Steady particulate flows in a horizontal rotating cylinder

PHYSICS OF FLUIDS

VOLUME 10, NUMBER 6

JUNE 1998

Steady particulate flows in a horizontal rotating cylinder K. Yamane Quality Control Department, Taiho Pharmaceutical Co., Ltd., Tokushima 771-0194, Japan

M. Nakagawaa) The Lovelace Institutes and New Mexico Resonance, Albuquerque, New Mexico 87108

S. A. Altobelli The Lovelace Institutes and New Mexico Resonance, Albuquerque, New Mexico 87108

T. Tanaka and Y. Tsuji Department of Mechanical Engineering, Faculty of Engineering, Osaka University, Suita, Osaka 565-0871, Japan

~Received 25 September 1997; accepted 23 February 1998! Results of discrete element method ~DEM! simulation and magnetic resonance imaging ~MRI! experiments are compared for monodisperse granular materials flowing in a half-filled horizontal rotating cylinder. Because opacity is not a problem for MRI, a long cylinder with an aspect ratio ;7 was used and the flow in a thin transverse slice near the center was studied. The particles were mustard seeds and the ratio of cylinder diameter to particle diameter was approximately 50. The parameters compared were dynamic angle of repose, velocity field in a plane perpendicular to the cylinder axis, and velocity fluctuations at rotation rates up to 30 rpm. The agreement between DEM and MRI was good when the friction coefficient and nonsphericity were adjusted in the simulation for the best fit. © 1998 American Institute of Physics. @S1070-6631~98!01906-0#

Woodle and Munro2 studied the rate of mixing of particles in both slipping and rolling modes7 by visually observing tagged particles in a rotary kiln. They concluded that particles mixed much more effectively when the bed was in rolling mode, that mixing was promoted by increased kiln loading, but was not dependent on rotation speed as long as the bed stayed in the same configuration. Broadbent et al.3 studied the motion of particles in mixtures with two Lodigetype ploughshare blades by positron emission tomography. A radioactive tracer which was 18 times larger than the rest of particles was used to relate their findings to the behavior of a lump of material in the system. The signal from the radioactive tracer particle revealed axial, radial and angular spatial information as well as velocity distributions of particles. Parker et al.4 used the same positron camera with a smaller radioactive tracer ~1.5 mm! which permitted a more detailed analysis of these flows. Metcalfe et al.5 studied mixing of materials in a rotating cylinder by visual experiments and numerical modeling. The effects of the end plates are greatly reduced in the two-dimensional disk flow experiments at the expense of ignoring information about motions of particles in the axial direction.6 Nuclear magnetic resonance ~NMR! imaging ~MRI! is another noninvasive measurement technique and has become accepted as a velocity measuring technique for fluid flows8,9 with special advantages in measuring optically opaque multiphase flows.10,11 No radioactivity is involved and it yields concentration, velocity-vector fields, and diffusion coefficients in two and three dimensions. MRI was first used to measure granular flow parameters in a steady horizontal rotating cylinder flow in 1993.12 Mustard seeds were used and

I. INTRODUCTION

Engineers and physicists interested in complex nonNewtonian behavior of granular flows have developed several continuum theories. Campbell has reviewed the state-ofthe-art on rapid shear flows of particles.1 However, it is difficult to measure bulk properties such as stress, strain, and void fraction which are necessary to develop a continuum theory. Recently, rotating cylinders have captured the imagination of many engineers and physicists and knowledge of the transport of granular materials in a kiln is expected to be utilized for engineering processes of particles such as granulation, mixing, milling, and coating. The knowledge of particle motion in the cylinder is essential in order to optimize designs of such process equipment. Unfortunately, granular flows are opaque, which prevents any optical and visual measurements inside the granular assembly. The available experimental methods have been limited to either using tracer particles or observing only the surface of the flow. For the present case of granular flows in a rotating cylinder, visual tracers that could be seen when on or near the surface2 or radioactive tracer studies in the form of positron emission tomography ~PET! ~Refs. 3, 4! have been used. Flows can also be studied through transparent end plates of short aspect-ratio cylinders.5 A variant of the short aspect-ratio cylinder is to do two-dimensional flow experiments by using thin disks that barely fit between closely spaced end plates.6 a!

Present address: Division of Engineering, Colorado School of Mines, Golden, Colorado 80401-1887.

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the signal from the protons in the mustard oils were imaged. The need to use particles with proton containing liquids arises because of the difficulty in getting NMR signals from solid samples and this restriction makes the MRI method not universally applicable to all granular samples. Along with the efforts in developing noninvasive experimental techniques, numerical simulation has made great progress partially due to increased computer capability. Among them the discrete element method, ~DEM! originally introduced by Alder and Wainwright13 to study molecular dynamics and reintroduced by Cundall and Strack14 to study rock-mass behavior, has established itself as one of the most versatile numerical tools. Walton and Braun15 simulated particulate motions in a rotating cylinder and obtained the variation of dynamic angle of repose with the interparticle friction, particle shape, and rotation speed. Mishra and Rajamani16 applied DEM to simulate particle crushing in a ball mill. Yamane et al.17 simulated the motion of pharmaceutical tablets in a coating drum. DEM a priori assumes a suitable collision operator for each simulation but it is easy to change particle properties such as size, density, and even shape,18,19 which are difficult to incorporate in other nondiscrete models. Furthermore, DEM has been extended to full three-dimensional data for trajectories, concentration, and velocity. Since the dynamics of all individual particles are calculated in DEM, detailed data on flow properties can also be exacted from results. In this paper, we report steady granular flow studies in a rotating cylinder by both DEM simulation and MRI. The connection between flow profiles including the thickness of flowing layer and the dynamic angle of repose and nonsphericity are compared. We find good correspondence between DEM and MRI results. II. DEM SIMULATION

Equations for translational and rotational motion for a single spherical particle, in the absence of interstitial fluids, are r¨5

f 1g m

~1!

FIG. 1. ~a! Model of contact force, showing springs, dashpots, and friction sliders. ~b! Two partially overlapping spheres of diameter d and separation l d to model nonspherical particles. Nonsphericity As is defined as d/(d 1l d ).

The normal component of the contact force fn between two particles is given by Hertzian contact theory as fn 5 ~ 2k n d 1/22 h n Vr •nˆ! nˆ,

where k n is the spring constant, d is the normal deformation defined as the overlap distance between particle centers, h n is the normal damping coefficient, Vr is the relative velocity of two particles, and nˆ is the unit vector directed from the center of one particle to the other. The tangential component of the contact force ft is given by ft 52k t˜d2 h t Vs ,

and

v ˙5

T , I

~2!

where r is the position vector of the particle center, m is the particle mass, f is the sum of all contact forces, g is the gravitational acceleration, v is the angular velocity, T is the net torque due to contact forces, I is the moment of inertia, and ~•! indicates the time derivative. Contact forces between spherical particles are modeled by springs, dash-pots, and a friction slider, as originally considered by Cundall and Strack14 @see Fig. 1~a!#. The springs account for elastic repulsion, dash-pots express the damping effect, and friction sliders express the tangential friction force in the presence of a normal force. The effects of these mechanical elements on particle motion appear through stiffness k, damping coefficient h, and friction coefficient m f .

~3!

~4!

where k t is the tangential spring constant, ˜d is the tangential deformation vector, h, is the tangential damping coefficient, and Vs is the slip velocity of the contact point. When the sliding condition u ft u . m f u fn u Vs

~5!

is satisfied, the Coulomb-type friction law applies and the tangential contact force of Eq. ~4! is replaced by ˆ , ft 52 m f u fn u V s

~6!

ˆ is the unit vector defined by V ˆ 5V / u V u . Detailed where V s s s s discussions of this model is given by Tsuji et al.20 In addition, we model the effect of nonsphericity by partially overlapping two spheres as shown in Fig. 1~b!. Nonsphericity As is defined by

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As5

d , d1l d

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~7!

where d is the particle diameter and I d is distance between the two centers. This is one of the simplest ways to include nonsphericity. Some other nonspherical or noncircular models include ellipse models,18 super quadric equations,19 and cubic particles21 but they are difficult to use as a test of particle–particle contacts and particle–boundaries contacts. The inclusion of nonsphericity as a variable in the model modifies the original rotation equation of motion, Eq. ~2!, to I X v˙ X 2 v Y v Z ~ I Y 2I Z ! 5T X , I Y v˙ Y 2 v Z v X ~ I Z 2I X ! 5T Y , ~8!

and I Z v˙ Z 2 v X v Y ~ I X 2I Y ! 5T Z ,

where (X,Y ,Z) defines the inertial system. In our computations, the particle and cylinder specifications generally match those used in MRI experiments. The parameters special to DEM are as follows. Each time step was 1024 s while Young’s modulus and the Poisson ratio were 1.03105 N/m2 and 0.3, respectively. We used periodic boundary conditions in the longitudinal direction to alleviate the severe computing demands of 3D simulations. Therefore, the simulation takes place in a 15 mm long, half-filled cylinder with two periodic end boundaries. About 12 000 particles were used for the calculation. III. MRI EXPERIMENT

Protons, nuclei of hydrogen atoms, are best suited for MRI because of their high sensitivity and abundance in nature. In the first MRI studies of granular flows, Nakagawa et al.12 used mustard seeds which yield excellent proton NMR signal from their oil. The average diameter of the seeds was 1.5 mm with a size range of 60.1 mm, the density 1.3 g/cm3, and the coefficient of restitution 0.75. The NMR image/spectrometer works with a 1.9 T superconducting magnet having a room-temperature bore diameter of 31 cm but the field of view is approximately a sphere of diameter 8 cm which is determined by the homogeneity of the magnetic field. A horizontal, acrylic cylinder of length 49 cm with an inner diameter of 6.9 cm was half-filled with seeds and rotated in the magnet. Concentration and velocity distributions from an 8 mm thick slice near the center of the cylinder, far from the ends, were obtained. No radial segregation is observed for this nearly monodisperse sample, as expected. The range of rotation rates studied was 4–31 rpm. Below this range, the flow is intermittent7 while the departure from a flat free-surface becomes significant at the upper end of the range. The main velocity measurement was made by the phase method8,9 but tagging22 also worked well. The former yields a complete velocity field that is limited only by the spatial resolution of the image whereas the latter, though much more robust and rapid, yields displacements on a coarse scale. Tagging is ideally suited for flow visualization while

FIG. 2. ~Top! A photograph of 5.95 mm diam particles in an 80 mm diam drum rotating at V550 rpm. Some of the particle can be seen through an opening in the conical end plate. ~Bottom! A DEM simulation result showing the same behavior.

the phase method yields quantitative velocity fields. A detailed discussion of these granular flow MRI measurements can be found elsewhere.12 IV. RESULTS AND DISCUSSION A. Dynamic angle of repose

In order to check the DEM code, we simulated flows of spherical particles with the coefficient of friction of 0.4 in a typical coating drum with tapered ends.17 In this calculation, there were 1000 particles with diameter 5.95 mm and density 2470 kg/m3 which matched the experiment. Figure 2 are snapshots of the experiment and simulation at an angular velocity of 50 rpm, showing that they agree, at least qualitatively. Figure 3 shows that the dynamic angle of repose varies almost linearly with the rotation speed in both simulation

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FIG. 3. The dynamic angle of repose vs rotation speed for the experiment and DEM shown in Fig. 2. The coefficient of friction m f was chosen to be 0.4.

and experiment. The dynamic angle of repose was measured though the open section of the end wall and this limited the accuracy of measurement because this opening was only about seven particle diameters. Nevertheless, this observation confirms that the DEM code can simulate the dynamic angle of repose for these spherical particles. Next, a straight cylinder is used in the simulation to investigate the effect of friction. Figure 4 shows the effect of friction coefficient on the dynamic angle of repose for spherical particles flowing in a cylinder rotating at 4.8 rpm. The dynamic angle of repose increases rapidly for very small coefficient of friction but begins to saturate even at a friction coefficient of 0.1 and reaches full saturation for friction coefficient less than 0.5. This tendency has also been observed at other rotation speeds. In this work, friction coefficient is taken to be 0.4, a realistic value; the simulation shows that the angle of repose and the velocity field are nearly insensitive to the friction coefficient at this value. Figure 5 shows the measured and simulated dynamic angle of repose as a function of the rotation rate for varying sphericity. The angle of repose increases linearly with rotation speed in all cases and increasing the nonsphericity makes the dynamic angle of repose steeper for a given rotation speed. The simulation agrees best with MRI for some value of nonsphericity between 0.75 and 0.79. For the faster rotation speeds, the free surface of particle bed becomes S-

FIG. 4. The effects of coefficient of friction m f on dynamic angle of repose at V54.8 rpm by DEM simulation, showing how the angle of repose saturates with increasing m f .

FIG. 5. The effects of sphericity on dynamic angle of repose as a function of rotation speed V with m f 50.4 for mustard seeds and the corresponding DEM simulation.

shape and we determined the angle of repose as that at the center of the free surface. Figures 4 and 5 indicate that the dynamic angle of repose is more strongly influenced by particle shape than friction. The dynamic angle of repose for mustard seeds near the end wall is steeper than that measured in the middle of the cylinder where there are no wall effects. We have compared the dynamic angle of repose near and far away from the end wall by experiment and simulation with As50.79 in Fig. 6. Both experiment and simulation show that the additional interaction between the particles and the end wall increases the dynamic angle of repose near the wall by about 5° for any rotation speed. This adds another validation that our DEM simulation can model the particle wall interaction and also demonstrates the unique ability of MRI to measure the angle of repose near the center of a long cylinder in order to avoid end-wall effects.

FIG. 6. Effects of end walls on dynamic angle of repose for both simulation (As50.79) and MRI, showing the enhancement of the angle by 5° at the end walls.

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FIG. 7. Evolution of grid pattern at rotation speed of 18 rpm. The times shown are delays from when the grid tags were applied to when the images were obtained.

B. Flows and rigid body rotation

Placing noninvasive grid tags and tracking their evolution is a robust visualization technique which supplies global information about the shear field. The MRI tagging is accomplished by a special rf pulse sequence22 followed by a delay during which the tags evolve according to the flow pattern of the granular material. The images that are taken after such delays, therefore, show the displacements of the particles during the delay. Figure 7 shows such tagging experiments and simulations at a rotation speed of 18 rpm for mustard seeds and particles with As50.79. A rectilinear grid was noninvasively applied to the sample and 2D images

were obtained after delays of 1, 10, 20, 50, and 100 ms. Both MRI and DEM clearly show the region of rigid body rotation near the cylinder wall and the region of high shear near the free surface. Figures 8~a!–8~c! are velocity vector plots from simulations with As51.0, As50.83, and MRI of mustard seeds, respectively, at a rotation rate of 30 rpm. Both simulations show reasonable agreement with MRI in estimating the depth of flowing layer, the region undergoing rigid body rotation, and the ratio of the maximum speed of the flowing layer to that of the rigid body rotation even though the simulation predicts the surface to be flatter than the slight S-shape

FIG. 8. Velocity vector plots from simulation with ~a! As51.0 and ~b! As50.83 at a rotation speed of 30 rpm. ~c! Velocity vectors from MRI of mustard seeds also at 30 rpm.

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FIG. 11. Normalized thickness of the sliding layer versus rotation speed from simulation with As50.79, 0.83, 1.0, and MRI of mustard seeds.

FIG. 9. Velocity vectors along the perpendicular bisector of the free surface from simulation with As50.83 at a rotation speed of 30 rpm. The outline of the cylinder and the idealized free surface are also sketched for orientation.

of the experiment. The As51.0 simulation underestimates the free surface angle while the simulation with As50.83 overestimates the measured slope slightly. Next, using the velocity plots in Fig. 8 we analyze the velocity profile along a line through the center of the cylinder and perpendicular to the free surface. Figure 9 shows the velocity profile along such a line AB from the simulation with As50.83. Near the cylinder wall, the velocity varies linearly with distance from the center of cylinder for rigid rotation without flow. The velocity deviates from linearity where the particles start to slip; this position defines the lower boundary of the sliding layer. As we move up within the sliding layer towards the free surface, the velocity crosses zero, where the sliding velocity is balanced by the cylinder rotation, then it increases rapidly until close to the free sur-

face. In this example, the velocity near the top of the flowing layer is almost constant, i.e., shear is almost zero. Figures 10~a!–10~c! show the x and y components of these velocities from simulations with As51.0, As50.83, as well as from MRI, respectively, for a rotation speed of 30 rpm. Every figure in Fig. 10 shows the same transition patterns. The flowing layer is thinner for non-spherical particles than for spherical particles at the same rotation speed. Due to flux conservation, this gives rise to a larger peak velocity at the free surface for the nonspherical particles and this, in turn, makes the As50.83 case correspond more closely with the experiment. The larger disagreement with experiment for y-component with As51.0 arises simply because of the smaller free surface angle when As51.0 which makes V x dominate over V y . The thickness of shear layer r 0 can be computed from Fig. 10. Figure 11 shows the relationship between normalized thickness of shear layer r 0 /R ~R5radius of the cylinder! and the rotation speed, for simulation with As51.0,

FIG. 10. Profile of x and y components along perpendicular bisector of the free surface for simulation with ~a! As51.0, ~b! As50.83 ~b!, and ~c! MRI of mustard seeds at 30 rpm. Figure 9 shows the relation between these velocity components and the flow geometry.

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TABLE I. Flow parameters. Angle of Vx repose ~m/s! V y ~m/s! r 0 ~mm! Simulation with spherical particles (As51.0) Simulation with nonspherical particles (As50.83) MRI measurements Simulation with nonspherical particles (As50.79)

32

23

11

21

35

26

19

18

38 37

27 32

23 30

16 13

As50.83, As50.79, and MRI result for mustard seeds originally shown in Ref. 12. Results of simulation for these values of As show that thickness of shear layer increases monotonically with rotation speed V but at an ever decreasing rate, reflecting the fact that the mass flux density along the free surface increases with the rotation speed. DEM results are similar to those from MRI. Furthermore, the spherical particles flow as a thicker shear layer than the nonspherical particles at every rotation speed. Thus, the thickness of shear layer depends on the shape of particle as did all other flow parameters that we already discussed. Although the agreement with the experiment seems best for simulation with As50.83, we need to remember that the experiment of Ref. 12 was performed with mustard seeds that were different from those used later. The comparison of the flow properties, angle of repose, V x , V y , and r 0 is presented in Table I. The simulation with As50.83 agrees reasonably well with the mustard seed experiments but the best agreement would be between As 50.83 and As50.79. As50.79 was chosen because the overall angle of repose was in good agreement with MRI experiment for mustard seeds at this value of As. The free surface being S-shape for this rotation speed makes it difficult to measure the angle of repose accurately. The maximum speed along the perpendicular bisector of the free surface occurs at the free surface for the data shown in Fig. 10, taken at a rotation rate of 30 rpm. For much faster rotation rates where the inertial effects dominate the surface shape, the maximum velocity will occur below the surface.23 Figure 12 shows the maximum speed V5 AV 2x 1V 2y of the flowing layer as a function of rotation speed up to 30 rpm. For high enough V, V becomes proportional to V. A further check of the simulation is based on the observation23 that the experimental velocity profile V(r) along a perpendicular bisector of the flowing surface is consistent with

S D

V 1 ~ r ! 5V 12

r r0

2

2Vr,

0