Couette grain flow experiments: The effects of the coefficient of restitution, global solid fraction, and materials

Powder Technology 211 (2011) 144–155

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Couette grain flow experiments: The effects of the coefficient of restitution, global solid fraction, and materials Martin C. Marinack Jr., Venkata K. Jasti, Young Eun Choi, C. Fred Higgs III ⁎ Mechanical Engineering Department, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213-3890, USA

a r t i c l e

i n f o

Article history: Received 29 September 2010 Received in revised form 10 March 2011 Accepted 17 April 2011 Available online 22 April 2011 Keywords: Granular flow Shear cell Granular material Digital particle tracking velocimetry Coefficient of restitution Solid fraction

a b s t r a c t Granular flows are complex flows of solid granular material which are being studied in several industries. However, it has been a challenge to understand them because of their non-linear and multiphase behavior. The present experimental work investigates granular flows undergoing shear, by specifically studying the interaction between rough surfaces and granular flows when the global solid fraction and the material comprising the rough shearing surface are varied. A two-dimensional annular shear cell, with a stationary outer ring and inner driving wheel, and digital particle tracking velocimetry (DPTV) technique were used to obtain local granular flow properties such as velocity, local solid fraction, granular temperature, and slip. A customized particle drop test apparatus was built to experimentally determine the coefficient of restitution (COR) between the granular and surface materials using high-speed photography. Results showed that wheel surface materials that produce higher COR values exhibit higher velocity and granular temperature values near the wheel, and lower slip velocities. The local solid fraction appears inversely related to the COR values. The global solid fraction seemed to correspond with velocity and granular temperature, while displaying an inverse relationship to slip. Results also showed an initial decrease in the kinetic energy of the flow as the global solid fraction increased, due to the formation of a distinct contact region. This was followed by a rise in kinetic energy as the global solid fraction continued to increase, based on the increase of particles present in the kinetic region of the flow. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Understanding granular flows has always been important for predicting natural phenomena such as avalanches, rockslides, and soil erosion. It is additionally important as granular flows have been proposed as a particulate lubricant alternative to traditional oil-based lubricants [1,2]. Gaining insight into granular flows is also important to key energy systems such as coal-based fossil fuel systems and industrial processes such as food manufacturing. Granular flows may exhibit solid, liquid, and/or gaseous behavior under certain conditions. This ability to display multiphase behavior has made predicting and understanding their behavior difficult. Granular flows have been studied extensively by means of both modeling/simulations and experimental techniques. In instances where granular flows behave like a fluid (continuum), Navier– Stokes based continuum modeling approaches have been examined as a means to predict and model the granular flow [1,3–9]. An alternative to continuum-based approaches are event-driven (discrete element method) simulations, which have been used to study granular flows undergoing shear [10–13], as well as numerous other granular processes and systems such as granular pile formation [14] and hoppers [15,16].

⁎ Corresponding author. E-mail address: [email protected] (C.F. Higgs). 0032-5910/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2011.04.012

Other modeling techniques include lattice-based cellular automata, which has modeled numerous granular flow problems including segregation [17], granular shear [18,19], pile (heap) formation [20,21], and flows inside hoppers and silos [22,23], and the explicit finite element method, which was introduced as an event-driven granular flow simulation approach by Kabir et al. [24,25]. In terms of experimental studies, many past experiments have been done to examine specific granular problems and applications, such as granular avalanches [26,27], rotating cylinders [28], hoppers [29–32], and silos [33–35]. A commonly studied granular flow situation is that of granular flows under shear [2,36], which is the focus of this study. The experimental setup used in this work is that of an annular shear cell, in which a granular shear flow is created between concentric cylinders where one is rotating relative to the other. GDR MiDi [37] gives a good account of a collection of such experiments, and also identifies some common flow features in their analysis. Similar to the current work, previous experimental work by Tardos et al. [38] sheared glass beads between concentric cylinders, where the inner cylinder rotated and the outer cylinder remained stationary. However, measurements were limited to global torque, providing no discrete particle or local flow measurements. More recent experimental works have also studied granular shear flows without obtaining discrete particle data, instead performing gross stress [39– 41] or segregation measurements [42].

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Nomenclature A HI HR N S T V g rb ro rw rp v ν

Area Initial height Rebound height Number of particles Slip velocity at inner moving wheel Granular temperature Average velocity Acceleration due to gravity Radius of bin Radius of outer rim Radius of wheel Particle radius Velocity Average solid fraction

Subscripts 1 Radial bin 1 G Granule P Plate Imp Impact Reb Rebound R Radial component T Tangential component i Radial bin i j Particle j w Wall

There are a large number of experimental studies which do include discrete particle and local flow measurements, such as recent work by Shearer et al. [43] and May et al. [44,45]. Their experimental work used digital particle tracking techniques to obtain local flow velocity profiles, while studying the size segregation of granular materials undergoing shear. However, unlike the experiments performed in this work, these studies [43–45] did not produce a shear flow between two concentric cylinders rotating relative to one another. Instead, their shear flow was formed in the gap between two stationary concentric cylinders through the relative rotation of the upper and lower plates/ disks which confined the flow (similar to the work Yu et al. [2]). Local granular flow measurements, for concentric cylinders rotating relative to one another, were included in the works of Veje et al. [46] and others [47–51], who used digital particle tracking techniques to obtain discrete particle velocities and rotations. Mueth et al. [52] performed experiments between relatively rotating concentric cylinders and extended the tracking of local flow properties into the third dimension through the combination of magnetic resonance imaging, X-ray tomography, and digital particle tracking techniques. A major difference between these prior works [43–52] and the experiments performed in this work is the magnitude of the nominal shear rate, as well as the rotation rate and linear speed of the shearing surfaces. The aforementioned shear cell experiments [43–52] were performed at much lower speeds and shear rates than those examined in the present work. Elliott et al. [53] performed shear cell experiments at high speeds and nominal shear rates which were similar to the current work. These experiments were performed in a rig similar to the one used in this work, and included local flow measurements obtained via digital particle tracking techniques. However, in their experiments [53] both the inner and outer cylinders were rotated in opposite directions which caused the entire granular flow to be maintained within the kinetic regime. This is unlike the current experimental work which only rotates the inner wheel, creating a flow comprised of both kinetic

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and contact (frictional) regimes. Characteristics of granular flow regimes have been discussed in many prior works [2,10,54–57], as well as later in this work. Liao et al. [58] and Hsiau et al. [59] also performed shear cell experiments at high speeds and rotation rates in line with the studies performed in this work. Yet unlike the experiments performed in this study, these works confined the granular material within an annular trough, and created a shear flow through the relative rotation of the end disks which confined the flow (similar to the works of May et al. [44,45] and Yu et al. [2]). Additionally, these studies [58,59] only performed tracking on a small percentage of tracer particles present within the flow. This is dissimilar to the current work, which tracks all the particles within a given image frame. Understanding the interaction and momentum exchange between the rough driving surface and the grains is important to understanding granular flows in shear. It is particularly important in gaining a better understanding of the transfer of energy to the granular flow. This importance was recognized in a previous work by the authors [54], who studied the effects of varying the roughness and rotation rate of a rough driving wheel on local granular flow properties. This work aims to further the understanding of the driving surfacegranular flow interaction by studying the effects of varying the material of the driving surface and the global solid fraction. Following the work of Bagnold [60], numerous granular shear experiments were performed as part of this study, using a two-dimensional granular shear cell (GSC), (see Ref. [54]). This granular shear cell consists of a stationary outer rim and a quantifiably rough inner rotating wheel. The annular gap was filled with granular material, and the local flow data – velocity, local solid fraction, granular temperature, and slip – were acquired by means of the digital particle tracking velocimetry (DPTV) approach used previously by the authors [54]. Particle tracking approaches similar to DPTV have also been used in many prior works studying granular shear [46,52,53,61,62]. A particle drop test apparatus, coupled with high-speed photography, was used to measure the COR values of the relevant granular materials from the shear cell experiments. The goal of this study is to determine how varying both the global solid fraction (i.e. the fraction of the total annular gap occupied by granular particles) and the wheel surface material affect the local flow properties (i.e. velocity, solid fraction, granular temperature, and slip) of the granular medium. Conclusions based on parametric tests with the shearing materials are formed based on the fact that changing the shearing materials changes the wheel-granule COR. 2. Experimental setup 2.1. Granular shear cell description The granular shear cell (GSC) used for the study of granular flow properties, was fabricated with an outer stationary ring and an inner rotating wheel with a prescribed roughness. The radii of the inner rotating wheel and outer stationary rim are 8.573 cm (3.375 in.) and 14.288 cm (5.625 in.), respectively. The top view of the rig is shown in Fig. 1. The frame is made of aluminum with a clear Plexiglas lid and base for observing the granular behavior. The spacing between the Plexiglas lid and base is 0.794 cm (0.313 in.), which provides for a slight gap such that the granules (0.476 cm (0.188 in.) in diameter) are not compressed between the base and the lid. Extensive details on the dimensions and parameters of the shear cell were presented in the authors' previous work [54]. A high-speed camera is used to record the GSC trials. The inner rotating wheel is given a macroscopic roughness factor by affixing granules, which create protruding hemispheres, about the wheel's periphery. The roughness factor R, similar but not identical to that of Jenkins and Richman [63], varies as 0 ≤ R ≤ 1, where R = 0 corresponds to a fairly smooth surface and R = 1 corresponds to a very

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Fig. 1. Granular shear cell (GSC). Top view of the granular shear cell; gravity acts into the page. The inner wheel rotates while the outer wheel remains stationary, creating a granular shear flow within the annular gap.

rough surface. On the inner wheel, R = 0 corresponds to a wheel with protruding hemispheres positioned adjacent to each other, whereas R = 1 corresponds to the hemispheres positioned such that a gap of exactly one granule diameter exists between consecutive hemispheres. In these studies, the roughness factor is held constant at R = 0.6, which corresponds to protruding hemispheres positioned such that a gap of 0.6 of a granule diameter is present between consecutive hemispheres along the wheel's periphery. 2.2. Drop test apparatus A drop test apparatus was designed and developed to measure the coefficient of restitution (COR) between test materials. A photograph and annotated diagram of the apparatus are shown in Fig. 2a and b, respectively. In Fig. 2b, the labeled components (A–G) are defined as a Plexiglas casing (G), air hose/pump (E/F), base plate and plate holder (A and B), granule being tested (D), and device for holding the granule being dropped (C). The casing (G) provides holes which fix the holding device (C) in place at different heights. The air hose/pump (E/F) provides a suction force to the holding apparatus (C), which is turned on to hold a granule (D) in place, and then switched off to allow the granule to drop from rest. This type of setup minimizes spin during the drop. The base plate (A), 7.62 cm square and 0.635 cm thick, is secured in place at the bottom of the apparatus by a holder (B). This plate is what the granule collides with during each trial. Consequently, a COR can be obtained between the granule and plate materials. A high-speed digital video camera, the same one used for the GSC trials, records each drop test trial.

wheel to form the roughness factor, R. The materials used in the study were S2 tool steel (from McMaster-Carr, part number: 1995T11), chromium (chrome) steel (Grade 24) (from Small Parts, part number: B000FN0OE4), tungsten carbide (grade 25) (from Small Parts, part number: B000FMYFI6), and polybutadiene (a composite rubber from Precision Associates, Inc., part number: 901-188 6743). Table 1b shows that for the global solid fraction (νg) study, νg was varied between 0.4 and 0.7, while all other parameters were held constant. Also shown in Table 1b is that S2 tool steel was used for both the wheel surface material and the granules filling the GSC. Table 2 displays specific data on the various particles used in both studies. It should be noted that diameter tolerance data for chrome steel and tungsten carbide was not available (n/a) from the manufacturer. For each trial performed, the GSC set-up was run for 2 min to establish a reasonable steady-state. A digital particle tracking velocimetry (DPTV) data retrieval scheme was used to track the position of each GSC particle. A highspeed digital video camera filmed the colliding granules during the operation of the GSC. This produced a series of images, represented by the image shown in Fig. 3a, where a light spot is formed on each particle from the reflection of a light source. A computational code was developed in Mathematica to threshold the images. During this process, each particle's light spot is displayed as white, while all nonparticle space (background) is displayed as black, as shown in Fig. 3b. Following the completion of this process, the code obtains the location of each particle's centroid from their respective white spots, and tracks the motion of the centroids between consecutive frames to obtain particle velocities. Each data point was obtained as the average of 5 trials, each of which was filmed at a minimum frame rate of 1000 fps and analyzed all the particles within the frame of interest (inside the solid white lines shown in Fig. 3c) for 450 sets of frames. These facts serve to minimize errors associated with the white spots of the particles being off-center, as well as errors associated with insufficient frame rate or sampling [64]. The local flow properties from the GSC were obtained by averaging discrete kinematic data of individual granules within the polar rectangle, which is outlined by the thick white line in Fig. 3c and divided into 6 radial bins, as shown by dashed lines. It is important to note that Fig. 3c represents the bin segmentation, which will be commonly referred to in this work. In this segmentation, bin 1 is adjacent to the moving inner wheel and bin 6 is adjacent to the stationary outer wall. Each bin was approximately two particle diameters wide. The local flow properties of tangential velocity, local solid fraction, and granular temperature were then plotted as a function of normalized bin radius ((rb-rw)/(ro-rw)). The slip and total translational kinetic energy of the flow were also obtained and plotted as a function of the parametric variable being studied. Additional details on data acquisition and DPTV can be found in Jasti and Higgs [54]. The equations used were as follows: VT;i =

1 Ni ∑ v Ni j = 1 T; j

ð1Þ

νi =

Ni πrp2 Ai

ð2Þ

3.1. Granular shear cell studies

Ti =

 2
2  1 1
Ni vT; j −VT;i + vR; j −VR;i ∑j = 1 Ni 2

ð3Þ

Two sets of experiments are shown in this work, which study the effect of wheel surface material and global solid fraction on the local flow properties of a granular flow. Table 1a and b summarizes the experimental inputs for the parametric wheel surface material and global solid fraction studies, respectively. For the wheel material study (Table 1a), all parameters were held constant except for the material of the particles which were glued around the periphery of the rotating

S = Vw −VT;1

3. Procedure and calculations

N

KE = ∑j = 1

ð4Þ  1
2 2 vT; j + vR; j : 2

ð5Þ

Eq. (1) solves for the average tangential velocity, VT, i, in bin i, where vT, j is the tangential velocity of a particle j within the prescribed

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Fig. 2. Drop test apparatus. (a) Photograph of drop test apparatus with dimensions of the Plexiglas casing (tower) included. (b) Schematic side view of the drop test apparatus where the labels correspond to the major components. A: Plate, B: Plate Holder, C: Apparatus for holding granules, D: Granule, E: Air hose, F: Air Pump (provides suction), G: Plexiglas casing.

bin i and Ni is the number of particles in bin i. Eq. (2) computes the average solid fraction in bin i, νi, where rp and Ai represent the particle radius and the bin area, respectively. Eq. (3) calculates the granular temperature Ti in bin i, where VR, i is the average radial velocity in bin i and vR, j is the radial velocity of a particle j within the prescribed bin i. Note that since the granular temperature in Eq. (3) is only in two dimensions, the factor 1/2 is used (similar to Hsiau et al. [61]). This is opposed to the commonly seen 1/3 which is used based on threedimensional kinetic theory of granular gases. Eq. (4) solves for the slip velocity, S, at the inner rotating wheel from the linear velocity of the wheel's outer edge, Vw, and the average tangential velocity of bin 1, VT, 1. Finally, Eq. (5) obtains the total translational kinetic energy, KE, in the annular gap, where N is the total number of particles residing in the entire annular gap (across all the bins).

Table 1 Summary of experimental inputs: (a) Wheel surface material study (b) Global solid fraction study. Parameter (a) Roughness Wheel RPM # of granules Global solid fraction Granule material Camera frame rate (fps) Camera resolution Wheel material

(b) Roughness Wheel RPM # of granules Global solid fraction Granule material Camera frame rate (fps) Camera resolution Wheel material

Value 0.6 240 1633 ~ 0.71 S2 tool steel 3000 512 × 384 S2 tool steel Chromium steel (Grade 24) Tungsten carbide (Grade 25) 70 Duro Polybutadiene

0.6 240 922–1613 0.40, 0.45, 0.50, 0.55, 0.60, 0.65, 0.7 S2 tool steel 1000 512 × 384 S2 tool steel

3.2. Drop tests For the experimental COR testing, the COR between steel, which represents the material of the granules filling the GSC's annular gap, and each of the wheel surface materials was determined. Using the drop test rig from Fig. 2, S2 tool steel, chromium steel, tungsten carbide, and polybutadiene granules, exactly matching those of the wheel surface materials given in Table 1a, were each dropped on an S7 tool steel plate (McMaster-Carr Part Number: 88815K219) with Rockwell C22–C23 hardness, to experimentally determine the COR for these collisions. At this point it should be noted that, due to material availability, the plate material (S7 tool steel) does not exactly match the material of the granules filling the GSC (S2 tool steel). An S7 tool steel plate was chosen due to its close match to S2 tool steel in material properties (identical elastic modulus and Poisson's ratio). A slight difference in hardness (and yield strength) is present which is further addressed and interpreted in the COR results in Section 4.1. For each collision pair, the granules were dropped from six fixed heights, with six trials being performed at each height. Each trial was recorded with the high-speed digital camera. The COR between the two colliding materials (the granule and the base plate in this instance) is given by Eq. (6), where vRebP, vRebG, vImpG, and vImpP represent the rebound velocity of the plate, the rebound velocity of the granule, the impact velocity of the granule, and the impact velocity of the plate, respectively. In the case of the falling granules striking the stationary plate, the velocity of the plate remains constant

Table 2 Particle data for the various particle materials used in both the wheel surface material study and global solid fraction study. Particle material S2 tool steel

Nominal (Mean) Diameter diameter tolerance

0.476 cm (0.188 in.) Chromium 0.476 cm steel (0.188 in.) Tungsten 0.476 cm carbide (0.188 in.) Polybutadiene 0.476 cm (rubber) (0.188 in.)

± 0.005 cm (0.002 in.) n/a n/a ± 0.008 cm (0.003 in.)

Sphericity

Hardness (or durometer)

0.0005 cm (0.0002 in.) 0.000061 cm (0.000024 in.) 0.000064 cm (0.000025 in.) 0.008 cm (0.003 in.)

Rockwell C55–C58 Rockwell C60–C67 Rockwell C77–C80 70 (Durometer)

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Fig. 3. GSC DPTV region of interest. (a) Image obtained from high speed video of the GSC during a Polybutadiene wheel trial. (b) This shows the same image from (a) after thresholding has been performed. The particles appear as white, while background space is black. (c) This shows the bin segmented region of interest, where data is obtained during DPTV within the polar rectangle outlined by the thick solid white line, and tabulated within six evenly divided bins across the annular gap. Bin 1 is adjacent to the rotating wheel, while bin 6 is adjacent to the stationary outer wall.

at zero, giving Eq. (7), where the rebound and impact velocity of the granule are given by vReb and vImp, respectively. The subscript “G” is dropped in Eq. (7), as both values now refer to those of the granule. COR =

COR =

vRebP −vRebG vImpG −vImpP −vReb vImp

ð6Þ

ð7Þ

From kinematics, the impact and rebound velocities of the granules can be given by Eqs. (8a) and (8b), respectively (where g is the gravitational acceleration, HI is the initial drop height, and HR is the maximum rebound height). Substituting these equations into Eq. (7) yields an expression for the COR in terms of the initial drop height and maximum rebound height of the granule (given as Eq. (9)). pffiffiffiffiffiffiffiffiffiffi vImp = − 2gHI vReb =

pffiffiffiffiffiffiffiffiffiffiffi 2gHR

sffiffiffiffiffiffi HR : COR = HI

ð8aÞ ð8bÞ ð9Þ

An image processing technique was developed, using high-speed photography and a Mathematica code. This technique was used to determine the COR between the granule and plate, by first obtaining the maximum rebound height of the granule during each trial. Since the initial drop height HI is known, the COR was calculated using

Eq. (9). The impact velocity, vImp, was calculated from Eq. (8a). From Eq. (7), it is clear that the COR could be calculated by directly obtaining impact and rebound velocities from the high speed image data. However, sensitivity analyses showed that being even half a pixel off when manually obtaining the location of the granule, can cause differences in velocities that lead to fluctuations in the calculated COR values of up to 20%. These same analyses showed the height based approach to be much more robust, as fluctuations in the COR values were b1% when obtaining the rebound height from the high speed images. This analysis obviates the use of Eqs. (8a) and (9) in obtaining the impact velocity and COR. 4. Results and discussion In Fig. 4a, the local flow properties versus normalized bin radius are plotted for an example case of S2 tool steel wheel surface material. As mentioned, the data points represent the average of five trials. The horizontal error bars display one standard deviation in either direction. These horizontal error bars are representative of the error for all of the experimental (GSC) results shown in this paper. In all the figures, (rb-rw)/(ro-rw) = 0 corresponds to the surface of the inner moving wheel, while (rb-rw)/(ro-rw) = 1 corresponds to the outer stationary rim. Fig. 4 shows a maximum tangential velocity near the driving wheel, which decreases while moving away from the wheel toward the outer (stationary) rim. This is intuitive since the driving wheel is where energy is imparted to the granules through momentum exchange during wheel-granule collisions. Fig. 4a also displays a distinct transition from a kinetic region (0 ≤ (rb-rw)/(ro-rw) ≤ 0.6) to a contact region (0.6 ≤ (rb-rw)/(ro-rw) ≤ 1). These regions are depicted

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in the image shown in Fig. 5, where Wc and Wk represent the width of the contact and kinetic regions respectively. In the kinetic region, the flow is sparsely populated, and characterized by high speed collisions with short contact times. In these types of collisions, extended sliding between granules, in which friction and COF become more significant, is unlikely. As a result, the coefficient of friction (COF) between colliding particles tends to have a much smaller influence on the flow characteristics than does the coefficient of restitution (COR). In contrast, the contact region is densely populated and depicted by extended frictional (sliding) contacts. This leads to the COF, as opposed to the COR, being more influential on the characteristics and properties of the contact region. This specific point has been shown in the work of Anand et al. [16] for a predominately contact/frictional flow in a hopper. This work [16] showed that because the flow existed as a contact flow, the system parameters were far more sensitive to changes in particle–particle COF than particle–particle COR. The solid fraction is a minimum near the wheel (Fig. 4b), since the particles are driven away by their collisions with the rough inner wheel. While it is expected that the solid fraction values would show a monotonic increase from the inner wheel to the outer wall, the solid fraction plot (Fig. 4b) actually shows a maximum (a spike in the solid fraction) at a normalized radius of 0.6. This maximum solid fraction actually exceeds the theoretical maximum packing fraction for these studies, which is 0.79. This value is equivalent to that of square packing (νmax = 0.79), as opposed to hexagonal packing (νmax = 0.91), due to the manner in which the DPTV procedure obtains solid fraction (a particle's entire area is counted within the bin where the centroid of its white spot is determined to reside). Ultimately, this maximum value at (rb-rw)/(ro-rw) = 0.6 is witnessed as a result of the DPTV analysis being done in 2D, while the GSC design leaves a small amount of space in the third dimension. This gives the potential for granules to overlap into the third dimension, resulting in increases in the 2D solid fractions being calculated. Overlapping granules tend to be most prevalent where the shift from the kinetic to the contact region occurs, which is near (rb-rw)/(ro-rw) = 0.6, explaining the spike in solid fraction at this location. Fig. 4c shows that the granular temperature decreases while moving away from the driving wheel, where it is a maximum. The low solid fraction and high velocities shown near the wheel promote a large number of high-speed particle collisions. This leads to larger velocity fluctuations, which in turn yields a high granule temperature near the wheel. Similar results for all the local flow properties have been observed in prior experimental [43–46,54] and computer modeling [25] studies. 4.1. Coefficient of restitution results Coefficient of restitution (COR) is a collision parameter, which is defined as the ratio of post-collision to pre-collision relative velocity between the two colliding materials. As such, COR is related to kinetic energy, where COR = 1 represents a perfectly elastic collision, COR b 1 represents an inelastic collision, and COR = 0 represents a perfectly inelastic collision. Fig. 6 shows the COR vs. impact velocity for the test materials, those used as granular particles on the wheel surface, as they collide with an S7 tool steel plate, meant to represent the S2 tool steel granules filling the GSC's annular gap. As mentioned previously, the plate material is not an exact match to the material of the granules in the GSC, since S2 tool steel was unavailable in plate form. Thus, the plate and granules filling the GSC were S7 and S2 tool steel, respectively. The elastic modulus and Poisson's ratio for the two steels match closely. However, their standard (static) hardness values differ. S2 tool steel has Rockwell C55–C58 hardness while S7 tool steel is softer, with Rockwell C22–C23 hardness. The difference in hardness can be interpreted through the work of K.L. Johnson [65] who

Fig. 4. Local flow properties for S2 tool steel wheel material: (a) Normalized tangential velocity vs. normalized radial position, the dotted line depicts the transition from the kinetic regime to the contact regime. (b) Local solid fraction vs. normalized radial position. (c) Normalized granular temperature vs. normalized radial position.

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Fig. 5. GSC regime snapshot. During a given trial, the granular flow through the annular gap of the GSC exists in two regimes; the kinetic regime (sparsely populated, short highspeed collisions, and COR dominant) exists in the region near the rotating wheel, while the contact regime (dense particle packing, enduring contacts, and COF dominant) presides in the outer region near the stationary outer wall.

provides a theoretical formulation for COR In this formulation COR is proportional to the dynamic hardness or pressure (pd) raised to the 5/8 power (COR ∝ p5/8 d ). The work of Tabor [66] has shown that this dynamic hardness can be related to the standard/static hardness (ps). This is done through the multiplication of ps by some constant scale factor (C) which can vary depending on material and impact velocity, making it difficult to ascertain. Ultimately, this means that the COR is proportional to this scale factor multiplied by the standard/static hardness raised to the 5/8 power (COR ∝ Cp5/8 s ). Based on this fact, if an S2, as opposed to an S7, tool steel plate were used, the difference in hardness between the two plates would provide for a shift in the steel-steel, chrome steel–steel, and tungsten carbide–steel curves in Fig. 6. This shift is difficult to quantify due to the difficulty in obtaining dynamic hardness values; however, the shift would be of the same factor for each curve, thus maintaining the same trend in COR across the various granule materials studied. The range of impact velocities for this experiment was limited based on the possible heights from which granules could be dropped. The drop heights were chosen such that the range of impact velocities for the COR measurements, would span the approximate range of impact velocities for collisions between the inner rotating wheel and granules in the GSC. For collisions between all three metallic test materials (steel, chrome steel, and tungsten carbide) and the steel plate, Fig. 6 shows an impact velocity dependence, where there is a general and slight decrease in COR with increased impact velocity. It should be noted that due to fluctuations in the average values and the size of the error bars for chrome steel and tungsten carbide, their trend of decreasing COR with increased impact velocity is less certain than that of steel. Nonetheless, this general trend was observed because as the drop height and impact velocity increases, there is a larger loss of kinetic energy due to deformation, and hence a decrease in rebound velocity. This same trend for metal spheres was shown in the theoretical formulations of Johnson [65] and Tabor [66]. The experimental work of Goldsmith [67], Okubo [68] as presented in Goldsmith [69], and Mangwandi [70] also predicted the trend of decreased COR with increased impact velocity. A closer examination also displays that the COR decreases as the composite elastic modulus of the plate and granules increase. This inversely proportional trend is also predicted in the COR theory of Johnson [65].

Fig. 7a shows that the tangential velocity values in the first bin (nearest to the inner rotating wheel) follow the same order as the COR values in Fig. 6, where the various wheel materials are all collided with steel. The wheel-granule collisions with higher COR values demonstrate higher tangential velocities in the GSC within the first bin nearest to the inner rotating wheel (0 ≤ (rb-rw)/(ro-rw) ≤ 0.1667). This relation can be explained by the fact that higher wheel-granule COR values lead to less post-collision kinetic energy loss and higher post-collision speeds for particle-wheel collisions, which occur within the innermost bin. Consequently, Fig. 7a shows that polybutadiene, which has the highest COR value (Fig. 6), displays the largest tangential velocity near the wheel; while tungsten carbide which has the lowest COR values, displays the smallest tangential velocity value in the region nearest the wheel. The steel and chrome steel curves fall on top of one another and in between the polybutadiene and tungsten carbide curves within the first bin. This observation is reasonable since their (steel and chrome steel) COR curves are nearly coincident and fall between the polybutadiene and tungsten carbide COR curves, as shown in Fig. 6. The tangential velocity plot (Fig. 7a) also shows a crossover between the steel/chrome steel curves and the tungsten carbide curve between bin 1 and bin 2. Tungsten carbide shows a more modest decrease in velocity between the first two bins than steel or chrome steel. This can be explained by the fact that in the second bin (0.1667 ≤ (rb-rw)/(ro-rw) ≤ 0.333), the flow is comprised of only

4.2. Local flow properties as a function of surface material In Fig. 7, the local flow properties versus normalized bin radius are plotted for the various wheel materials. The materials – steel, chrome steel, tungsten carbide, and polybutadiene – were chosen because they had widely varying mechanical properties and/or were tribologically relevant bearing materials. Recall, bin segmentation can be seen in Fig. 3c.

Fig. 6. Coefficient of restitution vs. impact velocity. COR data for collisions between the GSC's granule material (steel) and the various wheel surface materials, including steel, chromium (chrome) steel, tungsten carbide, and polybutadiene.

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particle–particle collisions which are steel–steel collisions for all of the varying wheel materials, whereas particle–wheel collisions occur in the first bin, which differ for varying wheel materials. The tungsten carbide wheel trials will have a higher COR for particle–particle (steel–steel) collisions than for wheel–particle (tungsten carbide– steel) collisions (as shown in Fig. 6). As such, as the granules move away from the innermost region near the tungsten carbide wheel, a larger portion of velocity is maintained post-collision since the collisions are all particle–particle as opposed to wheel–particle collisions. This increase in COR (from wheel–particle to particle– particle collisions) can explain the more modest decrease in velocity between the first two bins, and in turn the crossover between the tungsten carbide and steel curves. The inset of Fig. 7a displays the slip velocity between the granular flow and the wheel for the various wheel materials being studied. It is shown that for material combinations that result in higher COR, the slip velocity decreases. This is the case because of the increase in the velocity in the region nearest the wheel (0 ≤ (rb-rw)/(ro-rw) ≤ 0.1667) for wheel materials with higher COR values (shown in Fig. 7a). While this study is experimental, this same trend of decreasing slip velocity for higher COR values was previously demonstrated by both computational [71] and theoretical models [72]. Higher COR values allow for more velocity and kinetic energy to be conserved during granular collisions with the wheel, and in turn, larger post-collision velocities. These larger velocities result in better velocity accommodation between the wheel and granular terrain and hence a reduction in slip. Fig. 7b plots the local solid fraction against normalized bin radius. The low near-wheel tangential velocity for the tungsten carbide (Fig. 7a) leads to more particles remaining closer to the wheel, since the particles are being driven away at a slower speed. This results in the tungsten carbide wheel surface material demonstrating the highest solid fraction near the wheel, as shown in Fig. 7b. The plot of solid fraction shows little difference between the polybutadiene, steel, and chrome steel near the wheel, despite the higher COR and the higher tangential velocity for polybutadiene (shown in Fig. 7a). This similar solid fraction trend can be explained by mass conservation and system constraints. Since the GSC has fixed dimensions and a fixed global solid fraction, there is a limit on the maximum local solid fraction that can be reached in the contact region (Fig. 4a), leaving a certain number of granules present in the kinetic region near the wheel. Hence, although the polybutadiene displays a higher COR (Fig. 6) and higher velocity (Fig. 7a) than steel–steel and chrome steel–steel, their local solid fraction curves look almost identical in the kinetic region, as the regions away from the wheel become fully packed. Granular temperature (Fig. 7c) follows the same trend as velocity. Higher wheel–granule COR material combinations lead to higher granular temperatures in the innermost region (0 ≤ (rb-rw)/ (ro-rw) ≤ 0.1667). The material combinations with higher COR values exhibit greater velocities, which lead to faster moving particles, and more collisions. The fact that more collisions occur gives rise to larger fluctuations in speed and higher granular temperatures. Fig. 7c mimics what is seen in the velocity curves (Fig. 7a) by displaying the same crossover between the tungsten carbide and the steel materials between the first two bins. The larger velocity displayed by the tungsten carbide wheel, in the second bin away from the wheel, results in more high speed collisions, and larger velocity fluctuations. This leads to the granular temperature for the tungsten carbide wheel being higher than that of steel and chrome steel in the second bin (Fig. 7c). 4.3. Local flow properties as a function of global solid fraction In Fig. 8, the local flow properties versus normalized bin radius are plotted for global solid fraction (GSF) values (νg) which varied

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Fig. 7. Local flow parameters for varying wheel surface materials. (a) Normalized tangential velocity vs. normalized radial position as a function of wheel surface material with inset: Normalized slip velocity vs. wheel surface material. (b) Local solid fraction vs. normalized radial position as a function of wheel surface material. (c) Normalized granular temperature vs. normalized radial position as a function of wheel surface material.

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between 0.4 and 0.7 (922 to 1613 granules, respectively). The tangential velocity curves in Fig. 8a display a region with little to no flow (velocities near zero) between 0.6 ≤ (rb-rw)/(ro-rw) ≤ 1. Additionally, tangential velocities in the two regions (bins) nearest the wheel increase, as the global solid fraction increases. As the global solid fraction increases, more granules are present in the flow and hence more granules collide with the wheel. This results in a larger momentum transfer and thus higher velocities in the granular flow. Fig. 8b confirms the expectation that local solid fractions across the annular gap increase as global solid fraction increases. An interesting observation regarding the local solid fraction results is that there is a gradual change from a largely kinetic region across the annular gap, when νg = 0.4 and 0.45, to the presence of a very strong and distinct contact region (0.6 ≤ (rb-rw)/(ro-rw) ≤ 1) when νg = 0.65 and 0.7. For these curves (νg = 0.65 and 0.7), the local solid fraction reaches the theoretical maximum packing fraction for these studies (νmax = 0.79), while displaying low velocity (shown in Fig. 8a) in the region from 0.6 ≤ (rb-rw)/(ro-rw) ≤ 1. The combination of these characteristics depicts the highly dense and stagnant contact region (0.6 ≤ (rb-rw)/ (ro-rw) ≤ 1) present when νg = 0.65 and 0.7. For these two global solid fraction values (0.65 and 0.7), Fig. 8b actually shows a maximum solid fraction at (rb-rw)/(ro-rw) = 0.6., as opposed to the bin nearest the outer wall. This is the same trend seen and discussed for Fig. 4b

(νg = 0.71), and occurs due to the fact that the formation of a contact region is most distinct at these high global solid fraction values. As the global solid fraction increases, granular temperature (Fig. 8c) increases in the near-wheel regions, specifically the first two bins. Fig. 8a and b showed increased tangential velocity and local solid fraction for increasing νg, in this region. This implies that this region, which remains kinetic across all νg values in this experiment, is being filled with more and more granules with higher velocities, which results in a greater amount of high-speed collisions. The increase in high-speed collisions leads to larger velocity fluctuations, and in turn, increased granular temperatures in this near wheel region (0 ≤ (rb-rw)/(ro-rw) ≤ 0.333). Slip is defined as the difference between the linear velocity at the wheel and the average tangential velocity of the inner most bin (Eq. (4)). Fig. 8d shows that the slip decreases as global solid fraction increases, which is expected since the tangential velocity (Fig. 8a) shows an increase with an increase in global solid fraction. This also matches with the experimental results of Elliott et al. [53], which showed a decrease in slip for increased global solid fraction. Fig. 9 displays randomly chosen high speed images of the GSC during trial runs at global solid fraction values of 0.4, 0.45, 0.55, 0.65, and 0.7. These images display the packing behavior of the granules. As Fig. 9a shows, the annular gap does not show any distinct contact

Fig. 8. Local flow parameters at varying global solid fraction values. (a) Normalized tangential velocity vs. normalized radial position as a function of global solid fraction (b) Local solid fraction vs. normalized radial position as a function of global solid fraction. (c) Normalized granular temperature vs. normalized radial position as a function of global solid fraction. (d) Normalized slip velocity vs. global solid fraction.

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region formation at νg = 0.4. At this solid fraction value, easily discernable gaps and spacing exist throughout the particles filling the annular gap. As the solid fraction is increased to 0.45, a contact region begins to form near the outer rim, as shown in Fig. 9b. Fig. 9b displays two tightly, almost fully packed rows of particles against the outer stationary rim, which begin to comprise the formation of a contact region within the flow. As the global solid fraction then increases from νg = 0.45 (Fig. 9b) to νg = 0.55 (Fig. 9c) and on to νg = 0.65 (Fig. 9d), the images depict an increase in the width (Wc) of the contact region, but also a significant increase in the amount of particles present in the near wheel kinetic region. This results in increased collisions with the wheel and many more particles having momentum and energy imparted to them by the wheel. As the global solid fraction increases from νg = 0.65 (Fig. 9d) to its maximum value in this study of νg = 0.7 (Fig. 9e), the high speed images in Figs. 9d and e show a much more significant increase in the amount of particles present in the contact region than in the kinetic region. Within the kinetic region minimal to no increase in the amount of particles is shown. Fig. 10 plots the total normalized translational kinetic energy of the granular flow across the entire annular gap against the global solid fraction, where the data points are an average of 5 trials and the vertical error bars represent one standard deviation in either direction. The data points for global solid fraction values of 0.5, 0.55, and 0.6 are the average of only 4 trials, since in each case one of the trials presented itself as a clear outlier, not representative of the actual physical trend. Also included in the figure are schematic representations of how the particles tend to aggregate and pack in the annular gap for the varying global solid fraction values. Random inspection of the photographs (see Fig. 9) at the appropriate global solid fractions confirmed that the schematics are plausible and accurate representations. Fig. 10 shows that there is an initial decrease in kinetic energy as the global solid fraction increases from 0.4 to 0.45, which suggests a minimum kinetic energy value at νg = 0.45. This may appear counterintuitive as larger νg values would seem to result in more particles colliding with the rotating wheel, and hence more particles

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obtaining momentum and energy from the wheel. However, as νg is increased from 0.4 to 0.45 the granular flow begins to shift from a purely kinetic flow to a “two-phase” flow comprised of both a contact and kinetic region. This is demonstrated in the schematics in Fig. 10, where νg = 0.4 shows more particles present in the regions away from the inner rotating wheel and near the outer stationary rim, but does not show any sort of contact region formation. The flow for a global solid fraction of 0.4 appears to remain entirely in the kinetic regime. As the global solid fraction is increased to 0.45, a distinct contact region begins to form, which is depicted by the particles above the dashed line in the schematic of Fig. 10. This trend (decreasing kinetic energy from νg = 0.4 to νg = 0.45) can also be explained by the local solid fraction trends (Fig. 8b), which show that the increase in local solid fraction (particles) in the regions away from the wheel and near the outer rim (0.5 ≤ (rb-rw)/(ro-rw) ≤ 1.0) is much more significant than the increase in local solid fraction in the near wheel region (0 ≤ (rb-rw)/(ro-rw) ≤ 0.5). In fact, Fig. 8b shows that there is virtually no change in local solid fraction within the first bin (nearest to the rotating inner wheel), while there is a large increase in the outer two bins (5 and 6). This is indicative of the formation of the contact region as the global solid fraction is increased from 0.4 to 0.45. This contact region which becomes present at νg = 0.45, serves as an energy sink and results in the decrease in kinetic energy between νg = 0.4 and νg = 0.45 witnessed in Fig. 10. As the global solid fraction is increased from 0.45 to 0.65, Fig. 10 displays a steady increase in the kinetic energy of the flow across the annular gap. This can be explained by the fact, that while there is some increase in Wc, the width (size) of the contact region, there is also a significant increase in the number of particles present within the kinetic region near the wheel. This is demonstrated by the schematic representations in Fig. 10 as well as in Fig. 8b, where a considerable increase in local solid fraction (number of particles) is observed in the near-wheel region (0 ≤ νg ≤ 0.5), which includes the kinetic region. The schematic representations in Fig. 10 show that Wk, the width of the kinetic region, decreases as the global solid fraction progresses from 0.45 to 0.65. However, despite this fact, the amount of particles

Fig. 9. High speed video images of the GSC during trial runs. These images were taken at several different global solid fraction values: (a) 0.4 (b) 0.45 (c) 0.55 (d) 0.65 (e) 0.7. The images were randomly selected from trial runs at each of the global solid fraction values listed.

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in the kinetic region for each νg value still increase, as shown in the schematics and in Fig. 8b. This fact tells that there is a larger amount of particles for the rotating wheel to interact with and hence a greater medium for which to impart momentum and energy to the flow. This results in the kinetic energy increasing as the global solid fraction is increased from 0.45 up to 0.65. In increasing νg from 0.65 to its maximum value in this study of νg = 0.7, Fig. 10 shows that the kinetic energy still increases, but not as drastically as when the νg was increased incrementally from 0.45 to 0.65. As the schematic representations and local solid fraction trends show, the number of particles (local solid fraction) in the kinetic region increases minimally, while the increase in the outer regions away from the wheel (~ 0.6 ≤ (rb-rw)/(ro-rw) ≤ 1) is much more drastic as the global solid fraction increased from 0.65 to 0.7. In short, Wc and the amount of particles present in the contact region begins to grow more significantly than the amount of particles present in the kinetic region. And so, the energy sink created by the contact region's growth tempers the increase in kinetic energy seen from the slight increase in the amount of particles present in the kinetic region (those gaining energy from collisions with the wheel.) 5. Conclusion The motivation of this experimental work was to better understand the interaction between rough driving surfaces and granular flows. In this regard, this work studied the impact of the driving surface material and global solid fraction on local granular flow

properties (velocity, solid fraction, granular temperature, and slip). A granular shear cell and digital particle tracking velocimetry (DPTV) methodology were used to obtain the granular data, while a drop test apparatus was developed to measure the coefficient of restitution (COR) between the various driving surface (wheel) materials and steel. Parametric tests with varying materials showed that granules with higher COR displayed higher tangential velocity and granular temperature values in the kinetic region near the wheel, while displaying lower solid fractions in the region nearest to the driving wheel. Granules with higher COR values also showed a reduction in slip, or increase in traction, at the moving wheel. In further parametric tests, increasing the global solid fraction led to an overall increase in the tangential velocity and granular temperature. Increasing global solid fraction resulted in an increase in the local solid fraction everywhere except in the inner most bin where some crossover was present. Increased global solid fraction also caused the slip to decrease at the wheel boundary, which can be interpreted as increased traction performance at wheel-on-granular material interfaces. As the global solid fraction increased there was also an overall increase witnessed in the kinetic energy of the granular flow, based on the increase in local solid fraction and particles present in the kinetic region of the flow. However, a decrease in kinetic energy was witnessed as the global solid fraction increased from 0.4 to 0.45, due to the formation of a distinct contact region within the granular flow. These results can lead to a better understanding of the interaction between the granular medium and the driving surface for granular flows in shear, which is a common situation in rover traction, granular lubrication applications, and many natural and industrial processes.

Fig. 10. Normalized translational kinetic energy vs. global solid fraction. Schematic representations are included to show how the particles pack into the contact and kinetic regions at various global solid fraction values. The packing of the particles into the annular gap initially results in a fully kinetic flow. As global solid fraction increases the flow develops both a contact and kinetic region.

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