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Oct 23, 2008 - developed by Lun et al.1 is used to model kinetic-collisional ... + ∇ · (εsvs)]) 0. (1). Conservation of linear momentum: F s[∂ε...
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Ind. Eng. Chem. Res. 2008, 47, 8926–8932

Validation Study of Two Continuum Granular Frictional Flow Theories Sofiane Benyahia* National Energy Technology Laboratory, Morgantown, West Virginia 26505

Granular kinetic theories are valid in both kinetic (dilute) flow regime and dense collisional (intermediate) flow regime. This is true as long as contacts between two colliding particles are instantaneous. Empirical theories derived from soil mechanics have been traditionally used in continuum modeling for dense granular flows dominated by enduring contact between particles. This study focuses on validating two continuum theories commonly used to model dense frictional granular flows. The first validation example is conducted for a granular bin discharge, and model predictions are compared with a well-known empirical correlation for the discharge rate. The second example involves a more detailed comparison of flow variables with predictions obtained using a discrete technique for granular flow in a simple shear cell. The frictional flow theory that is only activated at the quasi-static flow regime above maximum packing does not show accurate predictions. Better predictions are obtained using a frictional theory that extends in the intermediate flow regime below packing where both collisional and enduring contact between particles occur. Introduction Granular flows present interesting scientific challenges (see, for example: http://www.phy.duke.edu/research/cm/behringer and http://jfi.uchicago.edu/∼jaeger/granular2) as well as practical engineering problems, such as flows in standpipes and silo discharge, that are relevant to many industries. Even in applications that use a relatively dilute flow of solids, it is possible to find regions of the flow that are dense and frictional. There are two approaches to model particulate flow systems: continuum and discrete. The discrete approach is usually more accurate as it describes the motion of each particle using Newton’s second law of motion, but it is limited to a relatively small number of particles because of computer time limitations. The continuum approach uses the kinetic theory of granular flows and is more relevant for industrial applications that use large amounts of solids. The objective of this study is the validation of two continuum frictional flow theories commonly used in the literature. We describe in this study a collisional-frictional model for granular flows in a vacuum. The granular kinetic theory developed by Lun et al.1 is used to model kinetic-collisional stresses. Two frictional flow theories are evaluated for dense granular flows. We assume, similar to Johnson and Jackson,2 that the total stresses acting on a granular assembly are the sum of stresses due to instantaneous binary collisions derived from kinetic theory and frictional stresses due to enduring contact between layers of particles. We describe a frictional model,3,4 which has been traditionally used in the MFIX computer code (https://mfix.netl.doe.gov/). We also use another frictional model developed by Srivastava and Sundaresan,5 who implemented it in the MFIX code and conducted validation studies of bin discharge. A simplified version of this model is described here. By implicitly expressing the divergence of solids velocity in this model, it is no longer necessary to relax the stresses by solving an additional transport equation for the ratio of frictional to critical pressure. The frictional stresses in this model affect the granular assembly at solids concentration lower than packing as proposed by Johnson and Jackson.2 Recently, Ng et al.6 proposed a similar idea and combined, using weight factors, * To whom correspondence should be addressed. E-mail: [email protected]. 10.1021/ie8003557

frictional and collisional stresses. Their frictional shear stress is proportional to normal stress as proposed by Schaeffer.3 However, the normal stress was the collisional pressure taken from kinetic theory. This contradicts their reference to Schaeffer,3 who mentions, “In a viscous fluid, dissipation is due to momentum transfer from collisions; in a granular material, an assembly of many small particles in frictional contact, dissipation is due to friction between sliding particles”. In contrast, the frictional stresses used in this study (and commonly in the literature) are largely empirical, from soil mechanics as described by Johnson and Jackson2 and Bouillard et al.7 Validation studies of dense frictional flow theories are available in the literature.8,9 Such studies include other complications in the flow physics, such as interstitial fluid, complex geometries, and cohesive particles. In contrast, this study focuses only on dense frictional flows in a vacuum, so no effect of the fluid is introduced in the model equations. Furthermore, we only study relatively simple systems such as bin discharge, similar to Srivastava and Sundaresan,5 and compare results of the discharge rate to the Beverloo et al.10 correlation. For a detailed quantitative comparison of the dense frictional models, we use further simplifications by studying a gravity-free granular shear flow between two parallel plates. Computer data generated using a discrete particle method with a soft-sphere contact model (DEM) are used for comparison with continuum models. Discrete techniques using both soft-sphere and hard-sphere contact models are powerful tools for validating continuum models, as shown by many studies in the literature.11-13 Although most of these studies are aimed at validating granular kinetic theories, discrete techniques are also useful to study dense frictional flows and are used here to validate two frictional flow theories commonly used in continuum modeling. Granular Flow Model The “dry” granular kinetic theory model used in this study is essentially the same as that derived by Lun et al.1 This includes conservation equations for flow of solids in a vacuum and the constitutive relations for the solids stresses based on granular kinetic theory. This theory applies to noncohesive and frictionless spherical particles and solves only for the translational granular energy. Jenkins and Zhang14 developed a kinetic theory for slightly frictional particles and showed that similar

This article not subject to U.S. Copyright. Published 2008 by the American Chemical Society Published on Web 10/23/2008

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results can be obtained by using the translational granular energy with a lower effective restitution coefficient (obtained from a simple formula in their eq 16). However, as discussed by Jenkins and Zhang,14 the assumptions used in their theory do not apply near-wall boundaries and near-packing where layering of particles occur, as we will show in the discrete element method (DEM) results presented in this study. Therefore, we only use a simple granular kinetic theory based on translational conservation of energy, and we focus mainly on the validation of frictional stresses that apply to dense granular flows. In this study, the total stress of the granular assembly is assumed to be the sum of collisional (derived from granular kinetic theory) and frictional stresses.2 Conservation of mass for constant solids density:

[

]

∂εs + ∇ · (εsvs) ) 0 ∂t Conservation of linear momentum: Fs

[

(1)

]

∂εsvs + ∇ · (εsvsvs) ) ∇ · (τk + τf) + εsFsg ∂t Translational granular energy conservation equation: Fs

[

(2)

]

3 ∂εsΘs F + ∇ · (εsΘsvs) ) - ∇ · q + τk: ∇ vs - FsJs (3) 2 s ∂t Solids kinetic-collisional and frictional stress terms: τk ) [-Ps + ηµb ∇ · vs]I + 2µsSs

(5)

1 1 Ss ) [∇vs + (∇vs)T] - ∇ · vsI 2 3 Solids pressure:

(6)

Ps ) εsFsΘs[1 + 4ηεsg0]

(7)

The compressibility factor (Z) derived by Carnahan and Starling15 can be used to express the radial distribution function at contact (g0) by the fact that16 Z ) 1 + 4εsg0: g0 )

1 - 0.5εs (1 - εs)3

)

εs εs2 1 + 1.5 + 0.5 (8) (1 - εs) (1 - ε )2 (1 - ε )3 s

s

Solids viscosity model: µs )

( 2 +3 R )[ g η(2µ- η) (1 + 58 ηε g )(1 + 58 η(3η - 2)ε g ) + s 0

s 0

0

]

3 ηµ (9) 5 b 256 2 5 F d πΘ , µb ) µε g 96 s p√ s 5π s 0 Granular energy flux and conductivity: µ)

(10)

q ) -κs ∇ Θs κs )

( )[( κ g0

1+

(11)

12 12 ηε g 1 + η2(4η - 3)εsg0 + 5 s 0 5 64 (41 - 33η)η2(εsg0)2 (12) 25π

)(

)

]

75Fsdp√πΘs 48η(41 - 33η) Collisional dissipation of granular energy: κ)

(13)

(14)

Frictional-collisional wall boundary condition:2 φπFsεsg0√Θs vsl · (τ + τf) · n + vsl + (n · τf · n) tan δ ) 0 |vsl| k 2√3ε max s

(15) n·q)

φπ|vsl|2Fsεsg0√Θs 2√3εsmax

-

√3πFsεsg0(1-ew2)√Θs 4εsmax

Θs (16)

Syamlal et al. (S-R-O) Frictional Model. This model has been traditionally used in the MFIX code and was described in detail by Syamlal et al.4 The model equations as presented in this study were first written by Schaeffer,3 who described the plastic flow of a granular material and related the shear stress to the normal stress. This model is activated at the critical state, where the solids volume fraction exceeds the maximum packing limit. In this model, I2D represents the second invariant of the deviator of the strain rate tensor, which is related to the norm of the square of the strain rate tensor used by Schaeffer3 simply by: I2D ) (Ss:Ss)/2. The Syamlal et al.4 (S-R-O) model expresses the frictional stresses by the following equations: Pf ) Pc )

{

(4)

τf ) -PfI + 2µfSs

εs2g0 3 ⁄ 2 48 η(1 - η) Θ dp s √π

Js )

µf )

1025(εs - εsmax)10 εs > εsmax

{

0 Pc sin(δ) 2√I2D 0

εs e εsmax εs > εsmax

(17)

(18)

εs e εsmax

As mentioned by Johnson and Jackson,2 the constitutive relations used to describe the frictional stresses are largely empirical. In this case, the critical solids pressure is a power law function4 of the solids volume fraction that allows for some compressibility near the packing limit similar to other plastic flow theories.7,17,18 With this formulation of solids pressure, the solids volume fraction only slightly exceeds the packing limit (εsmax). The frictional viscosity is proportional to the critical pressure as proposed by Schaeffer.3 Large values of frictional viscosity are expected because of the large values of critical pressure computed using eq 17 for the solids volume fractions higher than maximum packing. Numerically, an upper limit for the frictional viscosity is set by default in the MFIX code and corresponds to a value of 1000 poise. In this study, however, no limit was set for the frictional viscosity. Srivastava-Sundaresan (S-S) Frictional Model. The frictional model used in this study was proposed by Srivastava and Sundaresan,5 who gave expressions of the frictional stresses for a compressible granular assembly. The frictional stresses start influencing the granular flow at a minimum frictional solids volume fraction (εsmin), which is below the maximum packing (εsmax) as proposed by Johnson and Jackson.2 In this study, the critical state theory4 applies only when the granular assembly is incompressible (i.e., above maximum packing). Srivastava and Sundaresan5 also considered fluctuations in the strain rate, as was recognized by Savage,19 who expressed it as Ss∝Θs1/2/ dp, even for a quasi-steady granular flow. This formulation has the additional advantage of avoiding a singularity for the case where the strain rate becomes zero. They applied this model to a gravity-induced granular flow through an orifice and also to the rise of a single bubble in a fluidized bed. The SrivastavaSundaresan (S-S) model is expressed by the following equations:

{

8928 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008

Pc )

εs > εsmax

1025(εs - εsmax)10 Fr

(εs - εsmin)r

εsmax g εs > εsmin

(εsmax - εs)s

εs e εsmin

0

(

Pf ) 1Pc

(19)

∇ · vs n√2 sin(δ)√Ss:Ss + Θs/dp2

{

)

n-1

(20)

() }

Pf Pf sin(δ) µf ) n - (n - 1) P √2 √S :S + Θ /d 2 c s s s p

1 n-1

(21)

We should point out a typo in the Srivastava and Sundaresan5 article (this finding is credited to Dr. Dhanunjay S. Boyalakuntla, who first brought it to my attention): In their eqs 16 and 21, the exponent was incorrectly typed as 1/(n - 1) where it should be n - 1, as shown in our eq 20. However, we should add that this model was correctly coded by Srivastava and Sundaresan in MFIX. The coefficient n has different values depending on whether the granular assembly is experiencing a dilatation or compaction: n)

{

√3 sin(δ) ∇ · vs g 0 2 ∇ · vs < 0 1.03

(22)

Application of the Frictional Models to a Discharge of 1-mm Particles from a 2D Bin The first application of the frictional models presented in this study is to a 2D bin discharge. This is the same example that was studied by Srivastava and Sundaresan:5 a 2D rectangular bin 8 cm wide and 100 cm high with an open top and an orifice centered at the bottom. The height of the bin does not include an additional length of 5 cm below the orifice (Figure 1). The orifice width varied between 1.4 and 2 cm to study its effect on the solids discharge rate. All physical parameters are the same as those presented by Srivastava and Sundaresan5 in their Table 2. We also use the same numerical grid resolution of 1 and 2 mm along the horizontal and vertical directions, respectively. However, we do not assume symmetry along the center line of the bin. Two issues were reported by Srivastava and Sundaresan5 that are addressed in this section: First, the cause of the gridscale flutter in the solids volume fraction observed initially in the simulation was determined and eliminated in this study, and second, lower discharge rates were computed in this study, which resulted in better agreement with the Beverloo correlation. Figure 1 shows the distribution of the solids volume fraction in the lower section of the 2D bin using the S-S frictional model early in the simulation (after 0.03 s). The grid-scale flutter in the solids volume fraction distribution described by Srivastava and Sundaresan5 was not observed when the divergence of solids velocity was formulated implicitly in the solution algorithm. We found an explicit approach was used in the previous numerical implementation of this model in MFIX (https:// mfix.netl.doe.gov/) in the study of Srivastava and Sundaresan5 (i.e., the divergence of the solids velocity was computed at every time step). In this study, this term is now computed every nonlinear iteration, which we call implicit. Therefore, the source of this flutter in the solids fraction is simply numerical. An added benefit to the implicit approach is that there is no need to relax the frictional pressure by solving an extra transport equation for Pf/Pc as was done by Srivastava and Sundaresan,5 thus simplifying this model and saving computational time.

Figure 1. Initial (after 0.03 s) solids volume fraction and velocity profiles near the bin discharge orifice. Packed regions (εsmax ) 0.65) are represented with red color, and voids are shown in blue color. The flutter in solids volume fraction described by Srivastava and Sundaresan5 was not observed in the current simulations if the ∇ · vs term is treated implicitly.

Temporal Profiles of Discharge Rate. Figure 2 shows the temporal variation of the discharge rates for the four orifice diameters of 1.4, 1.6, 1.8, and 2 cm. A comparison of the discharge rate for the case of a bin with an orifice diameter of 1.4 cm with the simulation results of Srivastava and Sundaresan5 (see their Figure 4) shows that our computed discharge rate (shown in Figure 2) was lower. The reason for the disagreement could be a limitation in the frictional viscosity. The current model does not set any limit for computed values of the frictional viscosity for either S-S or S-R-O models. In their previous publication, Srivastava and Sundaresan5 may have used an upper limit to the frictional viscosity, but this value is not known. This is suspected because in one simulation using the S-S model, we limited the frictional viscosity to 100 P, which resulted in almost double the discharge mass flow rate. Figure 2 shows that lower values of discharge rate are computed using the Syamlal et al.4 (S-R-O) frictional model as compared to the S-S predictions. This is due to a fundamental difference between these two frictional models: the S-S model friction starts at a lower solids volume fraction (εsmin) so that the computed normal and shear stresses are lower than those computed using the S-R-O model where friction starts at maximum packing. Verification of Beverloo Correlation. Numerical data obtained using the S-S and S-R-O frictional models were compared to the Beverloo et al.10 correlation for estimating the discharge rate from hoppers and bins, which was written by Srivastava and Sundaresan5 for a 2D bin discharge as W ) CFBg0.5Do1.5H

(23)

This correlation gives the discharge rate (W) as function of the width of the orifice (Do). Here C is an empirical dimensionless constant expressed by Srivastava and Sundaresan5 in the range of 0.55 < C < 0.65, with the initial solids bulk density as FB ) Fsεs ) 2.9 × 0.6 g/cm3, the gravitational acceleration as g, and the depth of the bin as H, taken as 1 cm in this study. We should note that Beverloo et al.10 expressed the empirical constant C in s/min and thus gave a range of 33.2 < C < 38.8. This study assumes, similar to Srivastava and Sundaresan,5 that

Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8929

Figure 2. Temporal variation of the solids discharge rate for four different orifice widths using the Srivastava and Sundaresan5 (S-S) and Syamlal et al.4 (S-R-O) frictional models.

Figure 3. Verification of the Beverloo correlation using both S-S and S-R-O frictional models. The linear best fits of the computed data and the corresponding equations are also plotted.

the particle size is much smaller than the width of the orifice so that the correction10 to Do can be ignored. Figure 3 shows that the discharge rate is a function of Do1.4 and Do1.9 for the S-S and S-R-O models, respectively. This is in agreement with the results of Srivastava and Sundaresan,5 who found the discharge rate scales as Do1.4 using their frictional model. The S-R-O frictional model overpredicts the value for the exponent of the orifice width. According to Figure 3, the computed value of C (the empirical constant in the Beverloo correlation) was calculated as 1.08 and 0.61 for the S-S and S-R-O models, respectively. Srivastava and Sundaresan5 computed a larger value C ) 1.6 because of their larger computed discharge rate. This current study, however, found the value of C to be closer to the experimental measurements using the S-S model. The S-R-O frictional model predicted lower discharge rates for all widths of the orifice

and, thus, a value of C that is in better agreement with the Beverloo correlation. Both the S-S and the S-R-O models show some agreement with the Beverloo et al.10 correlation. For the S-R-O model, the exponent of the orifice width was within 27% and the S-S model was within 7% of the Beverloo correlation. For the S-S model, the constant C deviated the most (within 66%) from the range proposed by Beverloo et al.10 The next study provides a more detailed quantitative comparison of all flow variables between these models and the computer data generated using a discrete technique. Such a technique can be called a “controlled numerical experiment”, where all physical and numerical parameters are known. Thus, the next section is proposed to help us highlight more accurately the distinctions between these two frictional continuum models. Validation of the Frictional Models for 1D Granular Couette Flow We applied the frictional flow theories discussed previously to model granular flow (in a vacuum) in a Couette shear cell. To conduct the validation study, we generated computer data of the granular flow using the soft-sphere DEM20 available in MFIX. Before conducting the validation study of the continuum frictional flow theories, we proposed to first verify the DEM data generated using the MFIX code by comparing it to DEM data published in the literature. To our knowledge, such verification of the MFIX DEM code was not conducted in the past. Thus, it is reasonable to first verify our DEM code before conducting a comparison with continuum results. We used data published by Karion and Hunt21 for a simple granular shear flow of monodisperse particles (one particle size). The softsphere contact model was described in detail by Boyalakuntla,20 who implemented it in MFIX. In this model, normal contact is shown by using a linear spring that provides an elastic

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Figure 4. Verification of the soft-sphere DEM model implemented in the MFIX code is achieved by comparing our results with the results of Karion and Hunt21 (from their Figure 2) using the same physical and numerical parameters for the monodisperse system, with average solids area fraction νs ) 0.65 and h/dp ) 10.

restoration force after the two contacting particles are allowed to slightly deform, and a dashpot is used to dissipate energy during contact. The tangential contact is modeled using a linear spring along with a Coulombic friction sliding element.20 In this study, the mass, m, of a particle was equal to 1. The normal and tangential spring constants were equal, with a value of k/m ) 8 × 1010 s-2. Reducing these spring constants by 2 orders of magnitude had only a minor effect on time-averaged results while significantly decreasing the computer time. The same value for particle-particle and particle-wall friction coefficients was used in this study: µp ) µw ) 0.5. Other physical parameters such as restitution coefficients (e ) ew ) 0.8) were the same as those used in the study of Karion and Hunt.21 In this study, a gravity-free granular periodic flow was established between two parallel plates separated by a distance h/dp ) 10 with each moving at a velocity of (Vw/2 (Vw/h ) 250 s-1). A width-to-height ratio of 5 was used in this study, which fits about 413 particles. To average the quantities associated with discrete particles, a technique similar to that used by Karion and Hunt21 was adopted in this study: the domain was discretized using horizontal stripes of a height equal to one particle diameter, and statistics were gathered in each stripe. The results were averaged over a period of 80 s after an initial transient period of 10 s. Figure 4 shows a quantitative comparison of solids area fraction (νs ) (3/2)εs), translational granular temperature (T ) (3/2)Θs), and solids horizontal velocity of the results obtained in this study to those presented by Karion and Hunt21 in their Figure 2. Figure 4 shows that granular energy is produced at the moving walls due to friction, which pushes solids toward the center of the domain. Low granular energy at the center is due to larger dissipation by friction and inelastic collisions caused by the higher solids concentration. The MFIX DEM code solves for spheres in a pseudo-3D system where a depth of one particle diameter is used, which

differs from the study of Karion and Hunt,21 who used 2D disks. However, the flow constraint of 2D disks or spheres in a pseudo3D domain is similar, as collisions happen only along the circumference of the spheres that lay in the 2D plane similar to 2D disks. However, the deformation of a sphere upon collision could be slightly different from that of a 2D disk. Nevertheless, a reasonably good agreement for our computed results and those presented by Karion and Hunt21 is shown in Figure 4. In the following study, we change the geometry of the system to h/dp ) 40; the average solids area fraction is νs ) 0.75 (7632 particles were used); and the restitution coefficients are e ) ew ) 0.9. In this case, 10 horizontal stripes with a height of fourparticle diameters along the full horizontal length of the system were used to obtain average DEM data shown in Figure 6. The higher solids concentration and larger system geometry were chosen for comparison with the continuum dense frictional flow theories. The frictional continuum models use the same parameters as Srivastava and Sundaresan,5 except for the maximum packing (νsmax) and minimum frictional solids area fraction (νsmin), which have values of 0.9 and 0.6, respectively. Figure 5 shows a snapshot (after 20 s) of particle position and velocity in the periodic shear cell. Particles with high velocity are observed at the top and bottom walls due to friction with the moving walls. Near the walls, a relatively dilute flow is observed due to the high energy particles that tend to push nearby particles toward the center of the channel where the flow is clearly denser. Layering of particles is observed at the center of the channel, although some pockets of void can also be seen, perhaps due to large shear (and energy) applied at the frictional walls. Figure 6 shows a comparison of time-averaged DEM data and the results obtained using the continuum frictional flow theories discussed in this study. Although we used the transient continuum models, the numerical simulations quickly reached a steady state after only a few seconds of simulation. It was expected that transient continuum gas-solid simulations reach steady state for flows with no gravity as demonstrated by Benyahia et al.,22 who verified that

Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8931

Figure 5. Instantaneous particle position and velocity in the periodic shear cell. Red indicates higher velocity with maximum of about 9 m/s obtained at the top and bottom moving walls. Periodic boundaries apply to the right and left sides. Particles are monodisperse with an average solids area fraction of νs ) 0.75 and h/dp ) 40.

Figure 6. Comparison between the time-averaged DEM results and the steady continuum collisional-frictional models (using S-S and S-R-O frictional models) for the solids area fraction, granular temperature, and velocity.

the frequency of oscillations scales with g. Furthermore, we found identical results in both 1D and 2D periodic geometries, and therefore, only 1D continuum simulation results are reported in this study. Figure 6 shows that both the continuum collisionalfrictional models used in this study are able to reproduce the qualitative trends computed using DEM. Because of the high granular temperature gradient between the walls and center of the channel, solids move to the center to balance the collisional pressure. This migration of solids to the center can be limited with frictional pressure, which, in the case of the S-R-O model, does not occur until maximum packing is reached. This is why this model predicts packed regions (νsmax ) 0.9) at the center of the channel. This is not the case with the S-S model where friction starts at νsmin ) 0.6, so that a better agreement is obtained at the center. The continuum models predict a more dilute flow near the walls due to the large values of granular temperature computed in that region because of wall boundary conditions. We should mention that one parameter, the specularity coefficient φ in the granular boundary condition proposed by Johnson and Jackson,2 was adjusted to fit the near-wall velocity data obtained using the S-S model. A value of φ ) 0.052 was used to obtain the continuum data shown in Figure 6. It is possible that the cause of the disagreement observed for the granular temperature profiles is mainly because we are only solving for the translational granular energy, and, thus, a good agreement is not expected in the dilute regions of this highly frictional flow. Nevertheless, it is clear from

Figure 6 that a better quantitative agreement with DEM data is obtained using the S-S frictional model. Summary and Conclusions We presented in this study a granular kinetic theory model derived by Lun et al.1 and conducted a comparative study of two frictional flow theories: one proposed by Syamlal et al.4 and the other by Srivastava and Sundaresan.5 These two continuum frictional flow theories were validated for a granular bin discharge with an empirical correlation developed by Beverloo et al.10 and showed reasonable agreement for both the Srivastava-Sundaresan and Syamlal et al.4 models. We made a simple numerical improvement to the S-S model by using an implicit form of the solids velocity divergence that eliminated the grid-scale flutter observed by Srivastava and Sundaresan.5 With this improvement, there was no need to solve a transport equation for the ratio of frictional to critical pressure, which makes this model simpler to implement in a CFD code. A more detailed comparison of the two continuum frictional models was conducted in the second part of this study. We validated these two theories with computer simulation data obtained using a DEM for a granular Couette flow. The DEM code was first verified using data published by Karion and Hunt21 and was then used to generate data for comparison with continuum theories. Both frictional continuum theories showed trends similar to that of the DEM data. The solids area fraction

8932 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008

was lower at the walls because of the higher granular temperature generated from friction with the moving walls. Overall, the S-S frictional model showed better agreement with DEM data, which brings us to the conclusion that the model proposed by Srivastava and Sundaresan5 is better suited for dense frictional flows. Nomenclature dp ) particle diameter e, ew ) particle-particle and particle-wall restitution coefficients Fr, r, s ) constants in S-S model, equal to 0.5 dyn/cm2, 2, and 5, respectively g ) acceleration of gravity g0 ) radial distribution function at contact I ) identity tensor Js ) granular energy dissipation due to inelastic collisions k ) spring constant used in DEM simulations n ) coefficient in the frictional model n ) unit vector normal to wall surface Ps ) solids pressure q ) flux of granular energy Ss ) strain-rate tensor Vw ) wall velocity used in Couette shear flow study Vs ) velocity vector Greek Letters R ) constant in solids viscosity model, equal to 1.6 δ ) angle of internal friction of about π/6 εs ) solids volume fraction φ ) specularity coefficient used in wall boundary condition η ) constant depending on particle restitution coefficient equal to (1 + e)/2 κ ) solids phase dilute granular conductivity κs ) conductivity of solids granular energy µ ) solids phase dilute granular viscosity µb ) bulk viscosity of the solids phase µp, µw ) particle-particle and particle-wall friction coefficients µs ) granular viscosity Θs ) granular temperature Fs ) solids density τ ) stress tensor Indices c ) critical

f ) frictional k ) kinetic-collisional max ) maximum packing min ) minimum frictional solids fraction s ) solids phase sl ) particle-wall slip w ) wall

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ReceiVed for reView March 8, 2008 Accepted September 8, 2008 IE8003557