Chemical Engineering Journal 245 (2014) 217–227

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Chemical Engineering Journal

journal homepage: www.elsevier .com/locate /cej

Bubble formation at a central orifice in a gas–solid fluidized bedpredicted by three-dimensional two-fluid model simulations

http://dx.doi.org/10.1016/j.cej.2014.02.0261385-8947/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +31 40 247 3674.E-mail address: J.T.Padding@TUe.nl (J.T. Padding).

Vikrant Verma, Johan T. Padding ⇑, Niels G. Deen, J.A.M. KuipersDepartment of Chemical Engineering and Chemistry, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands

h i g h l i g h t s

�We present an extensive study ofbubble formation at an orifice in afluidized bed.� We use a highly efficient two-fluid

model based on kinetic theory ofgranular flow.� We quantify the effects of particle

properties, flow rate and bed size.� We compare our results with

experimental results andapproximate theoretical models.

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:Received 7 October 2013Received in revised form 31 January 2014Accepted 8 February 2014Available online 18 February 2014

Keywords:Fluidized bedTwo-fluid modelSimulationsEquivalent bubble diameterGas leakage

a b s t r a c t

We apply a recently developed two-fluid continuum model (TFM) based on kinetic theory of granularflow (KTGF) in three dimensional cylindrical coordinates, to investigate bubble formation through asingle central orifice in a gas–solid fluidized bed. A comprehensive study for Geldart D type particles,revealing the influence of particle diameter, jet injection flow rate, and bed size on bubble characteristicshave been investigated. At a given gas injection flow rate, the bubble diameter continuously increaseswhile gas leakage from the bubble to the emulsion phase decreases with time. With increasing particlediameter, leakage fraction increases and hence a smaller bubble diameter is predicted. These results areconsistent with DPM simulations, experimental results and approximate bubble formation modelsreported previously in the literature.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Gas–solid fluidized beds are extensively used in processindustries because of their excellent mixing and heat and masstransfer characteristics. They are currently used in separation,classification, drying and mixing of particles, chemical reactionsand regeneration processes. The performance of a fluidized bedin relation to solids motions, gas–solids contacting are majorlygoverned by its bubble characteristics [1], so understanding theformation and propagation of bubbles is of great practical interest.

The bubbles are formed at the gas distributor plate; thereafterthese bubbles propagate throughout the bed with different charac-teristics such as bubble size distribution, bubble rise velocity andthe bubble frequency distribution. Bubble formation in gas-fluid-ized beds is a fairly complicated process influenced by many fac-tors, including properties of the gas and particulate phases,orifice geometry, and gas flow rate. Many theories have beendeveloped and validated for bubble formation, both experimen-tally and numerically. However, most of the theories were limitedto, or validated for, 2D or pseudo 2D systems. 2D simulations havebeen performed and compared with experimental data gatheredfrom pseudo 2D beds [2–4]. Simulations with 2D cylindrical coor-dinates have also been used to predict bubble formation in 3D

Nomenclature

C fluctuation particle velocity, m s�1

g gravitational acceleration, m s�2

I unit tensor, –p pressure, Paq kinetic fluctuation energy, kg s�1

u velocity, m s�1

t time, se coefficient of restitution, –dp diameter of particle, mg0 radial distribution function, –D diameter, mV volume of the bubble, m3

Q volumetric flow rate, m3 s�1

Greek symbolsb interphase momentum transfer coefficient, kg m�3 s�1

c dissipation due to inelastic particles collisions, kg m�1 -s�3

e volume fraction, –q density, kg m�3

l shear viscosity, kg m�1 s�1

H pseudo particles temperature, m2 s�2

s stress tensor, PaW leakage fraction, –

Subscriptss solid phaseg gas phasee equivalentmf minimum fluidization

Operatorr gradientr� divergence

218 V. Verma et al. / Chemical Engineering Journal 245 (2014) 217–227

beds, but the validity of these simulations is questioned. Geldart[5] reported that in a 2D bed bubbles are restrained by the wallsin one dimension, transforming to slugs when viewed from theside of the bed, contributing to a greater bed expansion and consid-erable difference between particle and bubble characteristics.Three-dimensional studies are rarely found in the literature as theystill pose a challenge: numerically because of high computationalcosts, and experimentally because flow visualization and measure-ments are difficult to perform. Nevertheless, the most commonlyused geometry in industry is the cylindrical one. Therefore, in thiswork we consider a full three dimensional domain with cylindricalgrid structure to study realistic bubble formation at a central circu-lar orifice in a cylindrical fluidized bed. We investigate the influ-ence of particle diameter, gas injection velocity, bed size, orificesize and computational grid size on the bubble characteristics. Pre-vious studies [2–4] on bubble formation in 2D reported that bubbleformation is sensitive to the background gas velocity, i.e. Umf. Themost pronounced gas leakage effects take place for particles havingrelatively high Umf values, which is why in this study we focus onparticles falling in the Geldart D classification. Comparison is madewith experimental data reported by Nieuwland [4] and approxi-mate bubble formation models [6,7].

These models have been proposed based on different assump-tions to describe bubble formation in a gas–solid fluidized bed ata single orifice. Davidson and Schuler [8] were the first to providea theoretical solution to a single bubble formation in a viscous li-quid. On the same basis, with the assumption of no gas exchangefrom the bubble to the surrounding emulsion phase, Harrisonand Leung [7] provided a formula for the bubble volume at detach-ment and the time of bubble detachment. Nguyen and Leung [9]performed experiments on a 2D fluidized bed of alumina particlesand concluded that a considerable (47%) amount of gas leakagetakes place from the bubble to the emulsion phase. Later, Roweet al. [10] came to the same conclusion after observations in cylin-drical fluidized beds using X-ray cine-photography. Yang et al. [11]accounted for gas leakage from the bubble surface to the surround-ing emulsion phase, from evidence of their own experimental workon a large-scale (3 m diameter semicylindrical) cold flow construc-tion. They assumed that the gas leaks from the bubble at a rateequal to the minimum fluidization velocity, as was suggested byZenz [12]. However, this model is semi-empirical because itrequires the bubble frequency as an input obtained from experi-

ments. Caram and Hsu [6] applied Darcy’s law to obtain the expres-sion for the superficial gas leakage velocity at the bubble boundary.They reported satisfactory agreement of their model predictionwith available experimental data.

In recent years, powerful computational resources have enabledthe use of detailed computational models to investigate bubble for-mation on a feasible time span and in three-dimensional space.Two of the most common modeling approaches for dense gas-flu-idized beds are the two-fluid model (Euler–Euler) and the DiscreteParticle Model (Euler–Lagrange). In the latter, particle trajectoriesare obtained by integrating Newton’s equations of motion. Parti-cle–particle and particle–wall interactions are explicitly taken intoaccount using various physical models such as the soft spheremodel. To study large three dimensional systems a Lagrangian ap-proach becomes computationally too expensive, particularly forsmall particles. Therefore Euler–Euler approaches have beenadopted by several authors, where the gas and solid phase are trea-ted as fully interpenetrating continua. Incorporating the kinetictheory of granular flow is used to account for particle–particleinteraction. Kuipers et al. [2] used a two-fluid model (TFM) to sat-isfactorily predict the formation of a bubble at a single orifice in atwo dimensional (2D) bed. Pierrat and Caram [13] solved reducedtwo fluid model equations, for solid phase mass and momentumconservation in one dimension. Their model does not take into ac-count the equation of motion of the bubble so it can not predictbubble detachment time. Huttenhuis et al. [14] used TFM to studythe effect of gas-phase density on bubble formation at a single ori-fice in the 2D gas–solid fluidized bed. Nieuwland et al. [15] studiedexperimentally and numerically the effect of particle properties onthe bubble formation at a single orifice in a 2D bed and a semi-cir-cular bed. They compared results of a semi-circular bed with a 2Daxisymmetric simulation and found some discrepancy with theexperimental results due to absence of front wall in their simula-tion. Olaofe et al. [16] used an earlier developed discrete elementmodel to study Geldart D type particles and investigate the influ-ence of particle diameter and injection velocity on bubble forma-tion at a single orifice in 2D gas–solid fluidized beds. Rong et al.[17] used a soft sphere Discrete Particle Model to study the effectof various parameters on bubble formation due to a single jet pulsein a 2D coarse-particle fluidized bed. Other researchers [18,19]studied bubble formation in a 3D domain to assess certainnumerical aspects of the TFM. However studies on the influence

Table 1TFM simulations settings.

Bed diameter 15.0 cmHeight 22.5 cmParticle bed height 15.0 cmNumber of grid cells in radial direction

(0–r)30

Number of grid cells in radial direction(0–2p)

32

Number of grid cells in radial direction(0–z)

90

Flow solver time step 10�4 sDrag force model van der Hoef et al. [22]Frictional viscosity model Srivastava and Sundaresan [28]Flux limiter Superbee Roe [26]

Boundary conditionsWall of the cylinder No slip for gas and partial slip for

particlesAxis of the cylinder Free slip for particles and gasOutlet Prescribed atmospheric pressureInlet Gas influx through the central jet

Fig. 1. Schematic representation of the three-dimensional gas-fluidized bed.

Fig. 2. Schematic representation of computational grids in 3D cylindrical geometry.Note that the real number of grid cells is much larger than indicated here.

V. Verma et al. / Chemical Engineering Journal 245 (2014) 217–227 219

of particle size and injection velocity on bubble formation charac-teristics in a full 3D domain still remain to be reported. An impor-tant questions is whether these full 3D results will agree withaxisymmetric 2D simulations because it is not known at whichpoint the initial symmetry of the system will spontaneously break.

This paper is organized as follows. First a general description ofthe TFM is given, then we give the simulation settings for our 3Dsimulations, including the boundary conditions for the orifice.We then investigate and discuss our results to study the influenceof particle diameter, jet injection flow rate, and bed size on bubblecharacteristics. We show a comparison between TFM and DPM re-sults for a single test case, and additionally compare our resultswith published experimental work on semi-circular beds, andpoint out some important limitations of physical experiments.We end with our conclusions.

2. Two fluid model (TFM)

The two-fluid continuum model describes both the gas phaseand the solids phase as fully interpenetrating continua using ageneralized form of the Navier–Stokes equations [20,21]. Thecontinuity equation for the gas and the solids phase is given byEqs. (1) and (2), and the momentum equation is given by (3) and(4), respectively.

@ðegqgÞ@t

þr � ðegqg �ugÞ ¼ 0 ð1Þ

@ðesqsÞ@t

þr � ðesqs�usÞ ¼ 0 ð2Þ

@ðegqg �ugÞ@t

þr � ðegqg �ug �ugÞ ¼ �egrpg �r � ðeg��sgÞ

� bð�ug � �usÞ þ egqg �g ð3Þ

@ðesqs�usÞ@t

þr � ðesqs�us�usÞ ¼ �esrpg �rps �r � ðes��ssÞ

þ bð�ug � �usÞ þ esqs�g ð4Þ

The gas and solid phases are coupled through the interphasemomentum transfer coefficient b in the momentum equations.The modified Ergun drag force model of Van der Hoef et al. [22] isused in this work. To describe the particle–particle interactionsthe kinetic theory of granular flow is used, which expresses theisotropic and deviatoric parts of the solids stress tensor (i.e. the sol-ids pressure and solids shear viscosity) as a function of the granulartemperature as defined as:

H ¼ 13< �Cs � �Cs > ð5Þ

where Cs represents the particle fluctuation velocity and pointybrackets denote ensemble averaging. The derivation of these consti-tutive equations are discussed in the books by Chapman and Cowl-ing [23] and Gidaspow [21], and the papers by Jenkins and Savage[24], Ding and Gidaspow [25] and Nieuwland [4]. In this work theconstitutive equations by Nieuwland et al. [15] have been usedfor the particle phase rheology. The time evolution for the granulartemperature itself is given by:

32

@

@tðesqsHÞ þ r � ðesqsH�usÞ

� �¼ �ðps

��I þ es��ssÞ

: r�us �r � ðesqsÞ � 3bH� c ð6Þ

For the case of single bubble formation, a boundary condition forthe central jet is enforced by specifying the gas injection velocitythrough a specified orifice size and for a certain time durationbecause the time of bubble injection is limited.

3. TFM simulation settings

TFM simulation settings for the three dimensional fluidizedbeds considered here are presented in Table 1. A schematic of

220 V. Verma et al. / Chemical Engineering Journal 245 (2014) 217–227

the simulated 3D domain is shown in Fig. 1, and the 3D computa-tional grid structure is shown in Fig. 2. Details on the actual dis-cretization and implementation can be found in Verma et al.[27]. Initially, the particle phase occupies the bottom 15 cm ofthe cylinder. Before injecting gas through an orifice, the particlebed is simulated at its incipient fluidization velocity, typically fora real time of 0.1 s. Subsequently the gas is injected through thecentral circular orifice at the bottom. The influence of different par-ticle diameters varying between 1.5, 2.0 and 2.5 mm were com-pared at different gas injection flow rates corresponding to gasvelocities of 10, 15, 20 and 30 m/s. Longer and wider beds werealso simulated, by increasing the number of cells in the radialdirection for a wider bed and in the axial direction for a longer bed.

Fig. 4. Bubble contours in a bed of 1.5 mm particle at t = 0.075 s, for injection gasvelocity of (a) 10 m/s and (b) 30 m/s.

4. Results and discussion

When predicting the bubble size with time, it is important toclearly define the bubble boundary first. Ideally a gas fraction of1.0 should define the bubble boundary; however from laboratoryexperimental literature data it is clear that some particles do existinside the bubble, with a particle concentration gradient observednear the interface of the bubble and emulsion phase. So definitecriteria for defining the bubble boundary have not yet been estab-lished. Various authors define certain cutoffs to distinguish be-tween the bubble and emulsion phase. Kuipers et al. [2] showedthat for a 2D bubble, very sharp porosity gradients exist near thebubble base but these gradients are considerably weaker nearthe roof of the bubble. Considering the same approach, a particularchoice of gas fraction equal to 0.80 is defined as the bubble bound-ary in this work. The contours defining the sharp bubble boundarywas obtained by a linear interpolation in all directions using voidfraction value in the neighboring cells. To calculate the so-calledequivalent bubble diameter De, the volume V of a bubble definedby the contours is assumed to be spherical and De is calculatedaccording to the equation:

De ¼ffiffiffiffiffiffiffiffiffi

Vp=6

3

sð7Þ

Simulations were performed until the bubble detached from theorifice. In terms of the TFM, the bubble detachment time is defined

Fig. 3. Contours of the bubble formation in a bed of 1.5 mm particles with injection veloc0.075 s, and (d) 0.1 s.

as the time at which the bubble contours close above the orifice.The contours of bubble are determined by visualizing porosity plot.

4.1. Shape of the bubble

The shape of the bubble have been quantified using the bubbleshape factor. The bubble shape factor is defined as ratio of the max-imum diameter in vertical direction to the maximum diameter inthe horizontal direction. The shapes of bubble have been visualizeusing 3D contour plots, Fig. 3 shows how the bubble shape changesas it grows with time. The bubble initiates its growth in 3D with ahemispherical shape. Then it detaches from the orifice approxi-mately with a spherical shape (at low flow rate) or elongated shape(at high flow rate) as shown by the bubble contours in Fig. 4 andbubbles shape factor in Fig. 5. The bubble detachment time fromthe orifice was found to be independent of gas flow rate or particletype, which is approximately 0.1 s. In this work we focus on the

ity of 15 m/s, showing at four different times after injection: (a) 0.025 s, (b) 0.05 s, (c)

Fig. 5. Bubble shape factor as a function of time after start of gas injection in a bedof 1.5 mm particles, compared for different injection gas velocities. Fig. 7. Leakage fraction W as a function of time after start of injection for different

injected gas velocities into a bed of particles with diameter 2.0 mm.

V. Verma et al. / Chemical Engineering Journal 245 (2014) 217–227 221

bubble behavior until it detaches from the orifice. However we alsodiscuss our observation on the shape of bubble after detachment asseen in Fig. 6. As the bubble detaches its spherical shape is quicklydestroyed, molding into a kidney shape bubble with emulsionphase gas flowing into the bubble at its base, and bubble gas flow-ing into the emulsion from the roof. As the bubble rises upwardinside the bed, the kidney shape changes to a spherical cap, andthe bubble remains with this cap shape until it reaches the surfaceof the bed. The precise bubble shape in 3D was found to be highlydependent on the injected gas flow rate. The shape of the bubble isspherical at lower injected flow rates to elliptical at higher flowrates. Note that in literature studies using 2D geometries, the bub-ble size is relatively larger and more cylindrical in shape, due topresence of the back and front walls.

4.2. Model comparison

We now turn to the performance of theoretical models for bub-ble formation. Here we compare our TFM results with two differenttheoretical models, those of Harrison and Leung [7] and Caram andHsu [6], which differ with respect to the quantification of leakagefrom the bubble to the emulsion phase.

We define the leakage fraction W as the fraction of the injectedgas which has leaked from the bubble to the emulsion phase:

Fig. 6. Bubble shape after detaching from the orifice in a vertical plane in the mid

wðtÞ ¼ Qt � VðtÞQt

ð8Þ

In this expression Q denotes the injected volumetric flow rate and Vdenotes the volume of the bubble at any time t. The Harrison andLeung model estimates larger diameters because this models doesnot take into account gas leakage from the bubble to the emulsionphase (i.e. W = 0). The Caram and Hsu model does take this into ac-count, hence leading to lower bubble diameters and better fits withour TFM results. Fig. 7 shows how the leakage fraction decreaseswith time in our TFM simulations. A higher leakage is predictedduring the initial stage of the bubble growth. Stronger leakage isalso predicted at lower gas flow rates compared to higher gas flowrate.

Furthermore Fig. 8, reveals that the Caram and Hsu model givesbetter agreement with TFM at the highest gas injection velocity of30 m/s. At lower injection velocity, the agreement of the Caramand Hsu model with our TFM predictions seems to be less satisfac-tory. This apparent behavior at different velocities is due to differ-ent particle packing at the bubble boundary formed at differentinjection velocity. Discrete Particle Model (DPM) simulation for2D bed reported by Olaofe et al. [16] and Rong et al. [17] clearlyrevealed this behavior. Particles at the edge of a bubble are moretightly packed at higher gas velocities. This tight packing of

dle of the 3D bed at (a) t = 0.12 s, (b) t = 0.18 s, (c) t = 0.24 s and (d) t = 0.30 s.

Fig. 8. Equivalent bubble diameter at different injected gas velocities for a bed of 1.5 mm particles, compared with theoretical models with (dashed lines) and without(dotted lines) gas leakage.

Fig. 9. Azimuthally-averaged particle velocity vectors at 0.075 s after injection in abed of particles with diameter of 1.5 mm, for gas injection of (a) 10 m/s and (b)30 m/s.

Fig. 10. Gas leakage at the bubble boundary for bed of 1.5 mm particles, injectionvelocity = 20 m/s and t = 0.05 s. (a) Azimuthally-averaged particle velocity vectorsand (b) Azimuthally-averaged gas velocity vectors.

222 V. Verma et al. / Chemical Engineering Journal 245 (2014) 217–227

particles prevents leakage and most of the injected air entrappedcontributes to the bubble, resulting in a larger bubble diameter.This phenomenon is easily observed with DPM simulations, whereindividual particles are visualized. Because TFM simulations arebased on an interpenetrating continuum approach, it is moredifficult to visualize. Nevertheless our results show that this

phenomenon also occurs in TFM, and is consistent with resultsobtained from the Lagrangian model.

Note that the theoretical model is fully independent of theinjection rate through the orifice and assumes a uniform exchangevelocity over the spherical surface of the bubble. This assumptionis incorrect as discussed by Hailu et al. [29]. This is evidencedfrom Figs. 9 and 10(a) which show particle velocity vectors. The

Fig. 11. Azimuthally and time-averaged particle velocity vectors for a bed of (a) 1.5 mm particles and injection velocity = 15 m/s. (b) 1.5 mm particles and injectionvelocity = 30 m/s. (c) 2.5 mm particles and injection velocity = 30 m/s. The averaging time is the total time between the start of gas injection and detachment of the bubble(0–0.1 s).

Fig. 12. Equivalent bubble diameter as a function of time after start of injection fordifferent gas injection velocities in a bed of 2.0 mm particles.

Table 2Gas and particles properties.

Gas type AirInjection velocity 10, 15, 20, 30 m/s

Particle type Glass

Particle diameter Particle density Umf

1.5 mm 2526 kg/m3 0.94 m/s2.0 mm 2526 kg/m3 1.2 m/s2.5 mm 2526 kg/m3 1.4 m/s2.0 mm 1650 kg/m3 0.94 m/s2.0 mm 3370 kg/m3 1.4 m/sParticle coefficient of restitution = 0.97

Fig. 13. Equivalent bubble diameter as a function of time after start of injection fordifferent particle diameters at injection gas velocity of 20 m/s.

V. Verma et al. / Chemical Engineering Journal 245 (2014) 217–227 223

particles fall downwards from the side of the bubble due to grav-ity and these fallen particles move towards the base of the bubble.Fig. 10(b) shows that the emulsion phase gas in the bottom sec-tion near to the bubble prefers to bypass into the bubble fromits base and exits from the bubble roof. So there is an exchangeof gas from the emulsion phase to the bubble and from the bubble

to the surrounding emulsion phase. It can be expected that thereliability of theoretical models to predict bubble formation is lesswhen different characteristics of gas flow and particles are used.Earlier work of Kuipers et al. [2] showed that results obtained inbeds with particles of much smaller diameter (0.5 mm) tend toconverge to the theoretical models due to a lower minimum flu-idization velocity, resulting in lower gas leakage to the emulsionphase. The time-averaged particles velocity vector plot in Fig. 11shows that recirculation of particles takes place during the bubbleformation process. The center of the recirculation of particles vor-tex lies in the lower region (at the base) of the bubble. This showsthat the exchange of gas at the side bottom from the emulsionphase to the bubble is not high enough to provide sufficient dragon the sliding particles and carry them to the roof of the bubble.This is also evident from the instantaneous gas phase velocity vec-tor in Fig. 10(b). We have observed that this recirculation takesplace, irrespective of particle properties or injected velocity asshown in Fig. 11.

Fig. 14. Gas leakage fraction W as a function of time after start of injection fordifferent diameter particle compared at a gas injection velocity of 20 m/s.

Fig. 15. Equivalent bubble diameter for three different initial particle bed dimen-sions. The diameter of the particle is 1.5 mm and gas injection flow rate is 20 m/s.

Fig. 16. Equivalent bubble diameter from theory (dash–dotted and dash–double–dotted lines), determined from TFM in the current work (solid line) and determinedfrom TFM and experimental results of Nieuwland et al. [15] for semi-circular gasfluidized bed. Particle diameter is 275 lm, particle density is 3060 kg/m3, injectionvelocity is 20 m/s.

Fig. 17. Bubble shape factor determined from TFM and experimental results ofNieuwland et al. [15] for semi-circular gas fluidized bed. Particle diameter is275 lm, particle density is 3060 kg/m3, injection velocity is 20 m/s.

224 V. Verma et al. / Chemical Engineering Journal 245 (2014) 217–227

4.3. Influence of injection flow rate

The influence of the injection gas flow rates on the process ofbubble formation has been investigated as well. Fig. 12 shows theequivalent bubble diameter as a function of time after start of injec-tion for different gas velocities (10, 15, 20 and 30 m/s) for 2.0 mmparticles. All growth curves are qualitatively similar, with fasterexpansion occurring at higher injection rates. Similar conclusionscan be drawn for other particle sizes (not shown). The similarityis only qualitative because, as explained earlier in Section 4.2, at dif-ferent flow rates particles concentrate differently near the bubbleboundary, resulting in different leakage fractions. The same behav-ior was reported in the literature [15–17] for 2D fluidized bed.

4.4. Influence of particle properties

Next we study the influence of particle properties, meaning dif-ferent particle diameters and particle densities. The particle sizewas varied using 1.5, 2.0 and 2.5 mm diameter, resulting in differ-ent minimum fluidization velocities. The particle density of the2.0 mm particles was varied to match the Umf values of 1.5 and2.5 mm particles, as shown in detail in Table 2. We find a lowerequivalent bubble diameter for larger particle diameters, as shownin Fig. 13. This is caused by the fact that coarser particles allowmore leakage, as shown in Fig. 14. Close agreement could be seenfor particles of different diameter but the same Umf. This observa-tion shows that the minimum fluidization velocity, which is di-rectly related to the particle properties, is an importantparameter in the bubble formation process.

This is also evident from experimental observations of Nieuw-land et al. [15], when they compared particle sizes in the rangefrom 140 lm to 750 lm. Bubble detachment time was found tobe independent of particle diameter or particle density. It is animportant observation in 3D that the leakage fraction keepsdecreasing with time, however in 2D leakage reaches a minimumafter a certain time, which is due to more exchange area is avail-able in 3D. The bubble detachment time was defined as the timebubble stayed at the orifice observed from visualization of TFMdata. From the comparison of the computed detachment times in2D and 3D it was found that 3D bubble detach earlier then 2D bub-ble at a given particle diameter and injection velocity. In 3D bubbledetachment prevails approximately after 0.1 s whereas for 2D it

Fig. 18. Snapshot of the gas porosity in the cross-sectional plane in the center of the bubble predicted from the TFM at t = 0.095 s, showing the porosity gradient near thebubble surface and/or the wall for (a) a semi-circular bed, and (b) a full 3D bed.

Table 3DPM simulation settings.

Length/Width/Height (x/y/z) 0.1506/0.1506/0.5316 mNumber of grid cells in x/y/z direction 34/34/120Orifice area 0.008859 � 0.008859 m2

Particle diameter 0.0015 mNumber of particles 1,350,000Normal spring stiffness 7000 N/mTangential spring stiffness 2248 N/mNormal restitution coefficient 0.97Tangential restitution coefficient 0.33Friction coefficient 0.1Drag correlation Ergun [31], and Wen and Yu [32]Flow solver time step 2.5 � 10�5 s

Boundary conditionsWall No slip for gasOutlet Prescribed atmospheric pressureInlet Gas influx through the central jet

Fig. 19. 3D contours of a bubble at t = 0.75 s, obtained from DPM simulation forglass particles of diameter 1.5 mm and gas injection velocity of 20 m/s.

Fig. 20. Comparison between TFM and DPM simulations for bubble formation in a3D bed for glass particles of diameter 1.5 mm and injection gas velocity of 20 m/s.

V. Verma et al. / Chemical Engineering Journal 245 (2014) 217–227 225

takes approximately 0.23 s [2,15–17]. Stability of the bubble neckat the orifice during formation is less in 3D due to bubble expan-sion in all three directions. So the bubble tends to detach earlierwhen compared with 2D where expansion is majorly governed inthe direction of injection.

4.5. Effect of bed size

The effect of bed size on bubble properties was investigated aswell. We either widened the bed, or increased its height. Widening

the bed was achieved by increasing the number of cells in theradial direction from 60 to 120, while keeping the cell size andbed height constant, leading to a bed aspect ratio of 0.5. Whenincreasing the height of the bed, the number of cells in the heightdirection was increased proportionally, while keeping the cell sizeand bed width constant, leading to a bed aspect ratio of 2.0. Thesebed conditions are used to check whether the bubble expansion isinfluenced by the distance to the freeboard region or confiningwalls. Fig. 15 shows that a slightly higher bubble diameter is pre-dicted for both wider and higher particle beds. The former can beexplained because the emulsion phase displaced by the bubbleexperiences less hindrance from the wall. The latter is more diffi-cult to explain. During bubble formation the emulsion is displacedduring a very short time, so the net effect of the emulsion phasedoes not show any weight effects on the bubble however slightlyhigher expansion in the direction of injection.

4.6. Experimental comparison

In this section, we aim to validate our TFM simulations with 3Dexperimental results reported in the literature. In this work wehave focused on relatively large (Geldart D type) particles. Unfortu-nately, as far as we are aware, there is no reported experimentalwork quantifying bubble injection in a 3D bed for such particles.Therefore, in this section, we will focus on simulations of smaller(Geldart B type) particles for which experimental data is available.

226 V. Verma et al. / Chemical Engineering Journal 245 (2014) 217–227

Nieuwland et al. [15] compared their experimental work on semi-circular fluidized beds with 2D-axisymmetric TFM simulations.Fig. 16 shows a comparison of bubble diameter growth obtainedin these experiments (squares), our full 3D TFM results (in a cylin-drical fluidized bed, black solid line), and the two theoretical mod-els investigated before (dot–dashed lines). The TFM result isconsistently above the experimental results of semi-circular fluid-ized beds. Nieuwland et al. [15] attributed this to the presence of afront wall, which restricts the movement of the emulsion phaseand thereby induces penetration of the cavity in a direction awayfrom the flat front wall. So to make a better comparison we simu-lated a semicircular geometry with appropriate wall boundaryconditions applied for the front wall. The results (dotted line) arevery similar to the results obtained for the full 3D cylinder. We alsochecked a simulation using 2D axisymmetric coordinates (thusenforcing cylindrical symmetry). This simulation (circular dots)produced, as expected, no significant change in bubble diameter.Similar observation carry for the bubble shape factor as shown inFig. 17, where TFM (solid line) predicts slightly lower shape factorcompared to experiments (squares). So we suspect the disparitybetween TFM and experimental results is related to the lack ofhomogeneity of the gas flow entering the bed in the experiments.To promote a homogeneous gas flow distribution over the distrib-utor, Nieuwland et al. [15] used a 25 cm high chamber filled with3 mm glass beads, through which gas was allowed to flow. Thechamber was fed by three different inlets at the bottom and asquare wire mess was used as a distributor for the injected gas.Since the bed diameter is relatively large (30 cm), we suspect theseconditions may have led to a slightly non-uniform gas velocity.This may lead to disparities with the TFM results, which assumesa strictly uniform gas inlet.

We also explored the effect of the front wall on the bubbleshape in a horizontal cross-section of the semi-circular bed. Wefound that the bubble surface does not make a right angle withthe front wall. This is clear from Fig. 18(a) and (b) for a semi-circu-lar and a full 3D bed respectively. In a full 3D bed (Fig. 18b), thebubble shape is circular in the cross-sectional plane, with a uni-form porosity gradient near the bubble boundary. However, in asemi-circular bed (Fig. 18a) a notably different porosity gradientis observed at the bubble boundary near the front wall in ourTFM simulations. This effect is dominant in the radial directionand expected to be more pronounced in the experiments. Becauseimages from the front wall were used by Nieuwland et al. [15] tocharacterize the rising bubble, their bubble size measurementsmay be influenced by this wall effect. Therefore this may be an-other source of error introduced in the experimental measure-ments, leading to a disparity between the results.

4.7. Comparison with DPM simulation

For the experimental validation we are currently limited to rel-atively small (Geldart B type) particles. Another way to validate theaccuracy of the TFM model, also for the larger (Geldart D type) par-ticles of this study, is to make a comparison with more detailedDiscrete Particle Model (DPM) simulations. We carried out a soft-sphere DPM simulation using in-house code (Patil et al. [30]) ona 3D Cartesian grid for 1.5 mm glass particles and injection gasvelocity of 20 m/s at exactly the same conditions as used for ourTFM simulations. Other details of DPM simulation settings aregiven in Table 3; the gas and solid properties are the same as inTable 2. Fig. 19 shows the instantaneous 3D bubble contours ob-tained from the DPM simulation. Fig. 20 shows the equivalent bub-ble diameter as a function of time, comparing TFM and DPMresults. The excellent agreement shows that the TFM model canaccurately predict bubble growth, which suggests that the dispar-

ities with experimental results in Section 4.6 are due to experi-mental limitations.

5. Conclusion

A two fluid model has been used to study bubble formationfrom a single central orifice in a gas–solid fluidized bed in three-dimensional cylindrical coordinates. We have studied the influenceof particle properties (of Geldard D type particles) and gas injectionvelocities and compared our results with approximates bubble for-mation models. The Caram and Hsu model prediction for the bub-ble growth is in close agreement with our TFM simulations athigher gas velocities. This model predicts the bubble diameterassuming that part of the gas leaks through the spherical surfaceof the bubble as a consequence of a pressure difference across aporous medium. However, the Caram and Hsu model does not takeinto account explicitly the bubble shape, which depends on the in-jected gas flow rate, and the compressibility of the emulsion phase,both of which are important characteristics of a fluidized bed.Extensive data in the experimental and simulation literature nowpromote development of new theoretical models that include amore complete description of the phenomena observed during for-mation of a bubble at a single orifice. The bubble diameter is highlydependent on the emulsion phase velocity, irrespective of the par-ticle properties. The bubble shape is also dependent on the injectedflow rate, with a spherical shape bubble at lower gas flow rates andan elliptical shape at higher flow rates. Furthermore, a relativelyhigher leakage fraction was predicted at lower injection velocitiesand for coarser particles. The particle bed size does not show a sig-nificant effect on the bubble formation. The detachment time pre-dicted from TFM is independent of particle properties and inflowgas properties.

A comparison with DPM simulation and experimental resultsshowed that TFM predicts the bubble dimensions to a good extent,but also points out some important limitations of physical experi-ments. The 3D TFM simulations provide a better insight into thebubble formation process in three dimensional space, and facilitatethe study of life-size fluidized beds at low computational costswith refined grids and higher order numerical schemes.

Acknowledgment

The authors would like to thank the European Research Councilfor its financial support, under its Advanced Investigator Grantscheme, contract number 247298 (MultiscaleFlows).

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