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A new structural parameters model based on drag coefficient for simulation of circulating fluidized beds
Powder Technology 286 (2015) 516–526
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
A new structural parameters model based on drag coefficient for simulation of circulating fluidized beds Wenming Liu a,b, Hongzhong Li a,⁎, Qingshan Zhu a,⁎, Quanhong Zhu a a b
State Key Laboratory of Multi-phase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing 100190, PR China University of Chinese Academy of Sciences, Beijing 100049, PR China
a r t i c l e
i n f o
Article history: Received 13 January 2015 Received in revised form 27 August 2015 Accepted 31 August 2015 Available online 2 September 2015 Keywords: Fluidization Simulation Hydrodynamics CFBs Structure-based drag model
a b s t r a c t This work presented a new scheme to establish structural parameters model, and the model was used to solve structural parameters based on the available structure-based drag model. By combining with the Eulerian twofluid model, the hydrodynamics of circulating fluidized beds (CFBs) was simulated. Different combinations of clusters properties, including the cluster voidage and diameter, were adopted to fit for Geldart A and B particles and to close the insufficient solving equations, respectively. The simulated solid mass flux, radial and axial voidage profiles were in agreement with the experimental data. The dilute-top/dense-bottom and the coreannular flow structure were also captured. Moreover the spatiotemporal fluctuation of clusters can be observed from those simulations. Simulation results showed the combination of the structural parameters model with the available structure-based drag model can predict well the hydrodynamics for Geldart A and B particles in CFBs. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Circulating fluidized beds (CFBs) are widely used in various modern industrial processes, such as fluid catalytic cracking and coal combustion. Because of high flow rate of gas–solid and intensive contacting, CFBs have favorable hydrodynamics, reactions and heat transfer characteristics. Understanding of the hydrodynamic characteristics is the key to design and the scale-up of such reactors. With the increase of the computational ability, computational fluid dynamics (CFD) has been a valuable tool to predict the fluid dynamics, which is of vital importance to the scale-up of CFBs. For heterogeneous gas–solid flows in CFBs, drag models have a significant effect on the simulation of bed hydrodynamics [1–3]. The competition and compromise in different spatiotemporal scales lead to the formation of multiscale structures [4,5]. Conventional drag models, such as Gidaspow [6] and Syamlal-O’Brien [7], have been proved to over-predict the drag coefficient because they neglect the multiscale structures [5,8,9] in CFBs. Multiscale structures of CFBs are characterized by clusters, and heterogeneous flows can be divided into the cluster phase, the dispersed phase and the inter-phase which are usually defined by 8–10 structure parameters [10–12]. Because conventional two-fluid drag models assume homogeneous conditions inside the control volume, they cannot predict structure parameters. Structure-based
⁎ Corresponding authors. E-mail addresses: [email protected] (H. Li), [email protected] (Q. Zhu).
http://dx.doi.org/10.1016/j.powtec.2015.08.049 0032-5910/© 2015 Elsevier B.V. All rights reserved.
drag models have already been successfully developed to understand the multiscale nature of heterogeneous two-phase flows in CFBs [13]. The key to solve structure-based drag models is to solve the structural parameters. Due to the lack of enough number of equations to solve these parameters, many researchers adopted minimization stability condition to close the insufficient solving equations, such as the energy minimization multi-scale model (EMMS) [12] and the cluster structuredependent (CSD) drag coefficient model [14]. Adopting other measures to close the equations of structural parameters models has received much less attention and remains a challenge for the complex heterogeneous flows of CFBs. With respect to the system of multiscale structures, clusters play a dominant role in gas–solid interaction [15–17]. Clusters have two major properties, including diameter and voidage; equations of clusters properties have been used by many researchers to improve drag models. Gao et al. [15] adopted the cluster diameter, derived from the experimental particles terminal velocity, to replace the particle diameter of Gidaspow drag model, and the simulation results were in agreement with experimental data. Lu et al. [18] used equations of clusters properties to modify Gidaspow drag model and conservation equations of the kinetic theory of granular flow. As for EMMS model, Nikolopoulos et al. [19] claimed that the predicted cluster diameter was smaller than the diameter of a single particle or negative for averaged solid volume fraction εs less than 0.01 or greater than 0.44, respectively, and introduced Gu and Chen’s [20] correlation of the cluster diameter to confront these problems. Wang et al. [10] extended EMMS model from Geldart A to Geldart B particles by integrating the equation of the cluster voidage. Therefore, clusters properties are of prime importance to the simulation
W. Liu et al. / Powder Technology 286 (2015) 516–526
of CFBs, and an improved structural parameters model should be able to describe the heterogeneous multiscale structures caused by the clusters. In the present work, a new structural parameters model was established incorporating with equations of clusters properties, which can not only close the insufficient solving equations but also take into
517
consideration the effect of clusters. By incorporating it into an Eulerian two-fluid model, the gas–solid flow hydrodynamics of CFBs was simulated by using CFD software (FLUENT6.2.16). Simulation results were compared with the experimental data available in literature to validate its feasibility.
2. Mathematical model 2.1. Structural parameters model Fig. 1 shows multiscale resolution of structure and gas–solid interaction proposed by Wang et al. [10]. The complex heterogeneous flow structures of CFBs can be divided into three simple homogeneous phases, including the cluster phase, the dispersed phase and the inter-phase. Based on the homogeneous assumption and Matsen’s [21] investigation, the voidage in dispersed phase εd is defined as 0.9997. The heterogeneous structures have 9 unknown parameters, namely, Ufd, Upd and αd for the dispersed phase, εc, dc, Ufc, Upc, f and αc for the cluster phase. The solving equations can be detailed as follows, and the related variables involved in these equations are summarized in Appendix A. (1) Force balance in cluster phase The forces on the cluster include cluster-fluid drag force FDcn inside cluster phase, cluster-fluid drag force FDcf outside cluster phase, the collision force Fpdc from particles in the dispersed phase outside cluster phase and apparent gravity. The drag force FDcn equals to the drag force of single particle multiplying the effective number of particles inside cluster phase. F Dcn ¼ nF Dc
ð1Þ
The drag force of single particle FDc can be expressed by F Dc ¼
π 2 dp C Dc ρ f jU sc jU sc 8
ð2Þ
The force on the particles of outside surface of the cluster is different from those inside the cluster. The former force is caused by high gas velocity from the dispersed phase, and the latter force derives from low gas velocity of the cluster phase which means only half of particles of outside surface are relatively effective. Therefore, the effective number of particles inside cluster phase is calculated by π 3 3 dp ð1−εc Þ 1 πd 2 ð1−ε Þ dc dp c c n¼6 π − ¼ ð 1−ε Þ 1−2 c π 2 2 dp dc dp3 dp 6 4
Fig. 1. Resolution of structure and gas–solid interaction proposed by Wang [10].
ð3Þ
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Then, Eq. (1) is written as: F Dcn ¼
3 dp π 2 dc dp C Dc ρ f jU sc jU sc ð1−εc Þ 1−2 8 dp dc
ð4Þ
The drag force outside cluster phase FDcf is expressed by F Dc f ¼
π 2 dc C Di ρ f jU si jU si 8
ð5Þ
When the cluster moves upward at the velocity Upc/(1 − εc), particles in the dispersed phase move upward at the velocity Upd/(1 − εd). Because the cluster diameter is much bigger than the particle diameter, the particle velocity is faster than the cluster velocity. The particles below the cluster chase and bump the bottom of the cluster, then the particle velocity would reduced to be Upc/(1 − εc). The transverse collision would be offset because of symmetry, and the particles above the cluster would not be considered. Therefore, the mass flow rate of particles below the cluster is expressed as follows: Gpdc ¼
U pd U pc π 2 dc − ρ ð1−εd Þ 4 ð1−εc Þ ð1−εd Þ p
ð6Þ
According to the momentum conservation, the collision force Fpdc is written as: 2 π U pd U pc − F pdc ¼ Gpdc upd −upc ¼ d2c ρp ð1−εd Þ 4 ð1−εc Þ ð1−εd Þ
ð7Þ
Based on the force balance in the cluster phase, the equation is calculated by F Dcn þ F Dc f þ F pdc ¼
π 3 dc ð1−εc Þ ρp −ρ f ðg þ α c Þ 6
ð8Þ
Then substituting Eqs. (4), (5) and (7) into Eq. (8), the following equation can be obtained: 3 π d dp π ð1−εc Þ c 1−2 C Dc ρ f jU sc jU sc þ dc2 C Di ρ f jU si jU si 8 8 dp dc 2 U pd U pc π 2 π 3 − þ dc ρp ð1−εd Þ ¼ dc ð1−εc Þ ρp −ρ f ðg þ α c Þ 4 6 ð1−εc Þ ð1−εd Þ
ð9Þ
(2) Force balance in dispersed phase For unit volume of the bed, the forces on the particles in dispersed phase include the particles–fluid drag force FDdn, the collision force from the cluster Fpdcn and apparent gravity. The particles–fluid drag force FDdn can be expressed as a similar way with Eq. (1) F Ddn ¼ n F Dd
ð10Þ
The number of particles in the dispersed phase is written as: n¼
ð1−f Þð1−ε d Þ π 3 d 6 p
ð11Þ
The single particle drag force FDd is calculated by F Dd ¼
π 2 d C ρ jU jU 8 p Dd f sd sd
ð12Þ
Therefore, Eq. (10) is changed as: F Ddn ¼ F Dd
ð1−f Þð1−ε d Þ 3 ð1−f Þð1−ε d Þ ¼ C Dd ρ f jU sd jU sd π 3 4 dp dp 6
ð13Þ
Based on Eq. (6), the collision force Fpdcn is expressed as: F pdcn ¼ F pdc π 6
f 3
dc
¼
2 U pd 3 f U pc ρp ð1−εd Þ − 2 dc 1−εd 1−εc
ð14Þ
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According to the force balance in the dispersed phase, the following equation is obtained: 2 U pd U pc 3 ð1−f Þð1−ε d Þ 3 f C Dd ρ f jU sd jU sd − ρp ð1−εd Þ − 4 2 dc 1−εd 1−εc dp
ð15Þ
¼ ð1− f Þð1−ε d Þ ρp −ρ f ðg þ α d Þ
(3) Mass balance of gas and solid U f ¼ f U fc þ ð1−f ÞU f d
ð16Þ
U p ¼ f U pc þ ð1− f ÞU pd
ð17Þ
ε f ¼ f εc þ ð1−f Þε d
ð18Þ
(4) The superficial slip velocity In this work, particles and clusters were assumed to be homogeneously distributed in each phase, which was in accordance with the characteristic of particulate fluidization, so the well-known Richardson and Zaki equation [22] was selected. U sd ¼ U t εm d
ð19Þ
U sc ¼ U t εm c
ð20Þ
Ut is the terminal particle velocity, and the exponent m is calculated by the correlation suggested by Garside [23]. m¼
5:1 þ 0:28Re0:9 t
ð21Þ
1 þ 0:10Re0:9 t
(5) Equations for clusters properties Clusters manifest fluctuations in velocity, voidage, diameter and density over a wide range of space and time scales. The voidage and diameter of clusters are two major properties which affect the flow behavior of CFBs, and many researchers has performed lots of experiments and theoretical study [24–27] to propose equations of the cluster voidage and diameter. Different combinations of equations for the cluster voidage and diameter have a significant effect on the drag coefficient, so different combinations were adopted to fit for Geldart A and B particles, respectively. For Geldart A particles, adopting the similar way with Gao’s [1] investigation which proposed different drag force correlations by classifying the bed into four zones by the voidage differences, in this work only two zones (dispersed phase and dense phase) was divided, and the voidage of 0.933 was also used as the dividing criteria. The equations of the cluster voidage and diameter are as follows: For voidage εf ≤ 0.933, the equations of the cluster diameter dc of Harris et al. [28] and the cluster voidage εc of Gu [29] are dc jHarris ¼
εs 40:8−94:5εs
ð22 aÞ
ε c jGu ¼ 1−0:64 1−ð1−εs =ε sm Þ3:4
ð23 aÞ
For voidage εf N 0.933, the equations of the cluster diameter dc of Gu [20] and the cluster voidage εc of Harris et al. [28] are dc jGu ¼ dp þ 0:027−10dp εs þ 32ε6s ε c jHarris ¼ 1−
0:58ε1:48 s 0:013 þ εs1:48
ð22 bÞ ð23 bÞ
To avoid discontinuity of these equations, Lu et al. [30] used a switch equation as follows: φ¼
arctan½150 1:75ðεi −εs Þ þ 0:5 π
εi equals to the voidage for the dividing criteria 0.933. Therefore, equations of the cluster voidage and diameter are written as: dc ¼ φdc jHarr is þ ð1−φÞdc jGu
ð22 cÞ
εc ¼ φε c jGu þ ð1−φÞεc jHarris
ð23 cÞ
For Geldart B particles, the equations of the cluster diameter dc of Subbaro [31] and the cluster voidage εc of Harris et al. [28] are dc ¼
1=3 2 1−ε f = ε f −ε c 2ut
pffiffiffiffiffiffi 2 g 1 þ u2t 0:35 gDt þ dp
ð22 dÞ
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εc ¼ 1−
0:58εs 1:48 0:013 þ εs 1:48
ð23 dÞ
2.2 Computation scheme For a given system with specified Up, Uf, εf and physical parameters ρg, ρf, μf, dp, to solve 9 independent variables (Ufd, Ufc, Upd, Upc, f, dc, εc, αd, αc), we adopt the scheme as follows: 1, 2, 3, 4, 5,
Calculate εc, dc from Eqs. (22), (23). Combined with εd and εc, calculate f from Eq. (18). Calculate Usc, Usd, from Eqs. (19), (20). Combined with the definition of Usc, Usd, and Eqs. (16), (17), calculate Ufd, Ufc, Upd, Upc,then Usi can be calculated by its definition. Calculate αd, αc, from Eqs. (9), (15). The values of parameters of Usc, Usd, Usi, αd and αc as a function of voidage are summarized in Appendix B. According to the Hou et al. [2] analysis, heterogeneous drag coefficient can be expressed as follows: 2
3 ð 1−f Þ ð 1−ε Þ 1 π dp f ð 1−ε Þ 1 π f 1 π 6 c d 2 27 2 C Dd ρ f jU sd jU sd dp2 þ 1−2 4 π 3 π 3 C Dc 2 ρ f jU sc jU sc 4 dp þ π 3 C Di 2 ρ f jU si jU si 4 dc 5 ε f 2 4 dc dp dp dc 6 6 6 β¼ ug −us
ð24Þ
Theoretically, the values of Up, Uf, εf for the every local unit volume are substituted into the present structural parameters model, and every structural parameters can be calculated. Then the heterogeneous drag coefficient can be obtained and is incorporated into Fluent using the User-Defined Functions(UDF). But this way, it will lead to increasing of calculation for every local unit volume. Therefore, the heterogeneous index Hd is adopted which is correlated with local voidage in order to reduce the computational cost and is coupled with UDF to solve the drag coefficient. For comparison, the Wen and Yu drag coefficient is considered to be a contrast factor, that is f
3 1−εf ε f
ρ f u f −up C D0 ε−2:7 β0 ¼ f 4 dp
ð25Þ
The heterogeneous index is defined as the ratio between the heterogeneous drag coefficient β and the Wen and Yu drag coefficient β0, and that is Hd ¼
β β0
ð26Þ
3. CFD model and simulation method All the simulations were performed by CFD software (Fluent 6.2.16) in 2-D Cartesian space. By employing the kinetic theory of granular
flows, the solid viscosity, pressure and thermal conductivity can be expressed by the granular temperature. Appendix C shows the governing equations for two-fluid model and constitutive equations. Fig. 2 shows the geometry of the riser section, and Fig. 2(a) and (b) are used to simulate Geldart A particles and Geldart B particles, respectively. Inlet at the bottom was designated as velocity inlet for both gas and solid phases, and pressure outlet was fixed at a reference value (atmospheric) for the boundary condition outlet. The solids circulated from outlet to inlet with the same mass flux which was dynamically calculated. The no-slip wall condition was set for the gas phase, and the partially slip wall condition was set for the solid phase [33]. According to the literature, initially a certain amount of particles were filled within the bed with a given voidage 0.5, and the initial bed height was 1.225 m for riser (a) [12] and 2.64 m for riser (b) [10]. Table 1 summarizes the detailed simulation parameters. The simulation for each case is
Table 1 Parameters setting for the simulation.
Fig. 2. Geometry of 2D riser: (a) case 1 [10]; case 2 [12].
Particle diameter(μm) Particle density( kg/m3) Grid size Δx (m) Grid size Δy (m) Riser height (m) Superficial gas velocity Uf (m/s) Initial bed height H0 (m) Time step (s) Convergence criteria Maximum solid packing volume fraction
Geldart A particles
Geldart B particles
54 930 0.00225 0.035 10.5 1.52, 2.1 1.225 5.0e-4 10e-3 0.63
180 1420 0.00508 0.025 16.5 4.28, 4.78 2.64 5.0e-4 10e-3 0.63
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Fig. 3. Comparison of solid mass flux of Geldart A particles: (a) Uf = 1.52m/s; (b) Uf = 2.1m/s.
performed up to 40 s. In order to maintain sufficient statistical stabilization, the time-averaged samples are computed covering a period of 20– 40 s.
4. Results and discussion Figs. 3 and 4 present the fluctuation of solid mass flux with time, which can manifest the value of the drag coefficient, and the experimental data and simulated solid mass fluxes are summarized in Table 2. It can be seen that for drag model G (Gidaspow drag model), the average values of solid mass flux are greater than the experimental data [34,35]. This is mainly because drag model G is based on average approach and neglects the effects of multiscale structures, leading to the over-prediction of the drag coefficient [5,8,9]. Whereas, for drag model L (the present drag model), the average values of solid mass flux close to the experimental data. Besides, errors for drag model G are quite large, but errors for drag model L range from 7.9% to 40% which are very small, demonstrating that the improved structural parameters model can predict well the drag coefficient. Whether solving correctly drag coefficient or not greatly depends on considering the effects of multiscale structures. Therefore, the structural parameters model, considering the effect of heterogeneous flow structures, can
correctly compute multiscale parameters, and provide a suitable drag coefficient for CFBs. Fig. 5 shows the simulated distribution of instantaneous solids concentration for drag model G and L of Geldart A particles at Uf = 1.52m/s. Fig. 6 displays the simulated distribution of instantaneous solids concentration for drag model L of Geldart B particles at Uf = 4.78m/s. As shown in Fig. 5(a), the whole bed seems to be homogeneous whatever in the radial and axial direction because of the over-predicted drag coefficient calculated by drag model G. However, for drag model L from Fig. 5(b), the core-annular flow structure which means more particles gather near the wall than in the center can be clearly observed in the radial direction, and a dilute-top/dense-bottom flow structure can also be seen in the axial direction. Therefore, the conventional drag model G cannot simulate heterogeneous structures of CFBs, but the present drag model L can correctly simulate the dilute-top/dense-bottom and core-annular flow structure. Moreover, the spatiotemporal fluctuation of clusters can be observed from those simulations, indicating the formation and disaggregation of clusters with time. Clusters in the dense phase resist the up-flowing of gas and solid and result in the wriggling movement, which has been observed by Rhodes et al. [36] from the experiment. When clusters move into the upper dilute phase, they are blown into particles and leave out the bed or fall against the wall. During the fluctuating movement of clusters, the cluster
Fig. 4. Comparison of solid mass flux of Geldart B particles: (a) Uf = 4.28m/s; (b) Uf = 4.78m/s.
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Table 2 Summary of experimental data and simulated solid mass fluxes.
Uf (m/s) Exp. (Gs, kg/m2/s) Drag model L (c, kg/m2/s) Drag model G (Gs, kg/m2/s) Drag model L error (%) Drag model G error (%)
Geldart A particles
Geldart B particles
1.52 14.3 11.6 58.5 18.9 309.1
4.28 13.0 16.2 232.6 24.6 957.3
2.1 24.1 26.0 73.3 7.9 204.1
4.78 22.0 30.8 208.5 40 1503.8
diameter and voidage change with the variation of voidage in the dense or dilute phase, and it is agreement with the trend of the Eqs. (22) and (23) in the structural parameters model. Fig. 6 also shows the similar phenomenon which accords with the typical hydrodynamic characteristics of CFBs. It can be found that drag model L can correctly simulate the heterogeneous flow structures of CFBs. Figs. 7 and 8 show the axial profile of time-averaged voidage at different riser heights using drag model L, and their comparison with the experimental data [34,35], respectively. As can be seen from Figs. 7 and 8, drag model L can predict the flow structure of dilute-top and dense-bottom which can also be seen in Figs. 5 and 6, and is reasonable agreement with experimental results. The trends of curves are similar to other simulation results [3,10,12,37], although the values have quantitative discrepancies at the dense bottom bed which may be attributed to the inaccurate evaluation of solid inventory or incomplete 2D description [12,37]. The interaction between gas and solid is determined by drag force, and suitable drag coefficient could predict well the
Fig. 6. Simulated voidage distribution of Geldart B particles at U f = 4.78m/s for drag model L.
distribution of voidage for the whole bed. It can be seen that by considering multiscale structures, the computed drag coefficient can predict well the voidage in the axial direction for Geldart A and B particles at different riser heights and superficial gas velocity. Figs. 9 and 10 show the radial profile of time-averaged voidage at various riser heights between the simulation results and the experimental correlations of Tung et al. [38]. It can be found that the computed results of drag model L are in agreement with the experimental correlations, and core-annular flow structure, which can also be found in Figs. 5 and 6, can be reflected by the trends that the voidage in the core zone is greater than that in the annulus zone. Therefore, suitable structural parameters model and drag coefficient play a vital role in the prediction of voidage in the CFBs. 5. Conclusions The structure-based drag model has recently been an effective approach to understand the multiscale nature of heterogeneous twophase flows in CFBs, and the key to solve structure-based drag
Fig. 5. Simulated voidage distribution of Geldart A particles at Uf = 1.52m/s: (a) drag model G; (b) drag model L.
Fig. 7. Axial voidage profile of Geldart A particles: (a) Uf = 1.52m/s; (b) Uf = 2.1m/s.
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in CFBs were simulated by the CFD software with the structure-based drag model. The solid mass flux, radial and axial voidage profile were obtained. It was shown that the present drag model predicted the experimental data considerably better than the Gidaspow drag model. Moreover, the heterogeneous hydrodynamics of CFBs, including the dilute-top/ dense-bottom and core-annular flow structure, could be simulated by the present drag model. Besides, the formation and disaggregation of clusters with time can be observed. Understanding the properties of clusters is of prime importance to study the gas–solid interaction and establish drag models of CFBs. Therefore, the present structural parameters model incorporating the equations of clusters properties can compute correctly the drag coefficient, and is a valuable tool to simulate the hydrodynamics of gas–solid flows in CFBs.
Fig. 8. Axial voidage profile of Geldart B particles: (a) Uf = 4.28m/s; (b) Uf = 4.78m/s.
coefficient is to establish structural parameters model. An improved structural parameters model should be able to describe the heterogeneous multiscale structures caused by the clusters. In this work, a new structural parameters model based on the available structure-based drag model was established incorporating with equations of clusters properties. The hydrodynamic characteristics of gas–solid flow behaviors
Nomenclature CD drag coefficient CDc drag coefficient in cluster phase CDd drag coefficient in dispersed phase CDi drag coefficient between cluster and gas in dispersed phase CD0 drag coefficient for single particle in gas flow CD averaged drag coefficient dc diameter of cluster, m dp diameter of particle, m FD total drag force of flow gas on the particles in unit volume of flow, N/m3 FDc drag force of flow gas on single particle in cluster phase, N FDcn drag force of flow gas in cluster phase on the particles in single cluster, N FDcu drag force of gas flow on particles in unit volume of cluster phase, N/m3 FDd drag force of flow gas on the single particle in dispersed phase, N FDi drag force of flow gas in dispersed phase on the single cluster, N
Fig. 9. Radial voidage profile of Geldart A particles: (a) Uf = 1.52m/s, H = 8 m; (b) Uf = 1.52m/s, H = 5 m; (c) Uf = 2.1m/s, H = 8 m; (d) Uf = 2.1m/s, H = 5 m.
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Fig. 10. Radial voidage profile of Geldart B particles: (a) Uf = 4.28m/s, H = 10 m; (b) Uf = 4.28m/s, H = 5 m; (c) Uf = 4.78m/s, H = 10 m; (d) Uf = 4.78m/s, H = 5 m.
Fpdc Fpdcn f Gs HD Rep n Uf uf Ufc Ufd Up up Upc Upd Us Usc Usd Usi Ut
collision force, N collision force exerting the dispersed phase in unit volume, N/m3 volume friction of cluster phase solids circulation flux, kg/m2/s heterogeneity index Reynolds Number for particle the number of particles superficial gas velocity, m/s gas velocity, m/s superficial gas velocity in cluster phase, m/s superficial gas velocity in dispersed phase, m/s superficial particle velocity, m/s particle velocity, m/s superficial particle velocity in cluster phase, m/s superficial particle velocity in dispersed phase, m/s superficial slip velocity between gas and particles, m/s superficial slip velocity between gas and particle in cluster phase, m/s superficial slip velocity between gas and particle in dispersed phase, m/s superficial slip velocity between gas in dispersed phase and cluster, m/s the terminal particle velocity, m/s
Greek symbols αd the accelerated velocity of particles in the dispersed phase m/s2 αc the accelerated velocity of particles in the cluster phase m/s2 β drag coefficient including the heterogeneous flow structure β0 Wen and Yu drag coefficient ε voidage
εc εd εf εi εmf εmin εs μf ρf ρp
voidage in cluster phase voidage in dispersed phase averaged voidage voidage of inter-phase voidage at minimum fluidization minimum particle voidage at packing averaged solid volume fraction viscosity of gas kg/(m s) density of gas, kg/m3 density of particle, kg/m3
Acknowledgements The authors are grateful to the financial support from National Natural Science Foundation of China under Grant No. 21325628 and from the State Key Development Program for Basic Research of China (973 Program) under Grant 2015CB251402. Appendix A. Variables involved in the structural parameters model Variables involved in the structural parameters model are given below. Superficial slip velocity in different phases: U sc ¼ U fc −U pc U si ¼
εc 1−εc
ðA1Þ
U fd U pc − ε ð1−f Þ εd 1−εc d
ðA2Þ
εd 1−εd
ðA3Þ
U sd ¼ U f d −U pd
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Appendix C. Governing equations and constitutive equations
Reynolds number in different phases: Rep ¼
ρ f dp U sc μf
ðA4Þ
Rep ¼
ρ f dc U si μf
ðA5Þ
Rep ¼
ρ f dp U sd μf
ðA6Þ
Governing equations and constitutive equations are given below. Continuity equation (k = g, s): ∂ðεk ρk Þ þ ∇ðε k ρk uk Þ ¼ 0 ∂t
ðC1Þ
εs þ εg ¼ 1
ðC2Þ
Momentum equation (k = g, s; l = s, g):
Standard drag coefficient: 8 < 0:44 24 C D0 ¼ 1 þ 0:15Re0:687 : p Rep
Rep N 1000 Rep ≤ 1000
∂ðεk ρk uk Þ þ ∇ðε k ρk uk uk Þ ¼ −εk ∇pg þ εk ρk g þ ∇τ k þ βðul −uk Þ ∂t ðA7Þ
Effective drag coefficient in different phases: 8 −4:7 > < C D0 εc ð1−εc Þμ f 7 ¼ > : 200 ε3 ρ d U þ 3ε 3 c c f p sc
C Di
8 −4:7 > < C D0 εi ð1−εi Þμ f 7 ¼ > : 200 ε3 ρ d U þ 3ε3 p si f i i
C Dd
8 −4:7 > < C D0 εd ð1−εd Þμ f 7 ¼ > : 200 ε 3 ρ d U þ 3ε3 d f p sd d
εc ≥ 0:8 εc b 0:8
ðA8Þ
τs ¼ ½−ps þ λs ∇μ s δ þ 2μ s Ss
ðC5Þ
Deformation rate: εi ≥ 0:8 εi b 0:8
ðA9Þ
Sk ¼
i 1 1h ∇uk þ ð∇uk ÞT − ∇uk δ 2 3
ðC6Þ
Solid phase pressure: εd ≥ 0:8 εd b 0:8
ðA10Þ
ps ¼ εs ρs Θ½1 þ 2ð1 þ eÞεs g 0
μs ¼
Uf = 1.52 m/s, Uf =
(B1)
Usd = 0.0825
(B2)
U si ¼ −23:9ε f 3 þ 42:92ε f 2 −22:21ε f þ 4:146
2.1 m/s Uf = 1.52 m/s
(B3)
U si ¼ −23:85ε f 3 þ 38:16ε f 2 −14:53ε f þ 1:654 Uf = 2.1 m/s Uf = 1.52 m/s ε −0:5603 2 α c ¼ 9:79 105 ; exp½−ð f 0:2123 Þ Uf = 2.1 m/s ε f −0:5623 2 6 α c ¼ 1:863 10 ; exp½−ð 0:2109 Þ Uf = 1.52 m/s ε −0:4323 2 α d ¼ −7:691 109 ; exp½−ð f 0:2047 Þ 2 Uf = 2.1 m/s ε −0:4341 α d ¼ −1:454 1010 ; exp½−ð f 0:2041 Þ
pffiffiffiffiffiffiffi 5 ρ dp πΘ 96 s
ðC8Þ
ðC9Þ
Solid phase bulk viscosity:
Geldart A particles:
2.1 m/s Uf = 1.52 m/s, Uf =
rffiffiffiffi 2 4 2 Θ 2μ s:dilute 4 εs ρs dp g 0 ð1 þ eÞ 1 þ ð1 þ eÞεs g 0 þ 3 π ð1 þ eÞg 0 5
μ s:dilute ¼
λs ¼ 20:97ε f 2 −30:93ε f 2 þ11:55 3 −179:8ε 2 −150:2ε þ346:7 f f
ðC7Þ
Solid phase shear viscosity:
The values of Usc, Usd, Usi, αd and αc can be solved by the nonlinear equations under a given system with specified Up, Uf, εf and physical parameters ρg, ρf, μf, dp. Traverse εf within [εmf, 1], those values as a function of voidage εf can be obtained as follows:
f
ðC4Þ
Solid phase stress:
Appendix B. The values of Usc, Usd, Usi, αd and αc as a function of voidage
U sc ¼ ε
ðC3Þ
Gas phase stress: τg ¼ 2μ g Sg
C Dc
525
rffiffiffiffi 4 2 Θ εs ρs dp g 0 ð1 þ eÞ 3 π
ðC10Þ
Radial distribution functions: " 1=3 #−1 εg g 0 ¼ 1− εsm
ðC11Þ
(B4) (B5) (B6) (B7) (B8)
Granular temperature equation: 3 ∂ðεs ρs ΘÞ þ ∇ðε s ρs us ΘÞ ¼ τs : ∇us −∇q−γ þ βC g C−3βΘ 2 ∂t
ðC12Þ
Collisional energy dissipation: Geldart B particles: U sc ¼ ε
20:75ε f 2 −43:22ε f 2 þ23:05 3 þ723:7ε 2 −1455ε þ731:4 f f
f
Usd = 0.9707
Uf = 4.28 m/s, Uf =
(B9)
4.78 m/s Uf = 4.28 m/s, Uf =
(B10)
4.78 m/s U si ¼ −161:3ε f 3 þ 333:8ε f 2 −220:3ε f þ 49:12 Uf = 4.28 m/s Uf = 4.78 m/s U si ¼ −198:2ε f 3 þ 414:3ε f 2 −277ε f þ 62:48
(B11)
Uf = 4.28 m/s
(B13)
Uf = 4.78 m/s
(B14)
Uf = 4.28 m/s
(B15)
Uf = 4.78 m/s
(B16)
ε f −0:3778 2 0:2311 Þ ε −0:3838 2
α c ¼ 3:378 107 ; exp½−ð α c ¼ 4:076 107 ; exp½−ð α d ¼ −3:375 α d ¼ −3:894
f
0:2299
Þ
ε −0:1139 2 10 ; exp½−ð f 0:2449 Þ ε −0:1212 2 1012 ; exp½−ð f 0:244 Þ 12
(B12)
" rffiffiffiffiffiffiffiffiffi # 4 Θ 2 2 −∇us γ ¼ 3 1−e εs ρs g 0 Θ dp π
ðC13Þ
Flux of fluctuating energy: q ¼ −k∇Θ
ðC14Þ
Conductivity if the fluctuating energy: k¼
2 k 2k 1 þ 65 ð1 þ eÞεs g0 c þk ð1 þ eÞg 0
ðC15Þ
526 k
W. Liu et al. / Powder Technology 286 (2015) 516–526
k ¼
pffiffiffiffiffiffiffi 75 d ρ Θπ 384 p s
ðC16Þ rffiffiffiffi Θ π
c
k ¼ 2ε2s ρs dp g 0 ð1 þ eÞ
ðC17Þ
Drag coefficient for Gidaspow et al. [32]: 8
s > εg εs μ g ρg εs ug −us
> > > þ 1:75 εg b0:8 150 < 2 dp εg dp β¼
> > 3 εs εg ρg ug −us −2:65 > > CD εg εg ≥0:8 : 4 dp
ðC18Þ
Drag coefficient for this work is Eq. (24). References [1] J. Gao, X. Lan, Y. Fan, J. Chang, G. Wang, C. Lu, C. Xu, CFD modeling and validation of the turbulent fluidized bed of FCC particles, AICHE J. 55 (2009) 1680–1694. [2] B. Hou, H. Li, Relationship between flow structure and transfer coefficients in fast fluidized beds, Chem. Eng. J. 157 (2010) 509–519. [3] W. Wang, J. Li, Simulation of gas–solid two-phase flow by a multi-scale CFD approach – of the EMMS model to the sub-grid level, Chem. Eng. Sci. 62 (2007) 208–231. [4] J.D. Gibbon, E.S. Titi, Cluster formation in complex multi-scale systems, Proc. R. Soc. London, Ser. A 461 (2005) 3089–3097. [5] J. Li, M. Kwauk, Exploring complex systems in chemical engineering – the multiscale methodology, Chem. Eng. Sci. 58 (2003) 521–535. [6] D. Gidaspow, Multiphase flow and fluidization: continuum and kinetic theory descriptions, Academic Press, 1994. [7] M. Syamlal, O’Brien T.J., The derivation of a drag coefficient formula from velocityvoidage correlations, U. S. Department of Energy, Office of Fossil Energy, Technical Report 1994. [8] J. Li, M. Kwauk, Multiscale nature of complex fluid-particle systems, Ind. Eng. Chem. Res. 40 (2001) 4227–4237. [9] J. Ma, W. Ge, X. Wang, J. Wang, J. Li, High-resolution simulation of gas–solid suspension using macro-scale particle methods, Chem. Eng. Sci. 61 (2006) 7096–7106. [10] J. Wang, W. Ge, J. Li, Eulerian simulation of heterogeneous gas–solid flows in CFB risers: EMMS-based sub-grid scale model with a revised cluster description, Chem. Eng. Sci. 63 (2008) 1553–1571. [11] S. Wang, H. Lu, G. Liu, Z. Sheng, P. Xu, D. Gidaspow, Modeling of cluster structuredependent drag with Eulerian approach for circulating fluidized beds, Powder Technol. 208 (2011) 98–110. [12] N. Yang, W. Wang, W. Ge, J. Li, CFD simulation of concurrent-up gas–solid flow in circulating fluidized beds with structure-dependent drag coefficient, Chem. Eng. J. 96 (2003) 71–80. [13] J. Wang, A review of Eulerian simulation of Geldart A particles in gas-fluidized beds, Ind. Eng. Chem. Res. 48 (2009) 5567–5577. [14] W. Shuai, L. Huilin, L. Guodong, S. Zhiheng, X. Pengfei, D. Gidaspow, Modeling of cluster structure-dependent drag with Eulerian approach for circulating fluidized beds, Powder Technol. 208 (2011) 98–110.
[15] J. Gao, J. Chang, X. Lan, Y. Yang, C. Lu, C. Xu, CFD modeling of mass transfer and stripping efficiency in FCCU strippers, AICHE J. 54 (2008) 1164–1177. [16] J. Gao, J. Chang, C. Xu, X. Lan, Y. Yang, CFD simulation of gas solid flow in FCC strippers, Chem. Eng. Sci. 63 (2008) 1827–1841. [17] S. Wang, H.L. Lu, Z.H. Hao, F.X. Zhao, L.Y. Sun, Q.H. Zhang, Modeling of reactive gas– solid flows in riser reactors using a multi-scale chemical reaction model, Chem. Eng. Sci. 116 (2014) 773–780. [18] H.L. Lu, Q.Q. Sun, Y.R. He, Y.L. Sun, H.M. Ding, L. Xiang, Numerical study of particle cluster flow in risers with cluster-based approach, Chem. Eng. Sci. 60 (2005) 6757–6767. [19] A. Nikolopoulos, D. Papafotiou, N. Nikolopoulos, P. Grammelis, E. Kakaras, An advanced EMMS scheme for the prediction of drag coefficient under a 1.2 MWth CFBC isothermal flow-Part I: Numerical formulation, Chem. Eng. Sci. 65 (2010) 4080–4088. [20] W. Gu, J. Chen, A model for solid concentration in circulating fluidized beds, Fluidization IX, 1998 501–508. [21] J.M. Matsen, Mechanisms of choking and entrainment, Powder Technol. 32 (1982) 21–33. [22] J.F. Richardson, W.N. Zaki, Sedimentation and fluidization, Trans. Inst. Chem. Eng. 32 (1954) 35–53. [23] J. Garside, M.R. Al-Dibouni, Velocity–voidage relationships for fluidization and sedimentation in solid–liquid systems, Ind. Eng. Chem. Process Des. Dev. 16 (1977) 206–214. [24] M. Rhodes, H. Mineo, T. Hirama, Particle motion at the wall of a circulating fluidizedbed, Powder Technol. 70 (1992) 207–214. [25] A.K. Sharma, K. Tuzla, J. Matsen, J.C. Chen, Parametric effects of particle size and gas velocity on cluster characteristics in fast fluidized beds, Powder Technol. 111 (2000) 114–122. [26] R. Cocco, F. Shaffer, R. Hays, S.B.R. Karri, T. Knowlton, Particle clusters in and above fluidized beds, Powder Technol. 203 (2010) 3–11. [27] A.T. Andrews, P.N. Loezos, S. Sundaresan, Coarse-grid simulation of gas–particle flows in vertical risers, Ind. Eng. Chem. Res. 44 (2005) 6022–6037. [28] A. Harris, J. Davidson, R. Thorpe, The prediction of particle cluster properties in the near wall region of a vertical riser (200157), Powder Technol. 127 (2002) 128–143. [29] W. Gu, Diameters of catalyst clusters in FCC, AIChE symposium series, American Institute of Chemical Engineers, 1971–c2002, New York, NY 1999, pp. 42–47. [30] H.L. Lu, D. Gidaspow, Hydrodynamics of binary fluidization in a riser: CFD simulation using two granular temperatures, Chem. Eng. Sci. 58 (2003) 3777–3792. [31] D. Subbarao, A model for cluster size in risers, Powder Technol. 199 (2010) 48–54. [32] D. Gidaspow, R. Bezburuah, J. Ding, Hydrodynamics of circulating fluidized beds, kinetic theory approach, Fluidization VII, Proceedings of the 7th Engineering Foundation conference on fluidization 1992, pp. 75–82. [33] J. Ding, D. Gidaspow, A bubbling fluidization model using kinetic theory of granular flow, AICHE J. 36 (1990) 523–538. [34] J. Li, Particle-fluid two-phase flow: the energy-minimization multi-scale method, Metallurgical Industry Press, 1994. [35] E.R. Monazam, L.J. Shadle, L.O. Lawson, A transient method for determination of saturation carrying capacity, Powder Technol. 121 (2001) 205–212. [36] M.J. Rhodes, M. Sollaart, X.S. Wang, Flow structure in a fast fluid bed, Powder Technol. 99 (1998) 194–200. [37] N. Yang, W. Wang, W. Ge, L. Wang, J. Li, Simulation of heterogeneous structure in a circulating fluidized-bed riser by combining the two-fluid model with the EMMS approach, Ind. Eng. Chem. Res. 43 (2004) 5548–5561. [38] Y. Tung, J. Li, M. Kwauk, Radial voidage profiles in a fast fluidized bed, in: M. Kwauk, D. Kunii (Eds.), Fluidization’88: science and technology, Science Press, Beijing 1988, pp. 139–145.
Contents lists available at ScienceDirect
Powder Technology journal homepage: www.elsevier.com/locate/powtec
A new structural parameters model based on drag coefficient for simulation of circulating fluidized beds Wenming Liu a,b, Hongzhong Li a,⁎, Qingshan Zhu a,⁎, Quanhong Zhu a a b
State Key Laboratory of Multi-phase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing 100190, PR China University of Chinese Academy of Sciences, Beijing 100049, PR China
a r t i c l e
i n f o
Article history: Received 13 January 2015 Received in revised form 27 August 2015 Accepted 31 August 2015 Available online 2 September 2015 Keywords: Fluidization Simulation Hydrodynamics CFBs Structure-based drag model
a b s t r a c t This work presented a new scheme to establish structural parameters model, and the model was used to solve structural parameters based on the available structure-based drag model. By combining with the Eulerian twofluid model, the hydrodynamics of circulating fluidized beds (CFBs) was simulated. Different combinations of clusters properties, including the cluster voidage and diameter, were adopted to fit for Geldart A and B particles and to close the insufficient solving equations, respectively. The simulated solid mass flux, radial and axial voidage profiles were in agreement with the experimental data. The dilute-top/dense-bottom and the coreannular flow structure were also captured. Moreover the spatiotemporal fluctuation of clusters can be observed from those simulations. Simulation results showed the combination of the structural parameters model with the available structure-based drag model can predict well the hydrodynamics for Geldart A and B particles in CFBs. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Circulating fluidized beds (CFBs) are widely used in various modern industrial processes, such as fluid catalytic cracking and coal combustion. Because of high flow rate of gas–solid and intensive contacting, CFBs have favorable hydrodynamics, reactions and heat transfer characteristics. Understanding of the hydrodynamic characteristics is the key to design and the scale-up of such reactors. With the increase of the computational ability, computational fluid dynamics (CFD) has been a valuable tool to predict the fluid dynamics, which is of vital importance to the scale-up of CFBs. For heterogeneous gas–solid flows in CFBs, drag models have a significant effect on the simulation of bed hydrodynamics [1–3]. The competition and compromise in different spatiotemporal scales lead to the formation of multiscale structures [4,5]. Conventional drag models, such as Gidaspow [6] and Syamlal-O’Brien [7], have been proved to over-predict the drag coefficient because they neglect the multiscale structures [5,8,9] in CFBs. Multiscale structures of CFBs are characterized by clusters, and heterogeneous flows can be divided into the cluster phase, the dispersed phase and the inter-phase which are usually defined by 8–10 structure parameters [10–12]. Because conventional two-fluid drag models assume homogeneous conditions inside the control volume, they cannot predict structure parameters. Structure-based
⁎ Corresponding authors. E-mail addresses: [email protected] (H. Li), [email protected] (Q. Zhu).
http://dx.doi.org/10.1016/j.powtec.2015.08.049 0032-5910/© 2015 Elsevier B.V. All rights reserved.
drag models have already been successfully developed to understand the multiscale nature of heterogeneous two-phase flows in CFBs [13]. The key to solve structure-based drag models is to solve the structural parameters. Due to the lack of enough number of equations to solve these parameters, many researchers adopted minimization stability condition to close the insufficient solving equations, such as the energy minimization multi-scale model (EMMS) [12] and the cluster structuredependent (CSD) drag coefficient model [14]. Adopting other measures to close the equations of structural parameters models has received much less attention and remains a challenge for the complex heterogeneous flows of CFBs. With respect to the system of multiscale structures, clusters play a dominant role in gas–solid interaction [15–17]. Clusters have two major properties, including diameter and voidage; equations of clusters properties have been used by many researchers to improve drag models. Gao et al. [15] adopted the cluster diameter, derived from the experimental particles terminal velocity, to replace the particle diameter of Gidaspow drag model, and the simulation results were in agreement with experimental data. Lu et al. [18] used equations of clusters properties to modify Gidaspow drag model and conservation equations of the kinetic theory of granular flow. As for EMMS model, Nikolopoulos et al. [19] claimed that the predicted cluster diameter was smaller than the diameter of a single particle or negative for averaged solid volume fraction εs less than 0.01 or greater than 0.44, respectively, and introduced Gu and Chen’s [20] correlation of the cluster diameter to confront these problems. Wang et al. [10] extended EMMS model from Geldart A to Geldart B particles by integrating the equation of the cluster voidage. Therefore, clusters properties are of prime importance to the simulation
W. Liu et al. / Powder Technology 286 (2015) 516–526
of CFBs, and an improved structural parameters model should be able to describe the heterogeneous multiscale structures caused by the clusters. In the present work, a new structural parameters model was established incorporating with equations of clusters properties, which can not only close the insufficient solving equations but also take into
517
consideration the effect of clusters. By incorporating it into an Eulerian two-fluid model, the gas–solid flow hydrodynamics of CFBs was simulated by using CFD software (FLUENT6.2.16). Simulation results were compared with the experimental data available in literature to validate its feasibility.
2. Mathematical model 2.1. Structural parameters model Fig. 1 shows multiscale resolution of structure and gas–solid interaction proposed by Wang et al. [10]. The complex heterogeneous flow structures of CFBs can be divided into three simple homogeneous phases, including the cluster phase, the dispersed phase and the inter-phase. Based on the homogeneous assumption and Matsen’s [21] investigation, the voidage in dispersed phase εd is defined as 0.9997. The heterogeneous structures have 9 unknown parameters, namely, Ufd, Upd and αd for the dispersed phase, εc, dc, Ufc, Upc, f and αc for the cluster phase. The solving equations can be detailed as follows, and the related variables involved in these equations are summarized in Appendix A. (1) Force balance in cluster phase The forces on the cluster include cluster-fluid drag force FDcn inside cluster phase, cluster-fluid drag force FDcf outside cluster phase, the collision force Fpdc from particles in the dispersed phase outside cluster phase and apparent gravity. The drag force FDcn equals to the drag force of single particle multiplying the effective number of particles inside cluster phase. F Dcn ¼ nF Dc
ð1Þ
The drag force of single particle FDc can be expressed by F Dc ¼
π 2 dp C Dc ρ f jU sc jU sc 8
ð2Þ
The force on the particles of outside surface of the cluster is different from those inside the cluster. The former force is caused by high gas velocity from the dispersed phase, and the latter force derives from low gas velocity of the cluster phase which means only half of particles of outside surface are relatively effective. Therefore, the effective number of particles inside cluster phase is calculated by π 3 3 dp ð1−εc Þ 1 πd 2 ð1−ε Þ dc dp c c n¼6 π − ¼ ð 1−ε Þ 1−2 c π 2 2 dp dc dp3 dp 6 4
Fig. 1. Resolution of structure and gas–solid interaction proposed by Wang [10].
ð3Þ
518
W. Liu et al. / Powder Technology 286 (2015) 516–526
Then, Eq. (1) is written as: F Dcn ¼
3 dp π 2 dc dp C Dc ρ f jU sc jU sc ð1−εc Þ 1−2 8 dp dc
ð4Þ
The drag force outside cluster phase FDcf is expressed by F Dc f ¼
π 2 dc C Di ρ f jU si jU si 8
ð5Þ
When the cluster moves upward at the velocity Upc/(1 − εc), particles in the dispersed phase move upward at the velocity Upd/(1 − εd). Because the cluster diameter is much bigger than the particle diameter, the particle velocity is faster than the cluster velocity. The particles below the cluster chase and bump the bottom of the cluster, then the particle velocity would reduced to be Upc/(1 − εc). The transverse collision would be offset because of symmetry, and the particles above the cluster would not be considered. Therefore, the mass flow rate of particles below the cluster is expressed as follows: Gpdc ¼
U pd U pc π 2 dc − ρ ð1−εd Þ 4 ð1−εc Þ ð1−εd Þ p
ð6Þ
According to the momentum conservation, the collision force Fpdc is written as: 2 π U pd U pc − F pdc ¼ Gpdc upd −upc ¼ d2c ρp ð1−εd Þ 4 ð1−εc Þ ð1−εd Þ
ð7Þ
Based on the force balance in the cluster phase, the equation is calculated by F Dcn þ F Dc f þ F pdc ¼
π 3 dc ð1−εc Þ ρp −ρ f ðg þ α c Þ 6
ð8Þ
Then substituting Eqs. (4), (5) and (7) into Eq. (8), the following equation can be obtained: 3 π d dp π ð1−εc Þ c 1−2 C Dc ρ f jU sc jU sc þ dc2 C Di ρ f jU si jU si 8 8 dp dc 2 U pd U pc π 2 π 3 − þ dc ρp ð1−εd Þ ¼ dc ð1−εc Þ ρp −ρ f ðg þ α c Þ 4 6 ð1−εc Þ ð1−εd Þ
ð9Þ
(2) Force balance in dispersed phase For unit volume of the bed, the forces on the particles in dispersed phase include the particles–fluid drag force FDdn, the collision force from the cluster Fpdcn and apparent gravity. The particles–fluid drag force FDdn can be expressed as a similar way with Eq. (1) F Ddn ¼ n F Dd
ð10Þ
The number of particles in the dispersed phase is written as: n¼
ð1−f Þð1−ε d Þ π 3 d 6 p
ð11Þ
The single particle drag force FDd is calculated by F Dd ¼
π 2 d C ρ jU jU 8 p Dd f sd sd
ð12Þ
Therefore, Eq. (10) is changed as: F Ddn ¼ F Dd
ð1−f Þð1−ε d Þ 3 ð1−f Þð1−ε d Þ ¼ C Dd ρ f jU sd jU sd π 3 4 dp dp 6
ð13Þ
Based on Eq. (6), the collision force Fpdcn is expressed as: F pdcn ¼ F pdc π 6
f 3
dc
¼
2 U pd 3 f U pc ρp ð1−εd Þ − 2 dc 1−εd 1−εc
ð14Þ
W. Liu et al. / Powder Technology 286 (2015) 516–526
519
According to the force balance in the dispersed phase, the following equation is obtained: 2 U pd U pc 3 ð1−f Þð1−ε d Þ 3 f C Dd ρ f jU sd jU sd − ρp ð1−εd Þ − 4 2 dc 1−εd 1−εc dp
ð15Þ
¼ ð1− f Þð1−ε d Þ ρp −ρ f ðg þ α d Þ
(3) Mass balance of gas and solid U f ¼ f U fc þ ð1−f ÞU f d
ð16Þ
U p ¼ f U pc þ ð1− f ÞU pd
ð17Þ
ε f ¼ f εc þ ð1−f Þε d
ð18Þ
(4) The superficial slip velocity In this work, particles and clusters were assumed to be homogeneously distributed in each phase, which was in accordance with the characteristic of particulate fluidization, so the well-known Richardson and Zaki equation [22] was selected. U sd ¼ U t εm d
ð19Þ
U sc ¼ U t εm c
ð20Þ
Ut is the terminal particle velocity, and the exponent m is calculated by the correlation suggested by Garside [23]. m¼
5:1 þ 0:28Re0:9 t
ð21Þ
1 þ 0:10Re0:9 t
(5) Equations for clusters properties Clusters manifest fluctuations in velocity, voidage, diameter and density over a wide range of space and time scales. The voidage and diameter of clusters are two major properties which affect the flow behavior of CFBs, and many researchers has performed lots of experiments and theoretical study [24–27] to propose equations of the cluster voidage and diameter. Different combinations of equations for the cluster voidage and diameter have a significant effect on the drag coefficient, so different combinations were adopted to fit for Geldart A and B particles, respectively. For Geldart A particles, adopting the similar way with Gao’s [1] investigation which proposed different drag force correlations by classifying the bed into four zones by the voidage differences, in this work only two zones (dispersed phase and dense phase) was divided, and the voidage of 0.933 was also used as the dividing criteria. The equations of the cluster voidage and diameter are as follows: For voidage εf ≤ 0.933, the equations of the cluster diameter dc of Harris et al. [28] and the cluster voidage εc of Gu [29] are dc jHarris ¼
εs 40:8−94:5εs
ð22 aÞ
ε c jGu ¼ 1−0:64 1−ð1−εs =ε sm Þ3:4
ð23 aÞ
For voidage εf N 0.933, the equations of the cluster diameter dc of Gu [20] and the cluster voidage εc of Harris et al. [28] are dc jGu ¼ dp þ 0:027−10dp εs þ 32ε6s ε c jHarris ¼ 1−
0:58ε1:48 s 0:013 þ εs1:48
ð22 bÞ ð23 bÞ
To avoid discontinuity of these equations, Lu et al. [30] used a switch equation as follows: φ¼
arctan½150 1:75ðεi −εs Þ þ 0:5 π
εi equals to the voidage for the dividing criteria 0.933. Therefore, equations of the cluster voidage and diameter are written as: dc ¼ φdc jHarr is þ ð1−φÞdc jGu
ð22 cÞ
εc ¼ φε c jGu þ ð1−φÞεc jHarris
ð23 cÞ
For Geldart B particles, the equations of the cluster diameter dc of Subbaro [31] and the cluster voidage εc of Harris et al. [28] are dc ¼
1=3 2 1−ε f = ε f −ε c 2ut
pffiffiffiffiffiffi 2 g 1 þ u2t 0:35 gDt þ dp
ð22 dÞ
520
W. Liu et al. / Powder Technology 286 (2015) 516–526
εc ¼ 1−
0:58εs 1:48 0:013 þ εs 1:48
ð23 dÞ
2.2 Computation scheme For a given system with specified Up, Uf, εf and physical parameters ρg, ρf, μf, dp, to solve 9 independent variables (Ufd, Ufc, Upd, Upc, f, dc, εc, αd, αc), we adopt the scheme as follows: 1, 2, 3, 4, 5,
Calculate εc, dc from Eqs. (22), (23). Combined with εd and εc, calculate f from Eq. (18). Calculate Usc, Usd, from Eqs. (19), (20). Combined with the definition of Usc, Usd, and Eqs. (16), (17), calculate Ufd, Ufc, Upd, Upc,then Usi can be calculated by its definition. Calculate αd, αc, from Eqs. (9), (15). The values of parameters of Usc, Usd, Usi, αd and αc as a function of voidage are summarized in Appendix B. According to the Hou et al. [2] analysis, heterogeneous drag coefficient can be expressed as follows: 2
3 ð 1−f Þ ð 1−ε Þ 1 π dp f ð 1−ε Þ 1 π f 1 π 6 c d 2 27 2 C Dd ρ f jU sd jU sd dp2 þ 1−2 4 π 3 π 3 C Dc 2 ρ f jU sc jU sc 4 dp þ π 3 C Di 2 ρ f jU si jU si 4 dc 5 ε f 2 4 dc dp dp dc 6 6 6 β¼ ug −us
ð24Þ
Theoretically, the values of Up, Uf, εf for the every local unit volume are substituted into the present structural parameters model, and every structural parameters can be calculated. Then the heterogeneous drag coefficient can be obtained and is incorporated into Fluent using the User-Defined Functions(UDF). But this way, it will lead to increasing of calculation for every local unit volume. Therefore, the heterogeneous index Hd is adopted which is correlated with local voidage in order to reduce the computational cost and is coupled with UDF to solve the drag coefficient. For comparison, the Wen and Yu drag coefficient is considered to be a contrast factor, that is f
3 1−εf ε f
ρ f u f −up C D0 ε−2:7 β0 ¼ f 4 dp
ð25Þ
The heterogeneous index is defined as the ratio between the heterogeneous drag coefficient β and the Wen and Yu drag coefficient β0, and that is Hd ¼
β β0
ð26Þ
3. CFD model and simulation method All the simulations were performed by CFD software (Fluent 6.2.16) in 2-D Cartesian space. By employing the kinetic theory of granular
flows, the solid viscosity, pressure and thermal conductivity can be expressed by the granular temperature. Appendix C shows the governing equations for two-fluid model and constitutive equations. Fig. 2 shows the geometry of the riser section, and Fig. 2(a) and (b) are used to simulate Geldart A particles and Geldart B particles, respectively. Inlet at the bottom was designated as velocity inlet for both gas and solid phases, and pressure outlet was fixed at a reference value (atmospheric) for the boundary condition outlet. The solids circulated from outlet to inlet with the same mass flux which was dynamically calculated. The no-slip wall condition was set for the gas phase, and the partially slip wall condition was set for the solid phase [33]. According to the literature, initially a certain amount of particles were filled within the bed with a given voidage 0.5, and the initial bed height was 1.225 m for riser (a) [12] and 2.64 m for riser (b) [10]. Table 1 summarizes the detailed simulation parameters. The simulation for each case is
Table 1 Parameters setting for the simulation.
Fig. 2. Geometry of 2D riser: (a) case 1 [10]; case 2 [12].
Particle diameter(μm) Particle density( kg/m3) Grid size Δx (m) Grid size Δy (m) Riser height (m) Superficial gas velocity Uf (m/s) Initial bed height H0 (m) Time step (s) Convergence criteria Maximum solid packing volume fraction
Geldart A particles
Geldart B particles
54 930 0.00225 0.035 10.5 1.52, 2.1 1.225 5.0e-4 10e-3 0.63
180 1420 0.00508 0.025 16.5 4.28, 4.78 2.64 5.0e-4 10e-3 0.63
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Fig. 3. Comparison of solid mass flux of Geldart A particles: (a) Uf = 1.52m/s; (b) Uf = 2.1m/s.
performed up to 40 s. In order to maintain sufficient statistical stabilization, the time-averaged samples are computed covering a period of 20– 40 s.
4. Results and discussion Figs. 3 and 4 present the fluctuation of solid mass flux with time, which can manifest the value of the drag coefficient, and the experimental data and simulated solid mass fluxes are summarized in Table 2. It can be seen that for drag model G (Gidaspow drag model), the average values of solid mass flux are greater than the experimental data [34,35]. This is mainly because drag model G is based on average approach and neglects the effects of multiscale structures, leading to the over-prediction of the drag coefficient [5,8,9]. Whereas, for drag model L (the present drag model), the average values of solid mass flux close to the experimental data. Besides, errors for drag model G are quite large, but errors for drag model L range from 7.9% to 40% which are very small, demonstrating that the improved structural parameters model can predict well the drag coefficient. Whether solving correctly drag coefficient or not greatly depends on considering the effects of multiscale structures. Therefore, the structural parameters model, considering the effect of heterogeneous flow structures, can
correctly compute multiscale parameters, and provide a suitable drag coefficient for CFBs. Fig. 5 shows the simulated distribution of instantaneous solids concentration for drag model G and L of Geldart A particles at Uf = 1.52m/s. Fig. 6 displays the simulated distribution of instantaneous solids concentration for drag model L of Geldart B particles at Uf = 4.78m/s. As shown in Fig. 5(a), the whole bed seems to be homogeneous whatever in the radial and axial direction because of the over-predicted drag coefficient calculated by drag model G. However, for drag model L from Fig. 5(b), the core-annular flow structure which means more particles gather near the wall than in the center can be clearly observed in the radial direction, and a dilute-top/dense-bottom flow structure can also be seen in the axial direction. Therefore, the conventional drag model G cannot simulate heterogeneous structures of CFBs, but the present drag model L can correctly simulate the dilute-top/dense-bottom and core-annular flow structure. Moreover, the spatiotemporal fluctuation of clusters can be observed from those simulations, indicating the formation and disaggregation of clusters with time. Clusters in the dense phase resist the up-flowing of gas and solid and result in the wriggling movement, which has been observed by Rhodes et al. [36] from the experiment. When clusters move into the upper dilute phase, they are blown into particles and leave out the bed or fall against the wall. During the fluctuating movement of clusters, the cluster
Fig. 4. Comparison of solid mass flux of Geldart B particles: (a) Uf = 4.28m/s; (b) Uf = 4.78m/s.
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Table 2 Summary of experimental data and simulated solid mass fluxes.
Uf (m/s) Exp. (Gs, kg/m2/s) Drag model L (c, kg/m2/s) Drag model G (Gs, kg/m2/s) Drag model L error (%) Drag model G error (%)
Geldart A particles
Geldart B particles
1.52 14.3 11.6 58.5 18.9 309.1
4.28 13.0 16.2 232.6 24.6 957.3
2.1 24.1 26.0 73.3 7.9 204.1
4.78 22.0 30.8 208.5 40 1503.8
diameter and voidage change with the variation of voidage in the dense or dilute phase, and it is agreement with the trend of the Eqs. (22) and (23) in the structural parameters model. Fig. 6 also shows the similar phenomenon which accords with the typical hydrodynamic characteristics of CFBs. It can be found that drag model L can correctly simulate the heterogeneous flow structures of CFBs. Figs. 7 and 8 show the axial profile of time-averaged voidage at different riser heights using drag model L, and their comparison with the experimental data [34,35], respectively. As can be seen from Figs. 7 and 8, drag model L can predict the flow structure of dilute-top and dense-bottom which can also be seen in Figs. 5 and 6, and is reasonable agreement with experimental results. The trends of curves are similar to other simulation results [3,10,12,37], although the values have quantitative discrepancies at the dense bottom bed which may be attributed to the inaccurate evaluation of solid inventory or incomplete 2D description [12,37]. The interaction between gas and solid is determined by drag force, and suitable drag coefficient could predict well the
Fig. 6. Simulated voidage distribution of Geldart B particles at U f = 4.78m/s for drag model L.
distribution of voidage for the whole bed. It can be seen that by considering multiscale structures, the computed drag coefficient can predict well the voidage in the axial direction for Geldart A and B particles at different riser heights and superficial gas velocity. Figs. 9 and 10 show the radial profile of time-averaged voidage at various riser heights between the simulation results and the experimental correlations of Tung et al. [38]. It can be found that the computed results of drag model L are in agreement with the experimental correlations, and core-annular flow structure, which can also be found in Figs. 5 and 6, can be reflected by the trends that the voidage in the core zone is greater than that in the annulus zone. Therefore, suitable structural parameters model and drag coefficient play a vital role in the prediction of voidage in the CFBs. 5. Conclusions The structure-based drag model has recently been an effective approach to understand the multiscale nature of heterogeneous twophase flows in CFBs, and the key to solve structure-based drag
Fig. 5. Simulated voidage distribution of Geldart A particles at Uf = 1.52m/s: (a) drag model G; (b) drag model L.
Fig. 7. Axial voidage profile of Geldart A particles: (a) Uf = 1.52m/s; (b) Uf = 2.1m/s.
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in CFBs were simulated by the CFD software with the structure-based drag model. The solid mass flux, radial and axial voidage profile were obtained. It was shown that the present drag model predicted the experimental data considerably better than the Gidaspow drag model. Moreover, the heterogeneous hydrodynamics of CFBs, including the dilute-top/ dense-bottom and core-annular flow structure, could be simulated by the present drag model. Besides, the formation and disaggregation of clusters with time can be observed. Understanding the properties of clusters is of prime importance to study the gas–solid interaction and establish drag models of CFBs. Therefore, the present structural parameters model incorporating the equations of clusters properties can compute correctly the drag coefficient, and is a valuable tool to simulate the hydrodynamics of gas–solid flows in CFBs.
Fig. 8. Axial voidage profile of Geldart B particles: (a) Uf = 4.28m/s; (b) Uf = 4.78m/s.
coefficient is to establish structural parameters model. An improved structural parameters model should be able to describe the heterogeneous multiscale structures caused by the clusters. In this work, a new structural parameters model based on the available structure-based drag model was established incorporating with equations of clusters properties. The hydrodynamic characteristics of gas–solid flow behaviors
Nomenclature CD drag coefficient CDc drag coefficient in cluster phase CDd drag coefficient in dispersed phase CDi drag coefficient between cluster and gas in dispersed phase CD0 drag coefficient for single particle in gas flow CD averaged drag coefficient dc diameter of cluster, m dp diameter of particle, m FD total drag force of flow gas on the particles in unit volume of flow, N/m3 FDc drag force of flow gas on single particle in cluster phase, N FDcn drag force of flow gas in cluster phase on the particles in single cluster, N FDcu drag force of gas flow on particles in unit volume of cluster phase, N/m3 FDd drag force of flow gas on the single particle in dispersed phase, N FDi drag force of flow gas in dispersed phase on the single cluster, N
Fig. 9. Radial voidage profile of Geldart A particles: (a) Uf = 1.52m/s, H = 8 m; (b) Uf = 1.52m/s, H = 5 m; (c) Uf = 2.1m/s, H = 8 m; (d) Uf = 2.1m/s, H = 5 m.
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Fig. 10. Radial voidage profile of Geldart B particles: (a) Uf = 4.28m/s, H = 10 m; (b) Uf = 4.28m/s, H = 5 m; (c) Uf = 4.78m/s, H = 10 m; (d) Uf = 4.78m/s, H = 5 m.
Fpdc Fpdcn f Gs HD Rep n Uf uf Ufc Ufd Up up Upc Upd Us Usc Usd Usi Ut
collision force, N collision force exerting the dispersed phase in unit volume, N/m3 volume friction of cluster phase solids circulation flux, kg/m2/s heterogeneity index Reynolds Number for particle the number of particles superficial gas velocity, m/s gas velocity, m/s superficial gas velocity in cluster phase, m/s superficial gas velocity in dispersed phase, m/s superficial particle velocity, m/s particle velocity, m/s superficial particle velocity in cluster phase, m/s superficial particle velocity in dispersed phase, m/s superficial slip velocity between gas and particles, m/s superficial slip velocity between gas and particle in cluster phase, m/s superficial slip velocity between gas and particle in dispersed phase, m/s superficial slip velocity between gas in dispersed phase and cluster, m/s the terminal particle velocity, m/s
Greek symbols αd the accelerated velocity of particles in the dispersed phase m/s2 αc the accelerated velocity of particles in the cluster phase m/s2 β drag coefficient including the heterogeneous flow structure β0 Wen and Yu drag coefficient ε voidage
εc εd εf εi εmf εmin εs μf ρf ρp
voidage in cluster phase voidage in dispersed phase averaged voidage voidage of inter-phase voidage at minimum fluidization minimum particle voidage at packing averaged solid volume fraction viscosity of gas kg/(m s) density of gas, kg/m3 density of particle, kg/m3
Acknowledgements The authors are grateful to the financial support from National Natural Science Foundation of China under Grant No. 21325628 and from the State Key Development Program for Basic Research of China (973 Program) under Grant 2015CB251402. Appendix A. Variables involved in the structural parameters model Variables involved in the structural parameters model are given below. Superficial slip velocity in different phases: U sc ¼ U fc −U pc U si ¼
εc 1−εc
ðA1Þ
U fd U pc − ε ð1−f Þ εd 1−εc d
ðA2Þ
εd 1−εd
ðA3Þ
U sd ¼ U f d −U pd
W. Liu et al. / Powder Technology 286 (2015) 516–526
Appendix C. Governing equations and constitutive equations
Reynolds number in different phases: Rep ¼
ρ f dp U sc μf
ðA4Þ
Rep ¼
ρ f dc U si μf
ðA5Þ
Rep ¼
ρ f dp U sd μf
ðA6Þ
Governing equations and constitutive equations are given below. Continuity equation (k = g, s): ∂ðεk ρk Þ þ ∇ðε k ρk uk Þ ¼ 0 ∂t
ðC1Þ
εs þ εg ¼ 1
ðC2Þ
Momentum equation (k = g, s; l = s, g):
Standard drag coefficient: 8 < 0:44 24 C D0 ¼ 1 þ 0:15Re0:687 : p Rep
Rep N 1000 Rep ≤ 1000
∂ðεk ρk uk Þ þ ∇ðε k ρk uk uk Þ ¼ −εk ∇pg þ εk ρk g þ ∇τ k þ βðul −uk Þ ∂t ðA7Þ
Effective drag coefficient in different phases: 8 −4:7 > < C D0 εc ð1−εc Þμ f 7 ¼ > : 200 ε3 ρ d U þ 3ε 3 c c f p sc
C Di
8 −4:7 > < C D0 εi ð1−εi Þμ f 7 ¼ > : 200 ε3 ρ d U þ 3ε3 p si f i i
C Dd
8 −4:7 > < C D0 εd ð1−εd Þμ f 7 ¼ > : 200 ε 3 ρ d U þ 3ε3 d f p sd d
εc ≥ 0:8 εc b 0:8
ðA8Þ
τs ¼ ½−ps þ λs ∇μ s δ þ 2μ s Ss
ðC5Þ
Deformation rate: εi ≥ 0:8 εi b 0:8
ðA9Þ
Sk ¼
i 1 1h ∇uk þ ð∇uk ÞT − ∇uk δ 2 3
ðC6Þ
Solid phase pressure: εd ≥ 0:8 εd b 0:8
ðA10Þ
ps ¼ εs ρs Θ½1 þ 2ð1 þ eÞεs g 0
μs ¼
Uf = 1.52 m/s, Uf =
(B1)
Usd = 0.0825
(B2)
U si ¼ −23:9ε f 3 þ 42:92ε f 2 −22:21ε f þ 4:146
2.1 m/s Uf = 1.52 m/s
(B3)
U si ¼ −23:85ε f 3 þ 38:16ε f 2 −14:53ε f þ 1:654 Uf = 2.1 m/s Uf = 1.52 m/s ε −0:5603 2 α c ¼ 9:79 105 ; exp½−ð f 0:2123 Þ Uf = 2.1 m/s ε f −0:5623 2 6 α c ¼ 1:863 10 ; exp½−ð 0:2109 Þ Uf = 1.52 m/s ε −0:4323 2 α d ¼ −7:691 109 ; exp½−ð f 0:2047 Þ 2 Uf = 2.1 m/s ε −0:4341 α d ¼ −1:454 1010 ; exp½−ð f 0:2041 Þ
pffiffiffiffiffiffiffi 5 ρ dp πΘ 96 s
ðC8Þ
ðC9Þ
Solid phase bulk viscosity:
Geldart A particles:
2.1 m/s Uf = 1.52 m/s, Uf =
rffiffiffiffi 2 4 2 Θ 2μ s:dilute 4 εs ρs dp g 0 ð1 þ eÞ 1 þ ð1 þ eÞεs g 0 þ 3 π ð1 þ eÞg 0 5
μ s:dilute ¼
λs ¼ 20:97ε f 2 −30:93ε f 2 þ11:55 3 −179:8ε 2 −150:2ε þ346:7 f f
ðC7Þ
Solid phase shear viscosity:
The values of Usc, Usd, Usi, αd and αc can be solved by the nonlinear equations under a given system with specified Up, Uf, εf and physical parameters ρg, ρf, μf, dp. Traverse εf within [εmf, 1], those values as a function of voidage εf can be obtained as follows:
f
ðC4Þ
Solid phase stress:
Appendix B. The values of Usc, Usd, Usi, αd and αc as a function of voidage
U sc ¼ ε
ðC3Þ
Gas phase stress: τg ¼ 2μ g Sg
C Dc
525
rffiffiffiffi 4 2 Θ εs ρs dp g 0 ð1 þ eÞ 3 π
ðC10Þ
Radial distribution functions: " 1=3 #−1 εg g 0 ¼ 1− εsm
ðC11Þ
(B4) (B5) (B6) (B7) (B8)
Granular temperature equation: 3 ∂ðεs ρs ΘÞ þ ∇ðε s ρs us ΘÞ ¼ τs : ∇us −∇q−γ þ βC g C−3βΘ 2 ∂t
ðC12Þ
Collisional energy dissipation: Geldart B particles: U sc ¼ ε
20:75ε f 2 −43:22ε f 2 þ23:05 3 þ723:7ε 2 −1455ε þ731:4 f f
f
Usd = 0.9707
Uf = 4.28 m/s, Uf =
(B9)
4.78 m/s Uf = 4.28 m/s, Uf =
(B10)
4.78 m/s U si ¼ −161:3ε f 3 þ 333:8ε f 2 −220:3ε f þ 49:12 Uf = 4.28 m/s Uf = 4.78 m/s U si ¼ −198:2ε f 3 þ 414:3ε f 2 −277ε f þ 62:48
(B11)
Uf = 4.28 m/s
(B13)
Uf = 4.78 m/s
(B14)
Uf = 4.28 m/s
(B15)
Uf = 4.78 m/s
(B16)
ε f −0:3778 2 0:2311 Þ ε −0:3838 2
α c ¼ 3:378 107 ; exp½−ð α c ¼ 4:076 107 ; exp½−ð α d ¼ −3:375 α d ¼ −3:894
f
0:2299
Þ
ε −0:1139 2 10 ; exp½−ð f 0:2449 Þ ε −0:1212 2 1012 ; exp½−ð f 0:244 Þ 12
(B12)
" rffiffiffiffiffiffiffiffiffi # 4 Θ 2 2 −∇us γ ¼ 3 1−e εs ρs g 0 Θ dp π
ðC13Þ
Flux of fluctuating energy: q ¼ −k∇Θ
ðC14Þ
Conductivity if the fluctuating energy: k¼
2 k 2k 1 þ 65 ð1 þ eÞεs g0 c þk ð1 þ eÞg 0
ðC15Þ
526 k
W. Liu et al. / Powder Technology 286 (2015) 516–526
k ¼
pffiffiffiffiffiffiffi 75 d ρ Θπ 384 p s
ðC16Þ rffiffiffiffi Θ π
c
k ¼ 2ε2s ρs dp g 0 ð1 þ eÞ
ðC17Þ
Drag coefficient for Gidaspow et al. [32]: 8
s > εg εs μ g ρg εs ug −us
> > > þ 1:75 εg b0:8 150 < 2 dp εg dp β¼
> > 3 εs εg ρg ug −us −2:65 > > CD εg εg ≥0:8 : 4 dp
ðC18Þ
Drag coefficient for this work is Eq. (24). References [1] J. Gao, X. Lan, Y. Fan, J. Chang, G. Wang, C. Lu, C. Xu, CFD modeling and validation of the turbulent fluidized bed of FCC particles, AICHE J. 55 (2009) 1680–1694. [2] B. Hou, H. Li, Relationship between flow structure and transfer coefficients in fast fluidized beds, Chem. Eng. J. 157 (2010) 509–519. [3] W. Wang, J. Li, Simulation of gas–solid two-phase flow by a multi-scale CFD approach – of the EMMS model to the sub-grid level, Chem. Eng. Sci. 62 (2007) 208–231. [4] J.D. Gibbon, E.S. Titi, Cluster formation in complex multi-scale systems, Proc. R. Soc. London, Ser. A 461 (2005) 3089–3097. [5] J. Li, M. Kwauk, Exploring complex systems in chemical engineering – the multiscale methodology, Chem. Eng. Sci. 58 (2003) 521–535. [6] D. Gidaspow, Multiphase flow and fluidization: continuum and kinetic theory descriptions, Academic Press, 1994. [7] M. Syamlal, O’Brien T.J., The derivation of a drag coefficient formula from velocityvoidage correlations, U. S. Department of Energy, Office of Fossil Energy, Technical Report 1994. [8] J. Li, M. Kwauk, Multiscale nature of complex fluid-particle systems, Ind. Eng. Chem. Res. 40 (2001) 4227–4237. [9] J. Ma, W. Ge, X. Wang, J. Wang, J. Li, High-resolution simulation of gas–solid suspension using macro-scale particle methods, Chem. Eng. Sci. 61 (2006) 7096–7106. [10] J. Wang, W. Ge, J. Li, Eulerian simulation of heterogeneous gas–solid flows in CFB risers: EMMS-based sub-grid scale model with a revised cluster description, Chem. Eng. Sci. 63 (2008) 1553–1571. [11] S. Wang, H. Lu, G. Liu, Z. Sheng, P. Xu, D. Gidaspow, Modeling of cluster structuredependent drag with Eulerian approach for circulating fluidized beds, Powder Technol. 208 (2011) 98–110. [12] N. Yang, W. Wang, W. Ge, J. Li, CFD simulation of concurrent-up gas–solid flow in circulating fluidized beds with structure-dependent drag coefficient, Chem. Eng. J. 96 (2003) 71–80. [13] J. Wang, A review of Eulerian simulation of Geldart A particles in gas-fluidized beds, Ind. Eng. Chem. Res. 48 (2009) 5567–5577. [14] W. Shuai, L. Huilin, L. Guodong, S. Zhiheng, X. Pengfei, D. Gidaspow, Modeling of cluster structure-dependent drag with Eulerian approach for circulating fluidized beds, Powder Technol. 208 (2011) 98–110.
[15] J. Gao, J. Chang, X. Lan, Y. Yang, C. Lu, C. Xu, CFD modeling of mass transfer and stripping efficiency in FCCU strippers, AICHE J. 54 (2008) 1164–1177. [16] J. Gao, J. Chang, C. Xu, X. Lan, Y. Yang, CFD simulation of gas solid flow in FCC strippers, Chem. Eng. Sci. 63 (2008) 1827–1841. [17] S. Wang, H.L. Lu, Z.H. Hao, F.X. Zhao, L.Y. Sun, Q.H. Zhang, Modeling of reactive gas– solid flows in riser reactors using a multi-scale chemical reaction model, Chem. Eng. Sci. 116 (2014) 773–780. [18] H.L. Lu, Q.Q. Sun, Y.R. He, Y.L. Sun, H.M. Ding, L. Xiang, Numerical study of particle cluster flow in risers with cluster-based approach, Chem. Eng. Sci. 60 (2005) 6757–6767. [19] A. Nikolopoulos, D. Papafotiou, N. Nikolopoulos, P. Grammelis, E. Kakaras, An advanced EMMS scheme for the prediction of drag coefficient under a 1.2 MWth CFBC isothermal flow-Part I: Numerical formulation, Chem. Eng. Sci. 65 (2010) 4080–4088. [20] W. Gu, J. Chen, A model for solid concentration in circulating fluidized beds, Fluidization IX, 1998 501–508. [21] J.M. Matsen, Mechanisms of choking and entrainment, Powder Technol. 32 (1982) 21–33. [22] J.F. Richardson, W.N. Zaki, Sedimentation and fluidization, Trans. Inst. Chem. Eng. 32 (1954) 35–53. [23] J. Garside, M.R. Al-Dibouni, Velocity–voidage relationships for fluidization and sedimentation in solid–liquid systems, Ind. Eng. Chem. Process Des. Dev. 16 (1977) 206–214. [24] M. Rhodes, H. Mineo, T. Hirama, Particle motion at the wall of a circulating fluidizedbed, Powder Technol. 70 (1992) 207–214. [25] A.K. Sharma, K. Tuzla, J. Matsen, J.C. Chen, Parametric effects of particle size and gas velocity on cluster characteristics in fast fluidized beds, Powder Technol. 111 (2000) 114–122. [26] R. Cocco, F. Shaffer, R. Hays, S.B.R. Karri, T. Knowlton, Particle clusters in and above fluidized beds, Powder Technol. 203 (2010) 3–11. [27] A.T. Andrews, P.N. Loezos, S. Sundaresan, Coarse-grid simulation of gas–particle flows in vertical risers, Ind. Eng. Chem. Res. 44 (2005) 6022–6037. [28] A. Harris, J. Davidson, R. Thorpe, The prediction of particle cluster properties in the near wall region of a vertical riser (200157), Powder Technol. 127 (2002) 128–143. [29] W. Gu, Diameters of catalyst clusters in FCC, AIChE symposium series, American Institute of Chemical Engineers, 1971–c2002, New York, NY 1999, pp. 42–47. [30] H.L. Lu, D. Gidaspow, Hydrodynamics of binary fluidization in a riser: CFD simulation using two granular temperatures, Chem. Eng. Sci. 58 (2003) 3777–3792. [31] D. Subbarao, A model for cluster size in risers, Powder Technol. 199 (2010) 48–54. [32] D. Gidaspow, R. Bezburuah, J. Ding, Hydrodynamics of circulating fluidized beds, kinetic theory approach, Fluidization VII, Proceedings of the 7th Engineering Foundation conference on fluidization 1992, pp. 75–82. [33] J. Ding, D. Gidaspow, A bubbling fluidization model using kinetic theory of granular flow, AICHE J. 36 (1990) 523–538. [34] J. Li, Particle-fluid two-phase flow: the energy-minimization multi-scale method, Metallurgical Industry Press, 1994. [35] E.R. Monazam, L.J. Shadle, L.O. Lawson, A transient method for determination of saturation carrying capacity, Powder Technol. 121 (2001) 205–212. [36] M.J. Rhodes, M. Sollaart, X.S. Wang, Flow structure in a fast fluid bed, Powder Technol. 99 (1998) 194–200. [37] N. Yang, W. Wang, W. Ge, L. Wang, J. Li, Simulation of heterogeneous structure in a circulating fluidized-bed riser by combining the two-fluid model with the EMMS approach, Ind. Eng. Chem. Res. 43 (2004) 5548–5561. [38] Y. Tung, J. Li, M. Kwauk, Radial voidage profiles in a fast fluidized bed, in: M. Kwauk, D. Kunii (Eds.), Fluidization’88: science and technology, Science Press, Beijing 1988, pp. 139–145.