(deza 2013) a cfd study of pressure fluctuations to determine fluidization regimes in gas-solid beds

Mirka Deza1e-mail: [email protected]

Francine Battaglia

Department of Mechanical Engineering,

Virginia Tech,

Blacksburg, VA 24061

A CFD Study of PressureFluctuations to DetermineFluidization Regimesin GasSolid BedsReliable computational methods can provide valuable insight into gassolid flow proc-esses and can be used as a design tool. Of particular interest in this study is the hydrody-namics of a binary mixture of sandbiomass in a fluidized bed. Biomass particulates varyin size, shape, and density, which inevitably alter how well the particles fluidize. Ourstudy will use computational fluid dynamics (CFD) to interpret the hydrodynamic statesof a fluidized bed by analyzing the local pressure fluctuations of beds of sand and a bi-nary mixture of cotton stalks and sand over long time periods. Standard deviation of pres-sure fluctuations will be compared with experimental data to determine differentfluidization regimes at inlet gas velocities ranging from two to nine times the minimumfluidization velocity. We will use Bode plots to present the pressure spectra and revealcharacteristic frequencies that describe the bed hydrodynamics for different fluidizationregimes. This work will present CFD as a useful tool to perform that analysis. Otherimportant contributions include the study of pressure fluctuations of a fluidized bed inbubbling, slugging, and turbulent regimes, and the analysis of a binary mixture usingCFD. [DOI: 10.1115/1.4024750]

1 Introduction

Gassolid fluidized beds are widely used in many industrialapplications, e.g., coal and/or biomass gasification to produce pro-cess heat because the gassolid contact results in rapid heatingand mass transfer, and fast chemical reactions [1]. However, thehydrodynamics of gassolid fluidized beds require furtherresearch in order to improve existing processes and scale up newprocesses. Reliable computational methods can provide valuableinsight into gassolid flow processes and can be used as a designtool. Of particular interest in this study is the hydrodynamics of abinary mixture of sandbiomass in a fluidized bed. Biomass par-ticulates vary in size, shape, and density, which inevitably alterhow well the particles fluidize [2]. Deza and Battaglia previouslyused a two-fluid EulerianEulerian model to simulate biomass flu-idization and validated the computational models using experi-mental data [3,4]. A recent study by Pepiot and Desjardins [5]used an EulerLagrange approach to demonstrate the viability ofrepresenting the solids phase as discrete particles in large scalesimulations of a biomass fluidized bed reactor. The current studycontinues with interpreting the hydrodynamic states of a biomassfluidizing bed by analyzing the local pressure fluctuations andcompare with that reported in the literature.

Pressure fluctuation data obtained from fluidized bed combus-tors and gasifiers are a rich source of information on the hydrody-namic states of these systems [6,7]. The resulting time series datacan be analyzed by a number of different methods, includingstandard deviation, probability density functions, autocorrelationanalysis, and power spectral density (PSD) analysis [810]. Oneof the most common pressure fluctuation analyses is standarddeviation. It has often been used to identify different regimes influidized beds, where a maximum value with respect to inlet gasvelocity is associated with the transition from a bubbling to turbu-lent fluidization regime. Standard deviation has also been used to

determine minimum fluidization velocity [1114] and to detectthe onset of defluidization in operating fluidized beds [15]. Bi andGrace [16] and Bi et al. [1719] extensively studied the use ofstandard deviation to determine the transition to a turbulent fluid-ization regime. Two velocities were identified as delimiters for re-gime transition and will be discussed further in Sec. 2.3. Recently,Zhang et al. [20,21] used standard deviation to study the transitionfrom a bubbling to a turbulent regime in cotton stalksand fluid-ized beds of different mass ratios and this work will be discussedlater.

Another tool for studying pressure data is frequency domainanalysis, which is performed using a Fourier transform. A methodfor obtaining the frequencies associated with a system is a powerspectral density approach. The objective is to determine dominantfrequencies in the time-series and relate them to physical phenom-ena [22]. Other authors, such as Brue and Brown [7] and Nicastroand Glicksman [23] have used PSD to validate dynamic similitudeand scale-up by comparing spectra of models, prototypes, andfull-scale units. Using a Gaussian curve, Parise et al. [24] fittedpower spectra to detect defluidization of a bed using changes inthe average and the standard deviation. Brown and Brue [6] re-vised the sampling techniques commonly used for scaling beds.They determined that sampling of at least 20 min and as long as60 min was necessary to capture important low-frequency dynam-ical information. They also determined that at least 15 periodo-grams had to be averaged for an accurate estimate of the PSD.Another finding was that bubbling as well as circulating fluidizedbeds behaved as multiple second-order dynamical systems andthat unique peaks were identified by the PSD analysis caused bydifferent operation conditions.

Johnsson et al. [25] analyzed a circulating fluidized bed (CFB)at different ambient conditions and fluidization regimes. UsingPSD, they identified three regions: one macrostructure due to bub-ble flow and two other regions at high frequency due to finerstructures. Van der Schaaf et al. [26] determined bubble, gas slug,and cluster length scales from pressure fluctuation data measuredin the bed and the plenum. Gou et al. [27] compared the dynamicbehavior of a fluidized bed at high temperatures (1000 C) forthree different size ash particles using PSD, wavelet, and chaos

1Corresponding author.Contributed by the Fluids Engineering Division of ASME for publication in the

JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 27, 2012; finalmanuscript received May 20, 2013; published online August 6, 2013. Assoc. Editor:Michael G. Olsen.

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analysis. They found that PSD peaks coincided with the bubbleformation frequency based on the other two methods of analysis.Shou and Leu [28] compared standard deviation with PSD andwavelet analysis to find the critical gas velocities for fluidizationregime transition and found good agreement between the methodsused to analyze the dynamics of the bed.

The first CFD studies to predict pressure fluctuations were per-formed by Wachem et al. [29] and Benyahia et al. [30]. They bothused a 2D EulerianEulerian model and reported reasonableagreement with experiments. PSD of pressure fluctuations usingCFD predictions of slugging and turbulent regimes have onlybeen studied in spouted fluidized beds by van Wachem et al. [29].Van der Lee et al. [31] and Chandrasekaran et al. [32] used a 2DEulerianEulerian approach to investigate the behavior of linearlow density polyethylene (LLDP) particles in a fluidized bed andcompared their results with experimental data obtained with X-ray tomography. They studied the pressure fluctuation and bubbleproperties and the effects of bubble velocities, superficial gas ve-locity, and initial bed height on the hydrodynamics of the bed.Polyethylene is a particle with nonideal characteristics of shapeand size distribution, and therefore, the numerical model was notin good agreement with the experiments. It was suggested that thecoefficient of restitution and angle of internal friction be deter-mined experimentally and to properly develop a formulation formodeling the solid-phase stress tensor.

Johansson et al. [33] tested different closure models, compressi-bility, and inflow modes. They determined that modeling particlebehavior using kinetic theory of granular flow compared well withexperimental data and compressibility did not have a significanteffect on the results. On the contrary, the air feeding system wascrucial for predicting the overall dynamic behavior of the bed.This conclusion was also reported by Sasic et al. [34] when theyfound that under certain conditions, a strong interaction betweenthe inlet supply system and the gassolid fluidized bed in the formof pressure waves was propagated.

A rectangular bed with a single jet operating in a bubbling flu-idization regime was studied by Utikar and Ranade [35] usingglass and polypropylene as bed material. The dominant PSD fre-quency near the sparger was greater for the simulations than forexperiments but compared well at higher locations in the bed.They suggested quantifying sensitivity of solids viscosity, coeffi-cient of restitution, and frictional stresses to improve the model.Most recently, Mansourpour et al. [36] used a Lagrangian-Eulerian method to model a bubbling fluidized bed of polyethyl-ene particles. They investigated the influence of pressure onbubble characteristics (rise velocity, size, breakup rate) and foundthat pressurizing the bed promoted more uniform bed expansion.Wang et al. [37] concluded that pressure fluctuations originatedabove the distributor when a pulse of gas was injected in the bed.They also found that the amplitude of pressure fluctuationincreased with the inlet velocity for bubbling fluidized beds,where two peaks were identified in the spectrum. Acosta-Iborraet al. [38] found that using a 3D domain was necessary to model abubbling fluidized bed to obtain good predictions for power spec-tra and bubble behavior compared to the experiments. Using a 2DEulerianEulerian model to compare simulations with experimen-tal pressure fluctuation and acoustic emission energy, Sun et al.[39] studied a bubbling bed of LLPD particles. PSD for simula-tions using the SyamlalOBrien drag model compared betterwith results obtained from experimental pressure fluctuations.

In this work, the hydrodynamics of a fluidized bed will be stud-ied by analyzing the local pressure fluctuations above the inlet forbeds of sand and a binary mixture of cotton stalks and sand.Standard deviation of pressure fluctuations will be compared withexperimental data [20,21] to determine different fluidizationregimes at inlet gas velocities ranging from two to nine times theminimum fluidization velocity. Pressure spectra using Bode plotswill also be used to identify fluidized beds as second-order dy-namical systems and to reveal characteristic frequencies thatdescribe the bed hydrodynamics for different fluidization regimes.

The simulations will be performed using the software MultiphaseFlow with Interphase EXCHANGES (MFIX) [40] and will be quantita-tively compared with published experiments for pressure drop andstandard deviation of pressure. In addition, established correla-tions for the natural frequency for bubbling beds will be comparedto the simulations in this work.

2 Numerical Theory and Models

2.1 Governing Equations. A multifluid EulerianEulerianmodel is employed in MFIX [40] and assumes that each phasebehaves as interpenetrating continua with its own physical proper-ties. Volume fractions are introduced to track the fraction eachphase occupies. The solids phase is described with an effectiveparticle diameter dp and characteristic material properties. Conti-nuity and momentum equations are solved for both the gas phaseand the solids phases and have been presented in [40]. The mo-mentum equations include terms to describe the net rate of mo-mentum increase and the net rate of momentum transfer byconvection. The interaction force in the momentum equationsaccounts for the gassolids momentum transfer, which isexpressed as the product of the coefficient for the interphase dragforce between the gas and solids phases and the slip velocitybetween the two phases. The work herein has employed theGidaspow model for the interphase drag force [41] and previousstudies by Xie et al. [42], Deza et al. [3], and Hosseini et al. [43]have shown the validity of using the model. Further validation ofthe Gidaspow drag model for polydispersed mixtures have beenperformed by Gera et al. [44], England [45], and Kanholy et al.[46]. An equation for the granular temperature of the solids phaseis employed to model the specific kinetic energy of the randomfluctuating component of the particle velocity. Kinetic theory forgranular flow is used to calculate the solids stress tensor andsolidsolids interaction force in the rapid granular flow regime.Further details related to the constitutive relations can be found inRef. [40] and have previously been presented in detail by theauthors in Ref. [4].

2.2 Numerical Methodology. To discretize the governingequations in MFIX, a finite volume approach for a staggered grid isused to reduce numerical instabilities and ensure global conserva-tion of mass and momentum [47]. Discretization of time deriva-tives are first order, and a variable time-stepping scheme is usedto assist with convergence. Convection terms are discretized usingthe second-order MUSCL scheme to prevent numerical diffusionthat typically occurs with first-order schemes such as the upwindmethod. If first-order schemes are used to simulate a bubbling flu-idized bed, the predicted bubbles appear as a pointed shape[48,49]. A modification of the SIMPLE algorithm is used to solvethe governing equations that incorporate the solids volume frac-tion and solids pressure [47]. Further details of the numericalmodel can be found in Ref. [40].

2.3 Standard Deviation. As previously mentioned, standarddeviation has often been used to identify different fluidizationregimes in fluidized beds. The method of analysis has been widelyused by many researchers [1619]. Two velocities, UC and UK,are identified as delimiters for regime transition. UC marks the tran-sition from a bubbling to turbulent fluidization regime and occurswhen the standard deviation reaches its maximum. The transition isalso known as the slugging regime. UK marks the transition from aturbulent to a fast fluidization regime, where the standard deviationremains approximately constant. The standard deviation (r) for atime-series of pressure data, Pii 1; 2; :::; N, is:

r

1

N 1XNi1

Pi P2vuut (1)

where P is the average pressure for N time realizations.

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2.4 Power Spectral Density. Power spectral density isanother method to analyze pressure data and the frequencies asso-ciated with a system can be related to physical phenomena [22]. Acomplete formulation of spectral analysis for random processescan be found in Ref. [50] and a summary of the main equations isshown in this work. The power spectral density function (Syy) canbe defined as the Fourier transform of the autocorrelation function(Ryy) of a continuous time series (y(t)):

Syyix 11

Ryys expixs ds (2)

Ryys limT!1

1

T

T=2T=2

ytyt sdt (3)

where x is the angular frequency, s is a time shift, and T is thetotal length of the data set. However, time series are finite and anestimate can be obtained using the Fourier transform (=ix) ofthe continuous time series y(t):

Syyix limT!1

E1

2Tj=ixj2

(4)

=ix 11

yt expixs dt (5)

where E is the expected operator and the quantity 1=2Tj=ixj2is the periodogram of the time series y(t). To accuartely estimate aPSD, several periodograms will be required for averaging.

A fluidized bed can be described as a dynamical system, wherethe response, y(t), can be studied with linear system identificationtechniques for small inputs, x(t) [51]. The dynamics of a systemcan be represented by the linear relationship of the Laplace trans-form of the input and output time series, X(s) and Y(s), respec-tively, known as the transfer function (G(s)) that is related to thePSD functions of the input and output

Gs YsXs (6)

Syyix jGixj2Sxxix (7)

When the input is dominated by white noise, the PSD of the inputsignal (Sxx) can be found by calculating its statistical variance (r

2)and Eq. (7) can be rewritten using the variance

Sxxix r2 (8)10 log Syyix 20 log jGixj 20 log r (9)

The transfer function can be conveniently represented by plotting10 log Syyix versus log x and the representation is called theBode plot. For a second-order transfer function

Gix 11 x2=x2n i2xn=xn(10)

where xn is the characteristic frequency and n is the damping fac-tor with a high-frequency roll-off of 40 dB/decade. The Bodeplot will help determine x at the intersection of low and high fre-quency asymptotes and n will be noted in the trend of the plotaround xn. The value of n is greater than unity for overdampedsystems, which can be identified by a gradual decrease in the PSDas the frequency increases and a noticeable resonance at xn. Forunderdamped systems, n is less than unity and the PSD in theBode plot will show a sharp decrease as the frequency increases.

For a recorded data set, the PSD is calculated as the averagePSD of a series of overlapping segments. A way to compute thediscrete Fourier transform for a segment of Ns data points is byusing the fast Fourier transform (FFT) algorithm

FFTn XNs1k0

ykei2pkn

Ns (11)

for n 0; 1; 2; :::; Ns 1. FFT is applicable for Ns 2m, where mis an integer greater than unity. A PSD periodogram can be calcu-lated as follows:

PSDn 12T

E jFTTnj2h i

12T

E FTTnFTTn (12)

where FTT*(n) is the complex conjugate of the FTT(n). Theresulting PSD constitutes the frequency spectrum from zero toNs/2. FTT can be expressed in terms of frequency (f) by

f nNs

p (13)

where p is the sampling frequency. Notice that a conversion fromx to f can also be obtained by

f x2p

(14)

Similar to Eq. (9), the magnitude (M(n)) of a Bode plot can be cal-culated as follows:

Mn 10 log PSDn (15)

where M(n) is expressed in decibels (dB).

3 Results and Discussion

3.1 Problem Description. The rectangular fluidized bed ge-ometry used in this study is based on the work of Zhang et al.[20,21]. The lower section of the geometry is modeled in the nu-merical simulations and pressure measurements at locations (1)and (2), as denoted in Fig. 1, are taken to record the pressure dropfluctuations for all cases. Fluidizing air supplied through 48equally spaced holes in a tuyere distributor plate is modeled asuniform flow at the inlet. A single bed of sand and a binary bed of

Fig. 1 Schematic of the 3D rectangular cylinder representingthe experiments by Zhang et al. [20,21]

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cotton stalks and sand are used in this study and their propertiesare shown in Table 1. For the sand-only simulations, the initialvoid fraction is 0.423. Simulations of the binary mixture consid-ered sand and cotton stalks to be initially mixed with an initialvoid fraction of 0.392. Initial void fractions are calculated usingthe Ergun equation [52] with particle properties and minimum flu-idization velocity from the mentioned experiments. The cylindri-cal cotton stalks are simulated as spheres using the Sauter meandiameter dp [53], whose volume-to-surface area ratio is the sameas the nonspherical particle of interest

VpAp

4

3pdp=23

4pdp=22(16)

Therefore, the Sauter mean diameter is

dp 6 VpAp

(17)

Sphericity (w) is defined as the ratio of the surface area of a spherewith the same volume as the given particle to the surface area ofthe particle [54]

w p136Vp

23

Ap(18)

For these simulations, a uniform inlet air velocity is specified atthe bottom of the domain equal to the superficial gas velocity Ugand atmospheric pressure is specified at the exit. The no-slip con-dition is used to model the gaswall interactions and a partial-slipcondition for the particlewall interactions by Johnson and Jack-son [55]. Inlet gas velocities range from Ug 0.4 to 1.8 m/s; thelowest velocity corresponds to 2Umf and represents a mild bub-bling bed and the highest velocity corresponds to 9Umf and repre-sents a fluidized bed in the turbulent regime. A grid resolution of40 200 is used for the 2D domain and 40 200 40 for the 3Ddomain corresponding to square and cubic cells of 1 cm per side,respectively. The number of grid cells were found after beingcompared with a coarser grid of 2 cm per cell side and finer gridof 0.5 cm per cell side. The grid resolution study was performedfor the pressure drop of sand fluidized bed for inlet velocitiesbetween 0.05 and 0.3 m/s. The pressure drop for the three grid res-olutions is compared with experiments of Zhang et al. [20] andshown in Fig. 2. The numerical uncertainty due to discretization isestimated by analyzing the grid convergence index (GCI) asreported by Celik et al. [56]. The GCI values for the predictedpressure drop is presented in Table 2. The GCI at low flow ratesfor the unfluidized bed is the highest, with an uncertainty between3 and 5%; for the other flow rates, corresponding to the fluidizedbed, the GCI is less than 2%. Moreover, the approximate relativeerror (ea) is less than 3% for unfluidized beds and less than 2% forfluidized beds. We are confident that the use of the medium grid(40 200 40) provides results that are not sensitive to the gridresolution. Computational time was also taken into consideration,especially for 3D simulations, which required approximately 100

days to simulate 200 s using 16 processors in an Apple G5 with64 bit, 2.3 GHz IBM (PPC970) dual processors.

3.2 Pressure Drop Analysis. Initially, a bed of sand is usedto confirm that the CFD modeling predicts the same pressure dropthrough the bed using a two-dimensional domain. The pressuredrop across the sand fluidized bed versus the inlet gas velocity isshown in Fig. 3, comparing the experimental measurements [20]to those predicted using MFIX. Once the bed is fluidized aroundUmf 0.19 m/s, the measured pressure drop is approximately con-stant at 4.5 kPa, which also corresponds to the theoretical pressuredrop. The predicted pressure drop is approximately 4.5 kPa and isin very close agreement with the experimental and theoreticalvalues.

A bed of cotton stalkssand2 mixture is also simulated using atwo-dimensional domain and compared with experiments [21].Given the low mass ratio of cotton to sand (1%), it was of interestto compare simulations of the pressure drop for the binary mixturewith the sand-only data. The motivation is to continue with thestudy of pressure fluctuations with sand-only simulations todecrease the amount of time needed when 3D simulations are nec-essary. Figure 4 shows pressure drop for a bed of sand andcottonsand for both 2D and 3D simulations. For inlet gas veloc-ities higher than 0.8 m/s, the pressure drop predicted with 2Dslightly decreases to approximately 4.3 kPa, while predictionsusing 3D are approximately 4.5 kPa.

Table 1 Sand (S) and cotton stalks (CS) particle properties

S CS

dp (cm) 0.045 0.655qp (g/cm

3) 2.650 0.385qb,M (g/cm

3) 1.508 0.015W () 0.9 0.5m (g) 73,500 735e () 0.9 0.9

Fig. 2 Pressure drop versus inlet gas velocity comparingexperiments [20] and simulations of a sand fluidized bed forthree different grid sizes

Table 2 Grid convergence index (GCI) and approximate rela-tive error (ea) for pressure drop

Ug (m/s) GCI (%) ea (%)

0.05 4.81 2.560.10 3.97 2.910.15 1.63 2.350.20 1.85 2.110.25 0.14 2.070.30 1.25 1.20

2For the remainder of the discussion, the word cotton will refer to cottonstalks.



3.3 Standard Deviation Analysis. The next step is to com-pare standard deviation of pressure drop for inlet gas velocitiesranging from 1 9Umf. Standard deviation for sand and cottonsand fluidized beds are similar and follow the same trend, asshown in Fig. 5. A steadily increasing trend in standard deviationshows that for velocities greater than 0.8 m/s (4Umf), using a 2Ddomain is no longer appropriate. Taking into consideration thatinlet velocities are high (up to 9Umf), the use a of turbulencemodel and a 3D domain is considered next for the fluidized bedsimulations.

The sand and cottonsand fluidized beds are shown in Fig. 6with a best curve fit as a second-order polynomial for both 2D and3D simulations. Simulations using the Ahmadi turbulence model[57] are able to capture the trends reported in the experiments, as

seen in Fig. 6. Standard deviation shows a peak at 1.2 m/s(6Umf), which according to numerous studies by Bi et al.[1619], correlates to the transition from a bubbling to sluggingfluidization regime. The peak is captured with the CFD simula-tions for the sand-only fluidized bed. Standard deviation of thecottonsand bed, however, shows a lower value and is mostlikely due to difficulty in predicting the complex binary mixturetransitioning flow. However, an important conclusion is that theturbulence model is only necessary for velocities greater than6Umf, when the regime transitions to turbulent flow and thestandard deviation starts decreasing. Furthermore, it has been

Fig. 3 Pressure drop versus inlet gas velocity comparingexperiments [20] and 2D simulations of a sand fluidized bed

Fig. 4 Pressure drop versus inlet gas velocity comparingexperiments [20] and 2D and 3D simulations of sand andcottonsand fluidized beds

Fig. 5 Standard deviation of pressure drop comparing experi-ments [21] and 2D simulations of sand and cottonsand fluid-ized bed versus inlet gas velocity

Fig. 6 Best fit of standard deviation of pressure drop for sandand cottonsand fluidized beds compared with experiments[21]



shown that 2D simulations are valid for Ug 4Umf, which is thebubbling fluidization regime.

Some additional cases were considered between 9 and 10Umf totest the model capability to identify the transition to a fast fluid-ization regime as explained in Sec. 2.3. Both cottonsand andsand-only fluidized beds were simulated in three dimensions andthe values for rDP are shown in Fig. 6. Standard deviation of pres-sure fluctuation beyond 1.8 m/s slightly increases and remains rel-atively constant, showing agreement with the reported generaltrend [58]. The fast fluidization regime is out of the scope of thisstudy; however, it is encouraging that the numerical model canpredict a fast fluidization regime.

It is clear from the comparison shown in Fig. 6 that the sand-only fluidized bed can predict standard deviations of the cotton-sand bed for a mass ratio of 1%. The 3D simulations using a sandbed closely resemble predictions of the hydrodynamic behavior ofcottonsand beds, as shown in the time-averaged contour plots ofvoid fractions of Fig. 7. The slices correspond to a central verticalplane and only Z 0 80 cm is shown because no fluidizationoccurs beyond that height in the freeboard. Each set of subfigures7(a), 7(b), and 7(c) corresponds to the bubbling, slugging, and tur-bulent fluidization regimes represented by Ug 0.8, 1.2, and1.6 m/s, respectively. Gassolids distributions and bed heights forsand (top row) and cottonsand (bottom row) are in very closeagreement. Figure 7(d) represents the time-averaged void fractionhorizontally-averaged across the rectangular cross section of the

domain (xy plane), and is useful for identifying bed expansion. Itcan be seen that for both sand and cottonsand, the bed expansionfor the bubbling fluidization regime is around 40 cm. However,the bed surface is not well defined for the slugging and turbulentfluidization regimes.

In order to further identify salient features of each of the fluid-ization regimes, instantaneous void fraction contours at three timeintervals with Dt 0.1 s are shown in Fig. 8. The slices correspondto a central vertical plane and only 80 cm. The bubbling fluidizedbed (top row) shows small bubbles coalescing as they rise. Awell-defined bed surface can also be visualized, consistent withthe time-averaged void fraction horizontally-averaged in Fig.7(d). Void fractions for a slugging regime (middle row) show acoalesced large bubble rising to the bed surface. The bed surfaceis not well defined due to the slugging bubble rising periodicallyand exploding at the surface. The turbulent fluidization regime(bottom row) shows irregular voids and changing bed surface asgas is released and this fact is also evident in Fig. 7(d).

3.4 Power Spectral Density Analysis. PSD analysis hasbeen performed in sand and cottonsand fluidized beds for pres-sure drop fluctuations. Before analyzing results using PSD, someparameters such as data sampling frequency, number of overlap-ping segments for signal averaging, and time range were deter-mined for the available data sets. These parameters were tested

Fig. 7 Time-averaged void fraction of sand (top row) and cottonsand (bottom row) fluidized beds using a 3D domain with inletvelocities of (a) Ug5 0.8m/s, (b) Ug5 1.2m/s, (c) Ug5 1.6m/s, and (d) xy average void fraction for the three cases



according to recommended values by Brue and Brown [6] for lowfrequency sampling rates with long sampling intervals. Theynoted that a sampling frequency of 20100 Hz did not show anydifference in the high frequency of the PSD. To find an accurateapproximation of the PSD, data should be partitioned into a num-ber of overlapping segments that will be transformed into periodo-grams. The number of segments will depend on the amount ofdata collected. Brue and Brown [6] recommended collecting at

least 1200 s of data to capture phenomena occurring at low fre-quencies. A parametric study on a 2D case of the sand fluidizedbed at 2Umf (0.4 m/s) was simulated for 1205 s and the first 5 swere not considered in the analysis to remove initial transients.Data was recorded at 25, 50, and 100 Hz frequencies. As a result,it was found that 200 s with a sampling frequency of 50 Hz and 10overlapping segments was sufficient. Further details can be foundin Deza [59].

Fig. 8 Instantaneous void fraction of a sand fluidized bed at an inlet velocity Ug50.8m/s (toprow), Ug5 1.2m/s (middle row), and Ug5 1.6m/s (bottom row) at three different times



Continuing with the analysis performed for standard deviation,the same three cases are used to study PSD. The cases representbubbling, slugging, and turbulent fluidization regimes at inletvelocities Ug 0.8, 1.2, and 1.6 m/s, and correspond to 4, 6, and8Umf, respectively. Simulations using the 3D domain and sandas the bed material are analyzed with data collected for 195 s.Table 3 shows the number of processors and the CPU time in daysto simulate 200 s of the cases analyzed. CPU time for 20 s ofcottonsand bed simulations have been included in the table toshow how computationally expensive these simulations can be.Having confidence in the similarity of results between a fluidizedbed of cottonsand and sand, the following study will be per-formed using sand only.

Pressure data is presented in Figs. 911 using standard devia-tions of pressure fluctuations, PSD, and Bode plots to identify dis-tinct features for each fluidization regime. In Figs. 9(a), 10(a), and11(a), pressure drop for a 25 s time period is shown to demon-strate how the pressure drop fluctuates with time. Pressure dropfor all the cases have been shifted to zero to help quantify the fluc-tuation around the mean value by

DP DP P

For Figs. 9(b), 10(b), and 11(b), the PSD is shown for data col-lected over 195 s and the corresponding Bode plots are shown inFigs. 9(c), 10(c), and 11(c). The pressure drop has a minimum andmaximum of approximately 61 kPa for the bubbling fluidizationregime (Fig. 9(a)), but for other fluidization regimes (Figs. 10(a),11(a)) the pressure drop fluctuations are more pronounced.

PSD has commonly been used to find the dominant frequencyin the dynamic system; however, their Bode plot representationoffers an advantage in the identification of additional frequenciesand in determining the order of the dynamic system. As we com-pare Bode plots for each regime, it is evident that the dynamicsystem for all three cases is of the second order with a roll-off of40 dB/decade for high frequencies. On the other hand, there arenoticeable differences in the peak intensity and distribution amongall regimes, which is also evident in the PSD.

PSD and Bode plots for the bubbling regime (Figs. 9(b) and9(c)) show a series of peaks, which can be described as a broadpeak between 2 and 4 Hz, with a maximum at fB 2.6 Hz, wherethe subscript B denotes the bubbling fluidization regime. A similartrend for the PSD was obtained from the experiments of Johnsonet al. [25] for a fluidized bed in the bubbling regime. Brue [60]showed the same trend for PSD as well as Bode plots in many ofhis bubbling fluidized bed cases.

The slugging fluidization regime PSD and Bode plots(Figs. 10(b) and 10(c)) show two broad peaks at lower frequenciesover a range of 1.5 to 5 Hz, with a peak fS,1 2 Hz and anotherpeak at fS,2 3.5 Hz, where the subscript S denotes the sluggingfluidization regime. However, the magnitudes of these peaks arelower than the peak in the bubbling regime (Fig. 9). These distinc-tive broad peaks correspond to previous observations of two peaksat low frequencies, which were identified as characteristics of theslugging regime spectrum by van der Schaaf et al. [61].

For beds in the turbulent fluidization regime, one peak atapproximately fT,1 2 Hz and a broader peak at fT,2 4.5 Hz areidentified in the PSD and Bode plots (Figs. 11(b) and 11(c)). Brue[60] found that with increasing inlet velocity, the frequencies cor-responding to the higher peaks will decrease. For these cases, the

Table 3 Computational time for simulations using sand (S)and cottonsand (CS) beds in CPU (days) (Note: 2 processorsfor 2D and 16 processors for 3D simulations)

2D 3D

Ug (m/s) S CS S CS

0.8 4 22 102 45a

1.2 92 49a

1.6 101 45a

aFor 20 s of simulation.

Fig. 9 Pressure drop fluctuation (a) with time, (b) as a PSDanalysis, and (c) as a Bode plot for a sand fluidized bed withinlet velocity of 0.8m/s using a 3D domain



lower frequency peak has a greater magnitude. The existence oftwo peaks in the spectrum of a turbulent fluidized bed have alsobeen reported by Brue and Brown [6].

It is interesting to compare the frequency predictions (both 2Dand 3D) with correlations found in the literature for the bubbling

fluidization regime (Ug 4Umf). Correlations were developed topredict natural frequency of a second-order system in bubblingfluidized beds by Hiby [62], Verloop and Heertjes [63], Baskakovet al. [64], and Brue [60]. They found that natural frequencies willcoincide with the frequency at which the spectra shows a peak.





Table 4 presents the natural frequency calculated using correla-tions with the 2D and 3D simulations. When comparing 2D versus3D domains for sand-only fluidized beds, it can be seen that thePSD for the 2D case has a peak at fB 1.4 Hz and is low com-pared with the 3D case (fB 2.6 Hz) and the correlations. It isencouraging that the 3D case predicts a natural frequency that isin good agreement with the correlations by Hiby [62] and Brue[60]. The comparison suggests that using a 2D domain may under-predict the characteristic peak for a bubbling fluidization regime,which was also suggested by Acosta-Iborra et al. [38].

The corresponding frequency of the peak for a bed of cotton-sand using a 3D domain is determined based on a 20 s time rangesince the use of a 3D domain will take too long to simulate (seeTable 3). To determine if these simplifications are appropriate,dominant frequencies using 20 and 200 s for the 2D simulationwere compared, resulting in similar frequencies of 2.7 Hz for 20 sof simulation and 2.6 Hz for 200 s of simulation. Both resultsshow the same power spectra features. Therefore, the predicteddominant frequency for a bubbling bed of cottonsand using a 3Ddomain is approximately 2.8 Hz, which is in close agreement withthe correlations by Hiby [62] and Brue [60].

The peak for a cottonsand fluidized bed at 4Umf using a 2D do-main also shows a low frequency value of fB 1.5 Hz. The resultsof this comparison corroborate that using a 3D domain will appro-priately predict the hydrodynamic behavior of both sand-onlyand cottonsand fluidized beds, while using a 2D domain willunderpredict the dominant frequencies. They also confirm that asand-only fluidized beds can predict the behavior of cottonsandfluidized beds with a low cotton-to-sand mass ratio (1%).

4 Conclusions

Simulations have been validated for a cottonsand fluidized bedoperating at inlet velocities ranging from 1 9Umf with experimen-tal results of Zhang et al. [20,21]. Fluidization regimes for thesecases fall in the bubbling, slugging, and turbulent regimes. Thehydrodynamic features of fluidized beds in this study have beenmainly analyzed by means of pressure fluctuation analysis. Plotsof variations of pressure with time, standard deviation of pressuredrop, and power spectrum with frequency have been used to char-acterize fluidized beds at different inlet velocities for a single bedmaterial and a binary mixture.

Initially pressure drop for sand and cottonsand beds weretested using a 2D domain model, and were in very close agree-ment with those reported in the experiments of Zhang et al.[20,21]. Standard deviation of pressure drop was overpredictedfor inlet velocities greater than 4Umf when using a 2D domain;therefore, modeling with a 3D domain as well as the use of a tur-bulence model for higher velocities was studied. It was deter-mined that the fluidized bed can be appropriately modeled byusing MUSCL discretization and the Ahmadi turbulence modelfor velocities equal to and greater than 6Umf, which correspondedto the peak of standard deviation of pressure drop.

Three cases at 4, 6, and 8Umf were chosen to represent bub-bling, slugging, and turbulent regimes. Both PSD and Bode plotswere compared for all the cases studied. Fluidized beds for all the

regimes behaved as second-order dynamic systems. Bubbling flu-idized beds showed one peak while slugging and turbulent bedsshowed two distinct peaks. It was observed that the peak at lowfrequency increased in magnitude as the flow transitioned fromslugging to turbulent fluidization regimes. Comparisons of thepredicted frequency for a bubbling bed with correlations by Hiby[62] and Brue [60] were in very good agreement. CFD simulationsof fluidized beds with the purpose of studying pressure fluctua-tions have demonstrated to be a useful tool to obtain hydrody-namic information that will help determine the fluidizationregime. Furthermore, this was the first CFD study to predict slug-ging and turbulent regimes in a constant diameter (rectangular)fluidized bed. Therefore, the predictions reported in this study rep-resent an important advantage when designing a reactor and eval-uating different operation conditions without the need to test themin a pilot plant or prototype.

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(deza 2013) a cfd study of pressure fluctuations to determine fluidization regimes in gas-solid beds

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