Powder Technology 203 (2010) 70–77

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Quantitative validation of the discrete element method using an annular shear cell

J.J. McCarthy a,⁎, V. Jasti b, M. Marinack b, C.F. Higgs b

a Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, PA 15261, United Statesb Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, United States

⁎ Corresponding author.E-mail address: jjmcc@pitt.edu (J.J. McCarthy).

0032-5910/$ – see front matter © 2010 Elsevier B.V. Adoi:10.1016/j.powtec.2010.04.011

a b s t r a c t

a r t i c l e i n f o

Article history:Received 24 June 2009Accepted 18 February 2010Available online 18 April 2010

Keywords:Discrete element methodGranular materialsShear cell

The discrete element method (DEM) is often used as the “gold standard” for comparison to continuum-leveltheories and/or coarse-grained models of granular material flows due to its derivation from first-principalconstructs, like contactmechanics. Despite its prevalence, themethod ismost often validated against experimentin only qualitative ways – comparison of mixing rates, gross features of concentration profiles, etc. – for exactlythe reason it has found its popularity; detailed experimental measurements are difficult and often expensive. Inthis paper, we outline work aimed at using detailed, particle-level experimental measurements to quantitativelyvalidate DEM simulations. Specifically, we examine the flow in a horizontally-aligned annular shear cell.Measurements are performed using digital particle tracking velocimetry (DPTV) so that the velocity, granulartemperature, and solids fractions profiles may be extracted. Computationally, we attempt to match theexperimental measurements as closely as possible and study the impact of a variety of contact mechanics-inspired force laws aswell asperformsensitivity analysis ondevice andparticle geometry andmaterial propertiesemployed.

ll rights reserved.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Despite their clear industrial relevance, a fundamental understand-ing of the flow of granular materials is lacking. Possibly the biggesthindrance is that there is no accepted set of universal governingequations describing granular flow. This situation can be attributed, inpart, to the intrinsic physical complexity of the problems, as particleflows can exhibit behavior similar to traditional solids, liquids, or gases[1], depending on the state of agitation and imposed shear stresses,among other factors. Another complicating factor, however, is thedifficulty in experimentally measuring not only the bulk properties ofthe material (e.g., stress, strain, voidage, etc.), but also the macroscopicflow variables (e.g., velocity and granular temperature profiles). Whilethere have been notable advances in non-invasive experimentalmethods (positron tomography [2,3], nuclear magnetic resonance [4],gamma ray tomography [5], direct particle tracking [6–9], or combinedtechniques [10]) thus far no technique has been proven ideal as avalidation platform for granular flow descriptors, due to difficulties insimultaneously evaluating all of the information necessary to build and/or validate theoretical models.

These issues have led to an increasing reliance on computationalmodeling of particulate flows as a viable alternative or supplementto experiments as a complement to theoretical approaches; however,

using somewhat circular logic, these computational techniques them-selves are primarily “validated” only against gross external measure-ments (such as torque, mass flow rate, or heat transfer coefficient [11])or qualitative mixing patterns [12] or rates [13].

In this paper, we describe a detailed comparison of experimentaland simulation results in a horizontally-aligned annular shear cell.Specifically, we compare results obtained using digital particle trackingvelocimetry (DPTV) and a particle dynamics (PD, sometimes referred toasdiscrete elementmethod orDEM) simulation technique that employsa number of differing parameter values and forcemodel forms (Table 1).We measure solid fraction profiles as well as velocity and granulartemperature distributions and comment at the level of computationalfidelity necessary to capture both qualitative and quantitative aspects ofthe experimental flow.

2. System geometry and experimental specifics

The geometry of interest in this study is a horizontally-aligned annularshear cell (see Fig. 1, note that thedirectionof gravity is into thepage) thatis roughly one particle thick in the direction of gravity. It has an aluminumframewith a transparent Plexiglas body such that both thebottomsurfacethat the granules roll on and the top (confining) surface are smoothPlexiglas. The moving wheel is attached to a 1/16 hp motor capable ofachieving a constant rotational speed of approximately 53–280 rpm,which corresponds to a linear velocity of 0.55–2.89 m/s. Thewheel radiusis 8.5725 cm (3.375 in.) and the distance from the center of the wheel tothe stationary outer rim is 14.2875 cm (5.625 in.). Thus, the gap, which is

Table 1DEM simulation parameters.

Particle diameter 4.76±0.05 [mm]Density 7900 [Kg/m3]Young's modulus (Ei) 193 [GPa]Yield force 48 [N]Poisson ratio (σ) 0.29Friction coefficient (μ) 0.30Damping coefficient (β) 0.1 [kg/m s]Rolling friction coef. (μR) 0.5, 0.01

Fig. 2. Side-view photo showing a range of roughened inner rings with associatedvalues of the roughness parameter overlaid.

71J.J. McCarthy et al. / Powder Technology 203 (2010) 70–77

the distance between the moving wheel and outer rim, is approximately5.71 cm(2.25 in.). The granules aremade ofmilled shock-resistantwater-hardened S2 tool steel (from McMaster-Carr, part no. 1995T11)with Rockwell C55ÐC58 hardness and 290 000 psi yield strength.The diameter of the spherical granules in the base case is 0.476 25 cm(3/16 in.). The shear gap in thebase case isfilledwithanarea solid fractionof 0.708, which means that approximately 70% of the overall gap area isfilled with the granules during operation. While the granular shear cell(GSC) was built for a monolayer of granules, the spacing between thebottom and top Plexiglas surfaces was set to 0.762 cm (0.3 in.), which isslightly larger than a granule diameter, to allow the particles to freelymove in the gap without being constrained by the friction from both thetop and bottom GSC Plexiglas surfaces.

A somewhat unique aspect of this experimental arrangement isthe fact that the inner, moving wheel has a prescribed roughness [6].This roughness is characterized by the roughness factor R (see Fig. 2),which is inspired by the theoretical roughness factor of Jenkins andRichman [14], and varies as 0≤R≤1, where R=0 corresponds to amoderately smooth surface, and R=1 to a very rough surface. Thisroughness is implemented by spacing hemispheres on the movingsurface such that, for R=0, the hemispheres are directly adjacent toeach otherwith no spacing for a granule to fit between them,while forR=1 the gap between adjacent hemispheres is one granule diameterdp. This geometry is especially useful for validating particle dynamicssimulations as the proper method for handling smooth-walledboundary conditions has been a point of concern in the literature[15]. Moreover, the method for generating the well-defined boundarycan be exactly mimicked in both the experiment and simulation, thatis, we can “glue” hemispheres to themoving surface in both instances.

The digital particle tracking velocimetry (DPTV) data retrievalscheme used to acquire the experimental data has been reported indetail elsewhere [6], thus only the critical points are highlighted here.The granules are randomly arranged inside the GSC gap, which is thenclosed by placing the Plexiglas on the top before themotor is switchedon. Using an infrared tachometer, the rotation rate of the inner wheel

Fig. 1. (Left) Top-view photo of experimental rig, taken during rotation of inner ring (so thatof the experiment.

is measured and set to the required rpm. The setup is run for 2 min toestablish a reasonable steady state before the prescribed area ofinterest is recorded by the camera. The local granular flow propertiesto be measured (e.g., velocity, solid fraction, and granular tempera-ture) are assumed invariant in the tangential direction due to thesymmetry of the setup – thus, varying only in the radial direction – sothat the area of interest/measurement is an arch of the annulus.Approximately 3000 frames are extracted from the video and DPTV isperformed by post-processing 450 sets of consecutive frames selectedat equal intervals. Five trials were conducted for each experiment andthe averages of the trials are the data points presented in this work.

To obtain the local granular flow data as a function of radius, thearea of interest was divided into six radial bins with a width ofapproximately two particle diameters, with respect to the granulesused in this work. The local flow properties of each bin were thenobtained by averaging the discrete kinematic data of individualgranules within the prescribed area of interest according to thefollowing relations. For the solids fraction in radial bin i (i.e., atposition ri), νi, is given as

νi =Niπd

2p = 4

Ai; ð1Þ

roughening hemispheres are not visible). (Right) Snapshot of the computational analog

72 J.J. McCarthy et al. / Powder Technology 203 (2010) 70–77

where Ni is the number of particles in bin i, dp is the parti-cle diameter, and Ai is the bin area. Similarly, the average tangentialvelocity, VTi, and granular temperature, Ti, are evaluated using

VTi=

1Ni

∑Ni

j=1vTj ; ð2Þ

Ti =1Ni

∑Ni

j=1

13½ðvTj−VTi

Þ2 + ðvRj−VRi

Þ2�; ð3Þ

where VTjand VRj

denote the particle (as opposed to the average)tangential and radial velocity of particle j, respectively. Note that,when discussing the results, we normalize both VTi

and Ti with thelinear speed of the inner wheel and its square, respectively, andnormalized the radial position ri with its maximum value r0 (i.e., anormalized position of 0 corresponds to the inner, moving wheel,while 1 corresponds to the outer, stationary surface). Also, whilemolecular gas theory suggests the factor 1/2 should be used forcalculating the granular temperature in two-dimensional flows,here we use 1/3 as traditionally seen in granular temperaturerelations. Error bars are shown on experimental results based onthe standard deviation of the multiple trial measurements.

3. Discrete element method (DEM) and models

Discrete element method (DEM) simulations capture the macro-scopic behavior of a particulate system via calculation of the trajec-tories of each of the individual particles within the mass; the timeevolution of these trajectories then determines the global flow of thegranular material. The particle trajectories are obtained via explicitsolution of Newton's equations of motion for every particle [16]. Ina granular flow, the particles experience forces and torques due tointerparticle interactions (e.g., collisions, contacts, or cohesive inter-actions) as well as interactions between the system and the particles(e.g., gravitational forces). This section outlines the basics of the tech-nique as well as discusses the variety of force models and materialproperties used in our validation tests. In general, the collisional forcemodels range from pragmatic linear “spring and dashpot” techniquesto rigorously derived contact mechanics-inspired routines [17] andin this work we examine both ends of the spectrum. A thoroughdescription of normal interaction laws can be found elsewhere [18];therefore only the models employed will be reviewed here. As eachmodel is described, we will introduce the distinguishing notationused in the latter sections to denote the use of each respective model(a table of this notation is also included in Table 2).

3.1. Normal forces

Two approaches are used to model normal forces in this work: aHertzian spring-dashpot model and an elasto-plastic material model[19,20]. In both cases, the deformation of the particles is mimickedvia a computational “overlap” so that α=(R1+R2)−S12, where Ri is

Table 2Model variations notation.

Model variation Version 1 Version 2+

Normal force model Plastic [N] Spring-dashpot [n]Normal dissipation Fit to experiment [D] Larger than physical [d]Friction force model Nonlinear [F] Linear [f]Rolling friction Large [R′] Present [R] absent [r]Geometry Fit to experiment [3d] Larger head space [3D]

True two-dimensional [2D]Particle geometry Aspherical [S] Perfect spheres [s]

the particle radius and S12 is the distance between particle centersof particles 1 and 2. Again, in both cases, during the initial stages ofloading, the normal force, Fn, is purely elastic and is given by

Fn = knα3=2

; ð4Þ

where kn is the normal force constant from the Hertz theory [17]. Thisconstant is a function of the particle radii, Ri, and elastic properties(Young's modulus, Ei, and Poisson ratio, σi):

kn =43ET

ffiffiffiffiffiRT

p; ð5Þ

where R* and E* are given by

1ET

=1−σ2

1

E1+

1−σ22

E2ð6Þ

1RT =

1R1

+1R2

ð7Þ

respectively. At this point the two models differ in their mode ofenergy dissipation. In the case of the spring-dashpot model, adamping term that is proportional to the relative normal velocitybetween particles is linearly added to the repulsive force from Eq. 4.While the form of the damping term can take several forms, wechoose the form suggested by Oden and Martins [21] due to the factthat it qualitatively reproduces the experimentally observed depen-dence of the coefficient of restitution on impact velocity for manyengineering materials (i.e., a power law decrease). Combining therepulsive force and this dissipation term yields what we will refer toas our spring-dashpotmodel which will be denotedwith a lowercase nhenceforth, given as

Fn = knα3=2−βαα; ð8Þ

where α is the relative normal velocity of the particles, and β is adamping parameter that is assumed to be adjustable (discussedbelow).

In contrast, in our plastic model denoted with an uppercase N,dissipation is assumed to arise from the plastic deformation of thecenter of the contact spot. In this model, once the normal forceexceeds a yield force, Fy, further loading is given by the linearexpression

Fn = Fy + kyðα−αyÞ: ð9Þ

In this expression, ky is the plastic stiffness which is related to theyield force by ky=(3/2)(Fy/αy), and αy is the deformation at the pointof yield (i.e., where both Eqs. 4 and 9 give Fn=Fy). Unloading prior toexceeding the yield limit is purely elastic, while unloading after theyield limit is given by

Fn = Fmax−kn

ffiffiffiffiR

qðαmax−αÞ3=2; ð10Þ

where Fmax and αmax are the maximum force and deformation,respectively and R is dimensionless and given by the ratio of the newcontact curvature due to plastic deformation, Rp, to R*,

R =Rp

RT =Fy

Fmax

2Fmax + Fy3Fy

!3=2

: ð11Þ

Reloading after initial yield follows the same path as Eq. 10 up tothe maximum prior force, at which point the contact continues todeform plastically (Eq. 9).

While the yield force, Fy, can be loosely related to the yield stress ofthe bulk material, in this work, we treat it in the same manner as β

73J.J. McCarthy et al. / Powder Technology 203 (2010) 70–77

from the spring-dashpot model. In this way, we report results bothfrom arbitrary values of the damping parameters (that generate highdegrees of dissipation), denoted with d, as well as results where thedamping parameters were obtained by fitting to experimental droptests, denoted with D. The drop tests use a vacuum release techniqueto drop particles in a draft-proof tower onto plates of the same bulkmaterial (tool steel). The initial height of the drops are varied tobracket the range of impact velocities relevant to the annular shearapparatus and high speed video is used to extract the coefficient ofrestitution from the experiments. Results of the fitted simulations andexperiments are included in Fig. 3.

3.2. Tangential forces

Modeling the tangential forces that arise from oblique particleimpacts has elicited a considerably wider range of force models thanthose of normal interactions. For the purposes of this validation paper,we will refrain from examining models that elicit “creep” (i.e., modelswhose tangential force is based on the tangential velocity). Instead, wewill consider both the simplest [16] and most comprehensive [22] ofthe “history dependent” friction models. That is, for each time-step,the new tangential force acting at a particle–particle contact, Ft, isgiven as:

Ft = Fto−ktΔs; ð12Þ

where Fto is the old tangential force and ktΔs is the incrementalchange in the tangential force during the present time-step due torelative particle motion; i.e., Δs is the tangential displacement duringthe present time-step. This displacement is calculated from thecomponent of velocity tangent to the contact surface, vt (i.e., Δs=vtdtwhere dt is the time-step). In the simpler of our two friction models,we will assume that the tangential stiffness, kt, is a constant which wewill call kto . We will refer to this model as the linearmodel and denoteit with a lowercase f. In this model, we must impose a discontinuity inorder to limit the tangential force to the Amonton's Law limit (Ft≤μ fFnwhere μf is the coefficient of sliding friction).

In the more involved scenario, in order to mimic an annular regionof microslip at the edge of the contact as well as smoothly satisfy

Fig. 3. Coefficient of restitution – i.e., the ratio of final to initial impact velocity –

variation from drop tests of experimental materials (bullets) and for the plastic andspring-dashpot normal force models, with fitted damping parameters.

the Amonton's Law limit, the frictional stiffness, kt, is given by thenonlinear expressions obtained following the procedure of Ref. [22]

kt = ktoθFμð1−θÞΔs

ð13Þ

with

θL = 1− T + μΔFμF

� �13

θU = 1− TT−T + 2μΔFμF

" #13

θR = 1− T−TTT + μΔFμF

" #13:

ð14Þ

The subscripts L, U and R in Eq. 14 correspond to loading,unloading and reloading, respectively. The negative sign in 13 is onlyused for the unloading stage. The load reversal points T* and T** needto be continuously updated as T*=T*+μF and T**=T**+μF to accountfor the effect of varying normal force. This nonlinear friction force willbe denoted with an F. In both the linear and nonlinear case, the(initial/constant) tangential stiffness, kto, can be related to the normalstiffness kn by noting that the rates of change of displacement withloading should be similar for tangential versus normal interaction, sothat their ratio is given [17] as

ds= dFtdα= dFn

=1−σ = 21−σ

: ð15Þ

3.3. Rolling friction

Rolling friction is a rarely employed force in mainstream DEMsimulations [15]. Nevertheless, there are two potential reasons forinclusion of such a force when modeling real particles: to dissipateenergy when particles roll on a smooth surface, and in order toapproximate the behavior of slightly aspherical particles.

Looking first at the former case, in real particle applications, afriction-like effect is observed as particles that roll must both deformnew portions of their surface and release previously deformedportions of their surface. These “impact-like” deformation-releasecycles can dissipate some energy, if one assumes that some portion ofthe strain energy from these cycles is lost [17]. In DEM simulationsthat include only the previously discussed force laws, (perfectly) flatsurfaces allow a particle to roll without friction as none of the normalforce models distinguish what portion of a surface is being deformed.In many DEM simulations this problem is avoided by omittingperfectly flat surfaces, instead using immobilized particles to generatea “bumpy” wall [23]. If, however, one wanted to explicitly accountfor rolling friction and allow the use of perfectly smooth boundaries(as are present at the outer edge and the top and bottom plate ofthe present annular shear cell experiment), the simplest approachis to assume that the strain hysteresis from a rolling contact canbe quantified by a constant loss factor, γ, which corresponds to thefraction of energy lost during a single rotation. If we assume aHertzian pressure distribution on the contact spot (in accordancewith the models discussed in the previous sections), we get a torqueresisting the rolling motion [24], Mr, that is given by

Mr = −3γ16

aFnωjω j ; ð16Þ

where ω and |ω| denote the angular velocity and its magnitude,respectively, while a denotes the radius of the contact spot. If we

Fig. 4. Comparison of experimental (solid bullets) and computational results when varying the normal force model and dissipation fitting parameter. The models compared include:spring-dashpot with empirical dissipation fit (open diamonds), and plastic dissipation model both with an empirical fit (stars) and with an overly large dissipation (open squares).

74 J.J. McCarthy et al. / Powder Technology 203 (2010) 70–77

define a rolling friction coefficient as μr =3γa16R

, we can simplify thisexpression to

Mr = −μrRFnωjω j : ð17Þ

In the work reported here we vary the value of μr from zero(denoted with r), to an intermediate value (denoted with R), to anexceedingly large value (denoted with R′) in order to evaluate theimportance of this force as well as its ability to mimic the slightlyaspherical nature of the actual particle used in the experiments.

3.4. Computational geometry

While the geometry of the annular shear cell is ostensibly two-dimensional, we have simulated both a true two-dimensional (2D)annulus as well as a pseudo two-dimensional annulus that moreclosely mimics the experimental apparatus. Specifically, we examinethree separate system geometry cases: a true two-dimensionalgeometry, denoted as 2D; a pseudo two-dimensional geometry thathas a somewhat arbitrary head space above the particles (i.e., extraspace between the top and bottom plates), denoted as 3D; and ageometry that exactly matched the dimensions of the physicalapparatus, denoted as 3d.

Similar to our variation of the system geometry, we have studiedthe sensitivity of our results to the particle geometry. In other words,we havematched the particle mean size and size distribution width tothe values quoted by the sphere manufacturer, but have also usedboth true spheres (denoted as s) and particles that have a smallamount of asymmetry (denoted as S). In order to model the particleasymmetry we have used the standard multi-particle technique [25]that has been used for adapting the use of spheres to model non-spherical shapes. In particular, we use a two particle cluster in place ofeach true experimental “sphere”, where the center of mass mismatch

Fig. 5. Comparison of experimental (solid bullets) and computational results when varyingtriangles), and a nonlinear friction model (stars).

between particles of the cluster is chosen to match the degree ofasphericity quoted by the particle manufacturer.

4. Results and discussion

The experimental parameters that are varied include: inner wheelrotation rate, inner wheel roughness factor and total solids fraction.The “base case” of the experiment is taken as a rotation rate of240 rpm, a roughness factor of 0.6, and a total solids fraction of 0.75.This case is used for the comparison of the model parameters/detailsdiscussed in the previous section.

4.1. Base case

Our validation study involves comparison of three independentexperimental measurements: solids fraction profile, velocity profile,and granular temperature distribution. We should note that the solidsfaction profiles will often exceed the two-dimensional maximumareal fraction packing in both simulations and experiments due to thefact that the pseudo two-dimensional geometry allows particle to“overlap” slightly by occupying slightly different heights in the headspace.

Fig. 4 shows a comparison of the solids fraction, velocity, andgranular temperature profiles for the base case using two differingnormal force models and varying the normal dissipationmodel. Whileboth the plastic and spring-dashpot models yield reasonable resultsin the solids fraction profile, the spring-dashpot model dramaticallyunder-predicts both the velocity and granular temperature profiles.Surprisingly, the amount of dissipation makes essentially no differ-ence in the solids fraction profile, yet using an artificially large degreeof dissipation in the plastic model improves the velocity and granulartemperature agreement.

Fig. 5 shows a comparison of the solids fraction, velocity, andgranular temperature profiles for the base case using two differing

the tangential force model. The models compared include: linear friction model (open

Fig. 6. Comparison of experimental (solid bullets) and computational results when varying the degree or lack of rolling friction. The models compared include: no rolling friction(side-pointing triangles), an intermediate value of rolling friction (stars) and an overly degree of rolling friction (down-pointing triangle).

75J.J. McCarthy et al. / Powder Technology 203 (2010) 70–77

tangential force models. Surprisingly, friction model choice seems tohave almost no impact on the measured result. This observation maybe due to the fact that the easily discriminatory region of our resultscorresponds to a portion of the flow where very high speed collisionsoccur. This likely causes frictional forces in that region to quicklyreach the sliding limit so that the details of attaining that force areunimportant. In contrast, Fig. 6 shows that comparing two degrees ofrolling friction to a case without rolling friction does, in fact, make adifference; however, while one would be tempted to decrease thedegree of rolling friction – in order to more closely match the velocityprofile – closer inspection shows that any value of rolling frictiongreater than zero moves the granular temperature profile in thewrong direction. In other words, it seems that rolling friction may betuned to fit the velocity profile, but only at the expense of missing thegranular temperature profile.

Next, we varied the system and particle geometrical character-istics. Interestingly, while using a truly planar (two-dimensional)geometry improves the solids fraction profile somewhat, the corre-sponding velocity profile is now qualitativelywrong (see Fig. 7). That

Fig. 7. Comparison of experimental (solid bullets) and computational results when varying ttriangles), a pseudo-2d geometry with arbitrary (larger) head space (stars) and a pseudo-2

Fig. 8. Comparison of experimental (solid bullets) and computational results when varying tsymbols), and a slightly aspherical particle geometry (matching experimental particles) (o

is, there is significant slip even at the outer, immobile wall (unlikethe experiment and/or any of the pseudo two-dimensional simula-tions). At the same time, improving the accuracy of the head spacesize (i.e., matching the experiment quantitatively in gap height)improves both the velocity and temperature profiles. In order to getquantitative agreement for all three profiles, however, we also needto include in the simulation the (very small) degree of asphericityquoted by the particle manufacturer as can be seen in Fig. 8. Usingparticle “clusters” in order to capture the particle asphericity notonly makes the velocity and granular temperature profile fall withinthe experimental error bars, but also dramatically improves thesolids fraction profile which now captures both the magnitude andlocation of the peak correctly.

4.2. Parametric study

As a test of the robustness of thefit established for the base case, wenext compare the computational results from the “best fit” model(NDFR3dS)with experiments that vary rotation rate, roughness factor,

he system geometry. The models compared include: a truly 2D geometry (side-pointingd geometry with exactly matched head space (plus symbols).

he particle sphericity. The models compared include: a perfectly spherical particle (pluspen circles).

Fig. 9. Parametric study varying the inner wheel rotation rate as well as the inner wheel roughness factor. Experimental values are shown as symbols, while the results from the bestcase model (NDFR3dS) are shown as lines.

76 J.J. McCarthy et al. / Powder Technology 203 (2010) 70–77

and total solids fraction. Recall that our “best fit” model includes: aplastic deformation model whose normal dissipation parameters arefit with experiment, include a nonlinear tangential friction model, amodest degree of rolling friction, a “perfect” fit of the true pseudo-2Dgeometry of our system, and slightly aspherical particles (althoughneither the friction model choice nor presence of rolling friction werediscriminatory).

In Fig. 9, we show that the best case simulation model is capable ofcapturing both the values and the changing trends in each of the threedistribution whenwe vary both the rotation rate fromΩ=220 rpm toΩ=240 rpm to ga=270 rpm, as well as when we vary the roughnessfactor from R=0.0 to R=0.6 to R=1.0. Interestingly, the changes inrotation rate have a very small effect, while increasing the roughnessfactor increases both the granular temperature and the maximumtangential velocity. The weak dependence on rotation rate is possiblydue to the limited range of variation explored in this study, a pointthat requires further study.

Finally, in Fig. 10, we examine how changes in the total solidsfraction impact the three measured profiles. The agreement in thecase of the solids fraction profile is quite good both qualitatively andquantitatively. In the case of the velocity profile, again we capture thequalitative trends and the quantitative magnitude reasonably well.The only case where the “best case” simulation does not match as wellas expected is for the variation of granular temperature profile withtotal solids fraction. One should note that this disparity, in fact, seemsto increase with decreasing total solids fraction (recall that for thehigh total solids fractions of the previous sections, we matched thegranular temperature quantitatively). This discrepancy, and the trend

Fig. 10. Parametric study varying the total solids fraction. Experimental values are shown a

in the discrepancy, may be attributed to the fact that, as total solidsfraction decreases, the fraction of total particle interactions becomesskewed toward a higher fraction of particle–wall interactions(specifically, the top and bottom plates). As the material propertiesof thesewalls have not been incorporated into the simulation (insteadwe treat the walls as having the same properties as the particles), thismay lead to a lower rate of granular temperature dissipation thanobtained from the experiments; however, this hypothesis is not easilytested as the correct force models and material parameters to use in asteel–plastic impact is unclear.

5. Conclusion and outlook

In this work we use three experimentally measured profiles – forsolids fraction, tangential velocity, and granular temperature – as ameans of validating DEM model choices and parameter values. Weexamine a limited subset of packing fractions, materials properties,shear rates, etc. Thus, these conclusions are by no means general, butare nevertheless instructive for the regimes observed. Surprisingly,the choice of these profiles, while made for reasons of experimentalconvenience, has proven to be a powerful discriminator between thevarieties of force models typically used in the literature. It is foundthat two of the most significant characteristics of the “best case”model are matching the system geometry and particle shape exactly.This is quite an interesting observation as the particles used inthe experiments are as close to spherical as is typically availablefrom commercial sources, yet we needed to model the veryslight asphericity in order to obtain good quantitative matching to

s symbols, while the results from the best case model (NDFR3dS) are shown as lines.

77J.J. McCarthy et al. / Powder Technology 203 (2010) 70–77

experiment. Moreover, this suggests that even pseudo two-dimen-sional experiments are not easily related to truly planar simulations.Other findings are that a plastic deformation mode of dissipation issuperior to spring-dashpot, while the choice of friction model haslittle effect. Both of these observations, however, may be materialand system specific, such that polymeric materials and/or lowerimpact velocities may reverse these trends. A similar study with lowerrotation rates and/or different particulate materials would aid inclarifying this issue. Finally, while rolling friction had a measurableeffect on the simulation results, it was not capable of accounting forthe asphericity of the particles.

Acknowledgement

The authors would like to acknowledge the support of the NationalEnergy Technology Laboratory of the Department of Energy under theproject 41817.606.07.02C.

References

[1] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Granular solids, liquids and gases, ReviewModel Physical 68 (1996) 1259–1273.

[2] C. BROADBENT, J. BRIDGWATER, D. Parker, S. Keningley, P. Knight, A phenom-enological study of a batch mixer using a positron camera, Powder Technology 76(3) (1993) 317–329.

[3] Z. Yang, P.J. Fryer, S. Bakalis, X. Fan, D.J. Parker, J.P.K. Seville, An improvedalgorithm for tracking multiple, freely moving particles in a positron emissionparticle tracking system, Nuclear Instruments and Methods in Physics ResearchSection A—Accelerators Spectrometers Detectors and Associated Equipment577 (3) (2007) 585–594, doi:10.1016/j.nima.2007.01.089.

[4] M. Nakagawa, S. A. Altobelli, A. Caprihan, E. Fukushima, E. K. Jeong, Non-invasivemeasurements of granular flows by magnetic resonance imaging, Experiments inFluids 16 (1) (1993/11/01/) 54–60, URL http://dx.doi.org/10.1007/BF00188507.

[5] M.Nikitidis, U. Tuzun, N. Spyrou,Measurement of size segregation by self-diffusionin slow-shearing binarymixtureflows using dual photon gamma-ray tomography,Chemical Engineering Science 53 (13) (1998) 2335–2351.

[6] V. Jasti, I. Higgs, C. Fred, Experimental study of granular flows in a rough annularshear cell, Physical Review E 78 (4). doi:ARTN 041306.

[7] C.T. Veje, D.W. Howell, R.P. Behringer, Kinematics of a two-dimensional granularcouette experiment at the transition to shearing, Physical Review E 59 (1) (1999)739–745, doi:10.1103/PhysRevE.59.739.

[8] G. Chambon, J. Schmittbuhl, A. Corfdir, J. P. Vilotte, S. Roux, Shear with comminutionof a granular material: Microscopic deformations outside the shear band, PhysicalReview E 68 (1). URL http://link.aps.org/abstract/PRE/v68/e011304.

[9] N. Jain, J. M. Ottino, R. M. Lueptow, An experimental study of the flowing granularlayer in a rotating tumbler, Physics of Fluids 14 (2) (2002/02/00/) 572–582, URLhttp://link.aip.org/link/?PHF/14/572/1.

[10] D. M. Mueth, G. F. Debregeas, G. S. Karczmar, P. J. Eng, S. R. Nagel, H. M. Jaeger,Signatures of granular microstructure in dense shear flows, Nature 406 (6794)(2000/07/27/print) 385–389, URL http://dx.doi.org/10.1038/35019032.

[11] W.L. Vargas, J.J. McCarthy, Conductivity of granular media with stagnantinterstitial fluids via thermal particle dynamics simulation, International Journalof Heat and Mass Transfer 45 (2002) 4847–4856.

[12] J.J. McCarthy, D.V. Khakhar, J.M. Ottino, Computational studies of granular mixing,Powder Technology 109 (2000) 72–82.

[13] H.P. Kuo, P.C. Knight, D.J. Parker, Y. Tsuji, M.J. Adams, J.P.K. Seville, The influence ofDEM simulation parameters on the particle behaviour in a v-mixer, ChemicalEngineering Science 57 (17) (2002) 3621–3638, doi:10.1016/S0009-2509(02)00086-6.

[14] J.T. Jenkins, M.W. Richman, Boundary conditions for plane flows of smooth, nearlyelastic, circular disks, Journal of FluidMechanics Digital Archive 171 (1986) 53–69.

[15] Y. C. Zhou, B. H. Xu, A. B. Yu, P. Zulli, Numerical investigation of the angle of reposeof monosized spheres, Physical Review E 64 (2), URL http://link.aps.org/abstract/PRE/v64/e021301.

[16] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies,Géotechnique 29 (1979) 47–65.

[17] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1987.[18] J. Schäfer, S. Dippel, E. Wolf, Force schemes in simulations of granular materials,

Journal Physical I 67 (1991) 1751–1776.[19] S.T. Nase, W.L. Vargas, A.A. Abatan, J.J. McCarthy, Discrete characterization tools

for wet granular media, Powder Technology 116 (2001) 214–223.[20] C. Thornton, Z. Ning, A theoretical model for the stick/bounce behavior of

adhesive, elastic–plastic spheres, Powder Technology 99 (1998) 154–162.[21] J.T. Oden, J.A.C. Martins, Models and computational methods for dynamic friction

phenomena, Computer Methods in Applied Mechanics and Engineering 52 (1–3)(1985/9) 527–634.

[22] C. Thornton, C.W. Randall, Applications of theoretical contact mechanics to solidparticle system simulation, in: M. Satake, J.T. Jenkins (Eds.), Micromechanics ofGranular Material, Elsevier Science Publishers, Amsterdam, 1988, pp. 133–142.

[23] J.J. McCarthy, J.M. Ottino, Particle dynamics simulation: a hybrid techniqueapplied to granular mixing, Powder Technology 97 (1998) 91–99.

[24] J.A. Greenwood, H. Minshall, D. Tabor, Hysteresis losses in rolling and slidingfriction, Proceedings of the Royal Society of London, Series A Mathematical andPhysical Sciences 259 (Jan. 24 1961) 480–507 1299.

[25] V. Buchholtz, T. Poschel, Numerical investigations of the evolution of sandpiles,Physica A 202 (1994) 390–401.