A Specularity Coefficient Model and Its Application to Dense

Article pubs.acs.org/IECR

A Specularity Coefficient Model and Its Application to Dense Particulate Flow Simulations Yunhua Zhao, Tianqiang Ding, Liangyou Zhu, and Yingjie Zhong* Institute of Energy and Power Engineering, Zhejiang University of Technology, Hangzhou 310014, China S Supporting Information *

ABSTRACT: The specularity coefficient is a key parameter for the Johnson and Jackson boundary conditions used in a two-fluid model (TFM), but it is experimentally unmeasurable. This work proposes a semianalytical and flow-dependent model for the specularity coefficient based on measurable particle properties and the data of Louge [Louge. Phys. Fluids, 1994, 6, 2253−2269]. Through TFM simulations, the model was tested in a granular Couette flow, a spouted bed, and a circulating fluidized bed riser. A discrete particle model (DPM) simulation of the Couette flow was also performed for comparison. It was found that the TFM results of the present model agreed with the DPM results and experimental data and thus justified the potential further applications of the model. Moreover, the parameter studies indicated that the friction coefficient between particle and wall was crucial to the present model. ratios. (ii) ϕ plays an important role in dictating the particle phase flow, including local particle velocity and concentration, mesoscale cluster formation,23 and system-scale segregation12 and circulation.14 (iii) The tested values of ϕ are chosen somewhat empirically, and the recommended ϕ still differs by several orders for each kind of fluidization system. (iv) There is continuing research focus on the sensitivity of ϕ, and an effective expression for ϕ is urgently needed. In the literature, Jenkins26 proposed BCs for a flat wall by distinguishing between sliding and nonsliding collisions with three measurable collisional properties, namely the friction coefficient and both the normal and tangential restitution coefficients. For nonsliding collisions, the tangential restitution coefficient was introduced to characterize the tangential momentum and energy transfer. Because of the difficulty in analytical integration, Jenkins26 expressed the BCs for two asymptotic circumstances, which he called “small friction/all sliding” and “large friction/no sliding” limits. Jenkins and Louge27 further refined the calculation of the energy transfer at the wall and extended the expression for the “no sliding” limit to incorporate small sliding. Recently, Schneiderbauer et al.28 proposed new BCs by combining sliding and nonsliding collisions in one expression, and the authors also derived new BCs by additionally introducing particle rotational granular temperature.29,30 Within the same framework, Li and Benyahia31 recalculated the tangential momentum transfer at the wall and interpreted the specularity coefficient as a function of the measurable collisional properties. Soleimani et al.32 compared most of the current BCs and suggested that further analysis was needed for the collisional dissipation. However, when compared with the results from particle simulations by Louge,33 all of the

1. INTRODUCTION Dense particulate flows, including gas-particle multiphase flows, are commonly encountered in many industries, such as in food and pharmaceutical processing technologies, and in chemical and energy conversion technologies. Complex flow phenomena occur in these flows due to dissipative particle−particle interactions as well as nonlinear gas-particle interactions.1 To better understand the gas-particle flow behavior, computational fluid dynamics (CFD) has emerged as a valuable tool over the past two decades. Assuming that sufficient computational resources are available, one can track the motion of each particle with the Lagrangian approach and also solve the gas flow details around it.2−4 However, the reality is not so ideal, thus a Eulerian pseudofluid approach would be more preferable to deal with the huge ensemble of particles. The two-fluid model (TFM) integrating the kinetic theory of granular flow (KTGF) is nowadays well-known as a popular choice to simulate dense gasparticle flows.5−7 In KTGF, a particle−particle interaction is considered as a binary collision, and on this basis the particle phase pressure and viscosity closures are derived. In addition to KTGF, the wall boundary conditions (BCs) represent the overall performance of particle−wall interactions and measure the momentum and energy transfer between wall and particles. Johnson and Jackson8 proposed BCs by assuming that some particles enduringly slid and some instantaneously collided at the wall. Accordingly, the tangential momentum transfer of the sliding particles was described by Coulomb friction, and that of the colliding particles was characterized by a coefficient of specularity, ϕ. It is worth noting that ϕ depends on the wall roughness and varies between zero for perfectly specular collisions and unity for totally diffuse collisions. Moreover, ϕ is not measurable in experiments, and it is specified by empiricism or a cut-and-try method in practice. Table 1 lists some sensitivity studies of ϕ on the quantitative predictions of gas-particle flows, and the following points can be extracted: (i) ϕ is sensitive in modeling small size systems, and this might be explained by their large surface-area-to-volume © 2016 American Chemical Society

Received: Revised: Accepted: Published: 1439

October 10, 2015 January 5, 2016 January 12, 2016 January 12, 2016 DOI: 10.1021/acs.iecr.5b03792 Ind. Eng. Chem. Res. 2016, 55, 1439−1448

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Industrial & Engineering Chemistry Research Table 1. Sensitivity Studies of the Specularity Coefficient specularity coefficient, ϕ system chute bubbling bed

spouted bed

circulating fluidized bed/riser

width or radius (cm)

authors 9

tested

recommended

Chen and Wheeler (2013) Li et al.10 (2010) Loha et al.11 (2013)

height of outlet ∼10 W = 7.6 W = 15.5

0, 0.005, 0.01, 0.03, 0.05, 0.1, 0.2, 0.3 0, 0.005, 0.05, 0.5, 1 0, 0.01, 0.1, 0.3, 0.6, 1.0

Zhong et al.12 (2012)

W = 18.4

0, 0.0005, 0.005, 0.05, 0.5, 1

Li et al.13 (2010) Altantzis et al.14 (2015)

W = 30 W = 50

Lan et al.15 (2012) Bahramian et al.16 (2013) Béttega et al.17 (2009) Armstrong et al.18 (2010) Jin et al.19 (2010) Kong et al.20 (2014) Almuttahar and Taghipour21 (2008) Wang et al.22 (2010) Cloete et al.23 (2011)

R = 0.95−7.6 R = 6−12 R = 5−30 W = 3.2 W=6 R = 3.81 W = 7.62

0.001, 0.005, 0.05 0.0001, 0.0005, 0.001, 0.005,0.01, 0.05, 0.1, 0.5 0, 0.01, 0.05, 0.2, 1 0, 0.5, 1 0, 0.005, 0.1, 0.2 0, 0.25, 0.5, 0.75, 1 0, 0.0001, 0.001, 0.01 0, 0.0001, 0.0002 0, 0.1, 0.5, 1

0.005 0.05 (2.5umf) 0.5 (1.75umf) 0.05 0

W = 7.6 W = 7.6

0, 0.0001, 0.001, 0.01, 0.6 0.0001, 0.01, 0.1, 1

0 ⩾0.01

Zhou et al.24 (2013) Shah et al.25 (2015)

W=9 W = 9, W = 20

0, 0.00005, 0.0005, 0.6, 1 0.0001, 0.001, 0.01, 0.1

0 0.0001

validation data

0.2

air velocity

0.005−0.05 ⩾0.1

tracer concentration particle velocity and granular temperature particle segregation, particle concentration particle velocity particle circulation time and concentration map particle velocity and voidage particle velocity gas pressure (almost independent of ϕ) particle velocity gas pressure gradient particle concentration particle velocity and concentration

0.25 0 0.0001 0

particle velocity and concentration cluster formation cycle, particle velocity and concentration voidage voidage

maximum particle volume fraction in eq 10 (see Table S1) max depends on particle friction, i.e., εmax p = εp (μ). 2.2. Model for Specularity Coefficient. Physically, the specularity coefficient ϕ is defined to be the average fraction of relative tangential momentum transfer through particle−wall collisions. The relative tangential momentum of each particle is simply written as muslip, where m and uslip are the particle mass and mean slip velocity, respectively. If an isotropic Maxwellian velocity distribution is used for particles in the near wall region, the collision frequency can be calculated as n Θ/2π with n being the number of particles per unit volume. Then, the total tangential momentum transfer due to particle−wall collisions is expressed as

above-mentioned BCs fail to predict the momentum transfer for highly inelastic particle−wall collisions.30 Inspired by the actual need and the work of Li and Benyahia,31 the present study focuses on the specularity coefficient in the Johnson and Jackson BCs. Based on measurable particle properties, a new model is developed for the specularity coefficient for a wide range of granular particles used by Louge,33 including highly inelastic particles. The model is then integrated into the TFM using an open source code MFiX (http://mfix.netl.doe.gov), and numerical simulations of three systems are carried out to evaluate the applications.

2. MODEL DESCRIPTION 2.1. Two-Fluid Model. As mentioned above, TFM treats both the gas and particle phases as fully interpenetrating continua and describes each phase with separate conservation equations. Additionally, KTGF is integrated to provide additional closures for the particle phase, and the Johnson and Jackson BCs are preferred due to their relative simplicity and validity. A brief summary of the governing equations is given in Table S1. It is important to note that KTGF is theoretically valid for a kinetic-collisional regime, and an optional frictional stress, τf, can be added when multibody and enduring contacts of particles play a major role for a quasi-static regime. In this work, the frictional stress model proposed by Srivastava and Sundaresan34 is used (see Table S2). On the other hand, traditional KTGF assumes that particles are smooth spheres and the corresponding models are independent of the particle friction coefficient. In reality, granular particles are frictional, and even the microscopic frictional collisions have a significant effect on macroscopic granular flows.6,29 Following Jenkins and Zhang,35 the frictional collision is taken into account using an effective coefficient of restitution eeff, which is a function of particle collisional parameters (Table S3). Accordingly, the radial distribution function at contact, g0, is also modified by Chialvo and Sundaresan36 for frictional particles. In more detail, the

M t = ϕnm

Θ uslip 2π

(1)

Similarly, the average relative normal momentum of the incident particle is calculated to be m Θ/2π , and the corresponding magnitude of the normal momentum transfer can be expressed as M n = nm(1 + e w )

Θ 2π

(2)

where ew is the normal restitution coefficient of the particle−wall collision. Then, the ratio between the magnitudes of the tangential and normal momentum transfer is written as ϕuslip Mt = Mn (1 + e w )

2π Θ

(3)

In contrast, the momentum transfer can be calculated through integrating over all the particle−wall collisions with a detailed dynamics model. Taking both the translational and rotational particle motions into account, the authors29 have proposed the following expression: 1440

DOI: 10.1021/acs.iecr.5b03792 Ind. Eng. Chem. Res. 2016, 55, 1439−1448

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Industrial & Engineering Chemistry Research (1 + βw )uslip Mt = Mn 7(1 + e w ) +

2 ⎡ 1 ⎢ π Θ ⎣ (λ + 1) cot2 Φ + 1

well as the particle−wall collisional parameters. Additionally, the failures of current BCs to predict momentum transfer in Louge’s highly inelastic situations30 will be addressed by combining the present model with Johnson and Jackson’s BC.

arctan( λ + 1 cot Φ) ⎤ ⎥ ⎦ λ + 1 cot Φ

(4)

3. NUMERICAL TESTS AND RESULTS 3.1. System of Granular Couette Flow. The present model is first applied to a gravity-free granular flow in a simple Couette shear cell. In order to provide validation data, a discrete particle model (DPM) simulation is carried out prior to the TFM simulations. The basic setup of the DPM simulation is based on the work of Karion and Hunt37 with two main differences: (i) the original soft-particle approach is changed to a hard-sphere approach,38 and (ii) the original two-dimensional simulation is extended to a three-dimensional simulation. An illustration of the Couette flow can be seen in Figure 16; the upper and lower boundaries in the y direction are flat walls, which are moving with opposite velocities ±U/2 (U/dp = 2500 s−1) to generate shear. The other sides in the x and z directions are set as periodic boundaries. The material properties and domain size are given in Table 2. The number of particles in the present DPM simulation

where cot Φ = 2(1 + βw)/[7μw(1 + ew)] with βw and μw being respectively the tangential restitution coefficient and friction coefficient of particle−wall collisions, and λ is a particular ratio of the rotational to translational granular temperature. Comparing eq 3 with eq 4 then gives ϕ=

(1 + βw ) ⎡ 1 ⎢ 7π ⎣ (λ + 1) cot2 Φ + 1 +

arctan( λ + 1 cot Φ) ⎤ ⎥ ⎦ λ + 1 cot Φ

(5)

To calculate ϕ, the value of λ is needed first. Zhang and Jenkins35 have proposed a theoretical model, in which λ is dependent on particle properties but independent of the flow variables. However, on one hand, the authors29 have solved the conservation equations for both the translational and rotational granular temperatures, and λ has been obtained explicitly and shown to depend on the slip velocity of particles at the boundary wall. On the other hand, Louge33 performed discrete particle simulations and also reported that λ had strong dependence on both particle friction and slip velocity. Moreover, the theoretical model of Zhang and Jenkins was originally derived for nearly elastic particles, and it will fail to correctly predict for highly inelastic particles.30 Actually, after many particle simulations with different collisional parameters, Louge33 calculated the ratios of Mt/Mn and plotted them against the normalized slip velocities of uslip/ 3Θ . Thus, λ can be solved from eq 4 by using Louge’s data, and a fitting formula for λ is presented as λ = 93.15e w3 − 199e w2 + 121.4e w − 16.5 + 1.16 0.5 ≤ e w ≤ 1

2 uslip

3Θ

Table 2. Basic Settings for the Granular Couette Flow Simulation Karion and Hunt36 e = ew μ = μw β = βw εmax p εaverage p domain size (x × y × z)

present DPM

0.8 0.5

0.8 0.5 0.5

0.65 50dp × 10dp

0.433 50dp × 10dp × 3dp

present model 0.8 0.5 0.5 0.587 0.433 50dp × 10dp

is 1241 so that the overall particle volume fraction is about 0.433 and equivalent to 0.65 (3/2εp) in the two-dimensional simulation.39 It should be noted that the hard-sphere approach can only deal with instantaneous binary collisions, thus the frictional stress in the corresponding TFM simulations is ignored for this case. Figure 2 shows a comparison of time-averaged DPM and TFM results. In Figure 2a, the velocities obtained through the present DPM simulation have clear differences from those presented by Karion and Hunt.37 Generally speaking, without the particle motion in the third direction, the same wall velocity will produce larger shear in the two-dimensional granular flow. This implies that the present three-dimensional DPM simulation is necessary, and its results are more suitable for further validation. In the TFM simulations, three different models (the present model, Li and Benyahia model,31 and the constant model) for ϕ are tested. It can be seen from Figure 2a that the velocity profile obtained with the present model is close to that with ϕ = 0.05, and they both agree well with the DPM result. The Li and Benyahia model can also get a close velocity profile with a little overestimation in the near wall region. In fact, the effective values in the present and Li and Benyahia models are ϕ ≈ 0.051 and ϕ ≈ 0.062, respectively, and coincidently a value of ϕ = 0.052 was also recommended for the same Couette flow in the work of Benhayia.39 For the constant model, ϕ = 0.005 greatly underestimates the velocities; in contrast, ϕ = 0.5 greatly overestimates the velocities. In Figure 2b and c, the particle volume fraction profiles and granular temperature profiles obtained with the present model, Li and Benyahia model, and

, (6)

Figure 1 shows the comparison between the solved λ and fitting lines under different collisional parameters. It can be seen that the fitting lines resemble the discrete data very well for a wide range of normal restitution coefficients. Obviously, by using eqs 5 and 6, the present model for ϕ is related to the flow variables, as

Figure 1. Variation of the solved granular temperature ratio with the normalized slip velocity. 1441


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Figure 3 presents the effect of the tangential restitution coefficient on the granular Couette flow. It is known that βw can

Figure 3. Effect of the tangential restitution coefficient on the granular Couette flow.

range from 0 to 1, and a large value of βw corresponds to a relatively rough wall. This relationship is shown clearly in Figure 3, where the particle velocities gradually increase as βw is changed from 0 to 0.5 and then to 1. As further proof, the effective specularity coefficient is calculated to be ϕ ≈ 0.046 for βw = 0 and ϕ ≈ 0.053 for βw = 1. However, the influence of βw on this Couette flow seems to be slight, yet previous studies40,41 have also shown that the collisional parameters, except for the tangential restitution coefficient, could have a profound influence on the gas-particle fluidization behavior. The effect of the friction coefficient on the effective specularity coefficient is shown in Figure 4. It is observed that the effective ϕ

Figure 2. Comparison of the time-averaged results in the Couette flow for (a) particle velocity, (b) particle volume fraction, (c) granular temperature. Figure 4. Effect of the friction coefficient on the effective specularity coefficient.

ϕ = 0.05 are unsurprisingly very close to the DPM data, and the Li and Benyahia model performs best as far as the center region is concerned. For ϕ = 0.005, particles tend to accumulate in the center with extremely low granular temperature, since only small shear is caused by the walls to sustain the particle fluctuation. For ϕ = 0.5, the large shear caused by the walls generates too much granular heat so that overall profiles of both the particle volume fraction and granular temperature become inverted with respect to the other situations. Therefore, with the constant model, the value of ϕ should be specified carefully, or the results may be completely wrong. In addition, the DPM data in Figure 2c show that particle rotational granular temperatures are much higher than the corresponding translational granular temperatures in the near-wall regions, and this implies that the effect of particle rotation may be more significant near the walls.

can approach zero when μw is very small, and this is understandable since an extremely small μw means nearly specular particle−wall collisions. When μw becomes larger and larger, the effective ϕ increases continuously and tends to a maximum value, 2(1 + βw)/(7π). The maximum value is independent of μw, because in the “large friction/no sliding” limit, all collisions are nonsliding and characterized by βw. Thus, unlike βw, μw is much more sensitive in the present model. The sensitivity of μw has also been demonstrated in the modeling of a bubbling fluidized bed.41 3.2. Spouted Bed System. This testing system has been investigated by van Buijitenen et al.42 with both positron emission particle tracking (PEPT) and particle image velocim1442


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Industrial & Engineering Chemistry Research etry (PIV) measurement techniques. The geometry of the spouted bed is depicted in Figure 5. The bottom plane consists of

Figure 5. Sketch of the spouted bed geometry with dimensions in millimeters.

two parts to supply the spout and the background fluidization air, respectively. The top plane is prescribed as an atmospheric pressure outlet, and the other four sides are boundary walls. Following Goniva et al.,43 the windbox below the bottom plane to supply the spout gas in the experiment is ignored for this single-spout case, and the true height of 2.5 m is also reduced to 1 m to save computing time. The fluidized particles are glass beads, and the used material properties are given in Table 3. A grid size Figure 6. Comparison of the time-averaged particle phase vertical velocity in the spout bed at the height of (a) y = 0.05 m, (b) y = 0.1 m.

Table 3. Basic Settings for the Spouted Bed Simulation of van Buijtenen et al.41 property

value

unit

ρp dp e = ew μ, μw β = βw εmax p δ = δw usp g ubg g

2505 3 0.97 0.1, 0.3 0.33 0.6 28.5 43.5 2.4

kg/m3 mm

velocity profiles, and the present model performs slightly better in the off-spout region at y = 0.1 m. Figure 7 shows the time- and z-direction averaged particle phase volume fraction and velocity distributions in the spouted bed. Comparing Figure 7a with c, it can be found that the specularity coefficient has a great impact on the spouted bed expansion, fountain height, and inner circulation. When the present model is used for the specularity coefficient, the velocity vector field in Figure 7b matches well with the measured one in Figure 7d. Figure 8 plots the distributions of the time-averaged effective specularity coefficient on two plane walls. It can be observed that the overall value of the effective ϕ is around 0.08, and the variations show some relationships to the particle flow field. According to eqs 5 and 6, ϕ reduces monotonically with an increase of uslip/ 3Θ . Thus, comparatively large values of effective ϕ appear at wall regions where the particle velocities are small, e.g., the upper bed region has the largest ϕ since uslip is almost zero. Instead, the comparatively small values of effective ϕ seem to appear at wall regions where particles are accelerated to have relatively high velocity and small granular temperature, e.g., the spout channel region and the fall-down regions in Figure 8b. 3.3. Circulating Fluidized Bed Riser System. This last system is a circulating fluidized bed riser, which was installed at ETH Zürich and referenced by Lu et al.46 in their numerical studies. Figure 9 gives the relevant two-dimensional schematic geometry of the riser. The fluidization air is supplied from the bottom with a supercritical velocity of 7.76 m/s. The fluidized particles are glass beads with a diameter of 300 μm and a density of 2500 kg/m3 and supplied from the two sides with a total particle flux of 151.6 kg/(m2·s). The other used particle

deg. m/s m/s

of Δx × Δy × Δz = 5 × 5 × 4 mm3 is chosen for the present simulations according to a grid sensitivity study of the same system by Schneiderbauer et al.44 Additionally, the interphase momentum exchange is calculated with the drag law of Wen and Yu.45 Initially, particles are packed to a height of 0.15 m and a particle volume fraction of 0.39. All the simulations are conducted for 7 s of real time, with the last 5 s being used for the time-averaged analysis. Figure 6 compares the simulated time-averaged particle phase vertical velocity profiles with the measured data. It can be observed that the profiles using the constant model with ϕ = 0 and ϕ = 1 are very different from each other, especially at y = 0.1 m in Figure 6b, where the upward movement in the spout region and downward movement in the off-spout region are greatly overestimated with ϕ = 0 but underestimated with ϕ = 1. Thus, the constant value of ϕ needs further adjustment in order to obtain the best fit with the measurements. In contrast, the profiles using the present model show good agreement with the measurements at both heights. It is also observed that the present model and Li and Benyahia model31 can predict very close 1443


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Figure 7. Distributions of time-averaged particle volume fraction and velocity in the spouted bed using (a) ϕ = 0, (b) the present model, (c) ϕ = 1, (d) PEPT measurement.

Figure 10 shows snapshots of the particle volume fraction distribution obtained using the present model and different

Figure 8. Distribution of the time-averaged effective specularity coefficient at the spouted bed walls: (a) in the y−z plane, and (b) in the x−y plane.

Figure 10. Snapshots of the particle volume fraction distribution using (a) ϕ = 0, (b) the present model, (c) ϕ = 1.

constants for ϕ. As observed, particles can aggregate into clusters throughout the whole riser and tend to form densely packed clusters near the walls, especially in the bottom section of the riser. The cluster shapes are either undulating or in the form of thin, long, and vertical strands, just as those observed in the experiment by Shaffer et al.48 The comparisons of the time-averaged particle flow variables are presented in Figures 11−13. In addition to the present model and different constants for ϕ, the Li and Benyahia model31 is also used and compared. Figure 11 shows the cross-sectional averaged gas volume fractions along the riser height. It is observed that all the simulated profiles do not have significant differences and are close to the experimental data over most heights. Figure 12 shows the particle phase vertical velocity

Figure 9. Sketch of the riser geometry with dimensions in millimeters.

properties are the same as those in Table 3. According to Lu et al.,46 the present simulations use 80 × 420 grid cells and the Gidaspow47 drag law. 1444


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Figure 11. Comparison of the cross-sectional averaged gas volume fraction in the riser. The gray parts indicate the locations of inlet and outlet zones.

Figure 13. Comparison of the time-averaged particle volume fraction in the riser at the height of (a) y = 1.54 m, (b) y = 4.21 m.

of the models seems to match the experimental data well. Figure 13 shows the particle volume fraction across the two heights in the riser. Unsurprisingly, all the simulations are able to predict the same basic trend as the experimental data, and obvious differences only occur in the near wall regions. It can also be seen that the predicted particle volume fractions near the walls become much larger when ϕ is changed from 0 to 1, and a better agreement with the experimental data is obtained using ϕ = 1 or either of the models. To evaluate the effect of the specularity coefficient more clearly, Figure 14 presents the particle flux along the left wall of the riser. As observed, particle fluxes are negative at most points of the wall, and there are great differences between values Figure 12. Comparison of the time-averaged particle phase vertical velocity in the riser at the height of (a) y = 1.54 m, (b) y = 4.21 m.

across two different heights in the riser. As observed, all the simulations predict the typical core−annular structure; i.e., particles tend to move upward in the core region and fall downward in the near wall annular region. Using the models or different constants for ϕ, the predicted core flow velocities are very similar, but the predicted annular flow velocities are very sensitive to the choice of ϕ. Actually, Li and Benyahia49 have simulated this riser operating at a lower particle flux with eight different ϕ’s and reported the same finding. It can be seen that the near wall downward movement is greatly strengthened when ϕ is changed from 0 to 1. Comparing results with the experimental data, the downward movement is overestimated using ϕ = 0, and the velocity profile obtained using ϕ = 1 or either

Figure 14. Comparison of the time-averaged particle flux at the riser wall. 1445


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Industrial & Engineering Chemistry Research predicted by ϕ = 0 and those by ϕ = 1. This indicates that the choice of ϕ has a significant impact on the particle back-mixing behavior and hence the particle residence time and reaction performance inside the riser. Normally, the present model and Li and Benyahia model can predict similar particle flux profiles, and they both predict intermediate particle fluxes with respect to those by extreme constants. The corresponding effective ϕ calculated from the present model are shown in Figure 15. It can be found that the

good agreement with the experimental data. Moreover, the effective specularity coefficient on the spouted bed walls held an obvious relationship with the particle flow behavior. For the circulating fluidized bed riser, the near wall particle flow was greatly influenced by the choice of specularity coefficient, and in turn, the local effective specularity coefficient calculated with the present model was varied by the near wall particle clusters. When compared with the experimental data, the present model could also predict acceptable results in the riser. Frankly, the present model has not shown strong advantages over the Li and Benyahia model in the above cases. However, there are two points worthy of mentioning: first, the present model technically can be used for highly inelastic particles with a normal restitution coefficient no less than 0.5, and second, the methodology of the present model is a combination of theoretical analysis and discrete particle simulation, and the latter can not only offer a closure but also make up the deficiency of the theoretical analysis to some extent. Overall, the present model has provided a practical and meaningful tool to estimate the specularity coefficient.

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APPENDIX: DPM SIMULATION A hard-sphere-approach-based DPM is applied to model the granular Couette flow between two parallel flat walls. Detailed collision dynamics and the computing algorithm of the DPM are described by Hoomans.40 The present DPM simulation is implemented using homemade code. A volume-weighted average method is adopted to measure the local particle flow following Karion and Hunt.37 The final results are averaged over 20 s after an initial transient of 10 s. Figure 16 shows a snapshot of

Figure 15. Time-averaged and transient effective specularity coefficient at the riser wall.

spatial variation of the time-averaged effective ϕ along the wall is moderate, while the local temporal variation of the transient effective ϕ is intensive. At y = 4.2 m, the transient effective ϕ’s have often relatively lower values because of the large-scale particle clusters, which generally hold large falling down velocities and small granular temperatures. In contrast, the transient effective ϕ’s at y = 7.2 m experience frequent oscillations due to the small-scale particle clusters, which can move upward and result in a small magnitude of the timeaveraged particle flux as shown in Figure 14. In addition, the spatial distribution of the effective ϕ is very similar to those obtained by Li and Benyahia49 under a particle flux of 400 kg/ (m2·s). The mean value in Figure 15 is ϕ ≈ 0.0581 and thus very close to 0.0554, which is interpolated from four cases with different particle fluxes and friction coefficients in the work of Li and Benyahia.49

4. CONCLUSION Based on three measurable particle properties and the data of Louge,33 a semianalytical and flow-independent model has been proposed for the specularity coefficient used in the Johnson and Jackson boundary conditions. Combined in a TFM, this new model was numerically tested in three systems, comprising a granular Couette flow, a spouted bed, and a circulating fluidized bed riser. In addition, a DPM simulation was also carried out to provide better comparisons for the Couette flow. The TFM simulations have shown that the specularity coefficient is sensitive in modeling each of the three systems, and more importantly, confirmed the reasonable effectiveness of the present model in the tested applications. For the Couette flow, the results predicted by the TFM using the present model complied well with those by the DPM, and the parameter studies indicated that the friction coefficient between particle and wall play a crucial role in the present model, whereas the role of the tangential restitution coefficient was weak. For the spouted bed, the predicted particle velocities with the present model were in

Figure 16. Instantaneous (a) particle position and (b) velocity vector in the Couette shear cell.

particle position and velocity vector in the Couette shear cell. As observed, more voids exist near the top and bottom walls since energetic particles with relatively larger velocity tend to push nearby particles toward the center.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b03792. Table S1: Summary of the governing equations in the present two-fluid model. Table S2: Frictional model for the particle phase. Table S3: Effective restitution coefficient model (PDF) 1446


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AUTHOR INFORMATION

Corresponding Author

*Tel./Fax: +86-571-88320650. E-mail: [email protected]. cn. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors wish to acknowledge the financial support from the National Natural Science Foundation of China through Grant No. 51006089 and the 2014 open project program of the Key Laboratory of Enhanced Oil and Gas Recovery of Ministry of Education, Northeast Petroleum University. The authors also thank Professor H. Inaki Schlaberg for his assistance in English language editing.

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NOMENCLATURE dp = particle diameter e = coefficient of normal restitution g0 = radial distribution function m = mass of particle M = magnitude of momentum transfer rate between particle and wall n = number density of particles R = radius of the system u = velocity uslip = particle phase slip velocity magnitude at the wall U = wall velocity magnitude used in the Couette shear flow W = width of the system x, y, z = coordinate axes

Greek Symbols

β = coefficient of tangential restitution Δx, Δy, Δz = mesh size in the x, y and z directions ε = volume fraction ϕ = specularity coefficient λ = temperature ratio μ = Coulomb friction coefficient Φ = critical collision angle Θ = granular temperature τf = frictional part of particle phase stress tensor

Subscripts

n = normal component p = particle phase t = tangential component w = wall

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A Specularity Coefficient Model and Its Application to Dense

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