Eulerian simulation of a circulating fluidized bed with a new flow structure-based drag model

Chemical Engineering Journal 284 (2016) 1224–1232

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Eulerian simulation of a circulating fluidized bed with a new flow structure-based drag model Baolin Hou a, Xiaodong Wang a,⇑, Tao Zhang a, Hongzhong Li b,⇑ a b

State Key Laboratory of Catalysis, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, PR China State Key Laboratory of Multi-phase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing 100190, PR China

h i g h l i g h t s
A new model is developed to predict the heterogeneous flow structure parameters in CFB.
The drag coefficient based on these heterogeneous parameters can improve TFM model.
The collision of particles between the dispersed phase and cluster phase is analyzed.

a r t i c l e

i n f o

Article history: Received 21 April 2015 Received in revised form 10 September 2015 Accepted 17 September 2015 Available online 28 September 2015 Keywords: Circulating fluidized bed Euler–Euler Two flow model Multi-scale Cluster

a b s t r a c t The heterogeneous flow in a circulating fluidized bed (CFB) deteriorates momentum, combined with its heat and mass transfer between gas and solid, have been confirmed by several experiments. Due to no consideration of this heterogeneous flow in Euler–Eluer two phase flow model, there are the significant deviation of average voidage between the simulating results and experimental data. Therefore, we developed a new model to evaluate the local heterogeneous flow parameters, based on which gas–solid drag force coefficient in calculating mesh was calculated. This drag coefficient based on these local heterogeneous flow structure parameters improves the Euler–Eluer two phase flow model greatly, which further is confirmed by better description of average voidage in experiments. With this drag force coefficient model, the nature of axial S-shape and radial core-annular distribution of solid concentration in CFB can be captured. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Numerous experimental data and theoretical results on a circulating fluidized bed (CFB) have been obtained by academic researchers and engineers during the past decades because of its significance industrial applications [1–4]. Most of the conclusions in the literatures [5–13] suggested that the unresolved mesoscale structure, so-called as cluster, was critical for the gas–solid transport. Clusters in CFB would significantly reduce the gas–solid drag compared with the Ergun equation [14–16] derived from the pressure drop in packed bed. However, mass/heat transfer was deteriorated as well, and the mass and heat transfer coefficients were smaller than the prediction of semi-empirical equation [17–20]. Clusters also increase the terminal velocity of particles in CFB. Thus, resolving meso-scale structure is still one of the

⇑ Corresponding authors. E-mail addresses: [email protected] (X. Wang), [email protected] (H. Li). http://dx.doi.org/10.1016/j.cej.2015.09.073 1385-8947/Ó 2015 Elsevier B.V. All rights reserved.

focuses on the hydrodynamic of gas–solid fluid in CFB [21–25], experimentally and theoretically. With the assistance of computational fluid dynamics, two flow model (TFM) coupled with kinetics theory of granular flow (KTGF) [26–32] has been widely used to simulate industrial scale reactors based on coarser grid. The calculating results from TFM have shown that the drag force coefficient is critical for the simulation of gas–solid flow in CFB. Usually, Gidaspow’s drag force coefficient is used, but there is serious deviation between the simulation and experimental data [29,32,33], which is attributed to the omitted consideration of the heterogeneous flow structure in coarser calculating mesh grid. Yang et al. [16] firstly consider the effect of heterogeneous flow structure on the drag coefficient. TFM coupled with the developed drag force coefficient better described axial S-shape and radial core-annular distribution of solid concentration in CFB. This model is well known as EMMS, which was improved recently by Wang et al. [15]. In the EMMS model, due to the number of variables is counted more than that of independent equations, the stability condition of energy minimum principle

B. Hou et al. / Chemical Engineering Journal 284 (2016) 1224–1232

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Notation CD CDc CDd CDi CD0 dc dp dv FD FDc FDcn FDcu FDd FDi Fpdc Fpdcn f Gs HD Rep n Uf uf Ufc Ufd Up up

drag coefficient drag coefficient in cluster phase drag coefficient in dispersed phase drag coefficient between cluster and gas in dispersed phase drag coefficient for single particle in gas flow diameter of cluster, m diameter of particle, m diameter of circulating fluidized bed, m total drag force of flow gas on the particles in unit volume of gas-particles flow, N/m3 drag force of flow gas on single particle in cluster phase, N drag force of flow gas in cluster phase on the particles in single cluster, N drag force of gas flow on particles in unit volume of cluster phase, N/m3 drag force of flow gas on the single particle in dispersed phase, N drag force of flow gas in dispersed phase on the single cluster, N 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

has to be used. Especially, in the Wang’s model [15], ten variables need to be solved by solving only six equations. Thus, the global search for minimum energy has to be applied for multiple times. Therefore, although the EMMS model has been well known and widely accepted, its usage is limited because of this kind of complication. In addition, momentum exchange between the dispersed phase and the cluster phase is not considered yet in this EMMS model. In fact, the solid velocity in the dispersed phase is faster than that in the cluster phase. It is possible that the particles in the dispersed phase and cluster phase collide with each other. With these taken into consideration, a new model for prediction of heterogeneous flow structure is developed. Similar to EMMS model, the heterogeneous flow structure in CFB is divided into three homogeneous phases, namely, dispersed, cluster, and inter-phase phases. The voidage in cluster phase is calculated by Wang’s equation [34]. According to Matsen’s analysis [35], the voidage in dispersed phase is defined as 0.9997. In each homogeneous phase, the superficial gas velocity is correlated with voidage by Richard and Zaki Equation [36]. Momentum exchange between the dispersed phase and the cluster phase is included in our model because of considerable difference of velocity in the two phases. By assumption of being homogeneous, the drag force can be calculated by Wen and Yu’s [37] or Ergun’s [38] equation for each phase. The cluster size is calculated by the equation of pressure drop balance among the three homogeneous phases, and results are verified by the semi-empirical model in literatures. In this work, drag coefficient based on the heterogeneous flow

Upc Upd Us Usc Usd Usi Ut p

superficial particle velocity in cluster phase, m/s superficial particle velocity in dispersed phase, m/s superficial slip velocity between gas and particles 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 pressure, Pa

Greek letters ad the accelerated velocity of particles in the dispersed phase m/s2 ac the accelerated velocity of particles in the cluster phase m/s2 b drag coefficient including the heterogeneous flow structure bG drag coefficient in Gadispow model e voidage ec voidage in cluster phase voidage in dispersed phase ed ef local voidage ef axial average voidage voidage of inter-phase, i.e., the volume friction of gas ei except gas in clusters emin minimum voidage es solid volume fraction re standard deviation of solid phase concentration k heat conductivity of gas, J/(ms K) lf viscosity of gas, kg/(m s) qf density of gas, kg/m3 qp density of particle, kg/m3

structure parameters developed is calculated by applying our previously developed equations [17,18,33], and integrated into TFM to simulate the dynamics of fluid in CFB. Better agreement is correlated between the simulated results and the experimental data in literature [1]. 2. Models for the heterogeneous flow structure Eight parameters are commonly needed to describe the heterogeneous flow structure, which are listed as follows: the voidage in the dispersed phase and in the cluster phase ed and ec, respectively; the superficial gas velocities in the dispersed phase and in the cluster phase Ufd and Ufc, respectively; the superficial particle velocities in the dispersed phase and in the cluster phase Upd and Upc, respectively; the diameter of the cluster dc, and the volume friction of the cluster phase f [17,18,33]. The particle flow is characterized by the oscillating properties, hence, the fluid in the cluster phase and dispersed phase is not in equilibrium in each calculating grid; the relevant gas and particles can be accelerated or decelerated by the complex interaction. Therefore, two additional heterogeneous structure parameters have to be considered, denoted as the accelerated velocity in the dispersed phase ad and in the cluster phase ac. The calculation process for the above parameters and assumptions will be expatiated in the section below. In order to simply the process of calculating the parameters of heterogeneous flow structure, the following assumptions must be made:

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1. The effect of particle’s and cluster’s shape is not considered in the developed model and assumed to be sphere. 2. The transverse collision of particles in the cluster phase and dispersed phase is ignored in this model. 3. The effect of inlet structures for gas and solid are not considered, gas and solid are assumed to be uniformly flowing into the fluidized bed. 4. The voidage in the dispersed phase is assumed to be constant of 0.9997. 5. The diameter of cluster is limited to be smaller than the diameter of fluidized bed. 6. Due to the low fluid velocity, the turbulent flow is not considered in our developed model.

where CDi is the drag coefficient between the gas in the dispersed phase and the particles on the surface of the cluster phase, which can be expressed as

8 < C D0 e4:7 i

C Di ¼

ei P 0:8

ð1ei Þlf

: 200 e3 q d U þ 37e3 i

f c

si

ei < 0:8

i

ei ¼ ed ð1  f Þ

ð9Þ ð10Þ

where CD0 can be calculated by Eq. (4); however, the relative Reynolds number should be changed as

qf dc U si lf

Rep ¼

ð11Þ

In this paper, the Euler–Euler two phases model is employed to simulate the gas–solid fluid flow in the fluidized bed.

Usi is the superficial slip velocity between gas in the dispersed phase and cluster, which is defined as

2.1. Force balance for the cluster

U si ¼

The forces on the cluster include the gas–solid drag force inside the cluster FDcn, the drag force between the gas in the dispersed phase and the particles on the surface of the cluster FDcf, the collision force from the high velocity particles in the dispersed phase Fpdc, and the apparent gravity of the cluster. The drag force FDcn is equal to the drag force of the single particle FDc, multiplied by the particle number n inside the cluster phase

When the cluster moves upward at the velocity of U pc =ð1  ec Þ, the particles in the dispersed phase also move upward at U pd =ð1  ed Þ. Because the cluster size is bigger than the particle diameter, the cluster is slower than single particle in the dispersed phase. When the particles in the dispersed phase bump the cluster, the velocity would be reduced to U pc =ð1  ec Þ. Ignoring the transverse collision of the particles, the mass of particles involved in the relevant collision can be written as

F Dcn ¼ nF Dc

ð1Þ

FDc can be expressed as

F Dc ¼

p 8

2 dp C Dc

ð2Þ

where, CDc is the drag coefficient of gas–solid interaction inside the cluster phase, which can be obtained through the following equations:

C Dc ¼

ec P 0:8

ð1ec Þlf

: 200 e3 q d

f p U sc

c

þ 37e3 c

ð3Þ

ec < 0:8

C D0 ¼

0:44 Rep > 1000 24 Rep

ð1 þ 0:15Re0:687 Þ Rep 6 1000 p

ð4Þ

ð5Þ

The superficial slip velocity in the cluster phase can be calculated as

ec

ð6Þ

1  ec

The particles on the surface of the cluster phase are different from those inside the cluster. Half of that is exerted by the high velocity fluid in the dispersed phase and the other half is dragged by the low velocity fluid inside the cluster. Therefore, the valid number of particles can be obtained by subtracting half of the particles on the surface of cluster, which can be expressed as

n¼

p d3 ð1  e 6

cÞ

c

p d3 6 p



1 2

pd2c ð1  ec Þ p d2 4 p

¼ ð1  ec Þ

 3   dc dp 12 dp dc

ð7Þ

p 8

2

dc C Di qf jU si jU si

 U pd U pc q ð1  ed Þ  ð1  ec Þ ð1  ed Þ p

ð12Þ

ð13Þ

F pdc ¼

p 4

 2

dc qp ð1  ed Þ

2 U pd U pc  ð1  ec Þ ð1  ed Þ

ð14Þ

Based on the force balance theory, the equation can be written as

p 6

3

dc ð1  ec Þðqp  qf Þðg þ ac Þ

ð15Þ

Substituting Eqs. (1), (8) and (14) into Eq. (15), the following equation can be obtained:

 2 d 1  2 dpc C Dc qf jU sc jU sc þ p8 dc C Di qf jU si jU si þ  2 3 p d2 q ð1  e Þ Upc  U pd ¼ p6 dc ð1  ec Þðqp  qf Þðg þ ac Þ d 4 c p ð1ec Þ ð1e Þ d3c c Þ dp



d

ð16Þ 2.2. Force balance for particles in the dispersed phase The forces on particles in the dispersed phase includes the gas– solid drag force FDdn, the collision force from the cluster phases Fpdcn, and the apparent gravitation. Similar to be Eq. (1), the equation can be written as

F Ddn ¼ nF Dd

ð17Þ

where, FDd can be calculated as

F Dd ¼

p 8

2

dp C Dd qf jU sd jU sd

ð18Þ

Usd is the superficial slip velocity between gas and particle in dispersed phase, which is defined as

The drag force FDcf can be calculated as

F Dcf ¼

ed ð1  f Þ

The following equation can be obtained according to the momentum conservation:

8

qf dp U sc lf

U sc ¼ U fc  U pc

4





2

dc

p ð1  e

The Reynolds number is defined as

Rep ¼

p

U pc 1  ec



F Dcn þ F Dcf þ F pdc ¼

where, CD0 can be obtained through the following expressions:

(

ed

Gpdc ¼

qf jU sc jU sc

8 < C D0 e4:7 c

 U fd

ð8Þ

U sd ¼ U fd  U pd

ed

1  ed

ð19Þ

B. Hou et al. / Chemical Engineering Journal 284 (2016) 1224–1232

where, CDd is the drag coefficient in the dispersed phase, which can be expressed as

C Dd ¼

8 < C D0 e4:7 d

ed P 0:8

ð1ec Þlf

: 200 e3 q d d

f p U sd

þ 37e3 d

ed < 0:8

Rep ¼

qf dp U sd lf

ð21Þ

The number of particles in the dispersed phase in unit volume of CFB can be written as:

n¼

ð1  f Þð1  ed Þ p d3 6

ð22Þ

p

Based on Eq. (13), the collision force exerted on the dispersed phase in unit volume of CFB can be expressed as:

F pdcn ¼ F pdc

f p d3 6

c

¼

 2 U pd 3 f U pc qp ð1  ed Þ  2 dc 1  ed 1  ec

ð23Þ

Thus, the force balance equation for particles in the dispersed phase can be written as:

 2 U pd 3 ð1  f Þð1  ed Þ 3 f U pc  qp ð1  ed Þ  C Dd qf jU sd jU sd 4 dp 2 dc 1  ed 1  ec ¼ ð1  f Þð1  ed Þðqp  qf Þðg þ ad Þ

ð24Þ

2.3. Gas–solid mass conservations According to the gas–solid mass conservation, the average velocity of gas and solid can be expressed as, respectively:

U f ¼ fU fc þ ð1  f ÞU fd

ð25Þ

U p ¼ fU pc þ ð1  f ÞU pd

ð26Þ

The relationship among the voidages in each homogeneous phase can be written as:

ef ¼ f ec þ ð1  f Þed

ð27Þ

2.4. Pressure drop balance According to the theory of gas pressure equilibrium at the same cross-section in CFB, the gas pressure drop resulting from gas flow through the dispersed phase is equal to that of gas flow through the cluster phase. Thus, the equation of the pressure drop balance can be expressed as:

      dp dp dp þ ¼ dz d dz i dz c

ð28Þ

The equation of the pressure drop in the dispersed phase can be expressed as:

  qf dp ð1  ed Þ p 2 3 ¼ C Dd dp qf jU sd jU sd ¼ C Dd ð1  ed ÞjU sd jU sd p dp3 dz d 4 8 dp 6

ð29Þ   qf dp ð1  ec Þ p 2 3 ¼ C Dc dp qf jU sc jU sc ¼ C Dc ð1  ec ÞjU sc jU sc 3 p dz c 4 8 dp dp

¼

ð20Þ

where, CD0 can be calculated by Eq. (4). However, the relevant Reynolds number should be changed to:

ð30Þ

6

The equation of the pressure drop in the inter-phase phase can be expressed as:

1227

  dp f p 2 1 ¼ C Di dc qf jU si jU si dz i p d3c ð1  f Þ 8 6

qf f 3 C Dc jU sc jU sc 4 dc ð1  f Þ

ð31Þ

2.5. Voidage in each phase In the previous experiments [42–44], the vodiage in the cluster phase is identified when the local voidage is smaller than the timee -times the standard deviation mean voidage, and calculated as by n of voidage, which can be expressed as:

1  ec ¼ ð1  ef Þ þ nre

ð32Þ

In Eq. (32), the fitting parameter n is the uncertainty in the literature. Soong’s [39,40] result is n = 3, whereas Tuzla [41], Sharma [42] acquired n = 2; Manyele [43], Yan [44], and Qi et al. [45] have set n = 3. Wang’s results showed that the effect of n from one to three on the drag force can be ignored [34]. The corresponding standard deviation of solid concentration re was formulated by [34,46]:

re

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ es Sðes ; 0Þ ¼ es

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  es Þ4 ð1 þ 4es þ 4e2s  4e3s þ e4s Þ

ð33Þ

However, when the voidage approaches the packing maximum limitation, the discrepancy between the simulated results in Eq. (33) and the experimental data becomes serious, which will be discussed in detail in the Chapter 4.0. To obtain good agreement between the theoretical results and the experimental data under this case, Schneiderbauer [23] improved Eq. (32) by replacing re with re , calculated by the following equation:

e

re ¼ re tan h

 es  0:15

min

ð34Þ

According to the conclusions in literature [15,35], the voidage in the dispersed phase ed is assumed to be constant with the maximum value 0.9997. emin is defined as the minimum voidage of 0.4. 2.6. Relationship between the superficial slip velocity and the voidage Due to the homogeneous assumptions exist in each phase, the relationship between the superficial slip velocity and the local voidage can be determined by the well-known Richardson and Zaki equation, which can be expressed as:

emd ¼

  U sd U fd U pd ed ¼  Ut Ut U t 1  ed

ð35Þ

emc ¼

  U sc U fc U pc ec ¼  Ut Ut U t 1  ec

ð36Þ

where Ut is the terminal particle velocity. The exponent m can be obtained based on the m vs Ret curves, given by Kwauk [47,48]. 2.7. Solution of the flow heterogeneous structure In oder to characterize the heterogeneous flow structure in CFB, nine independent variables (Ufd, Ufc, Upd, Upc, f, dc, ec, ad, ac) must be determined by solving Eqs. (16), (24)–(28), (32), (35), and (36). The solution process is detailed as follows: (1) When Us and ef are specified in each calculating mesh grid, the voidage in the cluster phase can be calculated by Eq. (32). The volume fraction of cluster phase f can be obtained by Eq. (27).

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(2) Based on Eqs. (35) and (36), the gas–solid slip velocities in the cluster phase and the dispersed phase can be solved. The value Usi also can be obtained by solving Eqs. (25) and (26). (3) The cluster size dc is obtained by solving Eq. (28). However, when the Eq. (28) cannot converge to a value smaller than the diameter of CFB dv, dc is defined to be dv. (4) Based on the above the solution results, ad and ac can be easily obtained by solving Eqs. (16) and (24). 2.8. Integration with TFM The drag coefficient is critical for simulating the gas and solid flow in a circulating fluidized bed, which has been verified in literature [15,49]. The process of calculating drag in TFM model should consider the effect of heterogeneous flow structure. According to the previous analysis [17,3,18], drag coefficient considering the heterogeneous flow structure parameters can be expressed as the following:

b¼

ð1f Þð1ed Þ C Dd 12 pd3 6 p



qf jU sd jU sd p4 d2p þ 1  2 ddpc



f ð1ec Þ 1 pd3 C Dc 2 6 p

2 f dp

e

þ 1:75

ð1  ef Þqf juf  up j dp

ef < 0:8

ð40Þ

where CD0 can be calculated by Eq. (4), but the Reynolds Number is calculated as

Rep ¼

qf dp U s lf

ð41Þ

From the theoretical analysis, the drag coefficient considering the heterogeneous flow structure is smaller than the Gidaspow’s model [26] based on homogeneous assumptions. That is, HD is in a range (0, 1). However, when CFB lies in the operational region of dense phase with the average voidage smaller than 0.7, the flow structure tends to be homogeneous. The assumption of voidage in dispersed phase ed = 0.9997 deviates from the real value and results in HD bigger than 1. Given the Gidaspow’s drag coefficient obtained by correlating the experimental data of pressure drop in packed bed, the upper limitation of HD is 1. Moreover, the cluster

6 c

ð37Þ

ðug  us Þ

b bG

size should be smaller than CFB diameter dv because of the geometric structural limitation. When the calculating cluster size is bigger than dv, the flow structure becomes homogeneous, and the drag coefficient can be calculated based on Gidaspow’s model.

ð38Þ

where bG is defined as the Gidaspow drag coefficient[30], which can be calculated as:

bG ¼

ð1  ef Þ2 lf

qf jU sc jU sc p4 d2p þ pdf 3 C Di 12 qf jU si jU si p4 d2c  e2f

A heterogeneity index is defined to characterize the hydrodynamic difference between the homogeneous and heterogeneous fluidization, which can be expressed as

HD ¼

bG ¼ 150

3 ð1  ef Þef qf juf  up jC D0 ef2:7 4 dp

ef P 0:8

ð39Þ

3. Computing and boundary conditions This numerical simulation is based on the 2-D TFM, and the relevant equations are given in Table 1. The solid stress is calculated by the kinetics theory of granular flow (KTGF). The testing case is

Table 1 Governing equations for two-fluid model and its constitutive relations. Continuity equation @ ðek qk Þ @t

þ rðek qk uk Þ ¼ 0

Momentum equation ðk ¼ g; s; l ¼ s; gÞ @ðek qk uk Þ @t

þ rðek qk uk uk Þ ¼ ek rpg þ ek qk g þ rsk þ bðul  uk Þ

Gas phase stress sg ¼ 2lg Sg Solid phase stress   ss ¼ ps þ ks rls d þ 2ls Ss Deformation rate h i Sk ¼ 12 ruk þ ðruk ÞT  13 ruk d Solid phase pressure ps ¼ es qs H½1 þ 2ð1 þ eÞes g 0 

qffiffiffi l ¼ 4 e2 qs dp g 0 ð1 þ eÞ Hp Solid phase shear viscosity s2l 3 s 2 s:dilute 1 þ 45 ð1 þ eÞes g 0 þ ð1þeÞg pffiffiffiffiffiffiffiffi 0 ls:dilute ¼ 965 qs dp pH Solid phase bulk viscosity qffiffiffi ks ¼ 43 e2s qs dp g 0 ð1 þ eÞ H p Radial distribution functions h i e 1=3 1 g 0 ¼ 1  ðesmg Þ

Granular temperature equation h i 3 @ðes qs HÞ þ rðes qs us HÞ ¼ ss : rus  rq @t 2 c þ bC g C  3bH Collisional energy dissipation h qffiffiffiffiffiffiffiffi i c ¼ 3ð1  e2 Þe2s qs g 0 H d4p Hp rus Flux of fluctuating energy q ¼ krH Conductivity if the fluctuating energy

2 6 k 2k 1 þ ð1 þ eÞes g 0 5 c þk k¼ ð1 þ eÞg 0 p ffiffiffiffiffiffiffi ffi 75 k dp qs Hp k ¼ rffiffiffiffiffi 384 c

k ¼ 2e2s qs dp g 0 ð1 þ eÞ

H

p

Drag coefficient Gidaspow (Gidaspow and Bezburuah 1992) eg P 0:8: e e q ju u j b ¼ 34 C D s g gdp g s e2:65 g eg < 0:8 eg es l q e ju u j b ¼ 150 e d2 g þ 1:75 g s dpg s g p

In this work Eq. (37)

B. Hou et al. / Chemical Engineering Journal 284 (2016) 1224–1232

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Fig. 2. Comparisons for the standard deviation of solid concentration between the experimental fitting results and the predicted results.

Fig. 1. Schematic diagram of simulated 2-D riser.

3

Table 2 Parameters setting for the simulation. Particle diameter Particle density Grid size Dx Grid size Dy esm Riser height Superficial gas velocity Ug0 Solids flux Gs Initial bed height H0 Coefficient of restitution Time step Unsteady formulation Pressure–velocity coupling Momentum discretization

54 lm 930 Kg/m3 1.2 mm 5.0 mm 0.63 10.5 m 1.52 14.3 1.855 0.9 5.0e  5s First-order implicit Phase coupled SIMPLE Second-order upwind

adopted from Li’s [1] experiment. The only riser section in this paper is simulated because of the computing capacity limitation. The gas is uniformly flowed into CFB from the bottom at the superficial velocity and exited from the outlet. Solids are first stacked up to a certain height in the riser with incipient fluidization voidage, and then carried out by gas, and circulated into the sidewall inlets with a voidage of 0.5, as shown in Fig. 1 The initial bed height H0 is set at 1.855 m with an average voidage of 0.4, which is detailed by Lu et al. [6]. Fluent 6.3.26 is used to solve the relevant equations, whereas the matrix method developed by Wang [15] is employed to couple Eq. (37) into TFM. The calculation grid is generated by Gambit 2.2, and its effect on the simulated results is checked and proved to be ignored. The relevant boundary conditions and calculating sets are given in Table 2. 4. Results and discussion In this model, the terminal velocity is accurately predicted first, which can be calculated as:

Ut ¼

Ret lf dp qf

ð42Þ

where, Ret can be obtained based on the following expression:

8 Ar 6 18 Ret ¼ Ar=18 > <  Ar 2=3 18 < Ar 6 82500 Ret ¼ 7:5 > : Ar > 82500 Ret ¼ 1:74ðArÞ0:5

ð43Þ

where, Ar ¼ dp gðqp  qf Þ=l2f is Archimedes dimensionless number. With Eqs. (42) and (43), the terminal velocity is equal to 0.0810 m/s, which is close to the experimental data 0.0768 m/s. To calculate the exponent of the R-Z equation in Eqs. (35) and (36), the Kwauk’s [47] m vs Ret curve was correlated, and the following expressions can be given

8 Ret < 0:1 m ¼ 4:9 > > > > > 0:1 6 Ret < 1:0 m ¼ 4:9  0:18961Ret þ 0:05628Re2t > > > > > > 1:0 6 Ret < 10:0 m ¼ 4:8415  0:08496Ret þ 0:00337Re2t > > > 5 > 2 > > > 10:0 6 Ret < 100:0 m ¼ 4:469  0:01841Ret þ 8:182  10 Ret > > 5 < 100:0 6 Re < 1000:0 m ¼ 3:899  0:00603Re þ 1:33  10 Re2 t t t 8 12 3 4 > 1:36  10 Re þ 5:14  10 Re > t t > > > > 1000:0 6 Re < 10000:0 m ¼ 2:868  2:10  104 Re þ 4:26 > t t > > > > > 108 Re2t  4:18  1012 Re3t > > > > > > þ1:5  1016 Re4t > > : Ret > 10000:0 m ¼ 2:2 ð44Þ Issangya [50] studied the relationship between the timeaverage local voidage and the standard deviation of solid concentration. Based on their conclusions, the fluctuations were found to be close to zero when the local time-average voidage approaches emin or 1. When the local time-average voidage is smaller than 0.9, the experimental data for the standard deviation become scatter. The following equation was obtained by Issangya [51] by fitting the experimental data.

re ¼ 1:584ð1  ef Þðef  emin Þ

ð45Þ

Fig. 2 shows the comparison for the standard deviation of solid concentration between the correlated experimental equation and the results predicted by Eqs. (33) and (34). Based on the experimental data [51], when the solids reached the maximum packing limitation value, the heterogeneous structure disappears, and thus the standard deviation of solid concentration should be close to zero. Fig. 2 shows that in that region, the results of Eq. (33) obviously diverge from the experimental results, and the corresponding modified version Eq. (34) has an obvious improvement. However, when the solid concentration is bigger than 0.1, the simulated results of Eqs. (33) and (34) deviate from the experimental fitting results. Wang[34] thought that the local gas–solid flow in the Issangya’s [51] experiment should be in a bubbling or turbulent

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Fig. 3. Effect of various definitions of solid concentrations in cluster phase on the cluster size.

Fig. 4. Comparisons for the cluster size between the simulated results and the semi-empirical equations from the fitting experimental data in literature.

Table 3 Correlations of cluster size in literature. Author Zou et al. [53] Gu and Chen [54] Harris et al. [55] Subbaro [56]

Correlation   . 1:3889 Þ ðef  emf Þ2:41 þ1 dc ¼ dp 1:8543ðef1:5 e0:25 s dc ¼ dp þ ð0:027  10dpÞes þ 32e6s   dc ¼ 1  ef = 40:8  94:5ð1  ef Þ  pffiffiffiffiffiffi 2 

1=3 2 .  2ut g 1 þ u2t ð0:35 gDt Þ þ dp dc ¼ ð1  ef Þ=ðef  ec Þ

fluidized regime, in which the fluctuation of solid concentration is bigger than that in CFB [51], hence, violating the heterogeneous cluster assumptions in CFB. Due to the meso-scale cluster, the distribution of solid concentration in CFB is characterized with the axial S-shape and radial core-annular. In the experiment, the cluster is often identified by measuring the instant local solid concentration when its value is higher than the value of the artificially set parameter n times the standard deviation of solid concentration. However, the value of n is considered as a scatter in the range of 1–3, as described previously. Fig. 3 shows the effect of the definition of solid concentration in cluster phase on the cluster size based on the model developed in this paper. The value of n has a significant effect on the simulated cluster size. With an increase of n, the cluster size obviously decreases. When the solid concentration is smaller than 0.25, the effect of the different definitions of standard deviation using Eqs. (33) and (34) on the cluster size can be ignored. When the solid concentration exceeds 0.25, the cluster size based on Eq. (34) is smaller than that based on Eq. (33). The differences became more obvious with the solid concentration increases. Moreover, the comparison shows the simulated cluster size based on Eq. (45) correlated by Issangya [51] to be the smallest. The correlations of the cluster size in literature and the relevant calculating results are given in Table 3 and Fig. 4, respectively. The Subbaro’s [56] results are the smallest with compared to other model results. When the solid concentration is higher than 0.1, the cluster size based on the current model deviates from the results obtained from other models. With the solid concentration approaching 0.0, the clusters of all models are close to a single particle. The variation of the cluster size with the solid concentration in the limited range of 0.05–0.2 can be ignored in the current model. In addition, when the solid concentration reached the maximum limitation of packing value, the cluster size becomes equal to the diameter of the reactor that is 0.09 m.

Fig. 5. The influence of operational gas velocity on the cluster size.

Fig. 6. Comparison for the variation of solid flux with time between the simulated results and the experimental data [1].

Fig. 5 shows the influence of gas operating velocity on the cluster size based on the simulated results. When the superficial fluid velocity increases, a bigger cluster is developed at the same solid concentration, which can be explained by the stronger instability

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Fig. 7. Comparison for the axial distribution of average voidage between the experimental data [1] and the simulated results. Fig. 9. Snapshots (a and c at time 21 s) and time-average distributions of the solid concentration with the different drag coefficient models.

Fig. 8. Comparison for the radial distribution of solid concentration between the experimental correlations and the simulated results.

associated with the gas–solid relative motion and the higher dissipation of the fluctuating energy of particles by inelastic collisions between particles and viscous damping at higher superficial velocity. Fig. 6 shows the effect of drag model on the solid flux. The drag coefficient in TFM model has been known as a critical parameter in predicting the fluid flow in CFB [50]. Because the heterogeneous flow structure in CFB can reduce the gas–solid drag force, the solid mass flux would be seriously overestimated if the Gidaspow’s drag coefficient is used. However, with considering the heterogeneous flow structure parameters in a calculating mesh grid, for example EMMS model and the developed model in this paper, the solid flux is closer to the experimental data. The simulated results with EMMS model is slightly lower than the experimental data [1]. In this calculating process the outlet boundary is set as the atmosphere pressure. In the experiment, the outlet is directly connected with the inlet of gas–solid cyclone, and the relevant pressure should be higher than the atmosphere. Therefore, from the theoretical viewpoint, the simulated results for the solid flux should be slightly higher than the experimental data, which is consistent with our model’s results. Due the cluster in CFB, the axial distribution of solid concentration is featured with the S-shape, and is diluted at the top and dense at the bottom. Therefore, Gidaspow’s model, which did not

consider the effect of the heterogeneous flow structure cluster, cannot capture this kind of nature, as shown in Fig. 7. However, the feature can be captured by using the drag coefficient considering the heterogeneous flow structures, such as in EMMS model and the model developed in this paper. The simulated results agreed quantitatively with the experimental data. In addition, the curve of calculation results for the axial distribution of solid concentration is not smooth because there is insufficient time in obtaining the time-average axial solid concentration in simulating the fluid flow. Similar as the axial distribution of solid concentration, the radial distribution is characterized with the core-annular flow in CFB. Fig. 8 shows the comparisons for the radial distribution of solid concentration between the simulated results with or without the consideration of the effect of heterogeneous flow structure and the experimental data. The experimental data at the height of 3.5 m and 8 m are also given, which are obtained from the empirical correlation [52] 2

ef ð/Þ ¼ ef / þ0:191 / ¼ r=R < 0:75 ef ð/Þ ¼ ef 3:62/

6:47

þ0:191

/ ¼ r=R P 0:75

ð46Þ ð47Þ

As being shown from the simulated results of Gidaspow model, the difference for solid concentration at the height 3.5 m and 8 m is minimal, and the natural feature of core-annular radial flow in CFB is not captured. A good agreement exists between the simulated results and the experimental results obtained by using the drag coefficient, with consideration on the heterogeneous flow structure effect. However, there is still a minimal deviation from the experimental data at the center of CFB, which can be explained by the solid concentration in the dilute phase being assumed to be constant in the model developed in this paper. Fig. 9 shows the contours for the instantaneous and time-average distribution of solid concentration, with or without consideration on the effect of heterogeneous flow structure. The heterogeneous flow structure as the cluster can be clearly captured, and the natural feature of CFB with the axial S-shape and radial core-annular flow can be predicted by our developed model. However, the Gidaspow’s homogeneous drag coefficient can make the solid almost homogeneously distributed in CFB, as shown in Fig. 9c and d.

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5. Conclusion In this paper, a model for predicting the heterogeneous flow structure parameters is developed. The simulated results of the cluster size are given and compared with the correlation developed in the literature, showing that all of the results for the cluster size have almost the same order of magnitude and trends with the model developed in this paper. Using the drag coefficient with consideration of the heterogeneous flow structure predicted by the model developed in this paper, the gas–solid heterogeneous flow behavior in CFB can be quantitatively predicted, and the dynamics of the cluster can be qualitatively captured. However, there are a few the simulated results deviating from the experimental data because of some artificial assumptions. For instance, the voidage in the dispersed phase was taken as a constant of 0.9997. Therefore, efforts are still needed to further improve this model in the future. Acknowledgments The author is grateful to the supports on this work from the State Key Development Program for Basic Research of China (973 Program) under Grant No. 2015CB251402, and from National Natural Science Foundation of China under Grant No. 21206159. Reference [1] J. Li, M. Kwauk, Paticle-Fluid Two-Phase Flow: The Energy-Minimization MultiScale Method, Metallurgy Industry Press, Beijing, 1994. [2] D. Gidaspow, J. Jung, R.K. Singh, Hydrodynamics of fluidization using kinetic theory: an emerging paradigm, Power Technol. 148 (2004) 123–141. [3] L.B. Duan, H.Ch. Sun, Ch.S. Zhao, W. Zhou, X.P. Chen, Coal combustion characteristics on oxy-fuel circulating fluidized bed combustor with warm flue gas recycle, Fuel 127 (2014) 47–51. [4] B.L. Yang, X.W. Zhou, X.H. Yang, C. Chen, L.Y. Wang, Multi-Scale study on the secondary reactions of fluid catalytic cracking gasoline, AIChE J. 55 (2009) 2138–2149. [5] M. Kwauk, H.Z. Li, Handbook of Fluidization, Chemical Industry Press, Beijing, China, 2008. [6] B. Lu, W. Wang, J.H. Li, Searching for a mesh-independent sub-grid model for CFD simulation of gas–solid riser flows, Chem. Eng. Sci. 64 (2009) 3437–3447. [7] Y. Igci, T. Andrews IV, S. Sundaresan, S. Pannala, T. O’Brien, Filtered two-fluid models for fluidized gas-particle suspensions, AIChE J. 54 (2008) 1431–1448. [8] Y. Igci, S. Sundaresan, Verification of filtered two-fluid models for gas-particle flows in risers, AIChE J. 57 (2011) 2691–2707. [9] H.Z. Li, Important relationship between meso-scale structure and transfer coefficients in fluidized beds, Particuology 8 (2010) 631–633. [10] W. Ge, W. Wang, N. Yang, J.H. Li, M. Kwauk, Meso-scale oriented simulation towards virtual process engineering (VPE) – the EMMS paradigm, Chem. Eng. Sci. 66 (2011) 4426–4458. [11] H.Z. Li, Y. Xia, Y. Tung, M. Kwauk, Micro-visualization of clusters in a fast fluidized bed, Powder Technol. 66 (1991) 231–235. [12] B.K.N. Agrawal, P. Loezos, M. Syamlal, S. Sundaresan, The role of meso-scale structures in rapid gas–solid flows, J. Fluid Mech. 445 (2001) 151–185. [13] J. You, C. Zhu, B. Du, L.S. Fan, Heterogeneous structure in gas–solid riser flows, AIChE J. 54 (2008) 1459–1463. [14] M. Ye, J.W. Wang, M.A. van der Hoef, J.A.M. Kuipes, Two-fluid modeling of Geldart A particles in gas-fluidized beds, Particuology 6 (2008) 540–548. [15] 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. [16] N. Yang, W. Wang, W. Ge, CFD simulation of concurrent-up gas–solid flow in circulating fluidized bed with structure-dependent drag coefficient, Chem. Eng. J. 96 (2003) 71–80. [17] B.L. Hou, H.Z. Li, Relationship between flow structure and transfer coefficients in fast fluidized beds, Chem. Eng. J. 157 (2010) 509–519. [18] B.L. Hou, H.Z. Li, Q.S. Zhu, Relationship between flow structure and mass transfer in fast fluidized bed, Chem. Eng. J. 163 (2010) 108–118. [19] W.G. Dong, W. Wang, J. Li, A multiscale mass transfer model for gas–solid riser flows: Part 1-sub-grid model and simple tests, Chem. Eng. Sci. 63 (2008) 2798– 2810. [20] W.G. Dong, W. Wang, J. Li, A multiscale mass transfer model for gas–solid riser flows: Part 1-sub-grid simulation of ozone decomposition, Chem. Eng. Sci. 63 (2008) 2811–2823. [21] A. Vepsäläinen, S. Shah, J. Ritvanen, T. Hyppänen, Bed Sherwood number in fluidized bed combustion by Eulerian CFD modeling, Chem. Eng. Sci. 93 (2013) 206–213.

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