A Review of Eulerian Simulation of Geldart A ... - ACS Publications

Ind. Eng. Chem. Res. 2009, 48, 5567–5577

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REVIEWS A Review of Eulerian Simulation of Geldart A Particles in Gas-Fluidized Beds Junwu Wang* Faculty of Science and Technology, UniVersity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Although great progress has been made in modeling the gas fluidization of Geldart B and D particles and dilute gas-solid flow by standard Eulerian approach, researchers have shown that, because of the limitation of computational resources and the formation of subgrid-scale (SGS) heterogeneous structures, Eulerian model with a suitable SGS model for constitutive law is necessary to simulate the hydrodynamics of large-scale gas-fluidized beds containing Geldart A particles. In this article, a state-of-the-art review of Eulerian modeling of Geldart A particles in gas-fluidized beds is presented. The available methods for establishing SGS models are classified into six categories, that is, empirical correlation method, scaling factor method, structure-based method, modified Syamlal and O’Brien drag correlation method, EMMS-model-based method, and correlative multiscale method. The basic ideas of those methods, as well as their advantages and disadvantages, are reviewed. Finally, directions for future research are indicated. 1. Introduction Gas-fluidized bed reactors are widely used in various industrial processes, such as fluid catalytic cracking (FCC) reactors. Correct understanding of the hydrodynamic characteristics is essential for proper design and scale-up of such reactors. However, the analysis of such a system is notoriously difficult, because of the coexistence of disparate length and time scales and the conspicuous coupling between the different scales with their transport properties and the varying structures; the competition and compromise between members of different spatiotemporal scales inevitably results in the formation of multiscale structures.1,2 In such flow, physical phenomena occur at macroscopic spatiotemporal scales, whereas the fundamental causes of these phenomena can occur at the scale of the flow elements themselves; often, this challenge cannot be met by conventional, single-scale formulations, which therefore calls for a multiscale simulation approach to respond to it. Multiscale modeling strategies,2-8,69,102 have recently been developed to meet the needs in understanding the multiscale nature of the gas-solid two-phase flows that occur in gasfluidized beds. A review about variational multiscale methods for heterogeneous gas–solid flows can be found in ref 115. In the correlative multiscale modeling strategy, the constitutive laws, mainly particulate phase stresses and interphase interaction force, requested at higher scales can be obtained from smallerscale simulations, and the phenomena at higher scales can be formulated by analyzing the mechanisms at smaller scales.2 The mathematical models describing the gas-solid physical system are usually defined at four different levels of approximation, which are needed to obtain the requested precision on a fixed set of parameters: the most fundamental level of approximation is direct numerical simulation (DNS),9-11 where all the relevant spatiotemporal scales of fluid and particle phases are resolved and they are coupled through no-slip boundary condition at the fluid-particle interfaces; at a higher level, we have discrete particle method,12-14 where averaged equations are used to track the hydrodynamics of the fluid phase, whereas the particle phase * Corresponding author. E-mail: [email protected]. Phone: +31-53-4892370. Fax: +31-53-4892882.

is still resolved as in DNS. Molecular dynamics simulation are used to track the details of particle-particle and particle-wall interactions, and the interaction between fluid and particle phases is usually realized through an empirical interphase drag force term; at an even higher level, standard Eulerian model (SEM)15-17 is used to model the gas-solid flows, where the mesoscale structures (bubbles and/or clusters) in gas-fluidized beds are explicitly resolved. The constitutive laws, kinetic theory of granular flow (KTGF), which is usually obtained from systems without interstitial fluids or empirical correlations for particulate phase stresses and empirical correlations for interphase drag force, are based on the assumption of homogeneous structure inside computational cells; finally, in order to simulate the hydrodynamics inside large-scale gas-fluidized beds, filtered Eulerian model (FEM)3,18,19 is necessary due to the limitation of computational resources, where (part of) the mesoscale structures are no more resolved explicitly and the effect of those unresolved structures on the bed hydrodynamics is accounted for through suitable subgrid-scale (SGS) models for constitutive laws. It is well-known that particle properties have a significant effect on the bed hydrodynamics. Geldart20 was the first to classify the solid particles fluidized by air at ambient temperature and pressure into four different groups (Geldart A, B, C, and D particles), which are characterized by the mean particle size and the density difference between particle and air. Since the present study is about Eulerian simulation of Geldart A particles, a brief introduction of Geldart A particles is useful; the characteristics of other types of particles can be found in the literature.20-22 Geldart A particles have a small mean particle size and/or a low particle density, with FCC catalysts (dp ) 7.5 × 10-5 m, Fp ) 1500 kg/m3) being a typical example. The most distinct characteristic of Geldart A particles is that they exhibit an interval of homogeneous expansion after minimum fluidization and prior to the commencement of bubbling or of the formation of heterogeneous structures. Computational fluid dynamics has been successfully applied to model the hydrodynamics of gas-fluidized beds,5,7,17 especially for Geldart B and D particles20 and dilute gas-solid flows. However, realistic prediction of Geldart A particles using

10.1021/ie900247t CCC: $40.75  2009 American Chemical Society Published on Web 05/13/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 12, 2009

Eulerian method remains a great challenge for both engineers and scientists; many studies have shown that SEM fails to reproduce the hydrodynamic characteristics inside gas-fluidized beds as summarized in the next section. Different investigators use different approaches to remedy it, which is realized by considering the effect of unresolved structures on the constitutive laws. Therefore, they fall in the aforementioned FEM. In this study, we attempt to give an overview of the development of the FEM when applied to the fluidization of Geldart A particles in gas-fluidized beds. 2. Summary of Reported Failures of Standard Eulerian Model Gas-solid two-phase flows are usually destabilized because of the nonlinear interphase interaction and dissipative particleparticle and particle-wall interactions;23-26 as a result, mesoscale structures appear, which take the form of bubbles and/or clusters in gas-fluidized beds. Such mesoscale structures are characterized by multiple length and time scales. For example, the cluster size in risers varies from the size of particle diameter to the riser diameter,27,28 which means the computation will be extremely time-consuming if we attempt to capture all the relevant spatiotemporal scales by numerical simulation. On the other hand, the governing equations are usually discretized using coarse computational grids, with respect to the characteristic length scale of mesoscale structures, to reduce the required computational time. However, the theories and/or empirical correlations based on homogeneous fluidization are still used to close the physical model; such treatment will result in the effect of unresolved SGS structures on the constitutive laws being overlooked, which is actually why a lot of modelers report that SEM fails to predict the hydrodynamics of gas-fluidized beds,29-33 since the formation of mesoscale heterogeneous structures has a significant effect on the constitutive laws as proven in many studies.32-35 For a detailed discussion about the mechanics of the formation of the mesoscale structures in gas-fluidized beds, please refer to the reviews.2,24 It was shown that the bed expansion of bubbling fluidized beds of Geldart A particles was significantly overestimated by SEM; the reason is that the interphase drag force is significantly overestimated by standard drag correlations.36-47 Many other studies on fluidization of Geldart A particles in turbulent fluidized beds48-50 and circulating fluidized bed (CFB) risers18,32-35,51-68 also concluded that the interphase drag force is overestimated by standard drag correlations. 3. State-of-the-Art Review A key step toward FEM for the hydrodynamics of gasfluidized beds is the establishment of suitable governing equations. Fortunately, the studies by Zhang and Vanderheyden19 and later Igci et al.34 have shown that the governing equation of FEM is exactly the same as that of SEM, but now the constitutive laws must include both the microscale and mesoscale structural effects, which means much of the current knowledge of SEM, such as numerical algorithm, can be routinely extended to FEM. Therefore, the problem remaining is how to characterize the effect of the unresolved SGS heterogeneous structures on the constitutive laws.3,69 It is generally concluded that, in numerical simulations of Geldart A particles in dense gas-fluidized beds, the gravitational term and the drag term are the dominant terms. The particulate phase stresses only have a secondary effect on the simulation results;35,47 therefore, almost all of the studies paid their attention

to the interphase interaction force term, including present study, although several efforts have been devoted to the effect of mesoscale structures on the particulate phase stresses.70-74 3.1. Empirical Correlation Method. Several empirical correlations are proposed to consider the effect of clustering structure on the interphase drag coefficient as summarized in Table 1. The correlation of O’Brien and Syamlal57 is obtained from an air-FCC system with specific solid circulation fluxes; their simulations showed that, with the consideration of particle clustering effect, the axial pressure gradient of a CFB riser can be reasonably predicted. Wang and Li61 also concluded that the clustering structure, the radial solid volume concentration profiles, and the axial sigmoid voidage profile can be reasonably reproduced. Cruz et al.75 reported that, with their mixture viscosity and interphase drag force correlations, the radial solid concentration profile and solid mass flux in three high-density CFB risers can be reasonably predicted. However, those correlations differ significantly from each other as shown in Figure 1, where a correlation from homogeneous fluidization76 is also included for the purpose of comparison. It can be seen that the correlations from Wang and Li61 and O’Brien and Syamlal57 are 2 or 3 orders of magnitude lower than that of Wen and Yu76 in the fast fluidization regime; however, Cruz et al.’s correlation75 is larger than that of Wen and Yu. Plotted in Figure 2 is the solid and mixture viscosity correlations of Cruz et al.75 obtained from high-density CFB risers; it can be seen that the solid viscosity is usually >10 Pa · s, which indicates that the solid viscosity in high-density CFB risers is 3 or 4 orders of magnitude larger than the value measured by Gidaspow and co-workers in low-density CFB risers.77,78 Obviously, further studies are necessary to clarify the underlying mechanics that result in the transfer from the low solid viscosity and low drag force in low-density CFB risers to the high solid viscosity and high drag force in high-density CFB risers. It is interesting to note that the studies of Almuttahar and Taghipour79,80 showed that the hydrodynamics of highdensity CFB risers can also be predicted with low solid viscosity and low drag force; details will be given below. From the aforementioned analysis, it can be concluded that specific empirical correlation is only suitable for specific operating conditions; it is difficult to be extended to other systems and, therefore, loses its generalization. 3.2. Scaling Factor Method. The analysis of Zhang and Vanderheyden19 showed that the formation of mesoscale bubbling or clustering structures resulted in a significant change of both interphase drag force and the added mass force. However, to my best knowledge, no one has studied the effect of the added mass force in the transient simulation of heterogeneous gas-solid flows occurring in gas-fluidized beds. Zhang and Vanderheyden’s study suggested that the effect of mesoscale structures on interphase drag force may be corrected by rescaling the drag force obtained from homogeneous fluidization with a scaling factor (1 - Cr)2 where the value of Cr is between 0 and 1. Since then, the scaling factor method has been used by several authors to simulate the hydrodynamics of bubbling fluidized beds of Geldart A particles. With an empirically constant scaling factor, the bed expansion and/or bubble diameters and rise velocities in a freely bubbling fluidized bed,36,37,44 the hydrodynamics of a FCC stripper,45 the deaeration dynamics of FCC particles,38 the gas mixing characteristics of Gledart A particles,42 and the jet dynamics in a fluidized bed of FCC particles81 can be wellpredicted. However, different authors suggest different scaling factors, for example, for an extremely similar system (dp ) 7.5 × 10-5 m, Fp ) 1500 kg/m3, Ug ) 0.2 m/s), Mckeen and


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Table 1. Empirical Correlations for Drag Coefficient Considering the Particle Cluster Effect authors

form of drag coefficient

57

O’Brien and Syamlal

(

εsεgFg 3 1 βA ) CD0 |b ug - b u s | 2 , where CD0 ) 0.63 + 4.8 4 dP Vr

) Vr Re

2

1 Vr ) (K1 - 0.06Re + √(0.06Re)2 + 0.12Re(2K2 - K1) + K12)(1 + K3) 2

where K1 ) εg4.14, K2 )

{

Pεg1.28 εg e 0.85

εgQ εg > 0.85

K3 ) K4 · Re · εs exp[-0.005(Re - 5)2 - 90(εg - 0.92)2]

Wang and Li61

K4 )

{

Re )

ug - b u s| dpFg | b µg

250

Gs ) 147 kg/(m2 · s)

1500

, P ) 0.8, Q ) 2.65

εsεgFg 3 βA ) CD0 |b ug - b u s | × (0.0232εg-12.49) 4 dP

where CD0 )

Rep )

Cruz et al.75

Gs ) 98 kg/(m2 · s)

{

24(1 + 0.15Rep0.687) Rep < 1000 Rep 0.44 Rep g 1000

εgdpFg | b ug - b u s| µg

3 εsεgFg |b ug - b u s |εg-2.65 βA ) CD 4 dP

(

where CD ) 16.9 + εs

where Remix )

2 × 10-3 - 2.5 × 106Remix - 44.4 Remix

Fg | b ug - b u s |dp µmix

[

(

where µmix ) 0.806 0.47 + εs 19.3 - 3.7 Pugsley44 suggest a scaling factor of 0.2-0.3 and Ye36 suggests a value of 0.15, which suggests that the scaling factor method is case-sensitive and, therefore, requires an extensive case study for every application to obtain a suitable scaling factor, which significantly weakens its predictability. 3.3. Structure-Based Method. Since the main difficulty in simulating gas-solid flows in large-scale gas-fluidized beds arises from characterizing the mesoscale structure and its effect

)

)

]

|b ug - b u s| - 31.2εs2 Ug

on constitutive laws, several structure-based Eulerian models are developed to respond to this. Arastoopour and Gidaspow82 found that, for obtaining correct axial distribution of solids, it is necessary to adjust the particle diameter in simulations, which is larger than the real diameter of particles and may be regarded as the diameter of the cluster; their treatment can be considered as the first indication of the requirement of the SGS model. In their model, the fluidized bed is not a fluidization of individual

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Figure 1. Comparison of the gas-solid drag coefficients developed by different authors: b ug - b us ) 1 m/s, dP ) 75 µm, Fp ) 1500 kg/m3, Ug ) 4.7 m/s, Fg ) 1.225 kg/m,3 µg ) 1.7894 × 10-5 Pa · s.

Figure 2. Solid and mixture viscosities in Cruz et al.;75 operating condition is the same as for Figure 1. Table 2. Comparison of Gas-Cluster Model with Experimental Data, Cited from Gu51 unit number number of data points average relative error

1 9 8%

2 4 3%

3 12 11%

4 2 4%

5 1 9%

6 1 1.1%

7 1 9%

8 3 8%

particles but a school of clusters; therefore, we denote it as the gas-cluster Eulerian model. Such a concept is later extended by Gu and co-worker, where the cluster size is not a constant as in Arastoopour and Gidaspow’s study82 but a function of solid volume fraction, which is obtained from an empirical correlation52 or a simple analysis.51 By comparing the predicted pressure drop along the lift zones and reaction zones of eight different FCC commercial units as well as bed-density distributions and chemical samples with experimental measurements, very impressive results can be obtained with an average error of 8%, as detailed in Table 2. The simulated results of the gascluster Eulerian model are dominated by the correlation of the cluster size used. However, the knowledge about cluster sizes in risers is far from complete; it is almost impossible to obtain a universal correlation, because so many factors will affect the cluster size, such as gas and particle properties, the bed geometry, the operating condition, and the definition of cluster.35 Actually, the correlations of cluster size available in the literature27,28,52 showed a significant difference, the details of which and further discussion of the characteristics of the cluster can be found in previous studies.35,83 The model is further extended56,67 in the sense that different physical models should be used to calculate the cluster size in cases of concurrent up-

flow of gas and cluster and gas upward flow and cluster downward flow, respectively. The reported axial pressure, radial solid concentration, and velocity profiles agreed with the experimental data from a high density and high flux riser84 and benchmark test of eighth conference on fluidization. Their later study50 showed that such a method can be used to simulate the hydrodynamics of turbulent fluidized beds of FCC particles. Karimipour et al.85 reported that, with a cluster-based model, reasonable axial profiles of cluster velocity, solid holdup, and pressure in downers can be obtained. It is well-known that the mesoscale structures inside bubbling fluidized beds takes the form of bubbles. However, several recent experiments suggested the formation of clusters in the emulsion phase of bubbling fluidized beds.86-88 On the basis of the experimental results, the gas-cluster Eulerian model has been adopted to simulate the hydrodynamics of FCC particles in bubbling fluidized beds. Zhang and Lu89 found that the pressure fluctuation in a bubbling fluidized bed of FCC catalyst can be well-predicted, when the true particle diameter, 60 µm, is replaced by an equivalent cluster diameter, 280 µm. With the same approach, Cao,90 Gao et al.,41 and Gao et al.40 reported its predictability in the modeling of a FCC regenerator operating in the turbulent fluidization regime, a V-baffled FCC stripper, and mass transfer characteristics within FCCU strippers, respectively. In a recent study, a force balance model is established to predict the cluster size in the emulsion phase. It is then incorporated into Eulerian simulation to simulate bubbling fluidization of FCC particles; again, good agreements with experimental data are reported.91 In the gas-cluster model, the challenges not only lie in the establishment of suitable models for predicting cluster size but also lie in the characterization of cluster density or solid volume fraction inside the cluster. Current studies simply set the solid volume fraction inside the cluster as a constant,56,91 use an empirical correlation,52 or even assume that the value is 1, which is obviously inadequate or unphysical (in cases of setting the value equal to 1). It should be noted that the bubbles are still explicitly resolved in the gas-cluster modeling of bubbling fluidized beds, although the SGS clustering structures in the emulsion phase are modeled by using equivalent cluster diameters. The basic assumption, which is the existence of clustering or aggregating structures in the emulsion phase of bubbling gasfluidized beds containing Geldart A particles, is questioned by researchers. Recent discrete particle simulations,92,93 in which the particle-particle and particle-wall interactions are represented in a realistic way, did not display such clustering or aggregating structures in the emulsion phase of bubbling fluidized beds (BFBs), when no or only a slight cohesiveness is present, while the bed hydrodynamics is predicted reasonably well. The studies of Wang et al.29,31 even concluded that not the existence of clustering or aggregating structures in the emulsion phase of BFBs but the lack of scale resolution is the main origin of the reported failure of the two-fluid model when applied to simulate the hydrodynamics of Geldart A particles in bubbling gas-fluidized beds. Note that, if the particles do form clustering or aggregating structures in the emulsion phase, the application of current versions of kinetic theory of granular flow should be seriously questioned in any case. An alternative to the traditional Eulerian approach, in which gas and particle phases are described as two interpenetrating continua, is to describe the bubble and emulsion phases as the two interpenetrating continua; such a bubble-emulsion Eulerian model is developed by Krishna and van Baten94 to scale up


bubbling fluidized beds of FCC particles. The key point of this model is how to estimate the emulsion phase density and viscosity and the calculation of interaction force between the bubble and emulsion phases. As a first approximation, they suppose the density and viscosity of the emulsion phase are constants that are determined from experiments. The DaviesTaylor-Collins relation is used to calculate the interphase interaction force, where an empirical scale factor and an empirical acceleration factor are implemented in order to account for the influence of the bed diameter and the effect of bubble swarm on the rise velocity of the bubbles, respectively. Moreover, an empirical correlation is used to obtain the effective average bubble diameter, which is also needed in the calculation of interphase interaction force. There exists an Eulerian-Lagrangian approach for simulating the hydrodynamics of industrial-scale gas-fluidized beds. A discrete bubble model is developed by Bokkers et al.95 and later Laverman et al.96 to simulate bubbling fluidized beds, in which the emulsion phase is treated as a continuous phase as in the bubble-emulsion Eulerian model; its properties, such as density and viscosity, are obtained from experiments, whereas the motion of bubbles are tracked individually according to Newtonian equation of motion and the interaction forces between bubble and emulsion phase are derived from empirical correlations. Furthermore, a model for coalescence and break-up of bubbles has to be specified in the discrete bubble model, which is usually obtained from empirical correlation. It was shown that the large-scale circulation flow patterns inside industrialscale bubbling fluidized beds, up to 4 m × 4 m × 8 m, can be qualitatively predicted. A similar conceptual model, the discrete cluster model, is also developed by Liu et al.97 and Zou et al.98 to simulate the hydrodynamics of circulating fluidized bed risers. 3.4. Modified Syamlal and O’Brien Drag Correlation Method. Zimmermann and Taghipour47 recently proved that, with a little modification to the drag force correlation of Syamlal and O’Brien,99 which is obtained from homogeneous fluidization, the correlation can be extended to simulate the hydrodynamics of bubbling fluidized beds of FCC particles. The correlation can be obtained from the correlation of O’Brien and Syamlal,57 seen in Table 1, by setting the value of K3 ) 0, P ) 0.8, and Q ) 2.65. The modification is based on the criterion that, through adjusting the values of P and Q in the drag correlation, the drag force calculated at minimum fluidization condition should equal the particle weight and the velocity-voidage correlation must be continuous at εg ) 0.85, which is done as follows:47,99 (1) For a given system with known gas and particle properties, the Archimedes number (Ar) is calculated, Ar )

(Fs - Fg)dp3Fgg µg2

(2) The Reynolds number under terminal settling conditions for a single particle (Rets) is given by Rets )

(

√4.82 + 2.52√4Ar/3 - 4.8 1.26

)

2

(3) The Reynolds number at minimum fluidization condition (Remf) is calculated by Remf )

UmfdpFg εmfµg

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Figure 3. Effects of experimentally measured voidage and gas velocity at minimum fluidization conditions on the calculated drag coefficient in modified Syamlal and O’Brien drag correlation method: Fp ) 1560 kg/m3, dp ) 57.4 µm, Fg ) 1.225 kg/m,3 µg ) 1.85 × 10-5 Pa · s, Ug ) 0.4 m/s, Gs ) 0 kg/(m2 s).

(4) The value of P is then calculated through P)

Remf(1 + 0.06Rets)/Rets - K1 0.06Retsεg1.28

(5) Finally, the value of Q must be modified to ensure the continuity of the velocity-voidage correlation at εg ) 0.85; accordingly, the value of Q can be calculated through Q ) 1.28 + log(P)/log(0.85) With the modified drag correlation, Syamlal and O’Brien46 show that the catalytic decomposition of ozone in a bubbling fluidized bed can be captured quantitatively; Zimmermann and Taghipour47 reported that the overall trend of the time-averaged voidage profiles at different superficial gas velocities in a bubbling fluidized bed of FCC particles can be reasonably predicted. Two recent studies showed that such an approach can even be extended to simulate the hydrodynamics of circulating fluidized bed risers; it is shown that the coexistence of dense-phase upflow in the lower regions and core-annulus flows in the upper regions in a high-density CFB riser of FCC particles can be captured. The predicted solid volume fraction and radial particle velocity profiles are also in good agreement with the experimental data.79,80 The method looks like an ideal approach to make the twofluid model work when applied to the simulation of the hydrodynamics of Geldart A particles. However, we believe further validations are necessary, since Li et al.81 reported that the method failed in their studies. A simple parametrical study shows that the method is extremely sensitive to the experimentally measured voidage and superficial gas velocity at minimum fluidization conditions as shown in Figure 3, which are difficult to be measured very accurately. With respect to the model validation, the effect of gas viscosity on the hydrodynamics of bubbling fluidized beds may be severed as the best point of penetration, since the modified Syamlal and O’Brien drag correlation is also extremely sensitive to the variation of gas viscosity, as shown in Figure 4. These are possibly the reasons why Vejahati et al.100 reported that, in their cases, the drag forces calculated using the modified Syamlal and O’Brien method are even larger than the one obtained from homogeneous fluidization, which is totally different from the conclusion made by Taghipour and co-workers.47,79,80

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Figure 4. Effects of gas viscosity on the values of P and Q (inset) and the calculated drag coefficient in the modified Syamlal and O’Brien drag correlation method: Fp ) 1560 kg/m3, dp ) 57.4 µm, Fg ) 1.225 kg/m,3 Ug ) 0.4 m/s, Gs ) 0 kg/m2 s, Umf ) 0.0027m/s, εmf ) 0.45.

3.5. EMMS-Model-Based Method. Two-phase models for gas-fluidized beds, such as Davidson and Harrison’s model101 for bubble motion in bubbling fluidized beds and energy minimization multiscale (EMMS) model102 for riser flows, are developed in parallel to the Eulerian description of gas-solid flows. The two-phase model is based on the phenomenological nature of gas-solid flows, which is two-phase in structure; although the two-phase model is insufficient to describe the transient behavior of gas-solid flow, it captures the effect of mesoscale structures on the transport properties. On the other hand, the two-fluid model or Eulerian model considers the gas-solid flows as two interpenetrating continua, the derivation of which is mathematically strict, and it is naturally suitable for capturing transient and turbulent behavior. However, as emphasized above, the filtered Eulerian method must be used because of the limited computational resources, which needs a suitable SGS model to consider the mesoscale structural effect. Therefore, the combination of the two-phase model with the two-fluid model is expected to give a more realistic description of heterogeneous gas-solid flow in gas-fluidized beds, as pointed out by Li and Kwauk.69 With this inspiration, Yang et al.32 and Xiao et al.63 independently developed a SGS model to consider the effect of particle clustering structures on the interphase drag force, both of which are based on the EMMS model established by Li and Kwauk.69 The incorporation of the EMMS model into the Eulerian model is similar with that of the scaling factor method discussed in section 3.2, but now the scaling factor is a variable and is predicted by the EMMS model. It was shown that the combined EMMS/Eulerian approach can significantly improve the calculation accuracy in the sense that it improves the agreement with experimentally radial solid concentration profiles and axial pressure profiles,58,63 as well as with experimental solid circulation flux and the prediction of Sshaped axial solid concentration distribution.32 Yang et al.64 later found that the combined EMMS/Eulerian approach can predict the choking phenomenon in a CFB riser; Jiradilok et al.54 found that the hydrodynamics of a turbulent fluidization of FCC particles in a CFB riser can be predicted: the calculated granular temperature, solid pressure, FCC viscosity, and frequency of oscillations are close to measurements reported in literatures. In further studies, it was shown that the computed gas and solid dispersion coefficient103 and mass transfer coefficient104,105

agreed with experimental data, too. It should be noted that, in the combined EMMS/Eulerian approach, the heterogeneous index or correction factor, which represents the correction of drag force from the EMMS model to the one obtained from homogeneous assemblies of particles, is dependent on the gas and solid properties as well as the solid circulation flux and superficial gas velocity. Therefore, the correction factor reported by Yang et al.32 can only be applied for their specific cases, which is, unfortunately, wrongly cited by several subsequent studies.54,103-105 However, Wang and Li60 found that some problems still remain in the model of Yang et al.:32 (i) an empirical voidage of the dense phase is employed and (ii) the difference of the inertia associated with the dilute phase and the cluster phase is neglected. Therefore, they developed an updated version of the EMMS model, which solved, to a certain extent, the foregoing problems. With the updated EMMS model as the SGS model to correct the interphase drag force, it was found that flow regime transition in a CFB riser of FCC particles,62 the hydrodynamics of a novel dual-loop FCC riser,55 the effect of exit and gas distributor on the hydrodynamics of an industrialscale maximizing iso-paraffins (MIP) reactor,106 the effect of bed height on the choking phenomenon,68 mass transfer in CFB risers,65,66 and the hydrodynamics of full-loop CFB107 can be reasonably predicted. In the numerical practices, it was found that the combined EMMS/Eulerian approach is limited to simulating the hydrodynamics of CFB risers of Gledart A particles. The root of the problem is rapidly identified and lies in the cluster diameter correlation used in the EMMS model.108,109 In order to remove this deficiency, at least to a certain extent, Wang et al.35 recently proposed a new equation, by analyzing the generalized added mass force in heterogeneous gas-solid flows, to correlate the relationship between the acceleration of dilute phase and dense phase; the resulting equation is used to replace the original equation for the cluster size in the EMMS model, and it made the cluster size an implicit parameter. They showed that the hydrodynamics of both Geldart A and B particles in CFB risers35,59 can be predicted. Although the combined EMMS/Eulerian approach significantly improves the simulation accuracy of CFB risers, further improvements are absolutely needed. Up to now, the mathematical model used is highly arguable. The interphase drag force is calculated using the EMMS model, which is based on the assumption that the particles are primarily present in the form of clusters in riser flow. This means the particles are moving in a collective way and the motion of individual particles is strongly correlated. However, KTGF is still used to close the particulate phase stress, which is based on the assumption that the particles are primarily present in the form of individual particles; furthermore, in KTGF, one assumes that each particle is moving independently and that “molecular” chaos exists at the level of an individual particle, which means the motion of particles is not correlated. To my best knowledge, no study can justify the applicability of the physical model for the constitutive law, which is based on two conflicting assumptions, that is, using KTGF based on individual particles for particulate phase stresses while postulating that the particles are clustered when evaluating the interphase drag force. Note that this inconsistency of mathematical model also exists in many other studies, for example, the one by Shuyan et al.67 With respect to the EMMS models themselves, current available EMMS models neglect the effect of surface force and the mass exchange between dilute phase and dense phase on the force-balance analysis, which will,


to a certain extent, affect the characteristics of mesoscale structures and consequentially affect the constitutive laws used in FEM. 3.6. Correlative Multiscale Method. As indicated in the Introduction, the constitutive laws requested at higher scales can be obtained from smaller-scale simulations. Such a correlative multiscale method, according to the definition of Li and Kwauk,2 is naturally suitable for studying the heterogeneous gas-solid flow in fluidized beds. However, most of the currently available studies are focused on the validation and improvement of the constitutive laws used in the standard Eulerian model, using direct numerical simulation and the discrete particle model, which have been reviewed from both numerical and physical aspects.7,8 Few of the studies paid their attention to characterizing the effect of mesoscale structures on the constitutive laws used in the filtered Eulerian model, except for some preliminary studies.110-112 The studies by Sundaresan and co-workers18,33,34 suggested that the constitutive laws needed in FEM can be extracted from SEM simulation. Agrawal et al.33 revealed that, after the formation of mesoscale structures, the effective interphase slip velocities of the simulated domains are far smaller than that of in the corresponding homogeneous state and the particulate phase stresses are dominated by mesoscale contribution. Later studies8,18,34 extracted the constitutive laws suitable for FEM from fine-grid simulation and incorporated them into FEM simulations of gas-solid flow in risers; it was shown that qualitatively different results will be obtained with and without the SGS model for constitutive laws, which qualitatively agrees with the results obtained from other methods. The correlative multiscale method is conceptually simple and straightforward; at the same time, it is the only currently available method that can consider simultaneously the effect of mesoscale structures on interphase interaction force and particulate phase stresses as well as the effect of filter size and frequency. However, if the constitutive laws used in FEM are obtained from SEM as in the studies by Sundaresan and co-workers,30,33,34,113 a critical problem has be answered before its practical applications, that is, whether SEM with extremely fine grid size and time step can quantitatively capture the hydrodynamics of gas-solid flows in gas-fluidized beds containing Geldart A particles? The answer of this problem is of fundamental importance, because the success or failure of the method developed by Sundaresan and co-workers is fully dependent on it. If the answer is yes, it appears to be a perfect method to extract the constitutive laws needed in FEM; furthermore, with the rapid development of computer technology and parallel computation, it will be possible to directly simulate the hydrodynamics of large-scale fluidized beds with SEM in the near future. However, if the answer is no, we have to resort to other methods to obtain the constitutive laws used in FEM, such as direct numerical simulation. No unambiguous conclusion can be made to date; controversy exists in the literature. Some recent studies have suggested that, as long as the grid size and time step are small enough to capture the mesoscale structures, SEM can quantitatively capture the hydrodynamics of Geldart A particles in gas-fluidized beds: the experimental variation of solid volume fraction fluctuation in a CFB riser of Geldart A particles can be quantitatively predicted;114 the bed expansion characteristics of FCC particles in dense gas-fluidized beds simulated by TFM is in quantitative agreement with the values simulated by DPM;29 the simulated effective interphase slip velocity of Geldart A particles by TFM agrees well with the value obtained from direct numerical

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simulation, and the effect of gas and particle properties on minimum bubbling velocity of Geldart A particles can be reasonably predicted.31 Those studies support the method developed by Sundaresan and co-workers. However, based on two-dimensional simulation of the hydrodynamics of a circulating fluidized bed riser, Wang et al.115 and Lu et al.116 showed that SEM with extremely fine grid and time step failed to predict the experimentally measured S-shaped axial solid volume fraction profiles and solid circulation fluxes. They argued that the reported failure of SEM is due to the fact that the scale separation between “molecular” granular movement and continuum description of SEM, which is necessary when volume or time average is used to derive the basic equations of SEM, does not exist. Interestingly, no assumption of scale separation is needed when ensemble average is used, while the final form of governing equations is exactly the same as the one obtained from volume or time average. Obviously, further studies are needed to clarify this problem. 4. Conclusions The filtered Eulerian method with a SGS model to consider the effect of mesoscale structure on the constitutive laws is necessary for describing the hydrodynamics of large-scale gasfluidized beds containing Geldart A particles. Various approaches for establishing SGS models available in the literature are reviewed; it is evident that the filtered Eulerian model is still in its infancy. Therefore, we close the present article by discussing some of the possible directions of future research: (1) Many studies33,117-120 have shown the nonexistence of scale separation in rapid granular flow in general and gas-solid flow in particular, which suggests that the SGS model must be scale-dependent or filter-size-dependent. However, a preliminary study121 found that, with the increase of filter size, asymptotic values of interphase drag coefficient will be reached due to the saturation of mesoscale structures, which suggests the existence of a scale-independent plateau. The conclusion of Wang121 is only based on an air-FCC system at four different average solid volume fractions of simulated domain. Therefore, further studies are definitely needed to clearly clarify the relationship between filter size and constitutive laws used in the filtered Eulerian model, including the effect of various aspects, such as the effect of gas and particle properties, the effect of particle loadings, the effect of particle-size distribution and particle shape, and the effect of inlet, outlet, and wall boundary conditions. (2) Zhang and Vanderheyden19 found that the added mass force is unexpectedly large when FEM is used to simulate heterogeneous gas-solid flows. Latter studies122-125 based on the so-called linear wave propagation speed test confirmed their conclusion. The studies122-125 even concluded that the interphase drag force term should gradually give its way to the added mass force term when the filter size is gradually increased. The effect of the interphase drag force term disappears when the filter size is infinity, which corresponds to the steady-state simulation. However, one should be careful in interpreting their result, because the reported large added mass force is possibly due to the fact that the data obtained from the three-dimensional simulation are averaged to obtain the added mass force in a one-dimensional model as pointed out by one of the anonymous reviewers. The magnitude of added mass force is a linear function of the generalized added mass coefficient (Ca) with Ca )(〈εs′εs′〉)/([εs(1 - εs) -〈εs′εs′〉])[(εs(1 - εs)Fsg)/(Fm2)], where Fm ) εsFs + (1 - εs)Fg is the averaged mixture density and 〈εs′εs′〉 is the variance of the solid volume fraction.122 On the basis of the linear wave propagation speed test, De Wilde and

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Eulerian simulation of heterogeneous gas-solid flows covering all fluidization regimes are not available yet. (5) All of the currently available SGS models conclude that the interphase drag force is a function of voidage and gas and solid velocity as well as gas and particle properties. It is interesting to clarify whether other parameter(s) are also needed to correlate interphase drag force of heterogeneous gas-solid flows. In our viewpoint, parameter(s) that can characterize the dissipative particle-particle interaction might be the option, since the particle-particle interactions affect the mesoscale structures in gas-fluidized beds; meanwhile, mesoscale structures have significant effects on the interphase drag force and particulate phase stress. Other parameters might be the gradient of solid volume fraction and gas and solid velocity, which will include the effect of neighboring grids. Figure 5. Variation of maximal possible generalized added mass force coefficient with solid volume fraction in an air-FCC system: Fp ) 1500 kg/m3, Fg ) 1.2 kg/m3.

co-workers122,124 reported that Ca can be the order of 103 at low filter frequency, which is due to the fact that 〈εs′εs′〉 is approaching to εs(1 - εs). However, the numerical study by Wang121 showed that, because of the saturation of mesoscale structures, a plateau of 〈εs′εs′〉 will be reached with the increase of filter size, the value of which is much less than εs(1 - εs), which means the value of Ca at low filter frequency is much smaller than the value obtained by De Wilde and co-workers.122,124 From the fundamental point of view, the added mass force is mainly caused by the formation of mesoscale structures;19 therefore, we may analyze the added mass force from this aspect: in the heterogeneous gas-solid flow containing bubbles and/or clusters, the maximal possible value of 〈εs′εs′〉 can be obtained by supposing the formation of a clear two-phase structure,126-128 i.e., the dilute phase or bubble phase is empty of particles and the solid volume fraction of cluster phase or emulsion phase takes the value at minimum fluidization state. The calculated Ca of an air-FCC system is given in Figure 5. It can be seen that the maximal possible value of Ca is much smaller than the one obtained from the linear wave propagation speed test, which agrees with the numerical simulation of Wang.121 In summary, although the added mass force may not be as high as pronounced by some researchers,19,122,124 it may still play an important role. Research in this direction is needed to clarify the role of added mass force in the numerical simulation of heterogeneous gas-solid flows. (3) Although various methods have been suggested to consider the effect of subgrid-scale structures on the interphase drag force, they usually use the model, such as kinetic theory of granular flow, obtained from homogeneous fluidization to close the particulate phase stresses, which obviously conflicts with the true nature of heterogeneous structures inside the computational grid, as discussed in detail in section 3.5. A conceptually consistent subgrid-scale model, where the model for both the interphase interaction force and particulate phase stress is based on the heterogeneous nature of subgrid-scale structures, is urgently needed. (4) Currently available SGS models are usually established for a specific fluidization regime, for example, the EMMS model is for riser flow. Although the correlative multiscale method can in principle consider the effect of the variation of fluidization regimes, which is reflected by the fact that we get different subgrid-scale structures at different particle loadings or volume fraction rangessclusters and streamers at low particle loadings and transitioning to bubbles at higher loading, correlations and/ or physical models that can be directly applied in the filtered

Acknowledgment I would like to thank the anonymous reviewers for valuable suggestions and comments. This research was financially supported by NWO top grant “Towards a reliable model for industrial gas-fluidized bed reactors with polydisperse particles”. Note Added after ASAP Publication: The version of this paper that was published on the Web May 13, 2009 had an error in the author affiliation and minor text changes that were needed throughout the paper. The correct version of this paper was reposted to the Web May 21, 2009. Notation Ar ) Archimedes number dp ) particle diameter, m g ) gravity acceleration, m/s2 Gs ) solid circulation flux, kg/(m2 s) P,Q ) parameter in O’Brien and Syamlal’s correlation (1993) Re ) particle Reynolds number according to the definition of O’Brien and Syamlal (1993) Remf ) Reynolds number at minimum fluidization condition Remix ) Reynolds number according to the definition of Cruz et al. (2006) Rep ) particle Reynolds number according to the definition of Wang and Li (2004) Rets ) Reynolds number under terminal settling conditions for a single particle b ug ) gas-phase velocity vector, m/s b us ) solid-phase velocity vector, m/s Ug ) superficial gas velocity, m/s Umf ) superficial minimum fluidization velocity, m/s Greek Symbols βA ) drag coefficient, kg/(m3 s) εg ) voidage εmf ) voidage at minimum fluidization condition εs ) solid volume fraction εs,mf ) solid volume fraction at minimum fluidization condition µg, µs ) fluid and solid viscosity, Pa · s µmix ) mixture viscosity, Pa · s Fg, Fs ) fluid and solid density, kg/m3

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ReceiVed for reView February 15, 2009 ReVised manuscript receiVed April 15, 2009 Accepted April 30, 2009 IE900247T

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