Effect of bed thickness on a pseudo 2D gas-solid fluidized bed

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Jun 15, 2018 - pseudo 2D rectangular fluidized beds are commonly used. ..... fluctuations at heights 0.0413 m and 0.3461 m above the plate distributor. Fig. 5.
Powder Technology 336 (2018) 594–608

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Powder Technology journal homepage: www.elsevier.com/locate/powtec

Effect of bed thickness on a pseudo 2D gas-solid fluidized bed turbulent flow structures and dynamics Musango Lungu a,b, Haotong Wang b, Gershom Mwandila a, Jingdai Wang b,⁎, Yongrong Yang b, Fengqiu Chen b, John Siame a a b

Chemical Engineering Department, School of Mines and Mineral Sciences, Copperbelt University, Kitwe 21692, Zambia State Key Laboratory of Chemical Engineering, College of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, China

a r t i c l e

i n f o

Article history: Received 10 November 2017 Received in revised form 2 May 2018 Accepted 13 June 2018 Available online 15 June 2018 Keywords: CFD Coherence Fluidized bed Granular temperature Turbulence Wavelet analysis

a b s t r a c t This work explores the ability of the two-fluid-model (TFM) to model the dynamical and turbulent features of a pseudo 2D gas-solid fluidized bed operated under slugging conditions. 2D and 3D numerical simulations are performed to investigate the effect of the bed thickness on predicted quantities. Hi fidelity raw pressure drop and particle velocity data from the NETL small scale challenge problem is processed and used to validate the CFD model. Our work shows that the differences between 2D and 3D simulations in predicting the fluidized bed dynamics using pressure fluctuation data is minimal. However the effect of the bed thickness on turbulent properties namely the normal Reynolds stresses, turbulent kinetic energy, granular temperatures is significant. Taking into account the bed thickness does not necessarily improve the model predictions of all the dynamic and turbulent features. Furthermore mean profiles alone are not sufficient to validate TFM models as is quite common in the open literature. Mixing in the slugging bed is predominantly due to coherent meso-scale structures (voids and slugs) rather than individual particles as revealed from computed granular temperatures. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Fluidization was for the first time successfully applied on an industrial scale in 1929 for coal gasification and since then remains a popular technology finding applications in the chemical, metallurgical, and allied industries [1]. The technology offers several advantages over other gas-solid contactors including near perfect solids mixing and rates of high heat transfer [2]. Traditionally the design and scale-up of fluidized bed has heavily relied on empiricism and pilot plant experimentation and has often been described as being not an exact science [3]. Nowadays, computational fluid dynamics (CFD), a physics based approach offers an alternative method for designing and troubleshooting of multiphase reactors. The most widely used CFD models are the discrete element method (DEM) and two fluid model (TFM). One of the key steps involved in developing and adopting a CFD model for practical purposes is model validation. For this purpose pseudo 2D rectangular fluidized beds are commonly used. Such experimental set ups allow the easy use of non-intrusive techniques for the investigation of bed hydrodynamics [4–6], bubble motion [7, 8], and

Abbreviations: acf, autocorrelation function; CFB, circulating fluidized bed; CFD, computational fluid dynamics; DWT, discrete wavelet transform; FFT, fast Fourier transform; rms, root mean square. ⁎ Corresponding author. E-mail address: [email protected] (J. Wang).

https://doi.org/10.1016/j.powtec.2018.06.028 0032-5910/© 2018 Elsevier B.V. All rights reserved.

solids mixing and segregation [9, 10]. The agreement between experimental measurements from pseudo 2D experiments and model predictions from 2D CFD simulations are generally satisfactory with the exceptions of the solids velocity [4, 11–13] and the bubble rise velocity [11, 14] which have been observed to be over predicted in 2D simulations due to the negligence of friction from the front and back walls. Inclusion of the third dimension greatly improves the model predictions of the mean solids velocity profile and bubble rise velocity however it is demanding on computational cost. Li and Zhang [11] proposed a model for 2D simulations implemented in the open source code MFIX. It takes into account the friction effects of the front and back walls to cut down on the computational cost of 3D simulations. These studies have mostly focused on time averaged quantities in evaluating the differences between 2D and 3D geometry. For the interest of scale up and design of multiphase reactors it is of great interest also to compare dynamical and turbulent parameters. Dynamical features can be conveniently investigated by analyzing time series of quantities such as pressure, temperature or phase hold up [15]. Pressure signals are commonly used to characterize bed dynamics due to their relatively cheaper cost, ease of use and nearly non-intrusive nature of the pressure sensors. The signals can either be measured at a single point in the bed relative to the atmosphere (absolute) in which case the information obtained pertains to global behavior of the bed or between two vertical points in the bed (differential) reflecting local dynamics. Time series analysis

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provides a plethora of information such as the size and velocity of gas-solid flow structures [16], flow regime identification and characterization [17] and monitoring of the fluidization behavior to detect abnormal changes [18] to mention but a few. Excellent reviews by Johnsson et al. [19], Sasic et al. [20] and van Ommen et al. [17] give a treatise on the characterization of gas-solid fluidized beds dynamics using time series analysis. Multiphase reactors are operated under turbulent conditions with flow structures of varying length and time scales affecting the performance of the reactors in relation to mixing process and transport phenomena. Mixing in the respective phases is typically quantified by dispersion coefficients while heat and mass transfer coefficients are required to characterize the transport phenomena in the reactor. Estimation of these parameters heavily relies on empirical correlations sometimes leading to gross over predictions. Detailed knowledge of fluid mechanical turbulent parameters is needed to overcome this challenge and tools such as CFD can provide a great insight. These parameters include mean velocities and shear rates, turbulent kinetic energy and turbulent energy dissipation rate, eddy dispersion, energy spectra, size, shape, velocity and energy distribution of turbulent structures and so on [21]. CFD codes have several options of turbulence models that can be used in simulations depending on the requirements of the problem. Solution of these models yields the turbulent kinetic energy production and turbulent dissipation rate in the flow field. Alternatively the fluctuating particle velocity at different lateral and axial locations can be analyzed using statistical moments to reveal multi-scale flow structures. In addition, 1D, 2D or 3D energy spectra can be constructed from the particle fluctuating velocity from which information regarding length and velocity scales at different levels can be obtained [22]. Lastly the phenomena behind the turbulent fluctuations around the mean flow can be studied. The literature reviewed by the authors shows that although several modeling studies have been conducted to investigate the effect of the geometry and other parameters on thin gas-solid fluidized bed hydrodynamics including the solids velocity and bubble rise velocity, similar studies on the effect of the geometry on dynamic and turbulence parameters are rather scarce probably due to the absence of high quality experimental data. Unlike dilute gas-solid systems, experimental measurements of solids velocity in dense gas-solids like bubbling fluidized beds using conventional imaging techniques is challenging. Moreover from the particle velocity, turbulence parameters such as the flatness factor (kurtosis), Reynolds stresses, energy spectra, turbulent granular temperature and turbulent dispersion coefficients can be computed as previously demonstrated by several workers [23–27]. The objective of the present work is carry out a detailed investigation on the effect on the third dimension in a thin rectangular gas-solid bubbling fluidized bed on the slugging fluidized bed dynamics and turbulent parameters (Reynolds stresses, turbulent granular temperature, turbulent kinetic energy and turbulent diffusivity). The case study for this work is the small scale challenge problem (SSCP) posed by the US Department of Energy's (DOE) National Energy Technological Laboratory (NETL). In this challenge problem, particle velocity and pressure drop data from a pseudo 2D fluidized bed with Geldart D bed material is availed publicly to modelers to validate their models. Description of the experimental set up is provided in the experimental sub-section. For the interested reader, the background and details of the challenge problem can be obtained from the following website https://mfix.netl. doe.gov/experimentation/chalenge-problems/ or from the recent publication by Gopalan et al. [6]. The remainder of the paper is organized as follows; section 2 describes the experimental bubbling fluidized bed test rig, section 3 is a description of the CFD model and the simulation set up, results are reported and discussed in section 4 and finally the conclusions drawn from the work are given in section 5.

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2. NETL SSCP experimental setup As aforementioned in section 1, the experimental set up is a pseudo 2D bubbling fluidized bed, 0.075 m depth, 0.23 m width and 1.22 m length. Fig. 1 below displays the 2D schematic of the bed. Coarse sized (Geldart D) Nylon beads were employed as the granular media in the bed and fluidized with air at ambient conditions. The complete physical properties of the particles are presented in Table 1 while the cumulative particle size distribution is presented in Fig. 2. It is worth mentioning that the coefficients of restitution were measured experimentally using high speed imaging techniques described in the paper by Gopalan et al. [6]. The excel sheets in addendum 2 and 3 of the challenge problem from the website give detailed experimental measurement values of the particle-wall and particle-particle coefficients of restitutions respectively. Three experimental runs were carried out at superficial gas velocities of 2Umf, 3Umf and 4Umf respectively. Eulerian particle velocity measurements were taken at five lateral positions of 0.02356 m, 0.06928 m, 0.115 m, 0.16072 m and 0.20644 m at a height of 0.0762 m above the gas distributor using HsPIV [28] at sampling rates of 1, 1.2 and 1.5 kHz respectively for total sampling times of 21.096 s, 17.88 s and 14.665 s respectively for the three cases. Granular temperatures were calculated from the Eulerian particle velocity measurements following Gopalan and Schaffer [28] and provided by NETL. A low pressure transducer (Rosemount 1151DP smart transmitter) was used to measure the mean pressure drop at a frequency of 1 Hz meanwhile time series measurements were obtained using 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 1000 Hz for 300 s giving 300,000 samples.

X = 0.23 m

Y = 1.22 m

αs= 0.58

H = 0.173 m

Fig. 1. 2D schematic of the rectangular fluidized bed.

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Table 1 Particle properties.

Table 2 CFD model equations.

Physical property

Units

Nylon beads

Minimum fluidization velocity Bed height at minimum fluidization Void fraction (fluffed) Particle density Particle size (Sauter mean diameter) Sphericity Particle-wall coefficient of restitution (normal) Particle-particle coefficient of restitution (normal) terminal velocity [29]

m/s m – kg/m3 μm – – m/s

1.05 0.173 0.42 1131 3256 0.94 0.92 0.84 9.94

Mass conservation of phase k (k = g for gas and s for solid phase) ∂ ðα k ρk Þ ∂t Pn k¼1 α k

þ ∇  ðα k ρk uk Þ ¼ 0

¼1 Momentum conservation for solid phase:

∂ ðα s ρs us Þ ∂t

þ ∇  ðα s ρs us us Þ ¼ ∇  τ s −α s ∇p‐∇ps þ β gs ðug −us Þ þ α s ρs g Momentum equation for gas phase:

∂ ðα g ρg ug Þ ∂t

þ ∇  ðα g ρg ug ug Þ ¼ ∇  τ g −α g ∇p−βgs ðug −us Þ þ α g ρg g Granular Temperature equation

3 ∂ 2 ½∂t ðα s ρs Θs Þ

þ ∇  ðα s ρs us Θs Þ ¼ ð−ps I þ τ s Þ : ∇us þ ∇  ðκs ∇Θs Þ−γs −3β gs Θs

Constitutive equations for interphase momentum transfer Syamlal O'Brien drag model

3. CFD model and simulation set up GAMBIT 2.4.6 was used as the pre-processor for creating the computational meshes (2D and 3D) which were then exported into ANSYS Fluent 15, a commercial CFD code used to perform the numerical simulations. The Two fluid model (TFM) with closures from the KTGF available in the commercial code was utilized and the model equations are given in Table 2 below. As regards to the numerical scheme, the second order upwind scheme was used to discretize momentum and granular temperature equations; QUICK scheme for the spatial discretization of the volume fraction, the first order implicit scheme was used for the transient formulation and Phase Coupled SIMPLE for pressure–velocity coupling. The Syamlal-O'Brien parametric drag model [30] available in ANSYS Fluent 15 was used. Experimental values of the voidage and minimum fluidization velocity are supplied as inputs and the code recomputes and updates the default coefficients in the drag model. This model was selected after a detailed comparison with the Gidaspow model and the study is reported in our previous work [26]. An initial mesh sensitivity study revealed that a mesh size of 5 mm is sufficient to guarantee mesh independence for 2D simulations and 10 mm for 3D simulations. The full granular temperature partial differential equation was used. Case 1 i.e. Ug = 2Umf was initialized using the minimum fluidization condition while the second and third cases corresponding to Ug = 3Umf and Ug = 4Umf respectively were initialized from the corresponding previous final converged solutions and all the cases were run for a total of 40 s with data sampling for time statistics activated after 15 s at which the flow achieved pseudo steady state as monitored from the pressure drop fluctuations and flux report of the mass flow rate at the inlet and outlet. The velocity inlet and pressure outlet boundary conditions were specified at the bed inlet and outlet respectively meanwhile

β gs ¼ 34

α s α g ρg v2rs ds

C D ð vRerss Þjug −us j 2

ρ ju −u jd

4:8 ffi C D ¼ ð0:63 þ qffiffiffiffiffiffiffiffiffiffi Þ ; Res ¼ g gμ s s g Re s =v rs  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vrs ¼ 0:5 A−0:06 Res þ ð0:06 Res Þ2 þ 0:12 Res ð2B−AÞ þ A2

and B = Pα1.28 for αg ≤ 0.85 A = α4.14 g g and B = αQ A = α4.14 g g for αg N 0.85 P = .932 and Q = 1.713 Constitutive equations based on the KTGF Solid phase stress tensor τ s ¼ α s μ s ½∇us þ ð∇us ÞT  þ α s ðλs − 23 μ s Þ∇  us I Gas phase stress tensor τ g ¼ α g μ g ½∇ug þ ð∇ug ÞT − 23 α g μ g ð∇  ug ÞI Solids pressure ps = αsρs[1 + 2(1 + e)αsg0]Θs Bulk solid viscosity qffiffiffiffi λs ¼ 43 α 2s ρs ds g 0 ð1 þ eÞ Θπs Shear viscosity of solids μs = μs, kin + μs, col + μs, fr Kinetic viscosity, μs, kin pffiffiffiffiffiffi 2 10ρ ds πΘs μ s;kin ¼ 96αss ð1þeÞg ½1 þ 45 ð1 þ eÞg 0 α s  α s 0

Collisional viscosity, μs. col qffiffiffiffi μ s:col ¼ 45 α s ρs ds g 0 ð1 þ eÞ Θπs α s Frictional viscosityμs, fr ffiffiffiffiffi μ s;fr ¼ pspsinϕ 2

I 2D

Diffusion of granular temperature pffiffiffiffiffiffi qffiffiffiffi 2 150ρs ds Θs π κs ¼ 384ð1þeÞg ½1 þ 65 α s g 0 ð1 þ eÞ þ 2ρs ds α 2s g0 ð1 þ eÞ Θπs 0

Collisional dissipation of solid particle fluctuating energy pffiffi Þg 0 ρ α 2 Θ3=2 γ s ¼ 12ð1−e s s s d π 2

s

Radial distribution function

100

3 −1

s g 0 ¼ ½1−ðαs;αmax Þ 

cumulative distribution [%]

Nylon beads 80

Wall boundary conditions pffiffiffi pffiffiffiffiffiffi! ! s ρs g 0 Θs U s;∥ τ s ¼ − π6 3ϕ αs;αmax pffiffiffi α pffiffiffi 3 p ffiffiffiffiffi ffi ! ! s s qs ¼ π6 3 αs; max ρs g 0 Θs U s;∥  U s;∥ − π4 3 αs;αmax ð1−e2sw Þρs g 0 Θ2s

60

the Johnson and Jackson boundary condition [31] was imposed for the solid phase at the walls while the gas phase was modeled using the no-slip boundary condition.

40

20

0 3150

4. Results and discussion 4.1. Dynamic features 3200

3250

3300

mean particle size [µm] Fig. 2. Nylon beads cumulative particle size distribution.

3350

Unless new insights can be obtained from advanced analytical tools such as fractal or chaos analysis, it is always advisable to use the simplest method of analysis available [15, 20] and therefore in this study spectral and statistical analyses have been adopted for analyzing

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the absolute and differential pressure fluctuations.These analyses are relatively straightforward and easy to apply and of course less stringent on the level of noise available in the raw signal in comparsion to the more advanced techniques. Comparsions between numerical predictions from 2D and 3D simulations with experimental measurements are made in order to quantify the effect of the bed thickness on predicted quantities. Fig. 3 compares the spectral analysis of the differential pressure fluctuations between the taps at heights of 0.0413 m and 0.3461 m from the numerical predictions and experimental measurements on a log-log scale. The spectral analysis was computed using the Welch method [32] available as a subroutine in Matlab. The experimental time series were resampled at a rate of 200 Hz from the original 1000 Hz giving 60,000 resampled samples using the resample subroutine in Matlab which applies a low pass FIR filter. A segment length of 512 samples with 256 overlapped samples and Hanning window was used in the estimation of the PSD giving a frequency resolution of 0.3906 Hz. Meanwhile 5000 samples from the predicted time series was used for the PSD estimation using the same Welch subroutine settings. Following Johnsson et al. [19], by visual inspection the PSD plots can be split into three different regions based on the frequency. The macro flow structures typically voids and slugs exist below 5 Hz and this is well captured by both the 2D and 3D simulations however our study reveals the dominant frequencies predicted by the 2D simulations are in excellent

10

-1

10

-2

101

(a)

Ug/Umf = 2

10

-4

10-5 10-6

10-3 10-4 10-5 10-6

-7

10-7

10-8

10-8

10-9

10-9

10

10

Ug/Umf = 3 2D simulation 3D simulation Experiment

-1

10-2

10-3 10

(b)

100

2D simulation 3D simulation Experiment

-10

10-10 0.1

1

10

100

0.1

1

frequency [Hz]

10

frequency [Hz]

101 100

(c)

Ug/Umf = 4 2D simulation 3D simulation Experiment

-1

10

10-2

PSD [kPa2/Hz]

PSD [kPa2/Hz]

100

agreement with those from the experimental measurements in comparion to 3D simulations. For the frequency bands of 5–10 Hz and 10 Hz and above, a power fall off in the spectrum is observed due to finer flow structures. Morever the shape of the PSD plot is hardly changed with changing superficial gas velocity which is consistent with the observations of van der Schaaf [33] . For cases 1 and 3, the fall off from the 3D simulations are steeper in comparison with the 2D simulations suggesting that the pressure fluctuations from the 3D simulations are noisier in nature and thus tend to flattern the spectrum [17] . In addition to the PSD we computed and compared the coherence of the bivariate pressure fluctuations at bed heights of 0.0413 m and 0.3461 m from the 2D and 3D numerical simulations. A coherent structure is defined as three-dimensional region of the flow over which at least one fundamental flow variable (velocity component, density, temperature, etc.) exhibits significant correlation with itself or with another variable over a range of space and/or time that is significantly larger than the smallest local scales of the flow [34]. These coherent structures arise as a result of turbulent motion and are directly related to the performance of the chemical equipment in this case the fluidized bed vis-àvis heat and mass transport. A great detail of work has been done on coherent structures in bubble columns as reviewed by Joshi et al. [35] as opposed to gas-solid systems. Fig. 4 shows the magnitude squared coherence, γ2 computed using the Welch's averaged modified periodogram method for the 2D and 3D simulations at different

PSD [kPa2/Hz]

101

597

10-3 10-4 10-5 10-6 10-7 10-8 10-9

10-10 0.1

1

10

100

frequency [Hz] Fig. 3. Plot of measured and simulated pressure drop fluctuations PSD for different superficial gas velocities.

100

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Fig. 4. Magnitude squared coherence for (a) 2D and (b) 3D pressure fluctuations at heights 0.0413 m and 0.3461 m above the plate distributor.

superficial gas velocities. A magnitude squared coherence estimate of 1 at a particular frequency indicates that the bivariate times series are strongly correlated at that frequency although the PSD values in both time series might be different. On the other hand a magnitude squared coherence estimate of 0 indicates that the bivariate time series are not correlated at all though the power may be equally high in both PSDs. Coherence has been previously used to distinguish different sources in fluidized bed pressure fluctuations [16] and to compare different pressure fluctuation measurement techniques [36]. It can be seen from the figure that both 2D and 3D simulations show that the bivariate series are highly coherent at low frequencies between 0 and 3 Hz for all superficial gas velocities. This frequency corresponds to large scale and low frequency coherent structures and in this case votitical structures which are also equally captured at the lower end of the power spectrum. At higher frequencies the coherence reduces and this is consistent with the findings of van der Schaaf [16] . The shape of the coherence versus frequency plot is dependent on the measurement locations, probes which are in close proximity results in high coherence for nearly all the frequency range while for probes which are at a reasonable distance apart, the coherence will high only for the dominant strucutures at low frequencies like in our study. Decoupling of pressure fluctuations into so called coherent and incoherent components can be realized using the coherence function as demonstrated previously by van der Schaaf et al. [37]. The coherent component of pressure fluctuations represents pressure wave propagation generated by bubble coalescence, eruption and bed mass oscillation while the incoherent component is asscociated with local phenomenon such as void or cluster passage. From the incoherent component an estimate of the characteristic length scale, LB of flow structures i.e. voids or clusters can be made. Using this methodology we computed the charactristic length scales of voids from 2D and 3D simulations at different operating conditions and height of 0.3048 m as can be seen from Fig. 5. It is quite evident from the Figure that the bubble size (proportional to the characteristic length scale) from 3D simulations is smaller in

comparison to 2D simulations at all operating conditions. However both sets of simulations correctly show an increasing trend of LB with the superficial gas velocity. Our result is consistent with previously studies using other methodologies such as optical probes, digital image analysis (DIA) etc. From the cross correlation of the pressure fluctuations at heights 0.0413 m and 0.3461 m, the time lag between the signals is obtained. Knowing the spacing between the two numerical probes, an estimate of the average void/slug velocity can be made [38, 39] . Fig. 6 displays the cross-correlation functions of the pressure fluctuations from 2D and 3D numerical simulations at different superficial gas velocities. The time shift at the maximum correlation coefficient is taken to be the

Fig. 5. Predicted characteristic length scale versus normalized superficial gas velocity.

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Fig. 6. Cross correlation function from predicted pressure fluctuations at probes located at heights of 0.0413 m and 0.3461 m above the gas distributor for different cases.

time delay. The time lags are positive implying that the voids/slugs move upwards since the reference signal is upstream i.e. at 0.0413 m and the second signal is downstream at 0.3461 m. The oscillations from 2D and 3D simulations are quite similar implying that qualitatively both simulations capture the basic macro flow structures. Quantitatively however the amplitude of fluctuation from 2D simulations is somewhat higher in comparison to 3D simulations as deduced from the crosscorrelation coefficients. Specifically for case 2 the maximum correlation coefficients from the 2D and 3D simulations are 0.4670 and 0.2928 respectively while for case 3 the coefficients are 0.4563 and 0.3519 respectively. The deduction from this observation is that 2D simulations predict larger slugs due to the restriction of the flow to a plane.

The average slug velocities computed from the cross-correlation technique are presented in Table 3. The slug velocities predicted from 2D simulations are higher for all operating conditions as expected since void/slug velocity is dependent on the size (diameter) of the void/slug [1] . Moreover for both sets of simulations, the average slug velocity increases with increasing superficial gas velocity due to the increasing excess gas (Ug-Umf) passing through the bed as voids. The second order statistical moment is widely applied to multiphase reactor signals to detect regime change, minimum fluidization velocity and defluidization. Table 4 below compares the rms of the axial pressure drop taken between the pressure taps at heights of 0.0413 m and Table 4 rms of axial pressure drop [kPa].

Table 3 Average slug velocity from 2D and 3D numerical simulations. Geometry

Ug/Umf = 2

Ug/Umf = 3

Ug/Umf = 4

2D 3D

0.953 m/s 0.693 m/s

1.24 m/s 0.968 m/s

1.33 m/s 1.27 m/s

Case 1 Case 2 Case 3

2D simulations

3D simulations

Experimental data

0.24 0.27 0.27

0.23 0.24 0.22

0.18 0.32 0.23

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Fig. 7. 2D and 3D solid volume fraction contour plots for (a) Ug = 2Umf (b) Ug = 3Umf (c) Ug = 4Umf .

0.3461 m above the distributor plate with rms values from 2D and 3D simulations taken at numerical probes set up at equivalent positions with experiments for different superficial gas velocities. For the three cases, the difference between the experimental measurements and numerical simulations vary between 15 and 33% (2D simulations) and 4–28% (3D simulations). This finding demonstrates that inclusion of the third dimension improves the model prediction due to the extra space available for the solids fluctuations. The rms expresses the departure from the mean and is a measure of the strength of the fluctuations and including the bed thickness increases the anisotropy of the flow thereby predicting values closer to the experiment. Additionally from the experimental data in the table we observe that the rms goes through a maximum with increasing superficial gas velocity implying a change in the fluidization regime from slugging to turbulent [40]. In comparison to the 2D simulations, the 3D simulations captures this transition relatively better likely due to the unrestricted fluctuations in the depth of the bed unlike in 2D simulations where the flow is restricted. It is

worth mentioning that analysis of the pressure drop fluctuations from 2D simulations using the autocorrelation function (acf) was able to reveal the transition from slugging to turbulent regime as demonstrated in our previous work [26]. Therefore it is recommeded that several criteria be used in studying dynamical features before a firm conclusion is drawn. 4.2. Solids distribution Time averaged solids contour plots from 2D and 3D simulations are presented in Fig. 7 for the three cases. The 3D contour plots are taken at a X-Y plane located at the middle of the bed thickness i.e. z = 0.0375 m . The numerical simulations both 2D and 3D exhibit a flow structure in which the particles accumulate near the wall with a dilute core i.e. core-annular. Moreoever this core-annular like flow structure becomes more pronounced with with increasing superficial gas velocity. The excess gas above the minimum fluidization velocity that is Ug -Umf

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flows through the bed as bubbles and voids preferentially at the bed center and this increases with increasing superficial gas velocity thus forming the observed flow structure. A close visual observation of the 2D and 3D contour plots shows that the 3D simulations predicts slightly higher solids concentration and a reduced bed height for all the operating condition. 4.3. Probability density distributions of particle velocity The probability density function (PDF) of the instantaneous axial particle velocity from the numerical simulations and experimental measurements is compared at different radial positions at a height of y = 0.0762 m for case1 and plotted in Fig. 8. The conclusions drawn for case 1 equally apply to cases 2 and 3 therefore only case 1 results are presented for brevity. Moving from the bed center towards the wall region, the PDF from the experimental measurements shows an increasing tendency. This might be explained by appreciatng the fact that the flow structure in the bed is core-annular like as observed from the solids distribution in Figure. Therefore the PDF increases approaching the wall due to the accumulation of the solid particles in this region unlike at the center where the voids are predominanlty moving upwards with less particles. Moreover the velocity distribution shifts slightly to the left moving towards the wall indicating a predominatly negative velocity near the wall whereas at the center of the column the velocity is more less normally distributed, similar observations where made by Groen et al. [41] in a bubble column reactor. Suffice to say that the PDFs were estimated after steady state was achieved and therefore we can safely say that this picture reprsents the overall circulating pattern i.e. voids moving upwards carrying particles to the bed surface at the center and solid particles flowing downwards near the wall. At x/X = 0.0, predictions from both 2D and 3D simulations agree relatively well with the experimental profile however at x/X = 0.3976 and x/X = 0.7951 the comparison is poor with the simulations exhibiting a larger spread of the velocities. The discrepancies between the simulations and experiments points to the shortcomings of the modeling details including the drag interaction which has a significant effect on the solid velocity PDF as recently demonstrated by Lu and Holland [42]. 4.4. Flatness profiles of particle fluctuating velocity The values of second (standard deviation/rms), third (skewness), and fourth (kurtosis) statistical moments of the fluctuating velocity are useful characteristic features of turbulence in multiphase reactors [43]. Furthermore turbulent strucutues span different length and time scales and thus application of a multiscale methodology combined with statistiscal moments can provide a detailed insight into the different turbulence scales. Kulkarni et al. [44] applied the 1D discrete wavelet transform (DWT) in conjuction with kurtosis on bubble column liquid velocity time series data for different experimental conditions.

601

The authors went on to establish a relationship between the so called global flatness factor and gas hold up. Ellis et al. [45] used the 1D DWT to decompose local voidage data and then applied the third and fourth statistical moments on the decomposed signals to characterize the complex flow pattern in a turbulent gas-solid fluidized bed. They attributed higher kurtosis numbers (greater than 3) to intermittency associated asscoiated with passing voids. Breault et al. [46] identified and independently characterized the three characteristic scales (micro, meso and macro) by analyzing the raw and disrete wavelet decomposed voltage signal data from a gas-solid NETL riser. Characterization was done using average skewness, kurtosis, correlation dimenson and entropy. Lungu et al. [26] compared the flatness factors at different radial positions from wavelet decompsed experimental and 2D numerical simulation data for the same setup considered in this paper. Excellent agreement was observed betwwen simulations and measurements. Thus it can be seen that DWT is a powerful and promising tool for studying the multiscale nature of fluidization. Van Ommen et al. [17] in their review state that wavelet tool is yet to reach its full potential usage in multiphase reactors particularly. In this work for the first time the DWT and combined with kurtosis are applied on particle fluctuating velocity data from 2D and 3D simulations at different bed locations and superficial gas velocities. The goal is to investigate the effect of the third dimention on the profiles of the flatness and thus reveal the differences if any in the turbulent flow structures predicted from the 2D and 3D simulations. In our work, a 9 scale 1D DWT was performed on the predicted fluctuating velocity signals at different lateral positions in the axial direction thereafter we computed the kurtosis on each scale. There are many family of wavelets available however we opted to use the discrete Meyer wavelet following [17] which gives higher regularity in comparison to the Daubechies wavelet family. A sampling frequency of 200 Hz gives a Nyquist frequency of 100 Hz which is sufficient for the study of the multiscale phenomena in the gas-solid system under consideration here. Table 3 below gives the frequency bands for the 9 level wavelet decomposition. Fig. 9 shows the FF profiles for 2D and 3D simulations at different radial positions and superficial gas velocities. It can be observed from the Figure that for all operating conditions and positions, the FF remains relatively constant for the D4-D9 scales for both 2D and 3D simulations. Considering the multiscale demarcation of fluidization [47], these scales can be assigned to the macro and meso scales. The meso scale strutures (voids and slugs) contain the bulk of the energy and are relatively stable and therefore there is hardly any change in the FF profiles. This is consistent with the coherence function in Figure which shows a maximum peak at lower frequencies i.e. coherent power. At a scale of about D3 the FF profile starts to exhibit a sharp increase right up to scale D1 for all operating conditions and radial positions for both 2D and 3D simulations. The sharp increase is attributed to the irregular nature of energy dissipation known as intermittency. In these scales the primary scale of turbulence is in the order of a particle diameter [43] as opposed to

Fig. 8. Predicted and measured axial particle velocity PDFs at different lateral positions for case 1.

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Fig. 9. Comparison of 2D and 3D flatness factor profiles for different cases and lateral positions.

the macro and meso scales where the primary length scale is the equipment diameter. The differences between the 2D and 3D simulations are quite minimal demonstrating that 2D simulations are sufficient for computing FF and studying the multi-scale nature of fluidization. Moving from the bed center towards the near wall region, an increase in the FF is noticed which translates to an increase in the flow intermittency. This is expected since near the wall clusting of solid predominantly moving downwards occurs and creates an increase in the intermittency unlike at the bed center where the slugs and voids move predominantly upwards where a more less uniform flow due to enhanced mixing occurs leading to a reduction in the intermintency. 4.5. Reynolds stresses Stresses in turbulent flow arise due to random velocity oscillations. For 3D geometry the turbulent stress tensor per unit bulk density can be written as: 0

1 u0x u0x u0x u0y u0x u0z B 0 0 0 0 0 0 C τ ¼ @ uy ux uy uy uy uz A u0z u0x u0z u0y u0z u0z

ð1Þ

The nine stress components reduce to six due to the symettrical property of the stress tensor, therefore. the shear stresses per unit bulk density are u0y u0x ; u0y u0z ; u0z u0x since u0y u0x ¼ u0x u0y ; u0z u0y ¼ u0y u0z ; u0x u0z ¼ u0z u0x . The normal stresses per unit bulk density are given a su0x u0x ; u0y u0y ; u0z u0z . In a similar fashion for a 2D geometry, four stress components are obtained which reduce to three with two normal stresses and one shear stress that is u0x u0x ; u0y u0y and u0y u0x . The normal Reynolds stresses are computed indirectly from the Eulerian solids hydrodynamic velocities using the following equation: u0i u0i ¼

m 1X ðu ðx; t Þ−ui ðxÞÞðuik ðx; t Þ−ui ðxÞÞ m k¼1 ik

ð2Þ

where ui ðxÞ ¼

m 1X u ðx; t Þ m k¼1 ik

ð3Þ

and shear Reynolds stresses are computed from the equation: u0i u0j ¼

m   1X ðu ðx; t Þ−ui ðxÞÞ ujk ðx; t Þ−u j ðxÞ m k¼1 ik

ð4Þ

similarly

u j ðxÞ ¼

m 1X u ðx; t Þ m k¼1 jk

In this study only the axial and lateral normal Reynolds stresses have been computed to make the comparison straightforward. Contour plots of the predicted lateral and axial normal Reynolds stresses are shown in Figs. 10, 11 and 12. The lateral normal Reynolds stresses peak at the center of the column as can be seen from the magnitude of the contours in Fig. 10 and drop to zero at the walls, such profiles have been observed elsewhere in bubble columns [48] due to the lateral velocity component which attains its highest magnitude at the center. In comparison to 2D simulations, 3D simulations predict lower values of the lateral Reynolds stresses although both sets of simulations give similar trends. This can be explained by appreciating the fact that the contribution of the third dimension lowers fluctuating velocity and thus the normal Reynolds stresses. For both 2D and 3D simulations, an increase in the superficial gas velocity translates to an increase in the normal Reynolds stresses, this is expected since higher gas velocities energies the fluidization system more. As for the axial normal Reynolds stresses, they peak near the walls for both 2D and 3D simulations and this phenomenon becomes clearer with increasing superficial gas velocity as can be seen from Fig. 11. Near the walls the flow dynamically changes which results in large fluctuations and therefore the axial normal Reynolds stresses peak here [48]. From the ongoing it seems that similar flow structures govern the dynamics and circulation flow patterns in bubble column and gas-solid systems. From the 3D simulations, the cross sectional view of the lateral and axial normal Reynolds stresses were taken at a X-Z isosurface located at y = 0.0762 m to get a better understanding of their distributions in the fluidized bed and presented in Fig. 12. As aforementioned it can be clearly seen that the lateral profiles peak at the center while the axial profiles peak near the walls. The stresses are highly influenced by the large scale vortical coherent structures which give rise to such a distribution. A comparison of the normal Reynolds stresses from the numerical predictions and experimental measurements are presented in Table 5. As expected 3D simulations give better predictions over 2D simulations although both sets of simulations over predict the experimental values for all the cases. The ratio of the axial stresses to the lateral stresses gives a measure of the anisotropy of the flow field. From the table it is observed that the stresses from the experiments are highly anisotropic and although the CFD simulations are able to show the anisotropy in the flow, the differences with the experimental values are rather large. This rather large discrepancy can be explained by the CFD model to adequately capture the lateral transport mechanism accurately. The currently used Kinetic

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Fig. 10. Countour plots of 2D and 3D lateral normal Reynolds stresses for (a) Ug = 2Umf (b) Ug = 3Umf (c) Ug = 4Umf.

theory of Granular Flow is based on Maxwellian velocity distribution [49] which assumes homogeneous distribution of solid particles which is contrary to observed particle velocity probability distributions from experimental setups as seen from Fig. 5 . Development of an anisotropic kinetic theory will be expected to greatly improve model predictions. 4.6. Granular temperatures There exists two types of granular temperatures in gas-solid fluidized systems [50], the first kind is associated with individual particle oscillation and is termed laminar granular temperature or better known as the classical granular temperature. It is computed from the common CFD codes by solving the granular temperature equation in Table 2. In the experiments conducted by NETL, the granular temperature was computed from particle velocities measured using the HsPIV system. The second type of granular temperature is associated with particle clusters or bubbles and is known as the turbulent granular temperature and it is computed from the normal Reynolds stress per unit bulk density [26] . Similar to the laminar granular temperature, it is computed from PIV data in experimental fluid dynamics (EFD). Predicted and measured granular temperatures at different superifical gas velocities

taken at a height of 0.0762 m above the distributor plate are compared in Figs. 13. Interestingly laminar granular temperatures are underpredicted by both 2D and 3D simulations while opposite is true for the turbulent granular temperatures for all operating conditions. From Fig. 13 it can be observed that inclusion of the bed thickness lowers the predicted granular temperature values. This can be attributed to the predicted lower bed height by 3D simulations and therefore increased solids concentration see Fig. 7. With increasing solids concentration, there is a corresponding decrease in the mean free path of the particles and consequently the laminar granular temperature. The increasing trend of the laminar granular temperature with superficial gas velocity is well predicted by both 2D and 3D simulation although predictions from 2D simulations are somewhat slightly better. An increase in the superficial gas velocity translates into an increase of the energy supplied to the fluidization system and this results in the increased fluctuation of solids particles. As aforementioned the turbulent granular temperatures is due to particle cluster fluctuations and bubble/void oscillations. The overprediction of the turbulent granular temperature by both 2D and 3D in Fig. 14 underscores the importance of modeling the solid and gas interaction accurately because this ultimately dictates the size of the bubble/

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Fig. 11. Countour plots of 2D and 3D axial normal Reynolds stresses for (a) Ug = 2Umf (b) Ug = 3Umf (c) Ug = 4Umf.

voids and particle clusters. A recent study by Koralker and Bose [51] assessed the performance for different drag models using CFD-DEM simulations for the same experimental setup and concluded that no single drag model gives an all round good performance on the other hand Ayeni et al. [52] using a new drag law based on the mechanical energy balance which takes into account the energy dissipation demonstrated that improved predictions of velocity profiles (mean and rms) could be obtained. It would of great interest to test the drag model in the prediction of different dynamic and turbulent features and not only time averaged profiles before a conclusive decision can be arrived at. Consideration of the third dimension improves the numerical predictions this is due to the fact that the clusters and voids are no longer restricted to a 2D plane. In comparison to the laminar granular temperatures, the turbulent granular tempertures are much larger for both simulations and experiments. For instance at x/X = 0.0, the experimental turbulent granular temperature is 2.72, 2.88 and 1.96 times larger than the laminar granular temperatures at superficial gas velocities of 2Umf, 3Umf and 4Umf respectively. Meanwhile the turbulent granular tempemtures are 17.92, 26.75 and 29.96 times larger for 2D simulations and 13.15, 14.21 and 15.01 times larger in the case of 3D simulations for the same laterial position and operating conditions. It can be deduced

that mixing in the slugging and turbulent regime is dominated by the slugs and voids as opposed to the IIT CFB riser [53] where it was found that the laminar granular temperatures were larger than the turbulent granular temperatures due to the non formation of bubbles. 4.7. Turbulent dispersion The dispersion for fluid or solids, due to turbulence can be computed from the autocorrelation of the velocity fluctuations obtained from the velocity time series data. It is worth mentioning that the dispersion computed in such a manner represents meso scale values which is quite different from macroscale dispersion obtained from tracer studies [3]. This methodology has been utilized in the Lagrangian framework for liquid fluidized beds experiments [54, 55] using computer automated radioactive particle tracking (CARPT), bubble column experiments [56] using radioactive particle tracking (RPT), bubble column numerical simulations [57] and gas-solid riser experimental work [58] also using RPT. Later Jiradilok et al. [25, 59] adopted this method in the Eulerian framework for CFD simulations by approximating the Lagrangian time scale to the Eularian time scale. The product of the variance of the particle velocity (normal Reynolds stress) with the characteristic time

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Fig. 12. Cross sectional contour plots of 3D normal Reynolds stresses for (a) Ug = 2Umf (b) Ug = 3Umf (c) Ug = 4Umf at a height of y = 0.0762 m.

estimated from the autocorrelation of the fluctuating velocity gives an estimate of the turbulent dispersion and is expressed as: Di ¼ u0i u0i T L

ð5Þ

where Di is the dispersion coefficient or eddy diffusivity, u0i u0i is the normal Reynolds stress or the variance of the particle velocity at a particular location and TL is the Lagrangian time scale mathematically expressed as: Z∞ TL ¼

RL ði; τÞdτ

where u0 ðt Þu0 ðt þ τÞ u0 2

Table 5 Normal Reynolds stresses for different cases at a height of 0.0762 m above distributor plate.

ð6Þ

0

RL ði; τ Þ ¼

sufficiently small and the length of time under which the computation is done is 10TL [60], both criteria are satisfied in our work. Axial dispersion coefficients computed in this manner from simulations and experimental measurements are compared in Fig. 15. The coefficients were computed at a height of 0.0762 m above the gas distributor for different superficial gas velocities. The increasing trend of the coefficients with the superficial gas velocity can be observed

ð7Þ

That is the autocorrelation of the particle fluctuating velocity. TL is simply the area under the acf versus time lag curve up to the first zero crossing [60], other researchers like Bai et al. [57] estimated TL as the time for the acf to decrease to 1/e of its initial value. The assumption underlying the use of this method is that the time step is

u0y u0y

u0x u0x

u0y u0y =u0x u0x

Case1 2D simulations 3D simulations Experiment

0.101878 0.09645 0.038528

0.054772 0.033416 0.003752

1.860037976 2.886341872 10.26865672

Case 2 2D simulations 3D simulations Experiment

0.177584 0.127086 0.061532

0.131782 0.080872 0.009474

1.347558847 1.571446236 6.49482795

Case 3 2D simulations 3D simulations Experiment

0.235248 0.17749 0.087888

0.237356 0.153846 0.017408

0.991118826 1.153686154 5.048713235

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Fig. 13. Lateral profiles of measured and predicted classical granular temperatures for the three cases at y = 0.0762 m.

Fig. 14. Lateral profiles of the turbulent granular temperature computed from measured and predicted particle velocities at y = 0.0762 m.

from the figure for both the measurements and predictions. However the parabolic profile of the dispersion coefficients observed from the experimental measurements are not reproduced by the simulations and the reason for this mismatch is yet to be understood. In addition the coefficients are overpredicted by the CFD model particularly for cases 1 and 2. This is as a result of the larger normal Reynolds stresses predicted by the model from which dispersion coefficients are computed. It is encouraging however to note that the order of magnitude of the measured and predicted values are the same. Fig. 16 compares the variation of the turbulent dispersion coefficients with the superficial gas velocity with some literature values [24, 25, 27, 61–64] . Dispersion coefficients values in the literature vary widely albeit due to the different methods used in acquiring the coefficients, operating conditions, physical properties of the materials and so on. Nonetheless the dispersion coefficients computed in this study are in reasonable agreement with the various literature values and

moreover the differences between 2D and 3D simulations are not significant and therefore 2D simulations can be used to get first approximations of dispersion coefficients. 5. Conclusions In this work we have performed 2D and 3D CFD simulations using the Eulerian-Eulerian framework based on the experimental set-up of the National Energy Technology Laboratory small scale challenge problem. Particle velocity data from the experimental measurements and CFD simulations are processed and used to characterize and compare the dynamic and turbulence structures with the aim of testing the capabilities of the CFD code and to gain fundamental understanding of the role of the turbulent structures on mixing and transport phenomena. The study shows that 2D simulations are sufficient to model the dynamic behavior of a pseudo 2D gas-solid fluidized bed. The dynamic

Fig. 15. Lateral profiles of the axial turbulent dispersion coefficients computed from measured and predicted particle velocities at y = 0.0762 m for the three cases.

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607

Subscripts s solid g gas x lateral coordinate y vertical coordinate mf minimum fluidization

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 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. Balaji Gopalan of National Energy Technology Laboratory for providing the additional data and for attending to our queries. Fig. 16. Axial dispersion coefficient versus superficial gas velocity for different works.

behavior was studied using the PSD, coherence and rms of pressure drop fluctuations. As with regards to the turbulent and other properties, the inclusion of the third dimension yielded mixed results. For instance the prediction of normal Reynolds stresses and turbulent granular temperature improved using 3D simulations but the prediction of the laminar granular temperature was poorer in comparison to 2D simulations. As for turbulent dispersion, the differences between 2D and 3D simulations were not very significant however both sets of simulations failed to capture the parabolic lateral profile of the axial dispersion coefficients. In comparison to literature values, the values from this study were within acceptable limits. Therefore the debate as to whether 3D simulations should be used over 2D simulations especially for pseudo 2D beds still begs to be answered. Our study is a first attempt in answering this question. Lastly the key to improving the model predictions lies in developing a multiscale drag interaction model and an anisotropic kinetic theory. Nonetheless CFD model shows great promise and it is a matter of time before it will be fully adopted for design purposes. Nomenclature e particle-particle restitution coefficient [−] f frequency [Hz] g acceleration due to gravity [m/s2] radial distribution function [−] g0 H bed height [m] P pressure [pa] particle Reynolds number [−] Res t time [s] U superficial gas velocity [m/s] u′ fluctuating velocity in the ith direction [m/s] x time series of parameter of interest

Greek symbols α volume fraction [−] packing limit [−] αs,max gas/solid momentum exchange coefficient [kg/m2 s] βgs collisional energy dissipation [J/m3 s] γs bulk viscosity [Pa s] λs θ, Θ granular temperature [m2/s2] ρ density (kg/m3) μ viscosity (Pa s) stress tensor of phase k (pa) τk

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