Investigation of particle–wall interaction in a pseudo-2D fluidized bed using CFD-DEM simulations

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Investigation of particle–wall interaction in a pseudo-2D fluidized bed using CFD-DEM simulations Tingwen Li a,b,∗ , Yongmin Zhang c , Fernando Hernández-Jiménez d a

National Energy Technology Laboratory, Morgantown, WV 26507, USA AECOM, Morgantown, WV 26507, USA State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Beijing 102249, China d Universidad Carlos III of Madrid, Department of Thermal and Fluid Engineering, Av. de la Universidad, 30, 28911 Leganés, Madrid, Spain b c

a r t i c l e

i n f o

Article history: Received 2 April 2015 Received in revised form 15 May 2015 Accepted 18 June 2015 Keywords: Gas–solid flow Fluidized bed Computational fluid dynamics Discrete element method Particle–wall interaction Two-dimensional flow

a b s t r a c t We report on discrete element method simulations of a pseudo-two-dimensional (pseudo-2D) fluidized bed to investigate particle–wall interactions. Detailed information on macroscopic flow field variables, including solids pressure, granular temperature, and normal and tangential wall stresses are analyzed. The normal wall stress differs from the solids pressure because of the strong anisotropic flow behavior in the pseudo-2D system. A simple linear relationship exists between normal wall stress and solids pressure. In addition, an effective friction coefficient can be derived to characterize particle–wall flow interaction after evaluating the normal and tangential wall stresses. The effects of inter-particle and particle–wall friction coefficients are evaluated. Strong anisotropic flow behavior in the pseudo-2D system needs to be considered to validate the two-fluid model where the boundary condition is usually developed based on an isotropic assumption. The conclusion has been confirmed by simulation with different particle stiffnesses. Assumptions in the newly developed model for 2D simulation are further examined against the discrete element method simulation. © 2015 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

Introduction Pseudo-two dimensional (2D) fluidized beds, which are planar rectangular columns of limited thickness, are used extensively in experimental studies to improve understanding of complex flow dynamics in gas–solid systems. A benefit of the small thickness and typically transparent walls is that they allow direct observation of complex solids flow behavior, such as bubbles and clusters in gas–solid systems through non-intrusive imaging techniques, such as particle image velocimetry (PIV), and digital image analysis (DIA). Extensive fundamental research in pseudo-2D fluidized beds exists in the literature on bubble properties, jet penetration, particle clustering, solids flow patterns, and solids mixing and segregation, and has been used to improve the understanding of flow behavior in three-dimensional (3D) fluidized beds (Busciglio, ´ & Soler, Vella, Micale, & Rizzuti, 2008; Caicedo, Marqués, Ruız, 2003; Goldschmidt, Link, Mellema, & Kuipers, 2003; HernándezJiménez, Sánchez-Delgado, Gómez-García, & Acosta-Iborra, 2011;

∗ Corresponding author at: National Energy Technology Laboratory, Morgantown, WV 26507, USA. Tel.: +1 304 285 4538. E-mail addresses: [email protected], [email protected] (T. Li).

Hernández-Jiménez, Sánchez-Prieto, Soria-Verdugo, & AcostaIborra, 2013; Hernández-Jiménez, Li, Cano-Pleite, Rogers, & AcostaIborra, 2014; Laverman, Roghair, van SintAnnaland, & Kuipers, 2008; Lim, Agarwal, & O’neill, 1990; Müller, Davidson, Dennis, & Hayhurst, 2007; Pallarès & Johnsson, 2006; Rowe, Partridge, Cheney, Henwood, & Lyall, 1965; Sánchez-Delgado, Marugán-Cruz, Soria-Verdugo, & Santana, 2013; Xu & Zhu, 2011; Zhong & Zhang, 2005). The abundant qualitative and quantitative experimental data available and its relatively simple geometry make the pseudo2D system a good candidate for model development and validation. Continuous evolvements in computational hardware and numerical algorithms have led to computational fluid dynamics (CFD) becoming an important tool to help understand complex flow behavior in gas–solid fluidized beds. Several CFD modeling approaches exist for gas–solid flow simulations as reviewed by van der Hoef et al. (2006). Among them, the Eulerian–Eulerian (EE) method (also termed the two-fluid model or TFM) and the Lagrangian–Eulerian (LE) method is the most widely used approaches to simulate gas–solid flows. The former treats gas and solid phases as interpenetrating continua with appropriate constitutive correlations. The latter treats the gas phase as a continuum, but tracks the solid phase on the particle level by following the trajectory of each individual particle or swarm of particles. The

http://dx.doi.org/10.1016/j.partic.2015.06.001 1674-2001/© 2015 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

Please cite this article in press as: Li, T., et al. Investigation of particle–wall interaction in a pseudo-2D fluidized bed using CFD-DEM simulations. Particuology (2015), http://dx.doi.org/10.1016/j.partic.2015.06.001

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Nomenclature ε   g  ␴ ␻ A d F g I m N P S t T T u v U V x

volume fraction granular temperature, (m2 /s2 ) friction coefficient gas viscosity, (Pa s) density, (kg/m3 ) stress, (Pa) angular velocity, (rad/s) area, (m2 ) diameter, (m) force, (N) gravity, (m/s2 ) inertial momentum, (kg m2 ) mass, (kg) normal wall stress, (Pa) pressure, (Pa) tangential wall stress, (Pa) time, (s) stress, (Pa) torque, (J) particle velocity, (m/s) gas velocity, (m/s) velocity, (m/s) volume, (m3 ) position, (m)

discrete element method (DEM), which is also termed the discrete particle method (DPM), is used often. In the DPM, the solid phase is represented by individual particles and particle–particle and particle–wall collisions are resolved directly using a hard- or soft-sphere model. Given the detailed information provided by the DEM simulation, it has been used extensively to conduct fundamental research of granular flow to develop and validate constitutive sub-models for the continuum model (Chialvo & Sundaresan, 2013; Louge, 1994). TFM and DEM have been used to study flow hydrodynamics in pseudo-2D fluidized bed systems numerically. Particle–wall interactions play a significant role in affecting flow behavior in laboratory-scale gas–solid fluidized beds especially for pseudo-2D systems with limited thickness. The front and back walls restrict solids movement in two directions and exert a strong frictional force on the solids flow, which leads to flow behavior different from that in a 3D cylindrical system (Cranfield & Geldart, 1974; Geldart, 1970; Glicksman & McAndrews, 1985; Rowe & Everett, 1972). However, in most numerical simulations, the pseudo-2D fluidized beds are modeled in two-dimension, and the effect of the front and back walls is ignored. The wall effect in pseudo2D gas–solid systems has been investigated in several numerical studies (Cloete et al., 2013; Feng & Yu, 2010; Kawaguchi, Tanaka, & Tsuji, 1998; Li, Grace, & Bi, 2010; Li, Gopalakrishnan, Garg, & Shahnam, 2012). It has been demonstrated that the frictional effect from the front and back walls, which is not considered in most 2D simulations, leads to a deviation in solids velocity and bubble rising velocity (Li et al., 2010; Li, Gopalakrishnan, et al., 2012; Cloete et al., 2013). To account for the wall effect, a 3D simulation is needed for accurate prediction of pseudo-2D gas–solid fluidized beds. 3D DFM simulations are expensive, especially for systems with fine particles. The TFM is less computationally expensive but requires an accurate closure model and boundary conditions. In the TFM, the governing equations for the solids phase are mostly closed by kinetic granular theory (Gidaspow, 1994). Different wall boundary conditions for the solids phase, i.e., freeslip, partial-slip, and no-slip wall have been used in the literature.

Among them, the partial-slip wall boundary condition is believed to be most realistic and much effort has been made to improve its accuracy (e.g., Jenkins, 1992; Li & Benyahia, 2012; Johnson & Jackson, 1987; Schneiderbauer, Schellander, Loderer, & Pirker, 2012; Soleimani, Schneiderbauer, & Pirker, 2015; Zhao, Zhong, He, & Schlaberg, 2014). In the development of such wall boundary conditions, several assumptions have been made, such as steady flow state at the wall and isotropic particle velocity distribution. Given that an accurate wall boundary condition is critical for quantitative validation of a pseudo-2D system considering strong wall effects, detailed information on the particle–wall interaction in such a system is of great value to improve available boundary conditions. Alternatively, Li and Zhang (2013) developed a model for 2D simulations to account for the effect of front and back walls in a pseudo-2D gas–solid fluidized bed without the need for a 3D simulation. The model was developed based on the partial-slip wall boundary condition after introducing several assumptions. The results were improved compared with those of obtained when the system was modeled as a 2D plane, and the computational cost was reduced significantly compared with the 3D simulation. However, the validity of the new model relies on the appropriateness of the wall boundary condition and assumptions were used to develop the model. In this study, DEM simulations of a pseudo-2D gas–solid fluidized bed are reported to investigate particle–wall interactions. By taking advantage of the detailed particle–particle and particle–wall collisions resolved by the DEM simulations, the macroscopic flow field variables used in the TFM, including solids pressure, granular temperature and normal, and tangential stresses, are derived and examined. Specifically, the relationships between these field variables are analyzed and these can be used to improve the wall boundary condition for the solids phase in TFM simulations. Finally, the assumptions used in the 2D flow model proposed by Li and Zhang (2013) for pseudo-2D fluidized bed simulations are examined and the validity of that model is confirmed.

Numerical simulations Numerical model Open-source MFIX (multiphase flow with interphase exchange)-DEM code, developed at the US Department of Energy’s National Energy Technology Laboratory (NETL) was used to conduct numerical simulations of a 2D bubbling fluidized bed (Garg, Galvin, Li, & Pannala, 2012a; Garg, Galvin, Li, & Pannala, 2012b; Li, Garg, Galvin & Pannala, 2012). In MFIX-DEM, a DEM for the solid particles is coupled with a CFD flow solver to simulate gas–solid flow. In the DEM, the position and linear and angular velocities of each particle are tracked according to Newton’s laws. Inter-particle collisions are resolved directly using the soft-sphere model (based on a linear spring-dashpot model) of Cundall and Strack (1979), which treats the collision as a continuous process occurring over a finite time. The contact force is calculated as a function of the distance between colliding particles based on physically realistic interaction laws using an empirical spring stiffness, dissipation constant, and friction coefficient. The particle–wall interaction is treated in the same way as the particle–particle collision. Gas flow is simulated by solving the averaged Navier–Stokes equations for mass and momentum conservation, which account for solid volume fraction and additional coupling terms because of interactions between the two phases (Tsuji, Kawaguchi, & Tanaka, 1993). The governing equations and key closure models solved in the MFIX-DEM are summarized in Table 1. Details of the governing equations along with the numerical implementation and coupling procedure can be found in Garg et al. (2012a, 2012b).

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Table 1 Equations and closure models solved in MFIX-DEM for this study. Gas phase ∂(εg g ) + ∇ · (εg g vg ) = 0 ∂t d g vg ) = ∇ · S¯¯ g + εg g g (ε g dt

− Igs

S¯¯ g = −Pg I +  g DEM particles dxi dt

= ui

dvi dt d␻i Ii dt

mi

= Fi = Ti

Closures εs + εg = 1 Fi = mi g + Fci + Fdi Fci closed by soft sphere model (Cundall & Strack, 1979) Fdi = −∇ Pg (xi )Vi + Igs =

ˇi =

1 V

N 

ˇi Vi εs

(vg (xi ) − ui )

Fdi

⎧ i=1 ⎪ ⎨ 150 εs (1 − ε2g )g ⎪ ⎩ 

εg di

3 CD,i ε−2.65 g 4



+ 1.75



εs g vg (xi ) − ui 

di   εs εg g vg (xi ) − ui 

di 24 0.687 (1 + 0.15(Rei εg ) ) CD,i = Rei εg 0.44  |v (x )−u |d Rei = g g ig i i

ifεg < 0.8 ifεg ≥ 0.8

if Rei εg < 1000 if Rei εg ≥ 1000

Fig. 1. Schematic of a pseudo-2D gas–solid fluidized bed under investigation.

The MFIX-DEM code has been verified and validated for a wide range of applications (Garg et al., 2012a; Li, Garg, et al., 2012).

Simulated setup A 0.3 m × 1 m × 0.01 m pseudo-2D cold flow fluidized bed is shown schematically in Fig. 1. Detailed information on the experimental setup and test can be found in Hernández-Jiménez et al. 2013. The front and rear walls of the bed were made of glass to allow for visual observation. Air at room temperature was fed through a perforated distributor to fluidize the bed material. The distributor consisted of two rows of 30 1-mm-diameterholes arranged in a triangular configuration with 1 cm pitch. Experimental tests were simulated using ballotini glass beads (0.6–0.8 mm diameter with assumed average particle diameter of 0.7 mm, density of 2500 kg/m3 ). A static bed height of 0.3 m, which results in ∼3 million particles inside the system, was simulated. The bed is fluidized by air at a superficial gas velocity of 0.88 m/s, which corresponds to two times the minimum fluidization velocity of 0.44 m/s measured experimentally (Hernández-Jiménez et al., 2013). The numerical parameters used in the simulations are listed in Table 2. These parameters were maintained constant for all simulations unless noted in the parametric study. A parametric study for the coefficient of friction for inter-particle and particle–wall collisions has been carried out since it is believed to be the most critical wall frictional force parameter acting on the particles. A parametric study for the particle spring constant was conducted to investigate its influence on the simulation results. In the DEM simulation, rotation of each particle is tracked explicitly and its contribution to the particle–wall interaction is accounted for in the analysis. However, the effect of rolling friction is not considered in this study, because the numerical study of a similar system suggested that particle

Table 2 Parameters used in baseline numerical simulation. Bed height, width, thickness (m) Static bed height (m) Particles density (kg/m3 ) Particle diameter (mm) Gas density (kg/m3 ) Pressure (atm) Normal inter-particle spring constant (N/m) Tangential inter-particle spring constant (N/m) Inter-particle restitution coefficient Particle–particle friction coefficient Normal wall–particle spring constant (N/m) Tangential wall–particle spring constant (N/m) Particle–wall restitution coefficient Particle–wall friction coefficient Superficial gas velocity (m/s) Numerical grid size (mm)

1, 0.3, 0.01 0.3 2500 0.7 1.2 1 2000 571 0.95 0.6 2000 571 0.8 0.6 0.88 5

rotational kinetic energy is negligible compared with translational kinetic energy (Li, Gopalakrishnan, et al., 2012). A uniform grid size of 5 mm (7 times the particle size) was used for the CFD solver to discretize the 3D computational domain, which leads to two cells in the thickness direction. The wall effect is accounted for mainly by solids flow through particle–wall interactions. Ideally, the gas velocity at the wall is zero and a no-slip wall boundary condition should be used. As strong gradient exists for gas flow in the thickness direction, which is caused mainly by particle flow in the other two directions, a boundary layer with a thickness of the order of one particle diameter is expected. However, the detailed gas flow behavior near the wall in the 1- or 2-particle layer is beyond the current model capability of CFD–DEM solving average Navier–Stokes equations with a numerical grid greater than

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Fig. 2. Snapshots of solids distribution from (a) experiment and (b) numerical simulation (numerical simulation: Ug = 0.88m/s, fric = 0.6).

the particle size (Li, Gopalakrishnan, et al., 2012). In this study, the variation of gas flow in the thickness direction is ignored and a free-slip wall boundary condition is used for the gas phase at the front and rear walls. One case with 2.5 mm grid size in the thickness direction is conducted to confirm that the results are insensitive to the grid size in that direction. Gas flow is fed through dispersed cells in the bottom boundary to mimic the perforated distributor used in the experiment. Limited by the grid resolution, 20 holes in the distributor were simulated to promote bubble formation introduced by jet collapse observed in the experiment. The gas flow leaves the computational domain through the top boundary at constant pressure. A no-slip boundary condition was used for the gas phase at the left and right side walls for simplicity. The effect of gas phase wall boundary condition may become important for certain applications (Beetstra, van der Hoef, & Kuipers, 2007). No further investigation of the effect of left and right wall boundary conditions for the gas phase was conducted since this study focuses mainly on the particle–wall interaction from the front and back walls. All simulations were run in hybrid parallel mode by coupling distributed memory parallel (DMP) and shared memory parallel (SMP) using message passing interface (MPI) and open multiprocessing (OpenMP) on NETL’s supercomputer, HPCEE (high performance computer for energy and environment)(Liu, Tafti, & Li, 2014; Gopalakrishnan & Tafti, 2013). Transient simulation results were saved at 100 Hz and each simulation allowed for 10 s analysis time.

bed material. Overall, the numerical simulation predictions agree reasonably well with the experimental bubbling behavior. In addition to the qualitative comparison with experimental observations, DEM simulation results have been used to characterize the overall wall frictional force (Hernández-Jiménez et al., 2014). Based on the force balance model proposed by HernándezJiménez et al. (2013), the overall frictional force because of particle–wall interactions was estimated and found to be consistent with the experimental analysis. The DEM results corroborated that the overall frictional force can be considered equal to the velocity of the center of mass times, a particle–wall interaction coefficient, c. A local estimation of the coefficient c was obtained by analyzing the vertical component of the tangential force of the particle–wall collisions and the vertical particle velocity. Its most probable value in the DEM simulations is similar to the global value obtained experimentally. Given the qualitative and quantitative consistency to the experiment, the DEM simulation results reported in Hernández-Jiménez et al. (2014) were analyzed further to investigate the detailed particle–wall interaction in the pseudo2D column. This study seeks a more fundamental approach to characterize the particle–wall interactions to improve the boundary conditions used in the TFM simulations of pseudo-2D systems. Characterization of particle–wall interaction To improve the understanding of the particle–wall interactions, macroscopic flow variables typically used in the TFM, such as solids stress tensor and granular temperature, are extracted. To derive the continuous field variables from the DEM simulation results, an appropriate spatial window size is required for data analysis. For the CFD–DEM simulation, it is natural to choose the fluid cell as the window for calculating the macroscopic field variables. By summing the particle–wall contact force, Fiw , for each particle in contact with the wall, the stress exerted by the particles on the wall can be calculated as: TW =

i∈W

Tn = n · TW ,

(2)

and





Ts = TW − nTn  .

(3)

where, n is the unit vector normal to the wall. The granular temperature, , is calculated as 1  ¯ · (ui − u), ¯ (ui − u) 3N N

Results and discussion

Fig. 2 shows two snapshots taken from experimental tests and numerical simulations for qualitative comparison. The light color corresponds to a high solids concentration and the dark color corresponds to a bubble or freeboard with low solids concentration. Small gas jets formed above the distributor orifices can be observed from experimental images, which are captured by the numerical simulation. Small bubbles formed near the distributor grow and collapse as they rise. Large bubbles burst at the bed surface. The numerical result shows smoother bubble surfaces than the experimental image, mainly in the spherical capped region. The slight difference may be attributed to the particle size distribution in the

(1)

where, A is the surface area of the wall cell. The normal and tangential components of the wall stress are

=

Comparison with experimental observation

1 Fiw , A

(4)

i=1

where, N is the number of particles in the computational cell, ui ¯ is the instantaneous spatially is the particle velocity vector, and u averaged solids velocity in the cell, which is calculated as: 1 ui . N N

¯ = u

(5)

i=1

The macroscopic solid stress tensor, , is calculated in each computational cell as (Chialvo, Sun, & Sundaresan, 2012):

⎡ ⎤ N   1 1 ⎣ ¯ (ui − u) ¯ ⎦ = rij Fij + mi (ui − u) V

i=1

j= / i

2

(6)

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Fig. 3. Snapshots of distribution of different field variables: (a) solids concentration, (b) solids pressure, (c) kinetic contribution to the solids pressure, (d) normal wall stress, (e) tangential wall stress, and (f) granular temperature (numerical simulation: Ug = 0.88 m/s, fric = 0.6).

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Fig. 4. Raw data of normal wall stress versus solids pressure for emulsion phase together with a linear regression (numerical simulation: Ug = 0.88 m/s, fric = 0.6).

where, rij is the vector from center of particle j to center of particle i, Fij is the contact force between the colliding particles, mi is the particle mass, and V is the volume of the domain for analysis, i.e., the computational cell. The solids pressure is defined as the average of the diagonal terms of the solids stress tensor, i.e., Ps = 1/3 (11 + 22 + 33 ), and can be calculated as:

⎡ ⎤ N   1 1 ⎣ Ps = rij Fij ⎦ + p εs  3V

i=1

j= / i

2

(7)

The first term of Eq. (6) originates from the particle–particle interactions in the form of instantaneous collisions or long-time frictional contact. The second term stands for the kinetic contribution to solids pressure (Gidaspow, 1994). The calculated quantities may be affected by the number of particles inside the sampling window. For example, a granular temperature, and solids pressure may exhibits substantial statistical error for cells inside a bubble or in the freeboard region because of the limited number of particles present there. In this study, we focus mainly on the dense emulsion phase for which the number of particles in a

computational cell is believed to be sufficient for the calculation (Gopalan & Shaffer, 2013). Fig. 3 shows transient snapshots of distribution of the calculated field variables. Fig. 3(a) presents the solids concentration distribution inside the bed. The solids pressure as calculated based on Eq. (6) is shown in Fig. 3(b) and the kinetic contribution, i.e., the second term in the right-hand side of Eq. (6), is shown in Fig. 3(c). The solids pressure is high in the dense emulsion phase and low in the dilute bubble region. Because of higher solids velocities and stronger solids interactions, the solids pressure in the bubble wake is especially higher than in other areas. The kinetic component of solids pressure exists mainly in the lower bubble wake area where the granular temperature as shown in Fig. 3(f) is high. As revealed by the simulations, the kinetic part of the solids pressure shown in Fig. 3(c) is negligible compared with the first part in Eq. (6). In the following analysis, the kinetic part of the solids pressure is ignored and only the component from inter-particle contact is used. The normal and tangential components of the stress on the wall exerted by the particles are shown in Fig. 3(d) and (e). Transient snapshots show a strong correlation between solids pressure and wall stress.

Fig. 5. Averaged normal wall stress versus solids pressure for emulsion phase (the symbols are average data and error bars represent ± one standard deviation; numerical simulation: Ug = 0.88m/s, fric = 0.6).

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Fig. 6. Components of granular temperature in (a) x-, (b) y-, and (c) z-directions (numerical simulation: Ug = 0.88 m/s, fric = 0.6).

Fig. 7. Average tangential wall stress versus normal wall stress for emulsion phase (symbols are average data and error bars represent ± one standard deviation; solid line is a linear fit, dashed line represents Eq. (9) with slope of physical particle–wall friction coefficient; numerical simulation: Ug = 0.88m/s, fric = 0.6).

In the following analysis, mainly particle–wall interactions for the emulsion phase are investigated. For this purpose, the solids pressure, and wall stresses in each computational cell are extracted for the emulsion phase. To distinguish the emulsion and the bubble phases, a voidage of 0.7 is used. Different values for voidage have been used in the literature to demarcate bubble boundaries. Here a low value of 0.7 is used so that the analyses focus on the dense emulsion phase with lower statistical error. A large amount of data has been extracted from the transient simulation results that cover hundreds of frames for analysis. Fig. 4 shows plots of normal wall stress versus solids pressure for each computational cell collected over a 4 s simulation period. Approximately half a million points are plotted. The normal wall stress shows a clear dependence on solids

pressure. The calculated normal wall stress increases with solids pressure. According to the definition, the normal wall stress equals  33 at the wall, which is the solids stress component in the normal wall direction. Hence, the difference between normal wall stress and solids pressure shown here indicates strong anisotropy in the flow. A simple linear regression can be used to derive the relationship between these two variables as shown in Fig. 4. The reported R-squared value of 0.8161 indicates that linear fitting is reasonable for capturing the correlation between normal wall stress and solids pressure. In particular, the normal wall stress is of the order of 7% of the solids pressure. The anisotropic flow behavior is believed to be related to gravity, to the dominant flow direction, and to the wall effect. Particle movement in the thickness direction is limited

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Fig. 8. Plots of averaged normal wall stress versus solids pressure for emulsion phase for different friction coefficients: (a) fric = 0.3, (b) fric = 0.4, (c) fric = 0.5, and (d) fric = 0.7.

Fig. 9. Plots of averaged tangential wall stress versus normal wall stress for emulsion phase for different friction coefficients: (a) fric = 0.3, (b) fric = 0.4, (c) fric = 0.5, and (d) fric = 0.7.

because of the wall effect and is less vigorous compared with the other two directions. Hence, the normal force on the wall exerted by particle flow is much lower compared with the solids pressure, which is an average of three orthogonal normal stresses. To better quantify the relationship between normal wall stress and solids pressure, the large amount of data shown in Fig. 4 has been binned in small intervals with respect to the solids pressure, and then average and standard deviations of the normal wall stress are calculated as shown in Fig. 5. In Fig. 5, symbols are the

average normal wall stress and the error bars represent ± one standard deviation for data in each bin. A linear correlation can be obtained by fitting the average data, which yields a similar relationship to the linear regression of the raw data in Fig. 4. The equation is shown in Fig. 5, where x corresponds to the x-axis variable, i.e., the solids pressure and y corresponds to the y-axis variable, i.e., the normal wall stress. Results for high solids pressure are not included in Fig. 5 since there are insufficient data points for averaging (a cut-off criterion of 100 is used here).

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Fig. 10. Normal wall stress versus solids pressure in emulsion phase for simulation with spring constant of 200,000 N/m using different window sizes (numerical simulation: Ug = 0.88m/s, fric = 0.6).

Fig. 11. Plots of averaged normal wall stress versus solids pressure in emulsion phase for simulation with spring constant of 200,000 N/m using different window sizes: (a) 1 × 1, (b) 2 × 2, (c) 3 × 3, and (d) 4 × 4 (numerical simulation: Ug = 0.88 m/s, fric = 0.6).

To investigate the anisotropic flow behavior encountered in the current pseudo-2D system, the granular temperature is decomposed into three components:  = x + y + z ,

(8)

where, x , y , and z account for contributions from different velocity components. As shown in Fig. 6, the magnitudes of different components of granular temperature are different. Most granular temperature originates from fluctuating velocity in the vertical direction, which is the dominant flow direction. The presence of front and back walls confines particle movement and fluctuation in the thickness direction. Hence, the granular temperature contributed by the fluctuating solids velocity in the thickness (z-) direction is much less than that from the vertical (y-) and horizontal (x-) directions. This is consistent with the anisotropic behavior observed above for the solids stress tensor. One should be aware of the strong anisotropic flow behavior in pseudo-2D systems when conducting TFM simulations for which constitutive correlations

and boundary conditions for the solids phase are developed based on the isotropic flow assumption. Similar analysis is conducted for the tangential and normal wall stresses as shown in Fig. 7. A linear relationship was obtained, which is consistent with the standard friction equation. Hence, the slope of the fitted line can be interpreted as an effective coefficient of friction between particle flow and the wall. Ts = eff Tn .

(9)

The predicted effective friction coefficient of 0.2555 is lower than that of 0.6 for the particle and wall surface. The standard friction equation using the physical particle–wall friction coefficient is shown as a dashed line in Fig. 7 for comparison. According to the rigid-body theory, only two possible rebound behaviors corresponding to sliding and non-sliding collisions exist for a rigid spherical body colliding with the wall (Li & Benyahia, 2012). For a sliding particle–wall collision, the normal and tangential forces can be described by the Coulomb’s law.

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Fig. 12. Plots of averaged normal wall stress versus solids pressure for emulsion phase together with linear regression for different spring constants: (a) 2000 N/m, (b) 20,000 N/m, and (c) 200,000 N/m (numerical simulation: Ug = 0.88 m/s, fric = 0.6; using the sampling window of 3 × 3 fluid cells).

Fig. 13. Plots of averaged tangential wall stress versus normal wall stress for emulsion phase together with linear regression for different spring constants: (a) 2000 N/m, (b) 20,000 N/m, and (c) 200,000 N/m (numerical simulation: Ug = 0.88m/s, fric = 0.6).

For a non-sliding collision, a relative displacement of the contact point does not occur. The sphere rolls on the surface during the contact, which results in lower friction compared with the sliding collision. Hence the overall effective friction coefficient is lower than the coefficient of friction specified for the sliding particle–wall contact. The interaction between particle flow and wall is more complicated than depicted by the rigid body theory. The current DEM implementation is capable of capturing the aforementioned contacts, which makes it an ideal tool to investigate particle–wall interactions. The effective friction coefficient derived from fitting the transient data fluctuates slightly. To obtain a representative relationship between normal and tangential forces, sufficient sample

points are needed to reduce the statistical error. For the entire study, data collected in 2 s with a data saving frequency of 100 Hz were used after careful examination of the effective friction coefficients obtained from different sampling periods ranging from 0.1 to 8 s. Effective friction coefficient A parametric study has been conducted by varying the friction coefficient from 0.3 to 0.7. In this study, the same friction coefficients are used for particle–particle and particle–wall interactions since the bed and wall materials can be considered similar. Fig. 8 shows plots of normal wall stress versus solids pressure for

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Fig. 14. Results predicted by 2.5 mm grid in thickness direction: (a) normal wall stress versus solids pressure and (b) tangential wall stress versus normal wall stress (numerical simulation: Ug = 0.88 m/s, fric = 0.6).

the emulsion phase for simulations with different friction coefficients. For different conditions, the linear relationship fits the data reasonably well. Slopes of linear fitting are close and no general trend can be obtained. The results tend to indicate that the friction coefficient has no significant influence on anisotropy in the stress tensor. Fig. 9 shows the tangential versus normal wall stress for different friction coefficients. The data points become more scattered as the friction coefficient increases, which is visible from the increased standard deviations. Overall, the linear relationship still describes the data reasonably, although the fit becomes less accurate for high friction coefficients. The linear fitting slope, i.e., the effective friction coefficient, increases as the particle–particle and particle–wall friction coefficient increases for friction coefficients of 0.3, 0.4, and 0.5. It decreases slightly for friction coefficients of 0.6 and 0.7, which is attributed mainly to the less accurate fitting for both cases. A linear regression of the raw data points is also used to evaluate the dependence of effective friction coefficient on the physical particle–wall friction coefficient. Overall, the effective friction coefficient increases slightly as the particle–particle and particle–wall friction coefficient increases from 0.3 to 0.7. However, the low increase in effective friction coefficient compared with the change in physical friction coefficient suggests only a moderate dependence on friction coefficient. Effect of spring constant In the linear spring-dashpot soft-sphere model used in this study, the time for particle collision is expressed as a function of particle size and hardness. To avoid extremely small simulation time steps, a low spring constant is typically used in the CFD–DEM simulations, which usually does not affect simulation results (Tsuji et al., 1993). Typically, DEM simulations are accelerated by reducing the particle stiffness; less stiff or softer particles allow for a larger time step (Malone & Xu, 2008). Some studies have suggested that a reduction in particle stiffness can lead to undesirable effects. For example, properties such as bulk stiffness and bulk restitution of granular flow change as a result of stiffness reduction (Lommen, Schott, & Lodewijks, 2014). The spring constant in the contact force model affects the rebound and sticking behavior of the collision of cohesive fine particles in gas–solid fluidized beds (Kobayashi, Tanaka, Shimada, & Kawaguchi, 2013). Hence, it is important to verify that simulation results are unaffected substantially by particle stiffness. To investigate the effect of particle stiffness, two simulations with normal spring constants of 20,000 and 200,000 N/m for particle–particle and particle–wall interactions were conducted for a friction coefficient of 0.6. As the spring constant increases, the duration of collision between two particles decreases

proportionally to the square root of the spring constant. Hence, the time step required for accurate resolution of particle contact must be reduced, which results in a much longer CPU time. To alleviate the extreme computational cost associated with high spring constants, only 2 s simulations were executed by restarting from the run with the spring constant of 2000 N/m. The results from the last second are analyzed. During data analysis, it was found that calculated field variables, i.e., solids pressure and wall stresses for the runs with high spring constants, are highly dispersed compared with the results shown above. After careful examination, it is shown that the default window of one fluid cell for analysis is insufficient to derive macroscopic field variables without significant statistical noise. This is mainly because of the reduced particle contacts captured by an instantaneous snapshot of the simulation because of the reduced duration of collision. The increased contact force between two colliding particles introduces more statistical noises. To overcome this effect, the sampling window was extended from 1 fluid cell to 2 × 2, 3 × 3, and 4 × 4 fluid cells. As shown in Fig. 10, an increasingly evident correlation exists between normal wall stress and solids pressure as the sampling window is extended. To confirm this behavior, averaged data and its standard deviation after binning with respect to solids pressure are shown in Fig. 11 for different window sizes. Clearly, increasing the sampling widow reduces the standard deviation and leads to better fitting to the averaged data. The following results are based on the sampling window of 3 × 3 fluid cells. Fig. 12 shows the relationship between solids pressure and normal wall stress as predicted by different spring constants using the 3 × 3 fluid cells for analysis. Compared with the results in Fig. 5, the increased window size reduces the standard deviation of the data for the low spring constant of 2000 N/m in Fig. 12(a). However, the effective friction coefficients from data fitting are comparable, which suggests that a sampling window size of 1 cell is appropriate for simulations with low spring constant. In Fig. 12, the data become less correlated between solids pressure and normal wall stress for simulations with higher spring constants as indicated by the increased standard deviations. This is believed to be related to the statistical noise because of high particle stiffness. Overall, the linear relationship fits all data reasonably well and the slopes remain almost constant for different values of particle stiffness. Fig. 13 shows the tangential versus normal wall stress for simulations with different spring constants. Similarly, the data become less correlated for higher spring constants because of the increased statistical noise. A linear relationship is visible in the average data. The slope of a linear fitting, i.e., the effective friction coefficient, increases slightly as the particle spring constant is increased, which suggests that the particle stiffness affects the particle–wall interaction to some extent. Considering the small difference in fitting

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slope, the effect is deemed negligible. However, attention to the particle stiffness is required for the DEM study, when an accurate quantitative prediction of particle–particle and particle–wall contact forces is sought.

Effect of grid resolution in thickness direction To investigate the effect of grid resolution in the thickness direction, one simulation with refined grid in that direction was conducted for a friction coefficient of 0.6. In this case, the grid sizes in the other two directions were maintained the same. No other change was made except for the numerical grid. The mean flow hydrodynamics with respect to solid distribution and movement were compared with the results of the 5 mm grid and showed good consistency. Given the good 2D flow behavior of the pseudo-2D system, the results confirm that the grid resolution in the thickness direction has no significant effect on results. Several issues arise related to the DEM simulation when the computational cell dimension is too small compared with the particle size because of the large grid size required by the CFD–DEM coupling. Hence, no further grid refinement study was conducted. Similar analyses were carried out to extract the field variables. Fig. 14(a) shows the relationship between solids pressure and normal wall stress predicted by the simulation with a 2.5 mm grid in the thickness direction. The ratio between normal wall stress and solids pressure is ∼0.04, which is smaller than that of 0.0756 shown in Fig. 5. This tends to suggest that stronger anisotropy exists in the particle force network when approaching the wall. Fig. 14(b) shows plots of the tangential versus the normal wall stress. The fitted effective friction coefficient of 0.2471 is close to that of 0.2555 shown in Fig. 7, which indicates that the grid resolution in the thickness direction only has a negligible effect on particle–wall interaction for the current pseudo-2D system.

Revisiting Li and Zhang’s model Li and Zhang (2013) developed a 2D model for TFM simulations to account for front and back wall effects in a pseudo-2D gas–solid fluidized bed. In this model, collisions between particles and the front and back walls are assumed to be sliding. The shear stress S exerted by the wall is calculated as: S = N,

(10)

where, N is the normal wall stress. The normal wall stress is estimated from the solids pressure calculated in the 2D simulation. With the 2D flow assumption, the shear force imposed by the front and back walls can be interpreted as a body force acting on the solids flow. The DEM simulation results here can be used to verify the assumptions used in Li and Zhang’s model. Clearly, the normal wall stress is much lower than the solids pressure because of the strong anisotropic flow behavior. The anisotropy in solids stress should be considered when the solids pressure is used to approximate the normal wall stress. Furthermore, an effective friction coefficient should be used in Eq. (10) instead of the particle–wall friction coefficient. As demonstrated above, the effective friction coefficient is much lower than the particle–wall friction coefficient. There exists a linear relationship between the shear stress and solids pressure, which makes it possible to account for the wall effect in a 2D simulation through a single parameter lumping of the flow anisotropy and effective friction coefficient. However, certain calibration steps may be required to determine its appropriate value based on the experimental data.

Conclusions For the pseudo-2D gas–solid system investigated, there exists a strong correlation between normal and tangential stress at the wall in the dense emulsion phase. An effective friction coefficient can be used to describe the relationship. The effective friction coefficient is lower than the particle–wall friction coefficient. It increases slightly with particle–wall friction coefficient. Significant anisotropic behaviors exist in solids stress and granular temperature. The normal wall stress is much lower than the solids pressure and the granular temperature component in the thickness direction is lower than that in the other two directions. The strong anisotropic behavior in the pseudo-2D system poses challenges to TFM simulation for which the constitutive correlation and boundary condition for the solids phase are developed mainly based on isotropic assumptions. Assumptions in the 2D model for a pseudo2D gas–solid system by Li and Zhang (2013) were examined and the flow anisotropy and effective friction coefficient are suggested to be considered in the model. The simple 2D model is considered to be a reasonable simplification by lumping flow anisotropy and effective friction coefficient together through a single parameter. In this CFD–DEM study, the spring constant, which characterizes the particle stiffness, affects the force analysis to a certain extent, although it does not alter the qualitative conclusion. Special attention is needed for data analysis when using DEM to investigate particle–particle and particle–wall interactions, especially when quantitative prediction is sought. Overall, CFD–DEM has been demonstrated to be a very useful tool for detailed investigation of gas–solids flow. Acknowledgments This technical report was produced in support of the National Energy Technology Laboratory’s ongoing research in advanced numerical simulation of multiphase flow under RES contract DEFE0004000. References Beetstra, R., van der Hoef, M. A., & Kuipers, J. A. M. (2007). Numerical study of segregation using a new drag force correlation for polydisperse systems derived from lattice-Boltzmann simulations. Chemical Engineering Science, 62, 246–255. Busciglio, A., Vella, G., Micale, G., & Rizzuti, L. (2008). Analysis of the bubbling behaviour of 2D gas solid fluidized beds. Chemical Engineering Journal, 140, 398–413. ´ M. G., & Soler, J. G. (2003). A study on the Caicedo, G. R., Marqués, J. J. P., Ruız, behaviour of bubbles of a 2D gas–solid fluidized bed using digital image analysis. Chemical Engineering and Processing: Process Intensification, 42, 9–14. Chialvo, S., & Sundaresan, S. (2013). A modified kinetic theory for frictional granular flows in dense and dilute regimes. Physics of Fluids, 25, 070603. Chialvo, S., Sun, J., & Sundaresan, S. (2012). Bridging the rheology of granular flows in three regimes. Physical Review E, 85, 021305. Cloete, S., Zaabout, A., Johansen, S. T., van Sint Annaland, M., Gallucci, F., & Amini, S. (2013). The generality of the standard 2D TFM approach in predicting bubbling fluidized bed hydrodynamics. Powder Technology, 235, 735–746. Cranfield, R. R., & Geldart, D. (1974). Large particle fluidization. Chemical Engineering Science, 29, 935–947. Cundall, P. A., & Strack, O. D. L. (1979). A discrete numerical model for granular assemblies. Géotechnique, 29, 47–65. Feng, Y. Q., & Yu, A. B. (2010). Effect of bed thickness on the segregation behavior of particle mixtures in a gas fluidized bed. Industrial & Engineering Chemistry Research, 49, 3459–3468. Garg, R., Galvin, J., Li, T., & Pannala, S. (2012). Documentation of open-source MFIX-DEM software for gas–solids flows. Retrieved from https://mfix.netl. doe.gov/download/mfix/mfix current documentation/dem doc 2012-1.pdf. Garg, R., Galvin, J., Li, T., & Pannala, S. (2012). Open-source MFIX-DEM software for gas–solids flows: Part I—Verification studies. Powder Technology, 220, 122–137. Geldart, D. (1970). The size and frequency of bubbles in two- and three-dimensional gas-fluidised beds. Powder Technology, 4, 41–55. Gidaspow, D. (1994). Multiphase flow and fluidization: Continuum and kinetic theory descriptions. New York: Academic Press. Glicksman, L. R., & McAndrews, G. (1985). The effect of bed width on the hydrodynamics of large particle fluidized beds. Powder Technology, 42, 159–167.

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