Experimental Validation of Eulerian− Eulerian Simulations of Rutile

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Experimental Validation of Eulerian-Eulerian Simulations of Rutile Industrial Powders Olumuyiwa Owoyemi and Paola Lettieri* Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, U.K.

Roger Place Place Associates, 41 West End, Sedgefield, Stockton on Tees TS21 2BW, U.K.

This work seeks to validate two-dimensional CFD simulations of gas- solid-fluidized beds of two Geldart Group B industrial powders, natural and synthetic rutile used in the titaniumrefining industry, using a commercial code CFX-4.4. Experimental data on bed height, bubble diameter, and bed voidage are compared with those from simulations. Both qualitative and quantitative results are presented in this paper. Results obtained on bubble diameters from simulations are also compared with predictions using the semiempirical correlation by Darton et al.1 1. Introduction The use of gas-solid-fluidized beds spans across many industries ranging from the pharmaceutical, refining, and petroleum industries to the power generation industry. An example of an industry where the technology has found extensive use is the titanium-refining industry, where a fluidized-bed reactor is used for extracting titanium from naturally occurring ore. This widespread use of fluid-bed technology is due to the very good mixing that occurs within the bed, which in turn enhances solids and fluid contacting and improves heat and mass transfer.2 The early development of fluidizedbed technology was characterized by use of empirical correlations to understand observed physical phenomena; however, the last 40 years has seen a shift in paradigm toward gaining a fundamental understanding of the mechanisms governing the highly complex solid fluid flow phenomena. Scientists and engineers are increasingly employing computational fluid dynamics (CFD) modeling as a tool to study multiphase systems, including fluidization. CFD models that describe gas-solid flow systems can be formulated at different levels of mathematical detail. Simulating the individual solid particles, while the gas phase is simulated on a length scale larger than that of the particle phase, is referred to as the LagrangianEulerian approach; here, the Newtonian equation of motion for each particle is solved separately, taking into account direct collisions between particles when it is appropriate. This modeling approach for gas-fluidized beds was pioneered by Tsuji et al.,3 who developed a soft-sphere model of a gas-fluidized bed based on earlier work done by Cundall and Strack.4 In the soft-sphere approach, particles are allowed to overlap slightly, allowing contact forces to be calculated from the deformation history of the contact using a linear spring/dashpot model. Hoomans et al.5 presented a hard-sphere approach in their two-dimensional model for a gasfluidized bed. This approach assumes that particles interact via binary, quasi-instantaneous collisions where * To whom correspondence should be addressed. E-mail: [email protected].

contact occurs at a point. Some authors have also presented hybrid simulation techniques, which feature the former and latter approaches for modeling discrete particle systems, to study the microscopic properties of fluidized beds; some of these include studies by Kafui et al.6 and Xu and Yu.7 The aforementioned authors are only but a few of the many authors who have presented work using this simulation strategy. Simulating both phases as continuous and fully interpenetrating, a concept based on the two-fluid model (TFM), is termed as the Eulerian-Eulerian approach. Many researchers have successfully carried out bubbling-bed fluidization simulations of Group B particles using the Eulerian-Eulerian approach. Ding and Gidaspow,8 compared simulations of the formation, growth, and bursting of bubbles in a jet-bubbling fluidized bed qualitatively with experiments. Peirano et al.9 investigated the effect of the spatial computational domain on the simulated fluidized-bed dynamics. Van Wachem et al.10 investigated the bubble behavior in gas-fluidized beds at different superficial gas velocities and the resultant behavior of the bubbles in different column diameters; they validated their work both qualitatively and quantitatively by using semiempirical correlations. Boemer et al.11 computed the distribution of timeaveraged voidage and bubble formation at a jet within a bubbling bed. Gelderbloom et al.12 applied a multiphase CFD model to simulate the bubbling and collapsing behavior of Geldart Group A, Group B, and Group C powders. Computed bubble sizes were validated by comparison with the Davidson and Harrison13 bubblesize relation. Lettieri et al.14 used the EulerianEulerian approach to simulate the transition from bubbling to slugging fluidization of a Geldart Group B powder and used semiempirical correlations to validate the prediction of the transition velocity and maximum slugging bed height obtained from the simulation. The previously mentioned authors have used the two-fluidmodel approach that utilizes the kinetic theory of granular flow15 description for the solid phase as derived from the kinetic theory of gases.16 This approach is also adopted in this work.

10.1021/ie050784v CCC: $30.25 © 2005 American Chemical Society Published on Web 11/02/2005

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There are, however, alternative Eulerian-Eulerian models based on the TFM that have been developed for the modeling and simulation of fluidized beds. Lettieri et al.17 have used the particle-bed model (PBM) developed by Foscolo and Gibilaro18 to simulate in 2D the behavior of bubbling and slugging Group B gas-fluidized beds and the homogeneous expansion of a Geldart Group A powder. The particle-bed model, which is semitheoretical in origin, describes the particle-particle interactions based on the hydrodynamic forces involved in the gas-solid flow (gravity, drag, and buoyancy). In this model, direct collisions between particles are not a prerequisite for the particles to exchange momentum. The continuum solid phase is regarded as an elastic fluid, which is capable of opposing imposed deformations. A particle-phase elasticity is derived and an additional force, the so-called particle-phase elasticity force, dependent on the solids volume-fraction gradient, is introduced in the solid-phase momentum balance equation. Cammarata et al.19 investigated the consistency between two- and three-dimensional simulations of a Geldart Group B material using the granular kinetic and the particle-bed model. This study showed good qualitative and quantitative agreement between the models. Simulation results on bed voidage and bubble diameter were reported, and the latter were compared with empirical correlations. However, the same success story cannot be told for simulations involving a particle-size distribution. Particle-size distributions (PSD) are known to play an important role in the behavior of dense particlephase flows. In gas-fluidized beds, a wide size distribution results in a decrease in the minimum fluidization velocity as well as an increase in bed expansion.20 There are two modeling approaches used within the EulerianEulerian framework for simulating large-scale, dense particle-phase flow processes, which have a PSD. The first approach is characterized by the use of separate momentum equations to define each particle species. The difficulty, however, with this approach lies in an appropriate description for the exchange of momentum between particle species as well as a description for the particle-phase stress and granular energy balance associated with each species, which require closure for both the fluctuating particle velocities and the interaction between these fluctuating velocities.21 Currently applied closures based on the kinetic theory and which allow for a difference in granular energy between unlike particles assume a Maxwellian particle velocity distribution. In the second approach, averaged mixture properties from the individual particles are defined, and a mixture momentum and granular energy balance are employed. Each of these modeling approaches has been applied separately to simulate particle flow patterns within fluidized beds. The first approach based on separate momentum equations has been employed by Cooper and Coronella,22 and Gera et al.23 Both investigations report good model predictions with a limited set of experimental measurements. However, the above investigations assumed a Maxwellian particle-velocity distribution in allowing for a difference in granular energy. This assumption has been found to be highly inaccurate and results in an underprediction of the particle-phase stress in a binary mixture.24 This impasse has, however, been broken by Iddir and Arastoopour,25 who have extended the kinetic theory to multitype particles by assuming a

Figure 1. Natural rutile, particle-size distribution, dp ) 186 µm.

non-Maxiwellian particle-velocity distribution. The second approach based on the mixture momentum balance has been employed by van Wachem et al.26 to predict the flow of a binary mixture in a fluidized bed. Results from this work showed good predictions with regards to key qualitative flow features associated with a widening PSD. The relatively small amount of research conducted on simulations involving PSDs using the two mentioned approaches does not yet allow accurate predictions of the flow behavior of a particle mixture. In this study, an experimental validation of the simulation of a two-fluid CFD model for two Geldart Group B industrial materials, natural and synthetic rutile particles of diameter 186 µm and 156 µm and particle density 4200 kg/m3 and 3200 kg/m3, respectively, is performed. These powders are used in the titanium-refining industry. The initial step of the refining process is carried out in a fluidized-bed reactor, in which mixtures of natural and synthetic titaniumbearing ores are chlorinated in the presence of coke according to the following general reaction:

TiO2 + C + 2Cl2 f TiCl4 + CO + CO2 This work forms part of an initial study looking at the monocomponent behavior of the individual powders with the aim to subsequently model the binary mixture. The simulations are performed at different superficial gas velocities. Simulation results obtained for bed expansion, bubble size, and voidage profiles are compared with dedicated experimental results obtained using a 2D fluid-bed experimental rig, which is shown in Figure 1. Empirical bubble-size values obtained using the Darton et al.1 correlation are also compared with the simulated values and experimental bubble-size measurements. 2. Gas-Solid CFD Model According to the kinetic theory, mass and momentum equations for each phase, solid and gaseous, are solved. The momentum balance for the gas phase is given by the Navier-Stokes equations, modified to include an interphase momentum transfer term. The solid-phase momentum equation is also modified to include an

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Bagnold28 and modified by Ding and Gidaspow8 is used in this work. From prior knowledge of the granular temperature, the coefficient of restitution, the radial distribution function, the solid-phase pressure, and bulk and shear viscosities can be derived. The solid-phase pressure represents the solid-normal forces due to particleparticle interactions. Jenkins and Savage29 developed its description based on the kinetic theory of granular flow. In this approach, both the kinetic and the collisional influence are taken into account. The kinetic portion describes the influence of particle translations, while the collisional term accounts for the momentum transfer by direct collisions.15 The expression given by Lun et al.30 is used in this work. The solid stress tensor is modeled by considering the solid phase as a Newtonian fluid. The stress tensor is made up of two parts, the solids pressure, which is defined above, and the solid viscous shear stress tensor. The constitutive relation for the viscous shear stress tensor is modeled by the bulk viscosity, which describes the resistance of the particle suspension against compression and the shear viscosity, µs. Table 2 below summarizes the equations that describe the solid-phase stress tensor and the quantities required for the calculation of the solid-phase pressure and viscosities used in this work. The transfer of momentum between the two phases is accounted for via the interphase transfer coefficient, β, which can be computed from knowledge of the drag coefficient, CD, the Reynolds number, and the solids volume fraction, s. In this work, the interphase drag function of Di Felice31 is used. This drag function is based upon the drag coefficient by DallaValle.32

Table 1. Momentum Equation for Gas and Solid Phases

Gas phase:

Solid phase:

∂ j )+∇ h (gFgu j gu j g) ) ( F u ∂t g g g j ∇ h (gτg) - g∇ h P + gFggj - β(u jg - u j s) ∂ j )+∇ h (sFsu j su j s) ) ( F u ∂t s s s ∇ h ‚τjs - ∇ h P s - s ∇ h P + sFsgj + β(u jg - u j s)

interphase momentum transfer term, a solids pressure term, and an effective solid viscous stress term. We report in Table 1 the momentum equations used in this work. The closure of the solid-phase momentum equation above requires a description for the solid-phase stress tensor; this description is provided by the kinetic theory.15 Here, the effective stresses are said to originate from particle streaming (kinetic contribution) and direct collisions (collisional contribution) analogous to the gas kinetic theory.16 The thermodynamic temperature of the gases, the granular temperature, Θs, is introduced to describe the kinetic energy associated with the particle fluctuations, νs.

3 1 Θ ) (ν 2) 2 s 2 s

(1)

Since the effective stress tensor is dependent upon the magnitude of the particle fluctuations, a balance of the granular temperature associated with this fluctuation is also required; this is given by an algebraic balance, which is appended to the continuity and momentum equations.27 The coefficient of restitution, es, is introduced in the granular kinetic theory (where 0 < es < 1) to account for the inelasticity of the solid phase. We note here that a value of 0.9 was used for the CFD simulations in this work. This value is used by many authors.10,14,17,26 The radial distribution function, go, is also introduced in the granular kinetic theory to allow for tight control of the solid volume fraction, so that the maximum packing is not exceeded. It is worth mentioning that the radial distribution function given by

jg - u j s| 3 Ffs|u (1 - s)2-γ β ) CD 4 dp

(2)

γ ) 3.7 - 0.65 e-([1.5-log(gRep)] )

(3)

with 2

-0.5 2

CD ) (0.63 + 4.8Rep

)

(4)

Table 2. Constitutive Equations for Momentum Conservation Used in the CFD Model

solid stress tensor:

Solid-Phase Stress Equations Ts ) -PshI + τjs

Constitutive Equations For Solid-Phase Stress 2 τjs ) λs - µs ∇u j s hI + µs[∇u j s + (∇u j s)T] solids viscous shear stress tensor: 3 solids pressure: Ps ) sFsΘs(1 + 2g0s(1 + es))

[(

) ]

( )

bulk viscosity:

Θs 4 λs ) sFsdpg0(1 + es) 3 π

shear viscosity:

Θs 4 µs ) sFsdpg0(1 + es) 5 π

radial distribution function:

solid phase: dissipation of granular energy:

g0 )

[ ( )]

s 3 1 - max 5 s

0.5

( )

0.5

5π0.5 F d Θ 0.5 4 96 s p s + 1 + g0s(1 + es) 5 (1 + es)sg0

[

2

1/3 -1

Granular Temperature Balance j s ) γs Ts: ∇u γs ) 3(1 - es2)g0s2FsΘ

[

4 dp

xΘπ - ∇uj ] s

]

2

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Figure 3. Natural rutile, SEM at magnification X ) 250.

Figure 2. Synthetic rutile, particle-size distribution, dp ) 156 µm.

where Rep is the particle reynolds number given by

Rep )

dpufFf µf Figure 4. Synthetic rutile, SEM at magnification X ) 250.

and dp is the particle diameter. It should be noted here that the particles are assumed to be monodispersed. This assumption was necessary because the closure relationship adopted in this work, kinetic theory, is valid only for monodispersed particles.

Table 3. Particle Physical Properties material

dp (µm)

Fp (kg m-3)

σ/d50

Geldart Group

natural rutile synthetic rutile

186 156

4200 3200

20 37

B B

3. Experimental Section 3.1. Powder Properties. Analytical samples of the material investigated were obtained by riffling of large batches. The mean particle diameter and particle-size distribution were determined statistically by sieve analysis. The sieve test was repeated three times, and average values were taken. The relative diameter spread was used to compare the width of the size distribution of the two powders. The natural rutile material exhibits a fairly wide distribution, according to Geldart,33 while the synthetic rutile exhibits a wide size distribution based on the above classification. The particle-size distributions (PSD) of both powders are reported in Figures 1 and 2. Using a scanning electron microscope (SEM), the surface morphologies of the materials under investigation were observed and photographs were taken. SEM photographs of natural and synthetic rutile are reported in Figures 3 and 4, respectively. The natural rutile particle is more irregularly shaped, while the synthetic rutile, Figure 4, is more spherical. The physical properties of both powders are summarized in Table 3. 3.2. Experimental Setup. The experimental setup consisted of a two-dimensional plexiglass rectangular column, 600 mm high, 350 mm wide, and 10 mm thick. The distributor was a uniformly permeable sintered bronze rectangular plate with a thickness of 3.5 mm. The bed was filled to an initial height of 300 mm, for all experiments, corresponding to 1.89 and 1.92 kg of natural and synthetic rutile particles, respectively. The fluidizing gas, air, was supplied via rotameters. A

photograph of the setup is given in Figure 5. A mercury manometer was used to measure the pressure drop as a function of the superficial gas velocity to determine the experimental minimum fluidization velocity, umf. For both powders, the minimum fluidization velocity was also calculated using correlations given by Baeyens and Geldart34 and by Wen and Yu35 and compared with experiments; results are reported in Table 4. An agreement of 10% and 30% was found for the synthetic and natural rutile materials, respectively, when experimen-

Figure 5. 2D experimental fluid-bed vessel: 350 mm wide, 600 mm height, and 10 mm depth.

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Table 4. Comparison of Experimental Umf Values with Empirical Correlations material

exp. (m/s)

natural rutile 0.062 synthetic rutile 0.023

Baeyens and Geldart.34 Wen and Yu35 (m/s) (m/s) 0.045 0.026

0.046 0.025

tal measurements and empirical calculations were compared. Typical pressure drop profiles versus gas velocity are shown for both powders in Figures 6 and 7. A record of the bed height at increasing superficial gas velocity enabled the calculation of the bed voidage, which is used also for direct comparison to the CFD simulations. Digital video recordings of the fluid bed were made to analyze the development of bubble dynamics in the fluid bed and to determine the bubble size at different fluidizing velocities. Images were captured by means of a web camera at 14 frames/s; 80 s were recorded and were then subsequently analyzed using Optimas 6.0, image analysis software. Experiments and corresponding simulations were performed at three different fluidizing velocities to investigate fluidization of the two materials at (Case A) the same superficial fluidizing velocity, (Case B) the same excess

Table 5. Simulation Strategies Used in This Work Case A (m/s) same Uo

Case B (m/s) same Uo - Umf

Case C (m/s) same Umf

natural rutile 0.20 ) 3.2Umf synthetic rutile 0.20 ) 8.7Umf

0.20 ) 3.2Umf 0.16 ) 7Umf

0.20 ) 3.2Umf 0.074 ) 3.2Umf

material

Table 6. Computational Parameters Used in the CFD Simulations description

symbol

gas density gas viscosity restitution coefficient maximum solids packing bed height settled bed height grid cell size time step

Fg (kg/m ) µg (Pa s) es s,max Hbed (m) HS ∆x and ∆y (m) ∆t (s) 3

value

comments

1.2 1.85 × 10-5 0.90 0.60 0.60 0.30 0.005 square cells 10-4 time step

gas velocity, and (Case C) the same multiple of Umf. The inlet velocity at Case A was set equal to 0.20 m/s for both powderssinlet velocity for Case B was set at 0.20 m/s for the natural rutile, giving an excess gas velocity equal to 0.14 m/s and a corresponding fluidizing velocity of 0.16 m/s for the synthetic rutile. The inlet velocity at Case C was set equal to 3Umf, which corresponded to values for Uo ) 0.20 m/s for the natural rutile and 0.07 m/s for the synthetic rutile; operating conditions are summarized in Table 5. 4. Simulations

Figure 6. Measured pressure drop versus superficial gas velocity for synthetic rutile.

A rectangular geometry of dimensions equal to those of the experimental rig is used to set up the computational domain, with gas entering at a uniform velocity at the distributor plate. In this work, all simulations are carried out in a 2D computational grid in which front and back wall effects are neglected; the 2D grid was chosen based on earlier work done by Lettieri et al.36 The left and right walls of the domain were modeled using the no-slip velocity boundary condition for both phases. Dirichlet boundary conditions are employed at the bottom of the bed to specify a uniform gas inlet velocity. The boundary condition at the top of the bed is a so-called pressure boundary, which is set to a reference value of 1.015 × 105 Pa. The distributor is made impenetrable for the solid phase. The fluidization conditions used in this simulation are summarized in Table 6. All simulations were performed for 5 s of real time. The simulations used a Dell Xeon P4 3.1 Ghz machine. 5. Results and Discussion

Figure 7. Measured pressure drop versus superficial gas velocity for natural rutile.

5.1. Validation of CFD Simulations Using Experimental Data. 5.1.a. Voidage Profile. A major objective of this work is to validate results obtained from CFD simulations with experimental results. In this section, snapshots showing the fluidization of the natural and synthetic rutile are compared with voidage contour maps obtained from CFD simulations. Figures 8 and 9 show the voidage profiles for both powders when fluidized at the same gas velocity, Uo ) 0.2 m/s (Case A). The bubbles that form in the natural rutile are smaller in comparison to the synthetic rutile, as can be observed in the snapshots reported in Figure 8. This material is characterized by a higher Umf than the synthetic rutile and, thus, a smaller excess gas velocity at the same superficial gas velocity. The experimental and simulated snapshots of the synthetic rutile, Figure

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Figure 8. Comparison between (a) experimental and (b) simulated voidage profile for natural rutile material at Uo ) 0.20 m/s.

Figure 9. Comparison between (a) experimental and (b) simulated voidage profile for synthetic rutile material at Uo ) 0.20 m/s (Case A). Table 7. Comparison of Time-Averaged Macroscopic Fluidization Indicators for Both Powders powder natural rutile synthetic rutile

Case A Case B Case C

bed height (cm)

bed voidage

bubble holdup

37.6 39.2 37.3 32

0.521 0.540 0.516 0.438

0.053 0.048 0.035 0.007

9, show a more vigorous bubbling regime when compared to the natural rutile ones. This behavior is expected because of the lower minimum fluidization velocity of the synthetic rutile material. From the snapshots, it can also be seen that bubbles start to coalesce near the bottom of the bed and increase in size as the process of coalescence develops higher up in the bed. Table 7 shows the time-averaged macroscopic fluidization indicators. Quantities have been averaged after the first 2 s of the simulations to reduce the effect of large perturbations associated with the startup of the bed. The bubble holdup, shown in Table 7, is a measure of the fraction of gas that passes through the bed as bubbles. This value is higher for the natural rutile material, with the physical manifestation being the presence of more bubbles in the fluid bed, as shown in Figure 8. Experimentally observed bed expansion for

both powders has also been captured very well by the simulation. Figure 10 shows the voidage contour map for the synthetic rutile where 3Umf (Case C) has been used as the superficial velocity. It is clear from the snapshot of the bed that, in this case, the bubbling phase is much less prominent, unlike the natural rutile snapshot shown in Figure 8, where a superficial velocity of 3Umf resulted in a more-developed bubbling bed. This phenomenon is due to the different excess gas velocities available in the two-fluid bed systems at the velocities investigated, in light of the different physical characteristics of the two materials and the flow rate needed to achieve fluidization. 5.1.b. Bed Expansion. Figure 11 shows the fluctuation of bed height with time obtained from the simulations for both powders for Case A. The first 2 s of the simulation are ignored on account of the unsteady-state behavior, which characterizes the initialization of the fluid bed. The figure shows a higher bed expansion for the synthetic rutile powder, in agreement with the experimental results for bed height shown in Table 8. Figure 12, obtained by averaging the values for the bed height at different superficial velocities, shows a good match between experimental and simulated averaged bed expansion, with the simulated values being between

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Figure 10. Comparison between (a) experimental and (b) simulated voidage profile for synthetic rutile material at Uo ) 0.07 m/s (Case C).

Figure 11. Natural and synthetic rutile average bed height fluctuation with time at Uo ) 0.2 m/s. Table 8. Experimental Results material natural rutile synthetic rutile

Uo (m/s)

Hmf (cm)

Bed height (cm) at Uo

voidage at Uo

mb

0.20 0.20 0.16 0.07

30 33 33 33

36 42 40 34

0.66 0.59 0.57 0.50

0.58 0.48 0.48 0.48

2.5 and 11% lower than the experimentally measured bed height at the velocities investigated. A possible explanation might be that the actual powder has a wide particle-size distribution (as shown in Figure 3), while the simulation was performed assuming a system of monosized particles. Grace and Sun20 reported that the particle-size distribution can have a major influence on the bed expansion. Experimental and simulated results are reported in Tables 7 and 8, respectively. Case B in Table 7 shows results obtained using the excess gas velocity as a comparative criterion of fluidization for the two powders. The indicators show com-

Figure 12. Synthetic rutile: Comparison between experimental and simulated averaged bed expansion.

parable averaged values for bed height and bed voidage at the same excess gas velocity, however, with the natural rutile material having a higher bubble holdup; nevertheless, no immediate conclusion can be drawn regarding similar behavior between the two powders when the same excess gas velocity is applied as a limiting condition. 5.1.c. Bubble Properties. For Case A, where the same fluidizing gas velocity is used (Uo ) 0.2 m/s), the analysis of bubble-diameter data has been conducted by comparing simulated values with results obtained from empirical bubble growth correlations and experimental data analysis. In this work, the semiempirical model for bubble growth proposed by Darton et al.1 is used. This equation gives the bubble diameter as a function of the bed height as follows,

db ) 0.54(u - umf)0.4(h + 4xA0)0.8/g0.2

(5)

where db is the bubble diameter, h is the height above

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Figure 13. Comparison of experimental and simulated bubble diameters with Darton’s correlation for synthetic rutile material at Uo ) 0.2 m/s.

Figure 14. Comparison of experimental and simulated bubble diameters with Darton’s correlation for natural rutile material at Uo ) 0.2 m/s.

the distributor, and A0 is the “catchment area”, which characterizes the gas distributor. The constant 0.54 has been obtained experimentally. In the absence of available data on the distributor characteristics, Darton et al.1 suggested a value of A0 ) 0. This fairly corresponds to the inlet boundary conditions set up during this work, where no distributor has been simulated. In defining a bubble, an appropriate voidage has to be selected as the boundary between the emulsion and the gas phase. In this work, a voidage contour of 0.80 has been assumed for both experimental and simulation measurements. This arbitrary number is in line with values used in the literature.10,27,37,38 The analysis of experimental bubble diameter was carried out using Optimas 6.0, image analysis software. A total of 60 bubble diameters were collated from the experimental analysis. Bubble diameters obtained from the simulations were acquired assuming that the void areas represented the area of an equivalent circle; from this, the equivalent bubble diameter was calculated. Figures 13 and 14 show a comparison between the CFD simulations, experimentally obtained bubblediameter measurements, and predictions from Darton’s correlation for the synthetic rutile and natural rutile powders, respectively. The results show, in agreement with experiments and predictions from Darton, that the simulated bubble size increases as the bubbles travel up through the bed. The simulated values are, however, always below the experimental ones for both powders, with the scatter between experimental and simulated bubble size being greater at higher heights in the bed, in particular for the case of the natural rutile. On the other hand, the Darton’s correlation always overpredicts the bubble size for both materials. In previous work, Cammarata et al.19 had shown similar results when comparing bubble size obtained from 2D and 3D simulations of a Group B material with predictions using the Darton equation. In this case, results from 2D simulations were always lower than that predicted by Darton. That paper had also demonstrated that results from 3D simulations were always higher than those obtained from 2D runss

this may be seen as being in agreement with the results reported in this paper, where the “3D experimental bubbles” are always greater than the 2D simulated ones. 6. Conclusions CFD simulations of two industrial Group B materials have been performed using a commercial code, CFX 4.4. Simulations were carried out at three different superficial velocities. For all velocities investigated, a good qualitative match was found between the simulations and experiments. Simulated values for bed expansion were found to be in good agreement with experimentally measured values (within 11%) at the velocities investigated. Experimental results as well as an empirical bubble correlation were used to validate predicted simulation results for bubble size. Similar trends were observed for both powders; however, the Darton et al.1 empirical correlation was found to overpredict the bubble diameters when compared to experimental and simulated values; the simulated bubbles were found to be slightly smaller than those observed experimentally. Acknowledgment The authors wish to acknowledge financial support from the Engineering Physical Science Research Council (EPSRC). The authors wish to acknowledge financial support also from Tioxide Europe, Ltd.; in particular, many thanks are due to Mr. Mike Westwood and Dr. Stephen Sutcliffe. Nomenclature A0 ) catchment area, m2 C ) coefficient d ) diameter, m e ) coefficient of restitution E ) correction factor for interphase drag net force vector per unit volume, kg/(m2 s2) g ) acceleration of gravity, m/s2 go ) radial distribution function h ) bed height, m

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P ) pressure, kg/m s2 u ) superficial gas velocity, m/s u j ) velocity vector, m/s v′ ) velocity fluctuations, m/s V ) Volume, m3 Greek Letters β ) interphase drag, kg/m3s  ) volume fraction Θ ) granular temperature, m2/s2 γ ) dissipation of granular energy, kg/(m s3) F ) density, kg/m3 µ ) viscosity, kg/(m s) τ ) shear stress tensor, kg/(m s2) ν ) particle fluctuation term Subscripts and Superscripts area ) area of the equivalent circle b ) bubble D ) drag ex ) excess g ) gas phase gx ) gas phase, horizontal component gy ) gas phase, lateral component gz ) gas phase, vertical component mf ) minimum fluidization n ) exponent p ) particle S ) settled bed height s ) solid phase sx ) solid phase, horizontal component sy ) solid phase, lateral component gz ) solid phase, vertical component vol ) volume of the equivalent sphere 2D ) two-dimensional

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Received for review July 1, 2005 Revised manuscript received September 27, 2005 Accepted September 29, 2005 IE050784V