CPFD simulation of circulating fluidized bed risers

CPFD simulation of circulating fluidized bed risers

Powder Technology 235 (2013) 238–247 Contents lists available at SciVerse ScienceDirect Powder Technology journal homepage: www.elsevier.com/locate/...

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Powder Technology 235 (2013) 238–247

Contents lists available at SciVerse ScienceDirect

Powder Technology journal homepage: www.elsevier.com/locate/powtec

CPFD simulation of circulating fluidized bed risers Cheng Chen a, b, Joachim Werther a, Stefan Heinrich a, Hai-Ying Qi b, Ernst-Ulrich Hartge a,⁎ a b

Institute of Solids Process Engineering and Particle Technology, Hamburg University of Technology, D-21071 Hamburg, Germany Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, 100084 Beijing, PR China

a r t i c l e

i n f o

Article history: Received 22 May 2012 Received in revised form 26 September 2012 Accepted 11 October 2012 Available online 18 October 2012 Keywords: CFB Gas–solids flow Numerical simulations Eulerian–Lagrangian method CPFD

a b s t r a c t This study investigated the applicability of computational particle fluid dynamics (CPFD) numerical schemes for simulating flows in circulating fluidized bed (CFB) risers. Gas–solid flows were simulated in CFB risers containing Geldart A particles with both low and high solid fluxes as well as in a CFB riser containing Geldart B particles for three flow conditions using CPFD. The results are compared to experimental data and previous two-fluid model (TFM) simulations. The time-averaged axial and radial distributions of the solid concentration show that the bottom-dense, upper-dilute and core-annulus heterogeneous structures were successfully captured by the CPFD calculations, but only qualitatively. The results differ from experimental data for Geldart A particles and high solid fluxes, although they were more accurate than two-fluid simulations with conventional drag models. Two-fluid modeling with the EMMS (energy minimization multi-scale) drag model gave more accurate results than the CPFD simulations. The results indicate that the drag force in the CPFD scheme is still overestimated, although the cumulative method used to compute drag force is more accurate than the proportional method in the two-fluid model. An EMMS drag model which takes into account the intrinsic heterogeneity in the CFB risers is needed for the CPFD scheme. The effect of the realistic particle size distribution was seen in the by CPFD results. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Circulating fluidized beds (CFBs) play an important role in many industrial processes. Coal combustion for power generation, gasification for chemical production and fluid coal cracking for gasoline production are all typical, valuable CFB applications. Accurate modeling of CFBs is significant for their design, optimization, operation and scaling. Thus, reasonable numerical schemes with sufficient accuracy and low computational costs are needed. The computational particle fluid dynamics (CPFD) numerical scheme has been developed recently to supplement the conventional Eulerian–Eulerian and Eulerian–Lagrangian methods. However, the CPFD scheme is still being developed and needs further investigations. The Eulerian–Eulerian two-fluid model (TFM) has been widely used in many fields, especially for dense gas–solid fluidized beds. However, this method has some key limitations. As pointed out by Grace and Sun [1], the particle size distribution significantly influences on the performance of fluidized bed reactors. However, the two-fluid approach has trouble modeling flows with particle type and size distributions because separate continuity and momentum equations must be solved for each size and type [2,3]. In addition, the two-fluid approach cannot easily account for some characteristics of realistic particles such as shear stresses and inter-particle cohesive forces for Geldart A particles [4–6] when treated as a pseudo fluid. ⁎ Corresponding author. Tel./fax: +49 40 42878 3139. E-mail address: [email protected] (E.-U. Hartge). 0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2012.10.014

The Eulerian–Lagrangian approach using a continuum model for the fluid phase and a Lagrangian model for the particle phase provides analysis of flows with a wide range of particle types, sizes, shapes and velocities [7,8]. However, the high collision frequencies with particle volume fractions above 5% and the computational complexity of analyzing dense particle–particle interactions limit the number of particles in current Lagrangian calculations [9–11]. The Computational Particle Fluid Dynamics (CPFD) numerical scheme is an Eulerian–Lagrangian model for gas–solids flows. The fluid uses the Navier–Stokes equation with strong coupling between the discrete particles. The particle momentum model is based on the multiphase particlein-cell (MP-PIC) numerical description [12–14] which is a Lagrangian description of particle motion described by ordinary differential equations with coupling with the fluid. Thus, this scheme can readily handle particle type and size distribution. Furthermore, the interphase momentum transfer function in the present scheme is more detailed than in continuum models with summations of contributions from particles having different velocities and sizes, while the momentum transfer rate in two-fluid method is simply proportional to the difference between the mass-averaged velocities of the phases. In the present CPFD scheme, actual particles are grouped into computational particles each containing a set number of particles with identical densities, volumes and velocities located at a specific position. The computational particle is a numerical approximation similar to the numerical control volume where a spatial region has a single property for the fluid. With these computational particles, large commercial systems containing billions of particles can be analyzed using millions of computational

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particles. The CPFD scheme then allows for extremely efficient calculations of gas–solids fluidization in industrial units. With these attractive advantages, the CPFD scheme has been used for analyzing the fluidized beds [15], bubbling beds [16,17] and a fast fluidized bed steam coal gasifier feeding section [18] in recent years. The present study aims to evaluate the performance and applicability of the CPFD scheme for predicting the flow hydrodynamics in CFB risers. The CPFD modeling of CFB risers will use Geldart A and B particles since Geldart A particles are widely used for CFB catalytic reactors while Geldart B particles are common in coal-fired CFB combustors. Moreover, the CFB riser with the Geldart B particles will be investigated in more detail as a first step for large-scale simulations of CFB combustors with combustion and heat transport. The capability of the CPFD scheme to predict the effects of the particle size distribution will also be evaluated. The CPFD calculations used are the commercial code BARRACUDA [14,17].

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operator, and the summation is over all the numerical particles Np. The sum of the fluid and particle volume fractions must equal unity, εp + εg = 1. The conservation equations are approximated using finite volumes with staggered scalar and momentum nodes. The implicit numerical integration of the particle velocity equation is given by

nþ1 up

¼

h i nþ1 nþ1 1 1 unp þ Δt Dp unþ1 g;p − ρ ∇pp − ρ θ ∇τ p −g p

p p

1 þ ΔtDp

:

ð7Þ

2. CPFD mathematical model

n+1 is the interpolated fluid velocity at the particle locaWhere ug,p n+1 tion, ∇ pp is the interpolated pressure gradient at the particle location, and ∇ τpn + 1 is the interpolated particle stress gradient at the particle location. The new particle location at the next time step is then

2.1. Governing equations

xp

The fluid dynamics equations can be derived from kinetic theory for dilute gasses or the continuum points of view. The volume averaged fluid mass and momentum equations for a continuum are

The fluid momentum equation implicitly couples the fluid and the particles through the interphase momentum transfer. The interphase momentum transfer at momentum cell ξ is

  ∂ ε g ρg ∂t

  þ ∇⋅ εg ρg ug ¼ 0

  ∂ εg ρg ug ∂t

  þ ∇⋅ ε g ρg ug ug ¼ −∇p−F þ ε g ρg g þ ∇⋅ε g τ g :

nþ1

Δt:

"   1 1 nþ1 nþ1 nþ1 − ∇pp np mp : ¼ ∑ Sξ Dp ug;p −up Ωξ p ρp

ð2Þ

The inter-phase drag function, Dp is calculated using a drag model with the drag coefficient Cd,

ð3Þ

    3 ρg ug −up  : Dp ¼ C d 8 ρp rp

p 1 X V n S : V ξ 1 p p pξ

ð4Þ

ð5Þ

N

ð6Þ

Where Vξ is the element volume, Vp is the particle volume, np is the number of particles in a numerical particle, Spξ is the interpolation

ð10Þ

The drag coefficients calculated by the different drag models for spheral particles in the commercial CPFD code BARRACUDA for the conditions in this study are plotted in Fig. 1. The Richardson et al. model [20] and the Stokes model [21] are for a single particle in gas flow. Thus, their drag coefficients have no relationship with the solids concentration. The Wen–Yu model [22], Ergun model [23], Wen–Yu/Ergun model and Turton drag model [24] are for gas–solids

The particle properties are mapped from the Eulerian grid to the particle locations. The particle properties are also then mapped from the particles to the grid to get grid-based properties such as the particle volume fraction at cell ξ, εpξ ¼

ð9Þ

2.2. Drag model

The terms represent the acceleration due to aerodynamic drag, the pressure gradient, gravity and the gradient of the inter-particle normal stress, τp. The particle movement is given by dxp ¼ up : dt

ð8Þ

nþ1 Fξ

Where up is the particle velocity, ρp is the particle density and Dp is the drag function at the particle location. The particles are modeled using the Lagrangian method with the numerical particles each containing np particles with identical properties located at position, xp (xp, yp, zp). The particle acceleration is   1 dup 1 ¼ Dp ug −up − ∇p þ g− ∇τp : ρp ε s ρp dt

nþ1

ð1Þ

Where ug is the fluid velocity, εg is the fluid volume fraction, ρg is the fluid density, p is the fluid pressure, τg is the fluid stress tensor, and g is the gravitational acceleration. F is the rate of momentum exchange per volume between the fluid and particle phases. !   1 F ¼ ∬f m Dp ug −up − ∇p dmdv: ρp

n

¼ xp þ up

Fig. 1. Drag coefficients from different drag models.

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homogeneous flows, so their drag coefficients all increase with the solids concentration. The Wen–Yu equation is limited to particle volume concentrations from 0.01 to 0.61 [22] while the Ergun equation covers the range from 0.47 to 0.7 [23]. The Gidaspow drag model [3] is a combination of the Wen–Yu and the Ergun models with a transition point at a solids concentration of 0.2. However, this concentration is not in the range where these two models overlap which is between 0.47 and 0.61. To prevent possible numerical problems due to the sharp transition or discontinuity in the Gidaspow drag model, a linear transition between these two drag models is used in this study. The combined Wen–Yu/Ergun model is given by Dp ¼ DWen−Yu Dp ¼

εp <0:75εcp

 ε p −0:75ε cp  DErgun −DWen−Yu þ DWen−Yu 0:85ε cp −0:75εcp

ð11Þ

ð12Þ

0:75ε cp <ε p <0:85ε cp

Dp ¼ DErgun

εp > 0:85εcp :

ð13Þ

Where εcp denotes the particle volume concentration at the close packing condition. 2.3. Solids stress model Unlike the Discrete Element Method (DEM) which calculates the particle-to-particle forces using a spring-damper model and direct particle contact [25,26], the CPFD scheme models collision forces on particles as a spatial gradient. The particle stress gradient which is difficult to calculate for each particle in a dense flow is calculated as a gradient on the Eulerian grid and is then interpolated to discrete particles. Therefore, solids loadings from dilute to dense (close packed) can be modeled by the particle stress tensor formulation and interpolation. Particle-to-particle collisions are modeled by the particle normal stress, τp. The particle stress is derived from the particle volume fraction which, in turn, is calculated from the particle volume mapped to the grid. The particle normal stress model used here was developed by Harris and Crighton [27], γ

τp ¼

P s εp h  i : max εcp −εp ; θ 1−εp

ð14Þ

Where Ps is a positive constant that has units of pressure and εcp is the particle volume fraction at the close packing limit. The constant γ is recommended to be 2 b γZ b 5 [28]. The constant θ is a small number on the order of 10 −7 to remove the singularity at close packing [14]. The fluid density, velocity and pressure are coupled by a semiimplicit pressure equation [29]. The momentum and pressure equations are solved with a conjugate gradient solver. A quasi-second order upwind scheme is used for the convection terms [8].

3. Model setup and simulation conditions The two-dimensional CFB riser containing Geldart A particles is shown in Fig. 2a with the dimension parameters listed in Table 1. The three-dimensional CFB riser containing Geldart B particles which was based on a pilot-scale cold CFB experiment [30] is shown in Fig. 2b. The gas and solids properties are listed in Table 2. The Geldart B particles used the particle size distribution (PSD) shown in Fig. 3 which was used in experiment [30]. The CPFD calculations using this PSD are compared with using uniform particles. The input parameters for the CPFD simulations are listed in Table 3. The three dominant parameters in a CFB are the superficial gas velocity, U, the solids inventory in the riser, Iinv and the solids external mass flux, Gs. These three variables are not independent. Either U and Iinv or U and Gs are usually set with the other variable then calculated. U and Iinv are set in this study unless otherwise noted. U is set at the bottom of the riser and Iinv is given by the incipient height, H0. H0 is calculated based on the experimental pressure drop, Δpriser ¼ Iinv g=Ariser ¼ εs0 ρs gH 0 . An external loop is used to ensure a constant inventory at each time step. The simulation conditions are listed in Table 4.

4. Modeling of CFB risers with Geldart A particles Geldart A particles are widely used in CFB risers such as fluid catalytic cracking (FCC) reactor. Two CFB risers were modeled with low and high solid fluxes using the CPFD scheme. The material properties, grid sizes and time steps are the same as in a previous two-fluid study in the literature [31,32]. The particle-to-wall normal retention value is 0.3 as suggested by BARRACUDA. The cross-section averaged solids concentration in the following sections was calculated from the presdp sure drop, εs ¼ dh =ðρs ⋅g Þ, in both the experiments and simulations.

Fig. 2. Schematics of the simulation geometries. (a) 2D CFB riser with Geldart A particles. (b) 3D CFB riser with Geldart B particles.

C. Chen et al. / Powder Technology 235 (2013) 238–247 Table 1 2D geometry parameters.

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Table 3 Input parameters in the CPFD simulation.

Ht (m) Dt (m) h1/h1' (m) h2/h2' (m)

Case A1

Case A2

10.5 0.09 0.28 0.14

14.2 0.2 0.25 0.1

Table 2 Gas and solid properties.

Particle-to-wall interaction Particle normal stress model

Solver setting

Geldart A

Solid density, ρs (kg/m3) Surface mean diameter, dp (μm) Gas density, ρg (kg/m3) Gas viscosity, νg ( Pa·s ) Close-packing solid concentration, εcp

Geldart B

Case A1

Case A2

930 54 1.205 1.815 e−5 0.63

1712 76

2600 140 1.18 1.5e−5 0.55

For Case A1, the calculated solid flux is compared with experimental data [31] and two-fluid modeling results in Table 5. With the same conventional drag model, the CPFD calculations are more accurate than the two-fluid model (TFM) result, indicating the effectiveness of the cumulative drag force in the CPFD scheme. However, the relative error is still 200%, indicating overestimation of the drag force. By contrast, when the two-fluid model is coupled with the EMMS drag model which accounts for the intrinsic heterogeneity in CFB risers [31], the relative error is only −12.4%, indicating the significance of the drag model. The axial profile of the solid concentrations is plotted in Fig. 4. The solid concentration predicted by the two-fluid model distributed homogeneously along the height while the CPFD calculation predict a bottom-dense and upper-dilute heterogeneous structure with the same conventional drag model. Two-fluid modeling with the EMMS drag model provides a more axial heterogeneous structure. Therefore, incorporation of the EMMS drag model into the CPFD scheme should give better results. Unfortunately, the commercial CPFD code does not have user defined functions. The axial pressure drop for Case A2 with a high solid flux of 489 kg/m 2 s [32] is plotted in Fig. 5a with the radial profiles of the voidage at two axial locations (3.9 m and 8.1 m height) plotted in Fig. 5b and c. Fig. 5a shows that the TFM and CPFD calculations with the same drag model give similar results with higher pressure drops near the bottom and lower, nearly constant pressure drops along the height. The TFM with the EMMS drag model gives similar trends with higher pressure drops. This again indicates the overestimation of

Normal retention coefficient, en Tangential retention coefficient, en Diffuse bounce, Df Pressure constant of the solid-phase stress model, Ps Dimensionless constant of the solid-phase stress model, γ Dimensionless constant of the solid-phase stress model, θ Maximum volume iteration Volume residual Maximum pressure iterations Pressure residual Maximum velocity iteration Velocity residual Maximum momentum redirection from collision Gravitational acceleration, g Particle/fluid slip ratio Particle feed per average volume Time step, Δt Total time, t Beginning time for average

0.3 0.99 0 1 Pa 3 10−8 1 10−6 2000 10−8 50 10−7 40% −9.81 1 125 5 × 10−4 s 60 s 40 s

the drag force in the CPFD calculations with the conventional drag model compared to the TFM with the EMMS drag model. Fig. 5b shows that both the TFM and CPFD calculations successfully capture the trend in the experimental profile. The TFM results are in good agreement near the wall while the CPFD result is relatively accurate near the center. At 8.1 m height in Fig. 5c, both the TFM and CPFD calculations give better agreement with the experimental data than at 3.9 m in Fig. 5b, because the upper section in the riser is relatively dilute and the conventional drag model is more accurate. The bottom section is much denser with large heterogeneity, and so a drag model accounting for the intrinsic heterogeneity is needed. 5. Modeling of a CFB riser with Geldart B particles Geldart B particles are common in coal-fired CFB combustors and less likely to agglomerate than Geldart A particles due to their larger size. A three-dimensional CFB riser was modeled with Geldart B particles for three flow conditions. 5.1. Formulation study A formulation study was performed based on a simplified geometry to search for appropriate formulation parameters for the pilot-scale CFB modeling. The effect of grid size, drag model and particle-to-wall interaction parameters were analyzed. 5.1.1. Grid size An approach similar to that used by Kallio [33] was used to find an appropriate grid size for the simulation. The simplified geometry used only the first 4 m from the bottom and neglected the exit geometry to reduce the computing time. The three different meshes listed in Table 6 were used for this simplified geometry. Here, U was set to

Table 4 Simulation conditions.

Fig. 3. Particle size distribution.

Case

U (m/s)

H0 (mm)

εs0 (−)

Δpriser (kPa)

A1 A2 Β1 B2 B3

1.52 5.2 3 4 3

1170 2000 784.1 588.1 588.1

0.5

5.34 16.78 10 7.5 7.5

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Table 5 Comparison of computed and measured solid flux (Case A1). Numerical scheme

Drag model

Measured flux Gs,exp (kg/m2 s)

Calculated flux Gs,cal (kg/m2 s)

Relative error (%)

TFM CPFD TFM

Wen–Yu/ Ergun QL-EMMS

14.3

70.46 42.95 12.53

392.7 200.4 −12.4

3 m/s and Gs to 7.8 kg/m 2 s to eliminate solids looping to save computing time. The time-averaged pressure drop in the riser and the time-averaged local solids volume fraction at x=0.5 mm and y=0.15 mm along the height are plotted in Fig. 6. The difference between the medium and fine grids is much smaller than between the coarse and medium grids. The results for the medium and fine grids are in good agreement. Thus, the medium grid size was used for the following CPFD calculations. In contrast, the fine grid size was required for the TFM simulations to achieve a good grid-independent solution [19]. Therefore, the grid size for the CPFD calculations is coarser than for the TFM calculations.

5.1.2. Drag force The CPFD calculations with the conventional drag model for the Geldart A particles were described in Section 4. The applicability of the CPFD scheme for CFBs with Geldart B particles was investigated because the Geldart B particles are less likely to agglomerate than the Geldart A particles. To further reduce the simulation time, only one-third of the cross-section is used in calculations as suggested by Hartge et al. [19], in addition to using only the first 4 m vertically. The study uses the conditions for Case B3 (Table 3). The results in Fig. 7 show that the CPFD calculations with the Wen–Yu and Wen–Yu/Ergun drag models agree qualitatively. The results of the TFM simulation with the Gidaspow drag model for the same geometry and similar operating conditions [19] are added for comparison. The results are comparable although the drag models differ. The results are similar because both the Wen–Yu/Ergun model and the Gidaspow model are for homogeneous flows and the

Fig. 5. Hydrodynamics of Case A2. (a) Axial distribution of the pressure drop. (b) Radial distribution of the solids concentration at 3.9 m height. (c) Radial distribution of the solids concentration at 8.1 m height.

Fig. 4. Axial profile of solid concentration for Case A1.

TFM gives similar results with different homogeneous drag models [19,34–36]. Quantitative analyses are given in the full pilot-scale simulations described in the next section. The results in Fig. 7a show that, the CPFD predict an axial bottom-dense and upper-dilute heterogeneous structure while the TFM solid concentration is almost uniform along the riser height. This indicates the advantage of the cumulative method used to compute the drag force in the CPFD scheme for Geldart B particles. The results in Fig. 7b show that, the horizontal core-annulus heterogeneous structure is also successfully captured by the CPFD calculation. Therefore, the CPFD scheme is able to qualitatively capture

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Table 6 Grid characteristics. ID

Grid size (mm)

Number of cells

Coarse Medium Fine

62.5 × 37.5 × 80 50 × 30 × 62.5 31.25 × 20 × 50

6400 12,800 38,400

the axial and horizontal heterogeneous structures in CFB risers with Geldart B particles. The Wen–Yu/Ergun drag model is then used for the following simulations to investigate the ability of the CPFD scheme to give quantitatively accurate predictions. 5.1.3. Particle-to-wall interactions The particle-to-wall interaction parameters, including the normal momentum retentions, define how much a particle bounces if it hits a wall. The retention is material dependent, however, there is not much experimental data available to suggest reasonable values. The effect of the normal-to-wall momentum retention on the CPFD simulations with the Wen–Yu/Ergun drag model was investigated to search for the reasonable retention value. Table 7 shows that large normal-to-wall momentum retention factors reduce the solids external mass flux with the reduction decreasing with increasing retention value. The largest difference between the simulated fluxes is only 7.23% indicating the small effect of the retention. The results in Fig. 8a show that more solids are carried upwards with a large normal-to-wall retention. However, the difference between the various retention factors is small and gradually disappears in the upper region of the riser. The results in Fig. 8b show that all three retention factors are able to capture the radial heterogeneous structure. At x = 0.025 m, the solids concentration for a retention of 0.9 is larger

Fig. 7. Time-averaged solids concentration for different drag models. (a) Axial distribution. (b) Horizontal distribution.

than for the other two because more particles rebound in this region from the wall. For x > 0.225 m, the results with retention factors of 0.3 and 0.9 are almost the same. Therefore, the value suggested by BARRACUDA of 0.3 is reasonable. 5.2. Results and discussion for the CFB riser with Geldart B particles The following simulations are based on the geometry in Fig. 2b without simplification. The axial and horizontal distributions of the solid concentration predicted by the CPFD model are compared with experimental data and TFM results using the EMMS drag model [34], because the TFM with conventional drag model has been found to be inaccurate. The EMMS-based drag model is reasonable for CFB risers because it accounts for the intrinsic heterogeneity in the risers [34–38]. 5.2.1. Comparison of CPFD with TFM simulation The predicted external solid flux was found to be much larger than the measured data. Thus, the drag force is overestimated in the CFB riser with Geldart B particles. The drag overestimate is mainly due to the inaccurate drag model for these conditions and may be partially due to underestimates of the inter-particle collisions which have resistant effects on the solids acceleration [39–42]. The solid stress Table 7 Calculated solids mass flux. Fig. 6. Time-averaged properties with different element sizes (U=3 m/s, Gs =7.8 kg/m2 s). (a) Pressure drop. (b) Solids concentration.

Normal-to-wall momentum retention value Solids mass flux (kg/m2 s)

0.1 30.4

0.3 28.6

0.9 28.2

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Fig. 8. Time-averaged solids concentration for different wall retentions. (a) Axial distribution. (b) Horizontal distribution.

model in the CPFD scheme only prevents the solids concentration from exceeding the close-packing limit. An approach similar to Gibilaro et al. [43] was then used to reduce the drag force to improve

the simulation accuracy by multiplying the Wen–Yu/Ergun drag model by a scaling factor of 0.2. The results in Fig. 10 show that the CPFD simulations show agreement with experimental data [30] for the axial solids concentration profile. The TFM results with the EMMS drag model [19] are added for comparison. The EMMS drag correlation was calculated following the method of Yang et al. [34] for the specific conditions in this study. This comparison seeks to show whether the CPFD simulations with the conventional drag model are better than the TFM with the specially developed EMMS-based heterogeneous drag model, after the CPFD has been verified to be more reasonable than the TFM with the conventional drag model. The CPFD simulation shows good agreements with experimental data for Case B1 shown in Fig. 10a, near the bottom of the riser, especially the reduction of the solids concentration at about 0.5 m height which was not captured by the TFM simulation. This reduction is caused by the recycling inlet flow which can be seen in Fig. 9a. However, both the CPFD and TFM results are smaller than the experimental data above 6 m height. Thus, the drag force is properly predicted in the bottom of the riser but underestimated in the upper part. The resulting external solids mass flux is only 0.94 kg/m 2 s, much smaller than the experimental flux of 20 kg/m2 s. For Case B2, more particles are carried upwards as shown in Fig. 9b and the resulting external solids mass flux is 19.6 kg/m 2 s, agreeing well with the experimental value of 20 kg/m2 s. In Fig. 10b, the predicted CPFD solids concentration is accurate in the upper section but less than the experimental data below 0.6 m height in the riser. Thus, the drag force is reasonable in the upper part but overestimated in the bottom region. The results for Case B1 and B2 show that the drag force is quite important for CPFD simulations. Reducing the drag force with a fixed scaling factor can improve accuracy but still cannot predict accurate local heterogeneous structures in the riser. A reasonable drag force should be small in the bottom region to predict more solids accumulation and larger in the medium and upper regions to carry more solids upwards. The drag correlation in CPFD is not able to quantitatively predict the drag force even though it is more accurate than TFM method with the same drag model. A drag model is needed for heterogeneous flows which can account for the intrinsic heterogeneity at various locations in the riser. An EMMS drag model in the CPFD scheme may be very effective and needs future investigations.

Fig. 9. Transient and time-averaged solids concentrations in the riser. (a) Case B1: U = 3 m/s, Δpriser = 10 kPa. (b) Case B2: U = 4 m/s, Δpriser = 7.5 kPa.

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Fig. 10. Comparison between the TFM and CPFD simulations and experimental data. (a) Case B1: U = 3 m/s, Δpriser = 10 kPa. (b) Case B2: U = 4 m/s, Δpriser = 7.5 kPa. Fig. 12. Effect of particle size distribution on the axial solids concentration profiles.

5.2.2. Effect of particle size distribution In the CPFD, the solid particles are modeled using both Eulerian and Lagrangian approaches which is quite different from the original Eulerian–Lagrangian approach. The ability of CPFD to capture the particle size distribution (PSD) is then investigated here. Comparison of Fig. 11a and b shows that when the true particle size distribution is taken into consideration, more particles are carried upwards and the time-averaged solids concentration in the bottom is increased because small particles are easily to be carried up while larger ones tend to concentrate in the bottom region. Thus, the CPFD simulations are able to detect the effect of the particle size distribution. However, the particle size effect on the solids concentration distribution is not significant for these conditions here. As Fig. 12 shows, there are no obvious differences between the axial distributions of the solids concentrations simulated using the single particle size and that with the real particle size distribution.

corner at x=0.047 m. The simulated solids distributions show good agreement with the experimental profile, exhibiting the core-annulus flow structure. At x=0.247 m, the solids volume concentration decreases slightly in the middle with increasing height. The onset of the increase of the solids concentrations towards the wall moves closer to the center. This shows that the extension of the wall region in the bottom zone increases with height. At larger heights, the solids volume concentration decreases in the center as well as near the wall. The results close to the sidewall at x=0.047 m show basically the same trends as at x= 0.247 m although there is a slight influence of the wall zone along the sidewall at higher levels.

5.2.3. Horizontal flow dynamics Fig. 13 shows the horizontal solids concentration profiles at heights of 0.405 m and 0.67 m alongside the riser at x=0.247 m and close to the

The present study evaluates the applicability of the CPFD scheme for predicting the flow hydrodynamics in CFB risers. The CPFD was used to analyze flows in 2D CFB risers with Geldart A particles for

6. Conclusions

Fig. 11. The effect of the particle size distribution on the solids distribution inside the riser (Case B3: U = 3 m/s, Δpriser = 7.5 kPa). (a) Computed with average particle size. (b) Computed with particle size distribution.

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g H0 Iinv np Np p u up Vξ x

gravitational acceleration (m/s 2) initial height of solids accumulation in the riser (m) solids inventory in the riser (kg) number of particles in a numerical particle number of numerical particles gas pressure (Pa) gas velocity (m/s) particle velocity (m/s) grid volume particle position (m)

Greek letters εg gas volume fraction εs particle volume fraction εs0 particle volume fraction at the initial condition εcp particle volume fraction at close packing ρg gas density (kg/m 3) ρs particle density (kg/m 3) τg gas stress tensor τp particle normal stress (N/m 2) νg gas viscosity (Pa·s) ∇Pp pressure gradient at the particle location ∇τp particle stress gradient

References

Fig. 13. Horizontal solids concentration profiles (Case B3: U = 3 m/s, Δpriser = 7.5 kPa, PSD). (a) x = 0.247 m. (b) x = 0.047 m.

low and high solid fluxes and in a 3D CFB riser with Geldart B particles for three different flow conditions. The axial and radial heterogeneous flow structures in the risers were successfully captured by CPFD simulations with more accurate predictions than previous TFM simulations with the same drag model. Therefore, the cumulative method used to compute the drag force in the CPFD scheme is more precise than the proportional method used in TFM approach. However, quantitative comparisons with experimental data show that the CPFD calculations overestimate the drag force, especially for Geldart A particles and high solid fluxes. TFM simulations with the EMMS drag model gave more accurate results than the CPFD simulations. Therefore, an improved drag model, such as EMMS drag model which accounts for the intrinsic heterogeneities in CFB risers is required for CPFD calculations to give accurate results. Further work should incorporate an EMMS drag model into the CPFD scheme. The results further show that the CPFD scheme is able to predict the effect of realistic particle size distributions in CFB risers.

Nomenclature Ariser cross sectional area of the riser (m 2) Cd drag coefficient dp particle diameter (m) Dp drag function (kg/m 3 s) F rate of momentum exchange per volume between the gas and particle phases (N/m 3 s) Gs solids external mass flux (kg/m 2 s)

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