A bubble-based EMMS model for gas–solid bubbling fluidization

Chemical Engineering Science 66 (2011) 5541–5555

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A bubble-based EMMS model for gas–solid bubbling fluidization Zhansheng Shi a,b, Wei Wang a,b,n, Jinghai Li a,b a b

State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China Graduate University of Chinese Academy of Sciences, Beijing 100049, China

a r t i c l e i n f o

abstract

Article history: Received 5 April 2011 Received in revised form 3 July 2011 Accepted 15 July 2011 Available online 22 July 2011

An EMMS/bubbling model for gas–solid bubbling fluidized bed was proposed based on the energyminimization multi-scale (EMMS) method (Li and Kwauk, 1994). In this new model, the meso-scale structure was characterized with bubbles in place of clusters of the original EMMS method. Accordingly, the bubbling fluidized bed was resolved into the suspending and the energy-dissipation sub-systems over three sub-phases, i.e., the emulsion phase, the bubble phase and their inter-phase in-between. A stability condition of minimization of the energy consumption for suspending particles (Ns-min) was proposed, to close the hydrodynamic equations on these sub-phases. This bubble-based EMMS model has been validated and found in agreement with experimental data available in literature. Further, the unsteady-state version of the model was used to calculate the drag coefficient for two-fluid model (TFM). It was found that TFM simulation with EMMS/bubbling drag coefficient allows using coarser grid than that with homogeneous drag coefficient, resulting in both good predictability and scalability. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Meso-scale Fluidization Bubble Simulation Hydrodynamics Drag coefficient

1. Introduction Gas–solid bubbling fluidized bed has been widely used in chemical industries such as coal gasification, ethylene polymerization and fluid-catalytic-cracking regeneration (Kunii and Levenspiel, 1991). To understand the flow behavior inside these reactors, some pioneering work (to mention but a few of them, Toomey and Johnstone, 1952; Davidson, 1961; Jackson, 1963; Grace and Clift, 1974; Kunii and Levenspiel, 1991) have been presented to describe the hydrodynamics in terms of, e.g. the bubble size, the rising velocity and volume fraction of bubbles and so on. In recent years, with the rapid development of computational technology, more researches based on computational fluid dynamics (CFD) have been reported, in which the two-fluid models (TFM) have been found successful in predicting the hydrodynamics of Geldart B and D particles in bubbling fluidized beds (Ding and Gidaspow, 1990; Yuu et al., 2001; Taghipour et al., 2005; Mazzei and Lettieri, 2008; Parmentier et al., 2008; Wang et al., 2008). However, the queries about the accuracy of TFM and the scalability of simulations never cease (Wang et al.; 2009; Lu et al., 2009). Let aside these disputes, the so-called coarse-grid simulation with appropriate meso-scale modeling can be viewed at least a reasonable choice for industrial applications (Wang et al., 2010).

n

Corresponding author. Tel.: þ86 10 8254 4837; fax: þ86 10 6255 8065. E-mail address: [email protected] (W. Wang).

0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.07.020

The meso-scale structure is the key to drag modeling for CFD simulations. For a bubbling fluidized bed, the meso-scale structure may relate with bubbles (Sundaresan, 2000) or particle clusters (Mostoufi and Chaouki, 2000, 2004; Lettieri et al., 2002). To account for the effects of these structures, Mckkeen and Pugsley (2003) and Ye et al. (2008) proposed to use reducing factors of around 0.15–0.4, with which they predicted the bed expansion and the axial concentration profiles. By modifying the drag model based on Richardson and Zaki (1954) and Wen and Yu (1966) with an effective mean diameter of particle clusters, Lettieri et al. (2002) and Gao et al. (2008) predicted the expected bubbling behavior and bed expansion. Bubble is the typical meso-scale structure in low-velocity gas-fluidized beds, as is the role of clusters in circulating fluidized beds (CFB). In fact, it was reported that the bubble and the cluster actually belong to the same family of non-uniform solutions of hydrodynamic solutions (Glasser et al., 1998). To describe the effects of such meso-scale structures, the energy minimization multi-scale (EMMS) model (Li, 1987; Li and Kwauk, 1994) was proposed, originally for circulating fluidized beds. In the EMMS model, three different scales, i.e. the micro-scale of particles, the meso-scale of clusters and the macro-scale of the whole bed, were distinguished. The meso-scale parameters of clusters were closed by invoking a stability condition besides the hydrodynamic conservation equations, while the stability condition was derived through bi-objective optimization (namely, the compromise) between gas dominance and solids dominance (Li, 2000). Such a model has been generalized further into the multi-objective

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variational multi-scale methodology (for brevity, MOV (Li and Kwauk, 1994), or directly, the EMMS method), with extended applications to more multiphase systems (Ge et al., 2007). Since the EMMS method was originally a meso-scale modeling based on the concept of clusters, its application has been mostly limited to high-velocity circulating fluidized beds where clustering prevails. It is a natural choice, then, to explore a possibility to establish the EMMS-based models with some other characteristic structures, e.g., bubbles, thereby finding applications in more flow regimes that may range from bubbling to fast fluidizations. This work can be viewed as the first step to this long-term target. By replacing the clusters with bubbles as the meso-scale structure, we will propose a bubble-based EMMS model, namely EMMS/ bubbling, where the solids flux can be assigned values of non-zero for circulating fluidization or essentially zero for bubbling fluidization. In contrast, the original version of EMMS model is limited to circulating fluidization with non-zero solids flux. Further, the dynamic effects of bubbles will be taken into account by introducing accelerations for the emulsion and bubbles. This unsteady EMMS/ bubbling model will be used to modify the effective drag coefficient within TFM, and then, it will be tested through simulations of bubbling fluidized beds with comparison to experimental data.

2. Model formulation For a bubbling fluidized bed much higher than the transport disengaging height (TDH) (Kunii and Levenspiel, 1991), the gas flows upward and the solid particles are suspended (with internal circulation), with negligible transportation of particles out of the bed. The gas flow is separated into the emulsion phase and the bubble phase. Accordingly, the interactions within a bubbling fluidized bed consist of that between gas and solids in the emulsion, that between the emulsion and the bubble, and that between bubbles. Based on such physical image, in the following sections we will make resolution with respect to the phase, scale and energy consumption, respectively, together with hydrodynamic analysis of each phase at different scales. To distinguish from the classical EMMS model, this model is named after EMMS/bubbling. 2.1. Resolution of bubbling fluidized bed As shown in Fig. 1, the overall system is resolved into three sub-systems or phases, i.e., the emulsion phase, the bubble phase, and the inter-phase. The hydrodynamic parameters needed for

this system include six variables, namely, the superficial gas velocity in the emulsion phase (Uge), the superficial solids velocity in the emulsion phase (Upe), the volume fraction of bubbles (db ), the rising velocity of bubbles (Ub), the diameter of bubble (db) and the voidage of emulsion phase (ee). As a first approximation, the bubble phase is assumed only consisting of gas, omitting the contribution of particles, i.e., eb ¼ 1. 2.2. Multi-scale resolution Following the classification of the original EMMS model (Li, 1987), we may distinguish three characteristic scales: the microscale with respect to particles, the meso-scale with respect to bubbles and emulsion, and the macro-scale with respect to the whole bed. The micro-scale interaction here refers to that between the gas and particles in the emulsion. The meso-scale interaction refers to that between bubbles and the emulsion. The emulsion phase is assumed as a homogeneous mixture. As a first approximation, it may be viewed as a pseudo-fluid with mean density re, viscosity me (Thomas, 1965) and superficial velocity Ue, as follows:

re ¼ rp ð1ee Þ þ rg ee ,

ð1Þ

me ¼ mg ½1 þ2:5ees þ 10:05e2es þ0:00273 expð16:6ees Þ,

ð2Þ

Ue ¼

rg Uge þ rp Upe : rp ð1ee Þ þ rg ee

ð3Þ

The macro-scale interaction works through the outer boundary of the investigated system and it can be represented by the wall boundary conditions, inlets and outlets conditions and so on. The current model can be regarded as to a periodic domain within a fluidized bed, and thus, no macro-scale interactions will be dealt with, except that the diameter of the bed is used as a constraint for the maximum bubble diameter, as will be discussed in the following sections. 2.3. Energy consumption Following the original EMMS model (Li and Kwauk, 1994), the whole system is resolved into a suspending subsystem and an energy dissipation subsystem, as shown in Fig. 2. The total massspecific energy consumption rate, NT, is considered to be the sum of Ns, the mass-specific energy consumption rate for suspending particles, and Nd, the energy dissipation rate in particle collision,

Fig. 1. System resolution for bubbling fluidized bed.

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Fig. 2. Energy resolution of bubbling fluidized bed.

circulation, acceleration and breakup of bubbles etc. (i.e. NT ¼Ns þNd). Similar to that in original EMMS (Li, 1987), NT can be defined as follows: NT ¼ Ns þNd ¼

rp rg rpU g ¼ Ug g: ð1eÞrp rp

ð4Þ

Further, Ns can be split into that in the emulsion phase, Nse (suspending energy in the emulsion phase), and that between the emulsion phase and bubbles, Nbe, where Nse ¼

2 rg Use 3 C Uge : 4 de rp dp

ð5Þ

Nbe consists of the energy consumption rate due to the relative velocity between the emulsion and rigid bubbles, Ninter, and the energy consumption rate due to distortion of the bubbles, Nsurf. That is, Ns ¼Nse þNbe ¼Nse þNinter þNsurf. As the emulsion is approximated to be a homogenous pseudofluid, Nbe can be expressed by (Bhavaraju et al., 1978) lnðP1 =P2 Þ f Ug g, Nbe ¼ P1 =P2 1 b

ð6Þ

where P1 and P2 are the pressures at the lower and the upper parts of the fluidized bed, respectively. If the pressure difference between P1 and P2 is small, the above relation can be further reduced to Nbe ¼ fb Ug g,

ð7Þ

where fb is the ratio of gas in the bubble phase to that in total, that is, fb ¼

db eb : db eb þð1db Þee

ð8Þ

the gas suspends particles with least energy consumption and meanwhile flows through particles with resistance as minimal as possible. Thus, on one hand, the gas forms bubbles to pass through the particle layer with least resistance; on the other hand, the voidage of emulsion phase approaches emf to maintain least gravitational potential. Then, as in the original EMMS model (Li and Kwauk, 1994), Nst tends to minimum (Nst-min), where Nst ¼Ns þNt. In a bubbling fluidized bed, the particle flow rate approximates to zero, then, Nt ¼0 and Nst ¼Ns. So, the stability condition for a bubbling fluidized bed can be expressed by that the mass-specific energy consumption rate for suspending energy tends to minimum, that is Nst ¼ Ns ¼ Nse þ Nbe -min:

ð9Þ

3.2. Hydrodynamic equations For a bubbling fluidized bed under steady state, ignoring the accelerations of particles and bubbles, we can derive the momentum equations by balancing the effective drag force and buoyancy force, as follows: Force balance for particles in the emulsion

pd2p 4

Cde

1 p r U 2 ¼ d3 ðr rg Þg, 2 g se 6 p p

ð10Þ

where the drag force in the emulsion, Fde, counterbalance the effective gravity and the drag coefficient with particle–particle interactions, Cde, is given in Table 1. If the Ergun’s drag law (Ergun, 1952) is used, then, " # ð1ee Þ2 mg 7 ð1ee Þrg Use Use 150 þ ¼ ð1ee Þðrp rg Þg: ð11Þ 4 ee dp ee d2p e2e

3. Model equations

Force balance for bubbles

3.1. Stability condition

pd2b

When the gas velocity exceeds the minimum bubbling velocity, Umb, neither the particles nor the gas can dominate the other: they have to compromise with each other in such a way that

where the drag force on the bubble, Fdb, counterbalance the effective gravity and the drag coefficient with bubble-bubble interactions, Cdb, is given in Table 1.

4

Cdb

1 p r U 2 ¼ d3 ðr rg Þg: 2 e sb 6 b e

ð12Þ

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3.2.1. Mass conservation of gas

3.2.2. Mass conservation of particles

Ug Uge ð1db ÞUb db ¼ 0:

ð13Þ

Up Upe ð1db Þ ¼ 0:

ð14Þ

Table 1 Summary of the parameters and formula in the EMMS/bubbling model. Variables

Emulsion phase

Bubble phase

Inter-phase

Superficial velocity Superficial slip velocity

Uge,Upe

Ub

Uge ð1db Þ Usb ¼ ðUb Ue Þð1db Þ

Reynolds number

Use ¼ Uge Upe

Ree ¼

ee 1ee

rg dp Use mg

Energy consumption rate

a b

re db Usb me (

Drag coefficient for single particle or bubblea

Effective drag coefficient with multi-particle/bubble correction

Rei ¼

Cdb0 ¼

Cde ¼ 200

Nse ¼

ð1ee Þmg 1 7 þ e3e dp rg Use 3e3e

2 rg Use 3 C Uge 4 de rp dp

38Rei 1:5

0o Rei r 1:8

24 2:7 þ Re i

Rei 4 1:8

Cdb ¼ Cdb0 ð1db Þ1=2 b

Nbe ¼ Ninter þ Nsurf ¼ fb Ug g

Darton and Harrison (1974). Ishii and Zuber (1979).

Fig. 3. The variation of Ns and db with two traversed variables, ee and db for the EMMS/bubbling model.

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Fig. 3(c), and the solution of the model at the minimum of Ns corresponds to the maximum bubble diameter. In a bubbling fluidized bed, mean bubble size changes with height (Mori and Wen, 1975; Werther, 1983; Weimer and Quarderer, 1985; Darton et al., 1997), reaching its stable diameter when coalescence equilibrates with breakup of the bubbles. The maximum bubble diameter in the above algorithm is then not infinite and should be constrained by the maximum stable bubble diameter, dbe. Table 2 lists some of the correlations for bubble diameter in the literature. In this work, the correlation of Horio and Nonaka (1987) is selected for the later calculation. With the constraint of the maximum stable bubble diameter, dbe, the algorithm (named as scheme-2) is adjusted as follows:

As a first approximation, the amount of entrained particles is assumed negligible, so, Up ¼0, Upe ¼ 0, and then Eq. (14) can be omitted. Table 1 summarizes the relevant parameters and formula of this model.

3.3. Solution scheme From the above equations, six parameters (Uge, Upe, db, Ub, db,

ee) are solved by four conservation equations (Eqs. (11)–(14)) and one objective function (Eq. (9)). To solve this nonlinear programming problem, we adopt a global search scheme (named as scheme-1) to find the set of six parameters that satisfies both the conservation equations and the constraint of minimum Ns, as follows:

1. For the given operating conditions Ug, assign a big value for Ns. 2. Sweep ee within the range of [emf, 1.0). 3. Calculate Use from Eq. (11), then Uge from the expression of Use in Table 1. 4. Calculate db from the correlation of Horio et al. in Table 2. 5. Substitute Ub for Usb with the definition of Usb in Table 1, and then calculate db and Ub by solving Eqs. (12) and (13). 6. Calculate Ns from Eq. (9), store it and the relevant set of six parameters if Ns is smaller than the previous values, until finishing the sweep of ee.

1. For the given operating conditions (Ug, Up), assign a big value for Ns. 2. Sweep db within the range of (0, 1.0). 3. Calculate Upe from Eq. (14). 4. Sweep ee within the range of [emf, 1.0]. 5. Calculate Use from Eq. (11), then, Uge from the definition of Use in Table 1. 6. Calculate Ub from Eq. (13). 7. Calculate Usb from the definition in Table 1, then, db from Eq. (12). 8. Calculate Ns from Eq. (9), store it and the relevant set of six parameters if Ns is smaller than the previous values. 9. Complete the sweeps of db and ee, and then output the set of six parameters corresponding to minimum Ns.

4. Model verification 4.1. Bed expansion Expansion of freely bubbling fluidized bed is an important factor to quantify the fluidization quality (Johnsson et al., 1991; Geldart, 2004). Extensive experimental results show that the voidage of the bed increases with Ug, and some relevant literature are listed in Table 3. Fig. 4 shows the comparison between simulation and experimental data. For the two cases of ballotini

Fig. 3(a) and (b) shows the variation of Ns and db with two traversed variables, ee and db, respectively. It is found that Ns decrease with the decrease of ee and db, while db varies inversely. Accordingly, Ns is inversely proportional to db, as shown in

Table 2 Bubble diameter and maximum bubble diameter correlations in the literature. Author

Correlation

Mori and Wen (1975)

db ðhÞ ¼ dbm ðdbm db0 Þe0:3h=D , dbm ¼ 0:65½AðUg Umf Þ0:4

Darton et al. (1997)

db ðhÞ ¼ 0:54ðUg Umf Þ0:4 ðh þ 0:12Þ0:8 g 0:2

Werther (1983)

db ðhÞ ¼ 0:853½1 þ 0:272ðUg Umf Þ1=3 ð1 þ 0:0684hÞ1:21 "

Weimer and Quarderer(1985)

dbm ¼ 0:135

ðrp rg Þ2

rg m

#2=3 d2p g 1=3 , 0:4 o Rep o 500

dbe ¼ ½gM þ ðg2M þ 4dbm =dt Þ0:5 2 Udt =4, gM ¼ 2:56  102 ðdt =gÞ0:5 =Uge

Horio and Nonaka (1987)

dbm ¼ 2:59g 0:2 ½ðUg Uge ÞAt 0:4

Table 3 The parameters in the literature cited in Fig. 4. Authors

Abrahamsen and Geldart (1980) Cui et al. (2001) Hilligardt and Werther (1987) Glicksman et al. (1991)

Symbol Exp.

Cal.

~



rp (kg/m3)

dp (mm)

Umf (m/s)

dt (m)

System

2500 1673 2640 2640

0.06 0.07 0.13 0.48

0.0026 0.003 0.025 0.18

0.15 0.152 1.0 1.0

Air–ballotini Air–FCC Air–quartz sand

8090 8090

0.2 0.25

0.119 0.152

0.91 0.91

Air–grit

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and FCC particles, which belong to Geldart group A (Geldart, 1973), the simulation generally agrees well with the experimental data. With the increase of particle diameter, density as well as the gas velocity, the deviations between model prediction and experimental data are enlarged. That seems plausible since the fluidization behavior will gradually transit from somewhat regular bubbling to shapeless and violent turbulence with the increase of gas velocity, thus magnifying the deviation from the model assumptions. 4.2. Emulsion properties With the increase of superficial gas velocity, the voidage in the emulsion phase will increase. According to Abrahamsen and Geldart (1980), for fine powders, this can be correlated by  3   1emf Uge 0:7 ee ¼ : ð15Þ emf 1ee Umf Fig. 5(a) shows that the EMMS/bubbling model predicts an increase of emulsion voidage with gas velocity, which agrees quantitatively with the experimental data and correlation Eq. (15). Fig. 5(b) shows its variation with particle diameter, indicating that for larger particles, the voidage ee is closer to emf, which is also in agreement with that reported in the literature

Fig. 4. The variation of the bed-averaged voidage (e ¼ (1.0 db)ee þ db) with Ug.

(Grace and Clift, 1974; Abrahamsen and Geldart, 1980; Hilligardt and Werther, 1987). Fig. 6 further gives the variation of Uge with Ug. Generally, Uge is larger than Umf at high gas velocity for fine particles, while for coarser particles (e.g. dp ¼0.46 mm), Uge clings to Umf. That finding is in agreement with the experimental phenomena (Andreux and Chaouki, 2008). 4.3. Volume fraction of bubbles In a bubbling fluidized bed, the volume fraction of bubbles is an important factor affecting the inter-phase heat/mass transfer, since the gas in bubbles interacts less with particles than that in the emulsion. Fig. 7 shows a group of data in the literature, as listed in Table 4, and its comparison against the model predictions. In general, the trend is well predicted: db increases with the superficial gas velocity Ug. Similar to the trend in Fig. 4, for fine particles belonging to Geldart group A, the slope of db is smaller than that for coarser particles. That is because the fine particles are more aeratable compared to the coarse particles, and hence, more gas penetrates across the emulsion rather than forming bubbles.

Fig. 6. Prediction of gas velocity in the emulsion (larger particles (i.e., dp ¼ 0.15, 0.46 mm) are imaginary, with the same density as those of dp ¼ 0.06 mm).

Fig. 5. The variation of emulsion voidage with superficial gas velocity (experimental data of dp ¼ 0.06 mm are from Abrahamsen and Geldart(1980), larger particles (i.e., dp ¼ 0.15, 0.46 mm) are imaginary with the same density).

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4.4. Bubble velocity

5. CFD with the EMMS/bubbling drag model

For a single, isolated bubble with a diameter of db, its rise velocity Ub0 in an immense bed is given by Davies and Taylor (1950):

5.1. Unsteady-state EMMS/bubbling model

pffiffiffiffiffiffiffiffi Ub0 ¼ 0:711 gdb :

ð16Þ

In a freely bubbling fluidized bed, bubbles may coalesce and affect each other. Davidson and Harrison (1963) hence presented a modified correlation as follows: Ub ¼ ðUg Umf Þ þ Ub0 :

ð17Þ

That relation means the bubble velocity varies with the bubble diameter and, accordingly, the distance from the distributor. However, limited to the use of the correlation of maximum stable diameter of Horio and Nonaka (1987), our model so far predicts a constant bubble velocity for a given gas velocity. To apply the model for changeable db, we may free it from the correlations listed in Table 2, and assume it to be exactly the same value with the experimental. Thus, we can use the scheme-2 as discussed above to predict the variation of bubble velocity with bubble diameter. Fig. 8 shows the comparison between the simulation results and experimental data. In general, the model results show a power-function dependence on db, which is similar to that revealed by Eq. (17) and also the experimental data, as listed in Table 5. Their quantitative comparison is also approving except for the last case of coarse particles. With the above comparisons involving bubbles, emulsion and overall performance, it seems that the EMMS/bubbling model reasonably captures the effects of meso-scale structures in bubbling fluidized beds, and hence is encouraging to be used further for meso-scale modeling of structures in CFD. That will be the focus of our following section.

In a bubbling fluidized bed, bubbles form at the distributor and then rise to the top of the bed. Before reaching its maximum stable diameter, bubble coalesces or breakups with acceleration. Thus, for an unsteady-state and local description of the bubble motion, the acceleration effects should be considered. For simplicity, two inertial terms, ae and ab, are introduced besides the steady-state six variables. Following the system resolution in the steady-state model, the whole bubbling bed can be resolved into three sub-systems including the emulsion, the bubble, and their inter-phase, and these three sub-systems can be represented by eight variables: Uge, Upe, db, Ub, db, ee, ab and ae. The particle velocity approximates to zero and the acceleration of emulsion phase is negligible because of the large inertial difference between the gas and the solid particles (Up ¼0; ae ¼0). The bubble size is a variable, rather than a constant closed by the maximum stable diameter as in the steady-state model. In addition to the mass conservations of gas and particles, the conservation relations involved are as follows: Force balance for particles in the emulsion in unit volume

pd2p 4

Cde

1 p r U 2 ¼ d3 ðr rg Þðg þ ae Þ, 2 g se 6 p p

ð18Þ

If the Ergun’s drag law (Ergun, 1952) is used, then, # ð1ee Þ2 mg 7 ð1ee Þrg Use Use 150 þ ¼ ð1ee Þðrp rg Þðg þ ae Þ: 4 ee dp ee d2p e2e

"

ð19Þ Force balance for bubbles in unit volume

pd2b 4

Cdb

1 p r U 2 ¼ d3 ðr rg Þðg þ ab Þ, 2 e sb 6 b e

ð20Þ

Definition of mean voidage

eg ¼ ð1db Þee þ db :

ð21Þ

Accelerations and added mass force The inertial difference between the bubble phase and the emulsion phase results in the meso-scale added mass force, which is small compared to the drag force and is, therefore, neglected in the force balance equations above. However, it is useful to correlate the two inertial terms, i.e., ae and ab. According to Zhang and Vanderheyden (2002), it is proportional to the mixture density re of the emulsion phase, as follows: Fam ¼ Cb ð1ee Þdb re ðab ae Þ,

ð22Þ

where Cb is the coefficient of the added mass force and can be written according to Zuber (1964) as follows:   1 þ2db : ð23Þ Cb ¼ 0:5 1db

Fig. 7. Variation of volume fraction of bubbles with Ug (model prediction vs. experimental data).

Another form of the added mass force is used by De Wilde (2005, 2007), which reads Fam ¼ s2 ðrp rg Þg,

ð24Þ

Table 4 The parameters in the literature cited in Fig. 7. Authors

Krishna and van Baten. (2001) Johnsson et al. (1991)

Symbol Exp.

Cal.

’

&

rp (kg/m3)

dp (mm)

Umf (m/s)

dt (m)

System

1480 2600

0.06 0.15 0.46

0.0022 0.02 0.18

0.38 0.68

Air–FCC Air–silica sand

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Fig. 8. Variation of bubble velocity with the diameter of bubble, in which the operating parameters are offered in Table 5 ((a) Werther(1983); (b) and (c): Hilligardt and Werther(1987); (d) Glicksman and Mcandrews (1985)).

Table 5 The parameters in the literature cited in Fig. 8. Authors

rp (kg/m3)

dp (mm)

Umf (m/s)

System

Werther (1983) Hilligardt and Werther (1987)

1200 2640 2500

0.002 0.025 0.18 0.58

Air–FCC Air–quartz sand

Glicksman and Mcandrews (1985)

0.06 0.13 0.48 1.0

where the variance of local solids concentration fluctuation can be correlated by (Zenit and Hunt, 2000):

s2 ¼

2 4 gÞ g 2

ð1e

e

: 1 þ 4ð1eg Þ þ4ð1eg Þ 4ð1eg Þ3 þ ð1eg Þ4

Equivalence of Eqs. (6) and (8) gives ab ae ¼

s2 ðrp rg Þg : Cb ð1ee Þdb re

condition is hence needed to close these equations, as follows: Ns ¼

ð25Þ

ð26Þ

5.1.1. Stability criterion As in the steady-state model, this unsteady-state EMMS/ bubbling model has more variables than equations. The stability

Air–silica sand

2 rg Use 3 C Uge þ fb Ug ðg þ ab Þ-min: 4 de rp dp

ð27Þ

5.1.2. Structure-dependent drag coefficient Following the previous definition (Wang and Li, 2007), the drag coefficient with bubble-emulsion structures consists of the contributions from the emulsion and the bubble phases, respectively, and can be written as follows: e2 e2   bEMMS=bubbling ¼ g Fd ¼ g ð1db Þne Fde þ db nb Fdb Uslip Uslip i e2g h ð1db Þð1ee Þðrp rg Þðg þ ae Þ þ db ðre rg Þðg þ ab Þ : ð28Þ ¼ Uslip

Z. Shi et al. / Chemical Engineering Science 66 (2011) 5541–5555

For comparison, the heterogeneous index can be defined as a dimensionless drag coefficient scaled with the Wen and Yu drag coefficient, bWen & Yu, that is, Hd ¼ bEMMS=bubbling =bWen & yu

ð29Þ

where

bWen & yu ¼

3 rg eg es 9ug us 9 2:65 C eg 4 d dp

ð30Þ

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Table 6 Summary of parameters used in the simulation of Zhu et al. (2008). dp

rp rg mg g Umf

emf

6.5  10  5 m 1780 kg/m3 1.225 kg/m3 1.7894  10  5 Pa s 9.8 m/s2 0.003 m/s 0.4

e

es,max Ug Dt H0 uT

0.9 0.6 0.2, 0.3, 0.4 m/s 1.0  10  4 s 1.2 m 0.203 m/s

5.2. Model solution The model consist of eight variables (Uge, Upe, db, Ub, db, ee, ab, ae), which are closed by six equations, Eqs. (13), (14), (19)–(21) and Eq. (26) and one objective function, Eq. (27). To solve this nonlinear programming, we adopt a global search scheme as follows: 1. For a given system with physical parameters (rg, rp, mg) and the superficial gas velocity Ug,; traverse eg within [emf, 1.0] and then ee within [emf, eg]. 2. Calculate db from Eq. (21). 3. Calculate Use from Eq. (19). 4. Calculate ab from Eq. (26). 5. Calculate Upe from Eq. (14) and Uge from definition of Use in Table 1. 6. Calculate Ub from Eq. (13) and db from Eq. (20). 7. Calculate Ns from Eq. (27) and store all the other parameters if at minimum Ns. 8. Calculate bEMMS and Hd using Eqs. (29) and (30).

Table 7 Heterogeneous index at different velocities for Geldart A particles (Zhu et al., 2008). Gas velocity (m/s)

Heterogeneous index (Hd)

Application range(eg)

0.003 (Umf) 0.005 (1.67Umf)

1.0 Hd ¼ 0:03405eeg =0:1311 0:1315

emf  0.465

0.01(3.33Umf)

Hd ¼ 0:2453eeg =0:279 0:734

emf  0.535

0.02(6.67Umf)

Hd ¼ 0:00294eeg =0:1006 0:023

emf  0.57

0.05(16.7Umf)

Hd ¼ 7:317  105 eeg =0:0731 þ 0:0636

emf  0.675

0.2 (66.7Umf EuT)

Hd ¼ 0:0012eeg =0:12546 þ 0:01887

emf  0.85

0.3(100Umf)

Hd ¼ 0:03405eeg =0:1311 0:1315

emf  0.895

0.4(133.3Umf)

Hd ¼ 0:00182eeg =0:15619 0:00866

emf  0.985

5.3. Model results Fig. 9 shows the variation of the heterogeneous index Hd with voidage at different gas velocities. The relevant physical properties and fitting functions are listed in Tables 6 and 7, respectively, where the lower limit of emf is a natural limit of fluidization, while the various upper limits of voidage in Table 7 are determined from the values where the predicted Hd are equal to 1.0. Compared to the standard drag coefficient, this EMMS-based model of bubbling bed predicts lower drag coefficient, which is similar to our previous structure-dependent drag model (Yang et al., 2003; Wang and Li, 2007; Lu et al., 2009). When the gas

Fig. 10. The variation of heterogeneous index with voidage at different Ug for Geldart B particles.

velocity is near the minimum fluidization velocity Umf, Hd approaches unity in most of the range. Higher gas velocity results in lower Hd, reflecting the increase of the heterogeneity with the gas inflow. As shown in Fig. 10, similar trend is found for the cases of coarser particles (Geldart B, Table 8) and the relevant fitting functions of the heterogeneous index Hd are listed in Table 9. 5.4. Simulation setup Fig. 9. The variation of heterogeneous index with voidage at different Ug for Geldart A particles.

The numerical simulation was based on the two-fluid model (TFM) of Fluents 6.3.26, whose equations are listed in Appendix

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Table 8 Summary of parameters used in the simulation of Taghipour et al. (2005). dp

rp rg mg g Umf

emf

2.75  10  4 m 2500 kg/m3 1.225 kg/m3 1.7894  10  5 Pa s 9.8 m/s2 0.065 m/s 0.39

e

emax Ug Dt H0 uT

0.9 0.63 0.38, 0.46 m/s 1.0  10  4 s 0.4 m 2.0 m/s

Table 9 Heterogeneous index of different velocities for Geldart B particles (Taghipour et al., 2005). Velocity (m/s)

Heterogeneous index (Hd)

Application range (eg)

0.065(Umf) 0.195(3Umf)

1.0 Hd ¼ 0:017eeg =0:1256 0:0328

emf  0.515

0.38(6Umf)

Hd ¼ 0:0066eeg =0:112 0:057

emf  0.59

0.46(7Umf)

Hd ¼ 0:004eeg =0:1068 0:033

emf  0.61

0.78(12Umf)

Hd ¼ 0:000155eeg =0:0756 þ 0:0692

emf  0.664

¼ 0:000383eeg =0:1023 þ 0:0369

1.5(23Umf)

Hd

2.0(30.8Umf EuT)

Hd ¼ 0:000208eeg =0:01 þ 0:0307

emf  0.80 emf  0.84 Fig. 11. Schematic diagrams of the simulated 2D bubbling fluidized beds (a: rp ¼ 1780 kg/m3, dp ¼ 6.5  10  5 m, as shown in Zhu et al. (2008); b: rp ¼ 2500 kg/m3, dp ¼ 2.75  10  4 m, as shown in Taghipour et al. (2005)).

A. The solids stress in the momentum equations are closed with the algebraic form of the kinetic theory of granular flow (KTGF). The structure-dependent drag coefficient is incorporated into Fluent with User-Defined Functions (UDF). Two fluidized beds are simulated with particles belonging to Geldart A (Zhu et al., 2008) and Geldart B (Taghipour et al., 2005). Detailed simulation parameters are summarized in Tables 6 and 8. For the case of Geldart A particles, as sketched in Fig. 11(a), the bed is 2.464 m in height and 0.267 m in inner diameter, with its disengaging section neglected to save computation time. At the top outlet, atmospheric pressure is prescribed and the solid mass flux is monitored. The entrained solids are fed back into the bottom inlet with a solids concentration of 0.3. Gas enters the bed uniformly from the bottom inlet. The no-slip boundary condition is used to the gas phase while the partial slip boundary is used to the solid phase with a specularity coefficient of 0.6. Simulations lasted for 30 s in physical time and the time-average variables were obtained over the last 15 s. For the case of Geldart B particles, as sketched in Fig. 11(b), the bed is 1.0 m in height, and 0.28 m in inner diameter. The other settings were prescribed as is the case for Geldart A particles. 5.5. Simulation and discussion 5.5.1. Simulation results of Geldart A particles Fig. 12 shows the time-averaged axial profiles of solids concentration using different drag models. When using the EMMS/ bubbling model (drag B), the simulation results agree well with the available experimental data, and the grid refining test shows that such grid resolution (40  200) is sufficient to correctly predict the flow behavior. For simplicity, the later simulations take this mesh as the standard for comparison. When using the hybrid drag model of Wen & Yu and Ergun (drag G), the bed expansion is obviously overestimated under such grid resolution. Compared to the heterogeneous and

Fig. 12. Comparison of axial solid concentration for Ug ¼ 0.4 m/s.

turbulent distribution predicted using drag B, using drag G predicts an almost vertical curve across the bed height with uniform distribution of solids. Refining grids down to the size of 10 times the particle diameter (417  962) has little effects for this case. Compared to the other two cases with lower gas velocity, as shown in Fig. 13, similar trends can be found except that higher gas velocity results in higher bed expansion rate and more diffusive interface. According to the literature reports (Parmentier et al., 2008; Wang et al., 2009) and our own experience, fine-grid simulation of a smaller fluidized bed with drag G

Z. Shi et al. / Chemical Engineering Science 66 (2011) 5541–5555

Fig. 13. Comparison of axial solid concentration for Ug ¼0.2 m/s (a) and Ug ¼0.3 m/s (b).

Fig. 14. Comparison of time-average radial solid concentration profiles for Ug ¼ 0.4 m/s.

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also achieves a good prediction of the bed expansion. However, for current cases, it seems that using drag B is necessary to obtain reasonable predictions. This seeming contradiction may attribute to the oversized aspect ratio of the grid scheme used here, and square grid scheme may improve the prediction when using drag G. It should also be noted that, owing to computation limitation, we do not extend our comparison to the grid size of 3dp as in the case of Wang et al. (2009). We agree with that refining grid size will further improve the prediction. However, the merit of the EMMS/bubbling model lies in that it can greatly reduce the computation, and at least for the cases in this article, it allows using coarser grid scheme without losing accuracy. More practices are still needed to clarify this issue. Fig. 14 compares the radial profiles of solids concentration between simulation and experiments. At all heights, the solids concentration are lower in the center of the bed than near the wall, showing reasonable agreement with experimental data (Zhu et al., 2008). Fig. 15 shows the transient particle velocities at the center and near the wall, indicating that the particle velocity fluctuates at values around zero. That agrees with our model assumption that Up ¼0 and the experimental data, which fluctuate with magnitude of 0.3–0.4 m/s.

5.5.2. Simulation results of Geldart B particles As stated in the introduction, there are reports addressing the different applicability of TFM for Geldart A and B particles (Ding and Gidaspow, 1990; Yuu et al., 2001; Taghipour et al., 2005; Mazzei and Lettieri, 2008; Parmentier et al., 2008; Wang et al., 2008, 2009). In this section, we will evaluate the EMMS/bubbling model for hydrodynamic simulation of Geldart B particles. Fig. 16 shows the effect of grid resolution on the prediction of axial distribution of solids concentration under two different gas velocities. It is found that the predictions change little and converge to the same curve when the grid resolution is finer than 60  240 for using both drag G and the EMMS/bubbling model. It is easy to understand since the EMMS/bubbling model differs from the drag G model only at the dense region for this case, as shown in Fig. 2. Fig. 17 provides the radial profiles of time-mean voidage calculated under different grid schemes using both drag models. Under the two specified gas velocities (Ug ¼0.38, 0.46 m/s), the prediction using both drag models agree reasonably with the experimental data. So, it seems that the EMMS/bubbling model is reasonable for predicting bubbling fluidized beds for both Geldart A and B particles.

6. Conclusion A multi-scale model for gas–solid bubbling fluidized bed is proposed and named after EMMS/bubbling. It follows the method of EMMS for scale resolution and energy consumption criterion. This model has been evaluated with comparison to experimental data and empirical relations involving volume fraction of bubbles, bubble velocity, overall voidage and so on. The model generally agrees well with the available data. The unsteady-state EMMS/ bubbling model was further used to modify the drag coefficient and incorporated into TFM to simulate the hydrodynamics of Geldart A and Geldart B particles in bubbling fluidized beds. Current results show that the unsteady EMMS/bubbling model allows using coarse grid without losing accuracy.

Nomenclature

Fig. 15. Transient particle velocity signals obtained at two radial positions.

Cdb0 Cde Cdb

drag coefficient of single bubble (dimensionless) drag coefficient of multi-particle (dimensionless) drag coefficient of multi-bubble (dimensionless)

Fig. 16. Solids concentration distributions along the height for (a) Ug ¼ 0.38 m/s and (b) Ug ¼0.46 m/s.

Z. Shi et al. / Chemical Engineering Science 66 (2011) 5541–5555

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Fig. 17. Comparison of radial voidage profiles (a) Ug ¼ 0.38 m/s; (b) Ug ¼ 0.46 m/s.

dt db dbm dbe dp db(h) db0 e fb Fam Fd g g0 h H0 Hd I n Nbe Nd Ninter Ns Nse Nsurf

fluidized bed diameter (m) bubble diameter (m) maximum bubble diameter (m) stable bubble diameter (m) diameter of particles (m) bubble diameter of certain height (m) initial bubble size (m) elastic coefficient ratio of gas in the bubble phase to that in total added mass force per unit volume (kg m  2 s  2) drag force (kg m s  2) gravitational acceleration (m s  2) radial distribution function height of the reactor (m) initial packing height (m) heterogeneous index solid inventory (kg) number of particle or bubble per unit volume (m  3) mass-specific energy consumption for interaction between the emulsion phase and bubbles ( J s  1 kg  1) mass-specific energy consumption for dissipation (J s  1 kg  1) mass-specific direct interaction energy consumption between the rigid bubble and emulsion phase (J s  1 kg  1) mass-specific energy consumption for suspending particles (J s  1 kg  1) mass-specific suspending energy consumption in the emulsion (J s  1 kg  1) mass-specific distorted energy consumption of the bubbles (J s  1 kg  1)

Nst Nt NT p P1 P2 Re Dt U Ub Ub0 Uge Umb Upe Usb Use uT

mass-specific energy consumption for suspending and transporting particles (J s  1 kg  1) mass-specific energy consumption for transporting particles (J s  1 kg  1) mass-specific energy consumption for particles (J s  1 kg  1) pressure (Pa) pressure at the downer part of the fluidized bed (Pa) pressure at the upper part of the fluidized bed (Pa) Reynolds number time step (s) superficial velocity (m/s) superficial bubble velocity relative to bubble phase (m/s) superficial single bubble velocity (m/s) superficial gas velocity in the emulsion phase (m/s) superficial minimum bubbling velocity (m/s) superficial particle velocity in the emulsion phase (m/s) superficial slip velocity between bubble and emulsion (m/s) superficial slip velocity in the emulsion (m/s) terminal velocity (m/s)

Greek symbols

b

e ed es,max r m

drag coefficient (kg/m3 s) voidage voidage at the intersection of the fitting Hd function and 1 close packing density density (kg m  3) viscosity (Pa s)

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t

shear viscosity of solids

stress tensor (Pa) bubble holdup (dimensionless) granular temperature (m2/s2)

d Y

ms ¼ ms,kin þ ms,col þ ms,fr ,

ðA:11Þ

That is, pffiffiffiffiffiffiffiffiffiffi

 2 4 1 þ g0 es ð1 þ eÞ 5 96es ð1þ eÞes g0  12 4 Ys ps sin f þ es rs dp g0 ð1 þeÞ þ pffiffiffiffiffiffiffi 5 p 2 I2D ,

Subscripts

ms ¼ 10rs dp b e g mf p s

bubble emulsion phase gas phase minimum fluidization particle solid phase

Ys p

granular conductivity of fluctuation energy  2 150rs dp ðYs pÞ1=2 6 ks ¼ 1 þ es g0 ð1 þ eÞ 5 384ð1 þ eÞg0  1=2 Y s þ2e2s rs dp g0 ð1 þeÞ ,

Bold characters are for vectors or tensors.

p

Acknowledgement This work is financially supported by the Natural Science Foundation of China under the Grant no. 20821092 and MOST under the Grant nos. 2007AA050302-03 and 2007DFA41320, and Chinese Academy of Sciences under the Grant no. KGCX2-YW-222.

Appendix A. Hydrodynamic equations

collisional energy dissipation "   # 4 Ys 1=2 gs ¼ 3e2s rs g0 Ys ð1e2 Þ : dp p Drag coefficient of gas–solid 8 r e ð1e Þ m > 150 s eg g d2g þ 1:75es dpg 9ug us 9, > > p > > < eg es rg 9ug us 9 2:65 eg Hd , b ¼ 34 CD dp > > > e e r 9u u 9 > g s g g s > e2:65 , : 34 CD g dp

Continuity equations of gas and solids @ðeg rg Þ @t

þ rUðeg rg ug Þ ¼ 0,

@ðes rs Þ þ rUðes rs us Þ ¼ 0, @t

@t

ðA:1Þ

ðA:2Þ

þ rUðeg rg ug ug Þ ¼ eg rpg þ rUðeg tg Þ þ eg rg gbðug us Þ, ðA:3Þ

@ðes rs us Þ þ rUðes rs us us Þ ¼ es rpg rps þ rUðes ts Þ þ es rs g þ bðug us Þ, @t

ðA:4Þ Granular temperature equation   3 @ ðes rs Ys Þ þ rUðes rs us Ys Þ ¼ ðps I þ ss Þ 2 @t : ðrus Þ þ rUðks rYs Þgs 3bYs , gas phase stress h i 2 tg ¼ mg rug þðrug ÞT  mg rUug , 3

es

ðA:6Þ

ðA:7Þ

1=3

es,max

,

ðA:8Þ

solid pressure ps ¼ es rs Ys þ2ð1 þ eÞe2s g0 rs Ys ,

ðA:9Þ

bulk solid viscosity

ls ¼

 1=2 4 Y es rs dp g0 ð1 þ eÞ s , 3 p

eg o emf emf r eg o ed ðdrag BÞ, eg Z ed

where ed denotes the voidage at the intersection of the Hd function and unity. 8 rg es ð1eg Þ mg > < 150 eg d2p þ 1:75es dp 9ug us 9, eg r 0:8 b¼ ðdrag GÞ: ðA:16Þ e e r 9u u 9 > : 34 CD s g gd g s e2:65 , eg 4 0:8 g p where 8 < 0:44, Res Z 1000

CD ¼ 24 1 þ 0:15Re0:687 ,Re o 1000 , : Res s s

Res ¼

eg rg dp 9ug us 9 : mg

ðA:17Þ

ðA:18Þ

References

radial distribution function "   #1 g0 ¼ 1

ðA:14Þ

ðA:5Þ

solid phase stress

ts ¼ ms ½rus þ ðrus ÞT  þ ðls 2=3ms ÞrUus ,

ðA:13Þ

ðA:15Þ

Conservation of momentum of gas and solids @ðeg rg ug Þ

ðA:12Þ

ðA:10Þ

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