new residue is computed and updated by subtracting ck  t  from the previous residual signal, rk  t   rk 1  t   ck  t  .

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5. The residual signal, rk  t  obtained in step 4 is now used as the new raw data and the sifting procedure starts from step 1 to extract more IMFs until the resulting residue becomes a monotonic function from which no more IMFs can be extracted. Thus multi-scale resolution of the signal is realized and x  t  can be reconstructed by addition of the n IMFs and n

the nth residue i.e. x(t )   ci  t   rn  t  . i 1

1



C   d   t    

P

(8)

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H t  

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The Hilbert transform is expressed as:

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where P indicates the principal Cauchy value and therefore the transform exists for all functions of class Lp. Ci (t) and Hi (t) are complex conjugate pairs such that the analytical signal Zi (t) with the complex component is

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given by:

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Z  t   C (t )  jH  t   a  t  e ji t 

in which

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 H t   1/2 a  t   C 2  t   H 2  t   ,  t   arctan    C t  

(9)

(10)

and the instantaneous frequency is given by

d  t  dt

(11)

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

The Hilbert spectral analysis is now performed on the resulting IMFs to obtain the time-frequency-energy distribution of the signal. The signal is now expressed as: n



x  t    a j  t  exp i   j  t  dt j 1



(12)

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where a j and  j are the time dependent amplitude and instantaneous frequency of the IMFs respectively. The Hilbert-Huang Transform was implemented in Matlab R2018a environment using available subroutines with appropriate settings. The energy for the respective IMFs is computed by averaging the instantaneous energies i.e.

Ei  





ci  t  dt 2

(13)

Following the orthogonality characteristics of the EMD components, the total energy is simply the summation

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of the individual averaged IMFs energy given by

i 1

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Finally the entropy based on the Hilbert-Huang transform is given by

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n

E   Ei

n

H E   pi log pi

(15)

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i 1

(14)

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where pi  Ei / E i.e. the energy fraction of each IMF.

5.0 Results and discussions

The results are arranged as follows, firstly the general flow structure as visualized from the CFD simulations are

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presented to give an overall picture of the fluidization behaviour. Next coherent structures are characterized using spectral analysis of the measured pressure and axial particle velocity fluctuations. Time frequency behaviour of the flow structures is presented using the wavelet coherence analysis and Hilbert Huang Transform.

5.1 Fluidization behaviour

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The instantaneous solid volume fraction contours at different operating conditions after 40 s of running the simulation is presented in Figure 1 below. The slugging nature of fluidization is clearly evident and is pronounced for Ug/Umf = 2 as can be seen from the strands of solids formed that cover the entire cross section which is typical for Geldart D particles (Kunii and Levenspiel, 1991). The slugging fluidization of the bed has also been physically confirmed from the experimental measurements of the small scale problem (Gopalan et al., 2016) . For Ug/Umf = 3, the bed height increases and the strands of solids are now less pronounced due to the increased agitation of the system and the same conclusion can be arrived at for Ug/Umf = 4 and moreover the

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flow structure becomes more complex. The color bar indicating the solids concentration shows a decreasing

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trend due to the even distribution of solids with increasing superficial gas velocity. In our previous work (Lungu et al., 2016) we clearly demonstrated qualitatively and quantitatively that the CFD model captures the

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basic features of the fluidization pattern reasonably well.

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Ug/Umf = 3

Ug/Umf =

Ug/Umf = 4

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Fig.1. Predicted solid volume fraction for (a) Ug/Umf = 2, (b) Ug/Umf = 3, and (c) Ug/Umf = 4 after 40 s.

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Figure 2 compares the predicted time averaged velocity magnitude of the Nylon beads for different superficial gas velocities taken at z = 0.0375 m using numerical simulations with the experimentally observed flow pattern

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using HSPIV (Gopalan et al., 2016). The particle circulation pattern can be identified from the plots which becomes pronounced with increasing superficial gas velocity. The TFM reproduces with reasonable accuracy

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the experimental fluidization pattern. The particles are carried upwards at the column center by a stream of rising slugs and voids and downwards at the wall by the action of gravity and thus the gas-solid flows are gravity driven similar to the liquid-gas flows in bubble columns as described by (Mudde, 2005) . The observed

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gross circulation patterns contribute significantly to the mixing in the fluidized beds.

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(b)

(c)

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m/s m/s

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m/s

Ug/Umf = 2

Ug/Umf = 4

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Ug/Umf = 3

Fig.2. Comparison of predicted and experimental (Gopalan et al., 2016) solids mean velocity vectors for different operating conditions. 5.2 Frequency Analysis 17

5.2.1 Pressure fluctuations The pressure time series PSDs obtained using the Welch method for different operating conditions are displayed in Figure 3. For each PSD plot, 64 sub-spectra were averaged each with 8192 data samples giving a frequency resolution of 0.122 Hz. Low frequency coherent structures can be observed from the plots which appear as pronounced peaks in the frequency range < 3 Hz representative of the gross circulation patterns i.e. macro structures . Interestingly the power of the peak increases from 0.0439 to 0.138 kPa2/Hz when Ug/Umf is increased from 2 to 3 and then decreases again to 0.0619 kPa2/Hz when Ug/Umf is increased further to 4

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suggesting a change of regime from slugging to turbulent. There is also a corresponding change in the

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dominant frequency from 1.71 Hz through 2.56 Hz to 1.59 Hz for the three superficial velocities respectively. The frequency range 3 < f < 25 Hz is representative of the meso scale structures ( gas bubbles and voids) with

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a power fall off having a slope of -4.5±0.1 for the given operating conditions which is typical for fluidized bed

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dynamics (Johnsson et al., 2000; van der Schaaf et al., 2004, 2002). Micro-scale or finer flow structures dominate the PSD beyond 25 Hz up to the Nyquist frequency (500 Hz). A second pronounced peak can be seen

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at a frequency of around 60 Hz for the three superficial gas velocities most likely due to resonance in the air supply system which generates strong pressure fluctuations (van der Schaaf et al., 2002) but nontheless this

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does not change the interpretation of the results so far.

0

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-1

10

-2

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-3

10

2

PSD [kPa /Hz]

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Ug/Umf = 2 Ug/Umf = 3 Ug/Umf = 4

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10

-5

10

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10

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10

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10

-1

10

0

10

1 Frequency [Hz] 10

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10

2

Fig.3. PSD of measured pressure fluctuations for different operating conditions.

Following Parseval’s theorem, the ratio of the energy outside the macro-scale region to the total energy i.e. the wideband energy, EWB is defined as: f2

 P  f  df xx

EWB 

f1 fN



(16)

Pxx  f  df

f f

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where f1 and f2 are the lower and upper frequency limits respectively in the region of interest, ∆f is the

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frequency resolution of the spectrum and fN is the Nyquist frequency. For f1 = 3 Hz and f2=fN = 500 Hz, the EWB for the three superficial gas velocities is 0.21, 0.12 and 0.18 respectively. A minimum in EWB signals a

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transition from slugging to turbulent fluidization (Johnsson et al., 2000) which is consistent with the

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observations of the dominant frequency from the PSD. The change in the EWB is mostly due to the highly

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intermittent structures i.e. the finer structures which become significant with increasing superficial gas velocity.

5.2.2 Particle velocity fluctuations

The PSD estimates of the axial particle fluctuation velocities at different lateral positions at a height of 0.0762

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m above the distributor plate and for different operating condition are displayed in Figure 4. The PSD plots of the particle velocity fluctuations were also constructed by applying the Welch method on the detrended time series data in a similar fashion to pressure fluctuations. In comparison to the pressure fluctuation PSD plots, the particle velocity PSDs are not as smooth due to fewer averaged sub-spectra in the Welch estimate. Application of Parseval’s theorem i.e. equation 5 to PSD curves of particle velocity fluctuations yields the average turbulent kinetic energy. Equation 5 was evaluated using the band power subroutine in Matlab® and the average turbulent 19

kinetic energy calculated from the PSD plots in this manner agreed excellently well with computations directly from the particle velocity time series data validating Parseval’s theorem. Thus the PSD of the fluctuating component of the particle velocity can be viewed as an energy spectrum which contains a wealth of information pertaining to three distinct regions that is the low frequency flow structures, the inertial region and the high frequency dissipation region (Joshi et al., 2009). The low frequency region i.e. < 3 Hz is characterized by pronounced peaks particularly for fluidization numbers of 2 and 3 corresponding to the gross circulation patterns and gravity waves already visualized from Figure 2.

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Interestingly it has been observed that the HSPIV measurements capture the same information as the pressure

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fluctuations at low frequencies including the large scale oscillations and regime change from slugging to turbulent as observed from the changes in the dominant frequencies at the different lateral positions and

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operating conditions. In gas-solid fluidization, energy is supplied by the introduction the gas against the static pressure of the bed and is dissipated by at the particle scale and in the bulk of the continuous phase where the

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primary length scale is the column diameter (Joshi and Nandakumar, 2015) . Following the cascade theory of

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turbulence, energy is injected into the fluidization system where density differences generate vortical structures at the equipment scale at low frequencies. These large structures or eddies break up and transfer the energy to

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smaller eddies at higher frequencies until the particle scale where viscous effects start to dominate i.e. at low Reynolds number . The fact that the dominant frequency corresponding to the large scale oscillations is more less the same at all the 5 lateral positions particularly for Ug/Umf = 2 and 3 points to the presence of a large eddy

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spanning the width of the column consistent with (Joshi and Nandakumar, 2015).

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0

10

-1

10

(a)

-2

10

-3

2

PSD [m /s]

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-4

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Ug/Umf = 2 x/X = -0.795 x/X = -0.398 x/X = 0 x/X = 0.398 x/X = 0.795

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-4

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-5

10

x/X = -0.795 x/X = -0.398 x/X = 0 x/X = 0.398 x/X = 0.795

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Frequency [Hz]

0

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(c)

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Ug/Umf = 4

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2

PSD [m /s]

10

x/X = -0.795 x/X = -0.398 x/X = 0 x/X = 0.398 x/X = 0.795

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10

10

-1

10

0

10

1

10

2

Frequency [Hz]

Fig.4. Vertical energy spectra at different lateral positions for (a) Ug/Umf = 2, (b) Ug/Umf = 3 and (c) Ug/Umf = 4 at y = 0.0762 m.

At higher frequencies i.e. greater than 3 Hz, slopes in the vicinity of -5/3 are observed at all the lateral positions for the three operating conditions suggesting that the Kolmogorov -5/3 power law is obeyed i.e. 21

E  f   C r2/3 f 5/3 . The dependence of the power fall on the superficial gas velocity was also observed in this work. This can be explained by appreciating the fact that at constant system properties, there is a corresponding increase in the Reynolds number with an increase in the velocity scale i.e. the inertial effects dominate the viscous effects and therefore the fine scales move to lower values. The slopes were evaluated by linear regression of the log-log PSD plots from 3 Hz up to the respective Nyquist frequencies. Similar observations have been made in gas-solid risers (Chalermsinsuwan et al., 2009; Jiradilok et al., 2008) and gas-liquid bubble columns (R F Mudde et al., 1997; Sathe et al., 2013) . Limitations in the Kolmogorov -5/3 power law have been

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noted by (Joshi et al., 2009) in that it does not account for intermittent behaviour of energy dissipation which

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perhaps could explain the slight deviation of our computed slopes from -5/3 . The Levy-Kolmogorov law with a slope range of -3 to -5/3 has been proposed by (Sun and Yan, 2017b) to account for the flow intermittency.

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Other researchers such as (Kulkarni and Joshi, 2005) have developed intermittency models to describe the

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energy spectra.

5.3.1 Wavelet coherence analysis

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5.3 Time-frequency analysis

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The wavelet coherence analysis (WCA) based on the continuous wavelet transform (CWT) provides time localized characterization of coherent structures. It has been previously applied successfully in geophysical sciences (Grinsted et al., 2004) hydrology (Labat, 2005) and medical sciences (Cui et al., 2012) to mention a few for studying time localized phenomena . In the field of fluidization, (Dashtian and Sahimi, 2013) applied

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the cross wavelet transform coherence to pressure time series and demonstrated the time-frequency localization features between pairs of times series. More recently (Sánchez-Delgado et al., 2018) investigated the short and long term bubble dynamics vis-à-vis distributor design in a pseudo 2D gas-solid fluidized bed by applying wavelet coherence over the bubble void fraction time series. Their study revealed that neither the distributor type nor the bed height had any effect on the coherence between the short term and long term bubble dynamics. 22

In the current work, the WCA of numerical pressure drop signals at measurement positions 0.0413 m – inlet and 0.0413 – 0.3461 m showed the existence of spatially coherent structures. At measurement position 0.0413 m – inlet, coherent phenomenon due to bubble generation dominated most of the frequency and time scales due the proximity to the distributor. At a wider measurement spacing of 0.0413 – 0.3461 m, the fast traveling waves are attenuated due to gas bubble/void coalescence and acceleration and only pockets of highly coherent oscillations are visible mostly in the frequency range of 0.5-3 Hz at certain times corresponding to the large scale oscillations previously observed from the spectral analyses. The reader can refer to the supporting information

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of this paper for further details of the WCA.

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5.3.2 Hilbert-Huang Transform

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Axial particle velocity fluctuations at the five lateral positions for the different operating conditions were decomposed using EMD and checked for completeness and orthogonality. Typical multiresolution of the axial

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particle velocity at x/X = 0.0 for different operating conditions is displayed in Figure 5 for illustration and it is strikingly similar with multiresolution using DWT. However as reviewed in Table 1, wavelet analysis is non-

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adaptive in that the analysis of the data is based on a chosen basic wavelet which is not the case for HilbertHuang transform. The IMFs represent different transient coherent structures present in the fluctuating particle

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velocity signals exhibiting frequency and amplitude modulations. The modulations which are pronounced for fine scales corresponding to IMF 1 and IMF 2 reflect the intermittency of the flow field i.e. irregular energy dissipation strongly correlated to coherent structures. Previous studies with pressure fluctuation signals

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(Briongos et al., 2006; Llop and Gascons, 2018) have attempted to examine the physical meaning of the IMFs vis-à-vis fluidized bed dynamics but with much lower sampling frequencies compared to the HSPIV used in this study. The sifting process separates the finest local mode from the rest of the data and therefore IMF1 has the highest frequency most likely due to particle - particle and particle - wall interactions and the nth IMF the least which exhibits the trend in the data. A close inspection of the plots shows that for fluidization numbers of 2 and 4, a total of 10 IMFs are extracted from the data while only 9 IMFs were extracted for fluidization number equal 23

to 3. This can be appreciated by considering the fact that the Hilbert Huang Transform is an adaptive method based on the local characteristic time scales of the data and it will be therefore influenced by the local position and operating condition. Our study also shows that in addition to a particular position and operating condition, the choice of the interpolation method used in connecting the extrema affects the number of IMFs extracted. Matlab R2018a offers two options for the interpolation namely spline and pchip and the latter was adopted for our work as it is recommended for non-smooth signals.

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(a)

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(c)

Fig.5. Typical EMD decomposition of axial particle velocity at x/X = 0 for (a) Ug/Umf = 2 (b) Ug/Umf = 3 and (c) Ug/Umf = 4.

25

Figure 6 shows the Hilbert spectral analysis that is the instantaneous frequency spectrum of the IMFs in Figure 5 for different operating conditions. The plot is a time-frequency representation of the velocity fluctuation signals with a color bar indicating the instantaneous energy of the fluctuations ranging from dark blue (minimum) to yellow (maximum). The localized features vis-à-vis intra and inter wave frequency and amplitude modulations of the transient coherent structures embedded in the signal due to the non-linearity and non-stationary nature of the gas solid flow are evident. The lowest frequency resolution of the Hilbert spectrum

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for a uniformly sampled signal is given by 1/T Hz where T is the total data length and the highest frequency

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possible is given by 1/  n t  Hz where t is the digitizing rate and n is the minimum number of t required to define the frequency accurately (Norden E Huang et al., 1998). For fluidization number of 2, a rather strong

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quasi-periodic component in dark blue color can be noticed fluctuating between the ranges 0 and 5 Hz due to the motion of highly energetic and coherent structures most likely slugs which bring about strong velocity

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gradients in the dispersed phase. Other structures with lower intensity and higher instantaneous frequencies in

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light blue color are also visible corresponding to the small scale dissipative structures. There is a corresponding increase in the instantaneous energy of the coherent structures with an increase in the fluidization number to 3

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as evidenced from color bar. This is expected due to an increase in the supplied energy to the fluidization system by means of the continuous (gas) phase which is partially dissipated as dispersed (solid) phase fluctuations leading to an increase in the instantaneous energy. It can also be observed for fluidization number of 3 that the instantaneous frequency of energetic structures is somewhat constant hovering around 3 Hz

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attributed to the increased gas void coalescence and bed stability. With a further increase in the fluidization number to 4, the dominant energy structures previously seen for the other two operating conditions are now barely visible. This can be explained by appreciating the fact at this operating condition, the bed is in the turbulent regime characterized by predominant splitting up of voids as opposed to gas voids coalescence. This is consistent with the spectral analysis of the pressure and particle velocity fluctuations in Figures 3 and 4 respectively which show changes in the power and frequency of the dominant structure. It can also be deduced 26

from Figure 6c that the periodicity of the modes that relate to slugs/voids is significantly reduced implying that the slugs or voids have disappeared giving way to highly complex heterogeneous structures typical of turbulent fluidization. It has also been demonstrated in this work that for a fixed given operating condition, the Hilbert spectra captures the different flow structures at the different lateral positions. The local adaptability of the Hilbert Huang Transform for different operating conditions makes it a powerful tool for detecting changes in the flow structure as demonstrated here and elsewhere in an airlift reactor (Luo et al., 2012) , three phase bubble column (Li et al., 2013) and more recently in a vibrated gas-solid fluidized bed (Zhang et al., 2018) .

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Hitherto only a qualitative description of the Hilbert spectra has been given and a more quantitative analysis of

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the frequency and energy distributions follows in the later part of the results and discussions.

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(b)

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(c)

Fig.6. Fluctuating axial velocity Hilbert Spectra at x/X = 0.0 for (a) Ug/Umf = 2, (b) Ug/Umf = 3 and (c) Ug/Umf = 4.

28

The average instantaneous frequencies characterize the main physical time scales embedded in the original time series (Briongos et al., 2006). Figure 7 displays the average instantaneous frequency,  j of the IMFs at three different lateral positions covering half of the bed for different operating conditions. Considering the energy cascade hypothesis, high frequency modes i.e. IMF1 - 2 represents turbulence scales at which viscous dissipation of energy occurs due to particle-particle and particle-wall collisions, this observation is synonymous with the power fall off seen from the spectral analysis of the axial particle fluctuating velocity in Figure 4. Meanwhile medium frequencies belong to energy containing structures such as gas voids or slugs and finally

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low frequencies belong to dissipative structures in the bulk of the bed due to turbulent fluctuations. This is

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consistent with the typical frequency ranges given by (Drahoš and Čermák, 1989) for two phase flow that is 101 - 100 Hz for circulation, 100 - 102 Hz for gas voids/bubbles oscillation and 102 - 103 Hz for turbulence.

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To further illustrate the connection of the IMFs to the different fluidizations scales , two correlations due to

1/2

 g   H mf

  

  3 s /  g  2   s     s ,max  

1/2

and (Baskakov et al., 1986): 1/2

  

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1 g f     H mf

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1 f  2

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(Gidaspow et al., 2001):

(17)

(18)

for low frequency large scale oscillations are compared with the average frequencies where Hmf is the bed height at minimum fluidization conditions, αs is the solid volume fraction, αg is the fluid volume fraction, αs,max

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is the solid volume fraction at packing and g is the acceleration due to gravity. It can be seen that these two expressions correlate to IMFs 5 – 7 depending on the operating condition which lie within the meso scales. For equation 17, experimental values of the particle, αs and gas volume, αg fractions were not available and were thus obtained from the numerical simulations.

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(a)

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1

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0

(b) IMF 1 IMF 2 IMF 3 IMF 4 IMF 5 IMF 6 IMF 7 IMF 8 IMF 9 IMF 10 Gidaspow et al., 2001 Baskakov et al., 1986

 j [Hz]

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1

IMF 1 IMF 2 IMF 3 IMF 4 IMF 5 IMF 6 IMF 7 IMF 8 IMF 9 IMF 10 Gidaspow et al., 2001 Baskakov et al., 1986

 j [Hz] 10

0

x/X = -0.398

x/X = - 0.795 -1

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IMF 1 IMF 2 IMF 3 IMF 4 IMF 5 IMF 6 IMF 7 IMF 8 IMF 9 IMF 10 Gidaspow et al., 2001 Baskakov et al., 1986

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 j [Hz] 10

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Ug/Umf [-]

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x/X = 0.0 -1

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4

5

Ug/Umf [-]

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Fig.7. Plot of the average instantaneous frequency against fluidization number for (a) x/X = - 0.795 (b) x/X = - 0.398 and (c) x/X = 0.0 at y = 0.0762 m.

Having analyzed the frequency distribution of the IMFs at different operating conditions, the energy distribution of the coherent structures is now considered. Figure 8 displays the energy fraction for the different coherent structures extracted by multiscale resolution of the fluctuating axial particle velocity using emd at different lateral positions at a height of 0.0762 m and for different operating conditions. Different IMFs represent 30

different modes of oscillation due to the multiscale flow structures or eddies present in the data. The IMFs can be demarcated into three principal scales of fluidization i.e. micro scale associated with individual particle interactions and particle–wall interactions, meso-scale representative of the bubble oscillations and macroscale which represent the equipment scale (Llop and Gascons, 2018) but other researchers like (Briongos et al., 2006) instead identified the particle, bulk and local bubble dynamics scales. For Ug/Umf = 2, the energy content is concentrated in the scales IMF 4-7 associated with the slugs or gas voids. IMF 1-3 are high frequency and low energy structures representative of the microscale while the structures in bands IMF 8-10 have low energy

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and frequency suggestive of the macroscale. With the increase of the superficial gas velocity to Ug/Umf = 3 , the

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energy of the flow structures shifts from higher to lower scales and the majority of the energy content is now concentrated in the scales IMF 5-8, meanwhile the micro scale extends to IMF4. For this case, the energy for all

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the lateral positions with the exception of x/X = 0 peak at 1MF 5 i.e. within the range 1-5 Hz considering the different lateral positions. Interestingly when the superficial gas velocity is increased further to Ug/Umf = 4, the

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flow structure changes and different peaks can be observed for different oscillation modes unlike in the previous

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case where most of the peaks were concentrated in a single oscillation mode. The major energy content shifts further to lower modes which is most likely indicative of regime change from slugging to turbulent flow

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characterized by a reduction in the fundamental frequency due to splitting up of voids dominating gas void/slug

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coalescence. The meso scales at this velocity now cover IMFs 6-8 while the micro scale is IMF 9-10.

31

1.0

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(a)

0.6

x/X = - 0.795 x/X = - 0.398 x/X = 0.0 x/X = 0.398 x/X = 0.795 Ug/Umf = 3

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Energy fraction [-]

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Energy fraction [-]

(b)

x/X = - 0.795 x/X = - 0.398 x/X = 0.0 x/X = 0.398 x/X = 0.795 Ug/Umf = 2

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IMF [-]

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x/X = - 0.795 x/X = -0.398 x/X = 0.0 x/X = 0.398 x/X = 0.795 Ug/Umf = 4

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Energy fraction [-]

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Fig.8. IMF energy distribution for (a) Ug/Umf = 2 (b) Ug/Umf = 3 and (c) Ug/Umf = 4 at different lateral positions (y = 0.0762 m). As aforementioned in the introduction, fluid mechanical parameters required for reactor design consist of turbulent stresses among others. The stresses are due to phase (gas or solid) velocity fluctuations in the flow field and are referred to as normal Reynolds stresses if velocity fluctuations in the same direction are considered and shear Reynolds stresses if velocity fluctuations in different directions are taken into consideration. The normal stresses contribute to the turbulent kinetic energy, TKE, while the shear stresses are strongly related to 32

the gross circulation patterns in multiphase reactors. Characterization of turbulence in multiphase reactors including bubble columns and fluidized beds by means of spatial correlation (second order moment) of velocity fluctuations is well reported in the open literature. The equations for computing the Reynolds stresses are tabulated in Table 2 below for reference. The subscripts i and j represent the lateral and axial directions respectively and n is the total number of samples in the data.

It can be seen from Figure 9 that the normal stresses in the axial direction are significantly higher in comparison

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to the lateral direction due to the strong velocity gradients in the axial direction which is the main direction of

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flow given that the setup is a pseudo 2D column. Meanwhile the shear stresses are negligible (nearly zero) comparison to the normal stresses. The differences in the order of magnitude of the stresses in the axial and

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lateral directions highlights the anisotropic nature of the flow field which characterizes multiphase reactors. The figure also displays the lateral profile of the TKE per unit bulk density for different operating conditions which

1 uxux  uy uy  uz uz   2

(19)

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E

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is given as:

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The lateral profile of the kinetic energy peaks at the center and reduces approaching the walls. Such a profile indirectly reflects the profile of the solid particles which is low in the center and high near the walls i.e. core annulus flow structure. The reduced particle concentration at the bed center results in increased kinetic energy due to somewhat unrestricted particle motion whereas near the walls , the higher solids concentrations restricts

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the fluctuations thus reducing the turbulent kinetic energy. A reduction in the axial stresses and TKE is observed when the superficial gas velocity is increased from Ug/Umf = 3 to Ug/Umf = 4 which is consistent with the aforementioned regime change from slugging to turbulent fluidization. It must be appreciated that the normal Reynolds stresses contribute to the so called bubble-like or turbulent granular temperature due to oscillation of bubbles (Gidaspow et al., 2004). Therefore the reduction in the stresses with increase in the superficial gas velocity reflects the change in the dominant mechanism from bubble or void coalescence in the 33

slugging regime to splitting of bubbles and voids in the turbulent regime leading to a reduction in the axial

0.100

0.100

(b)

Ug/Umf = 2 Ug/Umf = 3 Ug/Umf = 4

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2

Ug/Umf = 2

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x/X [-]

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xx [m /s ]

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stresses.

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0.000 -1.00

Ug/Umf = 3 Ug/Umf = 4

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0.00

x/X [-]

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(c)

Ug/Umf = 3

Ug/Umf = 4

Ug/Umf = 4

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2 2

2 2

Ug/Umf = 2

Ug/Umf = 3

TKE [m /s ]

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xy [m /s ]

(d)

Ug/Umf = 2

0.00

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0.025

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x/X [-]

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x/X [-]

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Fig.9. Comparison of (a) lateral (b) axial (c) shear stresses and (d) TKE per unit bulk density profiles for different operating conditions.

The computation of stresses and TKE presented and discussed in the preceding paragraph has been reported by

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several researchers using different measurement techniques in different types of multiphase reactors. None the less, the contribution of the different coherent structures to the stresses and other turbulence parameters is not captured using such an approach. (R. F. Mudde et al., 1997) decomposed the liquid phase time series velocity in a bubble column into a sine like oscillation due to large scale oscillating structures and high frequency content

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associated with turbulence in an attempt to quantify the role of coherent structures on stresses. In this study the multiresolution of the axial particle velocity at five different lateral positions and height of 0.0762 m above the distributor plate for Ug/Umf of 2, 3 and 4 is performed using EMD instead after which the variance and covariance are computed on the resulting IMFs to obtain the normal and shear Reynolds stresses distribution. Figure 10 shows a typical example of the mode wise distribution of the axial normal Reynolds stresses this approach. It can be observed that the profile of the axial stresses resemble closely the profiles of the energy 35

fraction in Figure 8 implying that the meso scales arising from the voids and void-particle interaction are the main constituents of the axial normal Reynolds stresses. This is in agreement with the findings of (R. F. Mudde et al., 1997) who observed flat profiles of stresses in a 2D bubble column when the large scale structures ( meso and probably macro scale structures too) were neglected. The kinetic theory of granular flow as applied to fluidization distinguishes between particle granular temperature defined as the kinetic energy of the random motion of particles and bubble-like granular temperature due to oscillation of gas voids or clusters computed from normal Reynolds stresses as aforementioned. In this work, the laminar granular temperature was measured

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using kinetic theory based HSPIV (Gopalan and Shaffer, 2012) . Moreover the bubble-like granular

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temperature is an order of magnitide larger than the particle granular temperature and therefore we postulate that the flat profile of the stresses at low IMFs is correlated to the particle granular temperature while the meso-

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scale is correlated to the bubble-like or turbulent granular temperature. The research works of (Briongos et al., 2006) and (Llop and Gascons, 2018) revealed persistence in the mutual information function of IMF1 for

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slugging fluidization and attributed this to the oscillating motion of particles which is essentially granular

0.05

(a)

0.05

(b)

x/X = - 0.795 x/X = - 0.398 x/X = 0.0 x/X = 0.398 x/X = 0.795

Ug/Umf = 3

0.03

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2

2

yy [m /s ]

2

0.02

x/X = - 0.795 x/X = - 0.398 x/X = 0.0 x/X = 0.398 x/X = 0.795

0.04

Ug/Umf = 2

0.03

2

yy [m /s ]

0.04

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temperature as demonstrated in this paper.

0.01

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0.00 1

2

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1

IMF [-]

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IMF [-]

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7

8

9

10

0.05

(c)

x/X = - 0.795 x/X = - 0.398 x/X = 0.0 x/X = 0.398 x/X = 0.795 Ug/Umf = 4

0.03

2

2

yy [m /s ]

0.04

0.02

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3

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5

6

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10

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IMF [-]

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Fig.10. Mode wise axial normal Reynolds stresses for (a) Ug/Umf = 2 (b) Ug/Umf = 3 and (c) Ug/Umf = 4 at different lateral positions (y = 0.0762 m).

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It can be inferred from the mode wise distribution of energy and normal Reynolds stresses that mixing in the slugging and turbulent regime of the fluidized system currently under study is mainly due to meso-scale

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structures as the dispersion is directly proportional to the normal Reynolds stresses using Taylor’s expression i.e. Di  uiuiTL where Di is the dispersion coefficient or eddy diffusivity, uiui is the normal Reynolds stress of

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the particle velocity at a particular location and TL is the Lagrangian integral time scale taken as the first moment of particle fluctuating velocity autocorrelation.

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6.0 Summary

Coherent flow structures in a pseudo 2D rectangular gas-solid fluidized bed have been identified and characterized by means of spectral analysis and Hilbert-Huang transform applied to pressure and particle velocity fluctuations. The pressure drop fluctuations were measured by a differential pressure transducer with a temporal resolution of 1 kHz while the particle velocity fluctuations were measured with HSPIV with sampling rates of 1-1.5 kHz. Absolute pressure fluctuations inaccessible from the experiments were extracted from CFD 37

simulations using the Eulerian-Eulerian approach for the same experimental set up. Spectral analysis of the experimental pressure drop fluctuations for all the three operating conditions using Welch periodogram revealed energetic peaks in the low frequency region < 3Hz representing large scale oscillations or gravity waves. These peaks were also captured by the 1D axial energy spectra of the fluctuating component of particle velocity at the different lateral positions and height of 0.0762 m above the distributor plate demonstrating the ability of HSPIV to resolve the complete range of temporal information. A change in the frequency of the dominant eddy was observed from the spectral analysis of the pressure as well as the particle velocity fluctuations suggesting a

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regime change from slugging to turbulent fluidization. At higher frequencies of the energy spectra, the classical

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-5/3 Kolmogorov’s power law for isotropic turbulence was observed. The wavelet coherent analysis applied to pressure fluctuations revealed the spatial-temporal features of the coherent structures for different measurement

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and operating conditions.

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The Hilbert Huang transform comprising the empirical mode decomposition and Hilbert spectral analysis was applied to the fluctuating component of the axial particle velocity to extract the distribution of energy and

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frequency between different coherent structures. For the different operating conditions, the resulting IMFs are categorized into the micro scale related to particle-particle and particle-wall collisions, meso-scale correlated

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with gas voids and the interaction with the particles and the macro scale pertaining to the equipment. An inspection of the mode wise axial normal Reynolds stresses which contributes to the turbulent kinetic energy and dispersion coefficient follows a similar distribution to the energy fraction with the meso scales mostly responsible for the axial mixing in the bed. The energy content at low IMFs i.e.1-4 depending on the operating

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conditions is constant and almost zero corresponding to dissipation at the particle scale due to particle collisions and with the wall.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Acknowledgements The authors gratefully acknowledge the financial support provided by the Project of National Natural Science Foundation of China (91434205), the National Science Fund for Distinguished Young (21525627), the Natural

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Science Foundation of Zhejiang Province (Grant No. LR14B060001) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20130101110063). The authors would also like to thank Dr.

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Balaji Gopalan of the National Energy Technology Laboratory for making available the experimental particle

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velocity and pressure fluctuation time series data.

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Table 1 Comparison of Fourier, wavelet and Hilbert-Huang transforms (Huang and Shen, 2014) Fourier transform Wavelet transform Hilbert-Huang Transform Basis a priori a priori adaptive Frequency convolution: global convolution: regional differentiation: local uncertainty uncertainty certainty Presentation Energy-frequency Energy- timeEnergy- timefrequency frequency Nonlinear no no Yes Nonstationary no yes yes Feature extraction no discrete: no yes continuous: yes yes Theoretical base theory complete theory complete empirical

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Table 2 Equations for computation of stresses Mean particle velocity Normal Reynolds stress

1 n  uik  x, t  n k 1

1 n  uik  x, t   ui  x  uik  x, t   ui  x  n k 1 1 n uiuj    uik  x, t   ui  x    u jk  x, t   u j  x   n k 1 ui2 

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Shear Reynolds stress

ui  x  

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