Granular flow and pattern formation on a vibratory ... - Semantic Scholar

Granular flow and pattern formation on a vibratory conveyor Christof A. Kruelle1 , Andreas G¨otzendorfer1 , RafaÃl Grochowski2 , Ingo Rehberg1 , Mustapha Rouijaa1 , and Peter Walzel2 1 2

Experimentalphysik V, Universit¨ at Bayreuth, D–95440 Bayreuth, Germany Mechanische Verfahrenstechnik, Universit¨ at Dortmund, D–44227 Dortmund, Germany

Abstract. Vibratory conveyors are well established in routine industrial production for controlled transport of bulk solids. Because of the complicated interactions between the vibrating trough and the particles both glide and throw movements frequently appear within one oscillation cycle. Apart from the amplitude and frequency, the form of the trajectory of the conveyor’s motion also exerts an influence. The goal of our project is a systematic investigation of the dependence of the transport behavior on the three principle oscillation forms: linear, circular and elliptic. For circular oscillations of the shaking trough a non-monotonous dependence of the transport velocity on the normalized acceleration is observed. Two maxima are separated by a regime, where the granular flow is much slower and, in a certain driving range, even reverses its direction. In addition, standing waves oscillating at half the forcing frequency are observed within a certain range of the driving acceleration. The dominant wavelength of the pattern is measured for various forcing frequencies at constant amplitude. These waves are not stationary, but drift with a velocity equal to the transport velocity of the granular material, determined by means of a tracer particle. Finally, the fluidization of a monolayer of glass beads is studied. At peak forcing accelerations between 1.1 g and 1.5 g a solid-like and a gas-like domain coexist. It is found that the number density in the solid phase is several times that in the gas, while its granular temperature is orders of magnitude lower.

1

Introduction

Vibratory conveyors are highly used for discharging, conveying, feeding, dosing and distributing bulk materials in many branches of industry, for example in the chemical and synthetic materials industries, food processing (Fig. 1(a)), sand, gravel, and stone quarries, for small-parts assembly mechanics (Fig. 1(b)), the paper-making industry, sugar or oil refineries, and foundries [1–3]. In addition to transport, vibration can be utilized to screen, separate, compact or loosen product. Open troughs are used for conveying bulk materials, closed tubes for dust-sealed goods, and work piece-specific rails for conveying oriented parts. Some of the main advantages of vibratory conveyors are their simple construction and their suitability to handle hot and abrasive materials. In addition, they are readily used in the food industry, since they can easily be kept complying to hygienic standards by using stainless steel troughs. Some disadvantages of vibratory conveyors are their noisy operation, the induced vibrations on their

2

Christof A. Kruelle et al.

(a)

(b) Fig. 1. (a) Linear vibratory conveyors in the food industry, (b) Vibratory bowl feeder in an automated assembly chain at SUSPA company, Sulzbach-Rosenberg

surroundings and their limited transport distance. Furthermore, the granular material may be damaged when it is subjected to extreme accelerations normal to the trough. Important properties to be considered regarding the granular material (or bulk solid for engineers) are: particle size distribution and shape, friction between particles and trough and inter-particle friction, modulus of elasticity of the particles and/or the bulk, cohesion, layer thickness, and the permeability of air. Considering the many parameters involved, the performance of a vibratory conveyor is difficult to predict theoretically. Obvious design parameters are: vibration mode (linear with or without vibration angle α, circular, elliptic), amplitude A and frequency f of the oscillations, combined as dimensionless throw number Γ = A sin(α)(2πf )2 /g, inclination or declination of the conveyor, smoothness of the trough surface, modulus of elasticity of the trough’s inner surface, which can be coated with rubber, plastic, etc., and possible electrostatic charges.

2

Conveying principles

Three different principles of conveying have to be distinguished [3,4,7] (see Fig. 2): • Sliding: Here the deck is moved by a crankshaft mechanism only horizontally with asymmetric forward and backward motions. The material remains always in contact with the trough surface and is transported forward relative to the deck by a stick-slip drag. • Throwing: If the vertical component of the acceleration exceeds gravity, the material loses contact during part of the conveying cycle and is repeatedly forced to perform ballistic flights. Complicated sequences of a rest phase, a positive (or negative) sliding phase, and a flight phase have to be considered. The net transport in the forward (or even backward) direction depends sensitively on the coefficient of friction between the particles and the trough and on the coefficient of restitution for the collision with the deck. • Ratcheting: Motivated by advances in the investigations of fluctuation-driven ratchets a new transport mechanism has been proposed recently [5,6]: a horizontal transport of granular particles can be achieved in a purely vertically

Granular flow and pattern formation on a vibratory conveyor (a)

3

(c)

(b)

Fig. 2. Three different conveying principles: (a) ‘Sliding’ by asymmetric horizontal back-and-forth movements, (b) ‘Throwing’ by linear vibration with throw number Γ > 1, both images taken from [4]. (c) ‘Ratcheting’ by vertical vibration on a sawtoothshaped profile of the base. From [5,6]

vibrated system, if the symmetry is broken, instead of the direction of the vibration mode, by an asymmetric periodic sawtooth-shaped profile of the base.

3

Granular transport

Since the transport phenomena on vibratory conveyors involve the nonlinear interaction of many-particle systems with complex behavior, the investigation of their dynamical properties has become a challenging subject to physicists, too. In the past, most studies dealing with vibrated granular media were based on purely vertical or purely horizontal vibration. Only recently a few experimental explorations of the dynamics of granular beds subject to simultaneous horizontal and vertical vibration have been reported [3,8–14]. The observed phenomena include the spontaneous formation of a static heap, convective flow, reversal of transport, and self-organized spatiotemporal patterns like granular surface waves. The most important questions currently under investigation are: • How does the granular transport velocity depend on (i) external parameters of the drive like amplitude and frequency of the oscillation, the vibration mode, or the inclination of the trough, and (ii) internal bulk parameters like coefficient of restitution, friction coefficients, and the filling height? • Is it possible to optimize the transport effectivity by suitable modifications of the surface of the trough implying ratchet like profiles? • What kind of self-organized structures can be expected? Are there clearly characterized instabilities? Which physical mechanism underlie these structures? How is the granular transport effected? • Are there segregation effects in bidisperse or polydisperse systems? What are the analogies to vertical vibration? • Can these results eventually lead to optimized industrial devices like conveyor systems, metering devices, sieves, mixers, dryers, or coolers?

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Fig. 3. Annular vibratory conveyor with a toroidal trough of radius R = 22.5 cm and width w = 5 cm, suspended on adjustable columns via elastic bands: (1) Torus-shaped vibration channel, (2) Adjustable support, (3) Elastic band, (4) Vibration module with unbalanced masses, (5) Electric motor with integrated frequency inverter

4

Experimental setup

For this purpose, a prototype annular conveyor system has been constructed (Fig. 3) for systematic studies of the transport properties for different oscillation modes, i.e. linear, elliptical, and circular (see Fig. 4) for a long running time, without disturbing boundary conditions [10–12,14]. In principle, this conveyor can be excited in all six degrees of freedom individually, or by a combination of two of them. For the first experiments, a vibration mode has been chosen consisting of a torsional vibration φ(t) = A/R cos(2πf t) around the symmetry axis of the apparatus, superposed with a vertical oscillation y(t) = A cos(2πf t + ϕ) where ϕ is the fixed phase shift between the two oscillations. If, for example, this phase shift ϕ is chosen to be π/2, then each point on the trough traces a circular path in a vertical plane tangent to the trough at that point. In short, the support agitates the granules via a vertical circular vibration. For being able to adjust different vibration modes special driving units have been developed, equipped with four rotating unbalanced masses, as combinations of two unbalanced-mass linear vibrators oriented perpendicularly to each other [10–12]. Characteristic of the unbalanced-mass agitated system is the frequency dependence of the vibration amplitude: f2

A(f ) = At p

(f02 − f 2 )2 + (2ζf0 f )2

.

(1)

Granular flow and pattern formation on a vibratory conveyor

5

(a)

5 cm

(b)

Fig. 4. (a) Driving module with four unbalanced masses, (b) Side views of a driving unit with four rotating unbalanced masses, for three principal modes of oscillation: linear (ϕ = 0), elliptical (ϕ = π4 ), and circular (ϕ = π2 )

q keff 1 The resonance frequency f0 = 2π Mtot can be limited to a small value (through appropriate choice of spring constant keff ), so that in the over critical range, m f > 3f0 , an almost constant terminal amplitude At = ru Mu,tot arises which can tot be adjusted for fixed eccentricity ru by the out of balance mass mu . The frequency response (see Fig. 5) was measured experimentally before every measurement by determination of the position of an affixed LED on the vibrating device recorded with a CCD camera. The vibration amplitude is found by aligning a circular path with radius A(f ) to the image data. For excitation frequency f > 15 Hz, the vibration amplitude is nearly constant and therefore the dimensionless acceleration, i.e. the machine number K = A(f ) · 4π 2 f 2 /g is approximately proportional to f 2 . From the measured data, the damping constant ζ can be determined to be 0.08 ± 0.01 An important parameter of the bulk solid that should be chosen carefully for obtaining reproducible results is the layer height. For too thin layers the particles will dilate mutually and start to perform irregular motions across the trough bottom. In practice, as a rule of thumb, the layer of material should at least be 10 particles high. Under these conditions, the bulk solid moves more or less like a solid block, and the transport velocity is not very sensitive to the exact height. During the transport experiments, the average flow velocity is determined by tracking a colored tracer particle that is carried along with the bulk. This is done automatically with a PC based image processing system, which detects the passage of the tracer through a line perpendicular to the trough and stores each passage time on hard disk.

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Fig. 5. Amplitude vs. frequency A(f ) for different unbalanced mass mu . The inset q shows the dependence of the resonance frequency f0 = u amplitude At = ru MN/2·m on the mass mu 0 +N ·mu mass of the device M0 = 24.3 kg, eccentricity

1 2π

keff M0 +N ·mu

and the terminal

of the N = 32 unbalanced masses (net of the unbalanced masses ru = 19 mm,

effective spring constant keff = 177 N/cm

5 5.1

Results Flow reversal

The result for such an experiment is shown in Fig. 6. Below a critical value Γc ≈ 0.45 the grains follow the agitation of the tray without being transported. The onset of particle movements is hindered by frictional forces between grains and the substrate. By increasing the acceleration above this threshold a net granular flow with constant velocity is observed. For circular vibration of the trough, surprisingly, the transport velocity is not a monotonous function of Γ , but has two maxima at Γ = 1.2 and Γ = 4.2. In between, the granular flow is slower and even reverses its direction in the regime 2.6 < Γ < 3.8, whereas an individual glass bead is propagated in the same direction for all accelerations. The critical Γ values, at which the transport behavior changes qualitatively, are independent of the oscillation amplitude. In the frequency-amplitude parameter space (Fig. 7) the threshold values lie on f −2 hyperbola of constant acceleration (‘isoepitachs’). However, for Γ > 4, i.e. beyond the second reversal of flow direction, this scaling behavior is not observed anymore. Depending on the vibration amplitude and the filling height, the third reversal occurs in the range 5 < Γ < 7 [11].


7

Transport

Fig. 6. Normalized transport velocity v ∗ = vtr /Aω of a granular flow (≈ 300 000 glass beads with 1 mm diameter, see inset) on the vibratory conveyor with circular vibration, compared with the mean velocity of one single glass bead

5.2

Linear vibration mode

Complementary studies [9] applying linear vibrations have shown, that the flow reversal is a special property of the circular vibration. For linear vibrations, also a non-monotonous dependence of the transport velocity is seen (Fig. 8), with a dip at Γ ≈ 5, but no reversal of the flow. Note that the maximum transport velocity in this case arrives at about 90% of the oscillation velocity Aω, while in the circular case only about 50% can be attained. 5.3

Sand bag test

Worth mentioning is the so-called ‘sand bag test’ [7] routinely performed by the manufacturers of vibratory conveyors. Since an individual grain, such as a single glass bead, dropped onto the substrate will rebound with a high coefficient of restitution and therefore gives an unrealistic transport characteristic, a better approach for modelling the bulk transport is achieved by use of a small fabric bag, filled with the same kind of beads. The strikingly different collective behavior arises from the large number of rapid inelastic collisions of neighboring grains. However, it takes some experience to find a suitable single object for producing reliable data for comparison with an effective one-particle model.

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Fig. 7. Phase diagram of the transport behavior at clockwise circular forcing. The alternating arrows correspond to the transport direction of the bulk solid

5.4

Theoretical description

Such a theoretical approach is based on the following initial assumptions [3,15,16]: • The granular material behaves like a solid body and can be represented by a point mass. • When the layer of granular material hits the trough after a flight phase a fully non-elastic collision is assumed. • Rotations of the particle and interactions with the side walls of the trough are neglected. • The kinetic and static coefficients of friction are set equal or the distinction between them is neglected. • The air resistance during the flight phase is negligible. According to these assumptions, Sloot and Kruyt [3] obtained fairly good agreement between theory and experiment for slide conveyors but observed large deviations for linear throw conveyors with vibration angle α. Hongler et al. [17] described the dynamics of a vibratory feeder by a set of coupled, nonlinear and strongly dissipative mappings and identified the transport behavior to be determined by periodic and chaotic solutions. First simulations for conveyors with variable vibration mode (linear, circular, and elliptic) by Landwehr, Lange, and Walzel [18,19] applied various, more complex collision models taking also a finite coefficient of restitution into account. More recently, an effective singleparticle model by El hor and Linz [13,15,16] that includes only dynamic friction forces and collisions with complete dissipation of the vertical velocity component in order to understand the theoretical basics of the transport process led to rather good agreement with the experimental results shown in Figs. 6 and 8.


9

a a

61° 46° 31° 21° 11°

Fig. 8. Normalized transport velocity v ∗ = vtr /A cos(α)ω of a granular flow for linear vibration at different vibration angles α

5.5

Onset of particle motion

Despite the complex interactions between the particles and the vibrating trough during the transport process, which up to now can only be handled via numerical simulations, it is interesting that the onset of motion for a single block subject to static friction can be derived analytically. A linear harmonic motion of the conveyor with amplitude A and vibration angle α (see inset of Fig. 9(a)) can be expressed as x(t) = A cos(α) cos(2πf t) for its horizontal and y(t) = A sin(α) cos(2πf t) for the vertical component, respectively. Due to the periodic acceleration a particle with mass m lying on the trough experiences a horizontal force Fh (t) = m¨ x(t) and a modulated effective weight N (t) = m(g + y¨(t)). The mass is hindered from sliding if the resulting frictional force F (t) = µs N (t) is larger than |Fh (t)|, where µs is the static coefficient of friction. At the onset of particle motion both forces are equal, which leads to the balance equation |m¨ x(t)| = µs m(g + y¨(t)) .

(2)

This condition is met first at the highest point of the conveyor’s trajectory where the horizontal acceleration is maximal while the friction force is minimal, yielding

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a

(a)

(b)

Fig. 9. (a) Critical throw number Γonset for the onset of particle motion on a linearly vibrating conveyor at various vibration angles α. (b) Comparison of the critical throw number Γonset for the onset of particle motion for the circular and the (α = 45◦ ) linear vibration modes

a critical throw number Γonset for which the particle starts to move: · ¸−1 µs tan(α) 1 Γonset = = 1+ . 1 + µs tan(α) µs tan(α)

(3)

Figure 9(a) shows the monotonous but nonlinear dependence of Γonset as a function of both the vibration angle α and the static friction coefficient µs . A similar calculation for the circular motion of the conveyor, where both vibration amplitudes are set equal at a fixed phase shift ϕ = π/2, i.e. x(t) = A cos(2πf t) and y(t) = A cos(2πf t + π/2), leads to the expression µs 1 + µ2s

Γonset = p

(4)

for the slipping threshold. A comparison of both vibration modes, linear and circular, is made in Fig. 9(b), which shows that it is easier to overcome static friction by linear vibrations of the trough. The circular mode seems to be ‘softer’, i.e. less effective to set particles in motion. The most general, i.e. elliptic case is characterized by both parameters, inclination angle α and phase shift ϕ, with the corresponding horizontal and vertical orbital components x(t) = A cos(α) cos(2πf t) and y(t) = A sin(α) cos(2πf t + ϕ), respectively, which yields the general solution of this problem as µs tan(α)

Γonset = p

1 + 2µs tan(α) cos(ϕ) + µ2s tan2 (α)

.

(5)

These considerations may be seen as a starting point for the systematic study of the transport behavior for all parameters of a vibratory conveyor.


5 cm

6

11

Fig. 10. Granular surface waves (λ = 3.8 cm) inside a vibratory conveyor with vertical circular motion of the annular trough oscillating with Γ = 3.0 at a frequency f = 22.4 Hz

Surface waves

If Γ exceeds 1, the vertical component of the circular acceleration will cause the grains to detach from the bulk followed by a flight on a ballistic parabola. This results in a much less dense-packed, ‘fluidized’, state with highly mobile √ constituents. In a certain driving range between Γ1 = π 2 + 1 ≈ 3.3 and Γ2 ≈ 3.7 a locking of the time-of-flight between successive bounces and the period of the circular vibration occurs [20]. The initially flat bed becomes destabilized, and undulations of the granular surface occur in the range 2.4 < Γ < 4.5, see Fig. 10 [21,22]. High-speed CCD imaging showed that they oscillate with half the excitation frequency (‘f/2 waves’). In contrast to previous work [23–27] done at purely vertical vibration the present waves are not stationary but are transported

Fig. 11. Space-time diagrams of amplitude h(ϕ, t) of surface waves at maximum wave amplitude every other forcing cycle, for Γ = 2.84 (a) and Γ = 4.22 (b). High amplitude appears bright

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lmin (a)

(b)

Fig. 12. (a) Scaled velocity v ∗ = vtr /Aω of surface waves (•) and granular flow (¦). (b) Circles represent the measured wavelength λ plotted over the normalized acceleration Γ . Throughout the measurement the driving amplitude was kept constant at 1.47 mm. The solid line is the graph of the function λ = 1.3 cm + 20 cm · Γ −1.9 , the best fit to the data

along the annular trough. The drift velocity of the wave pattern can be measured by applying a phase-locked imaging technique with fixed time delay ∆t = 2T . This is done with a camera which is triggered by a pick-up signal taken from the rotating unbalanced masses. From these images space-time diagrams as shown in Fig. 11 can be constructed. The deviation of the striped pattern from the vertical is taken as a measure for the wave speed. A comparison of the wave speed with the bulk velocity of the transported particles shows that both velocities are identical (Fig. 12(a)). In particular, a reversal of the wave velocity is also possible by adjusting the vibration frequency alone. In the range Γ ≈ 4.5, when the granular flow is reversed a second time, surface waves disappear. The granular surface becomes flat but still oscillates at twice the vibration frequency. Above Γ ≈ 5.5 standing waves are observed again but this time repeating their patterns after four shaker periods (‘f/4 waves’). This scenario is reminiscent to the wave patterns found by Bizon et al. [28] for vertical vibration of a laterally extended system. A more detailed analysis of the Γ dependence of the wave length λ shows an algebraic decay as λ(Γ ) = λmin + Λ · Γ α ,

(6)

see Fig. 12(b). For the minimum wavelength λmin we obtain a value of 1.3 ± 0.3 cm, which is approximately the depth of the granular layer. The values for the other parameters are Λ = 20±4 cm and α = −1.9±0.3. This is consistent with the results of Metcalf et al. [26] who examined the dependence of the wavelength on the peak acceleration Γ at constant frequency.


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Fig. 13. Experimental setup with transparent trough and conical mirror placed in the center of the ring. The reflected image of the surface profile is captured with a high-speed CCD camera on top of the mirror

For comparison with theoretical models it is necessary to determine the dynamic surface profile h(ϕ, t) during the transport process with high spatial and temporal resolution. This task has been solved by Pak and Behringer [24] only for a small section of an annular trough. Our container consists of a 2 cm wide annular channel with open top, 7 cm high Plexiglas walls, and a radius of R = 22.5 cm giving a circumference of L0 = 141 cm (see Fig. 13). The granular system is observed from the top via a conical mirror placed in the center of the ring, similar to Ref. [29]. Thus a side view of the whole channel is captured with a single high-speed digital camera (resolution: 1280 × 1024 pixels at rates up to 500 images per second). Figure 14(a) shows an anamorphotic image reflected from the conical mirror. The wavy granular surface is seen as a jagged ring around the tip of the cone. For reconstructing the true shape of the profile h(ϕ, t) digital image processing is performed which delivers 360◦ panoramic side views of the granular profile in the channel as presented in Figs. 14(b) and 15. The spatial resolution is sufficient for detecting single particles of 2 mm size. The channel is lit from outside through diffusive parchment paper wrapped around the outer wall, hence particles appear dark in front of a bright background.

(a)

(b)

Fig. 14. (a) Anamorphotic image reflected from the conical mirror. (b) Section of the reconstructed granular surface

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Christof A. Kruelle et al. Fig. 15. Snapshots through the inner side wall of the channel covering 360◦ taken at y(t) = 0 during the downwards motion of the container. Time increases from top to bottom by 1.72 seconds (20 cycles) between consecutive snapshots. For clarity all images are stretched in the vertical direction by a factor of four (f = 11.6 Hz, Γ = 1.23)

7

Coexistence of condensed and fluidized phases

Finally, a rather surprising pattern has been observed in a single layer of monodisperse beads (see Fig. 15). At peak forcing accelerations between 1.1 g and 1.5 g a solid-like and a gas-like domain coexist. The solid fraction Ls /L0 decreases with increasing acceleration and shows hysteresis (Fig. 16). The sharp boundaries between the two regions travel around the channel faster than the particles are transported. Complementary to our experimental studies a molecular dynamics simulation is used to extract local granular temperature and number density [30]. It is found that the number density in the solid phase is several times that in the gas, while the granular temperature is orders of magnitude lower. This system shows that equipartition of energy can be violated by coexisting gaseous and solid domains, even though particle motion is fully three-dimensional and not restricted by guiding partitions. The rotation of the solid phase in the annular conveyor demonstrates that the coexistence of solid and fluid regions is not caused by small potential inhomogeneities in the forcing, particle container interactions or a tilt of the apparatus.

Fig. 16. Solid fraction Ls /L0 as a function of the peak container acceleration Γ in experiment (filled circles) and simulation (open squares). Arrows indicate how the system evolves in the hysteresis loops. The inset is a space-time diagram of the granular density. Solid regions appear dark


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15

Concluding remarks

The vibratory conveyor system presented here opens up the possibility to investigate the transport properties of granular materials in a systematic way. Our results show that under certain conditions not even the direction of the granular flow can be predicted a priori. The delicate interactions of the particles with the support as well as among themselves have to be taken into account. For industrial applications, the observed reversal effect is relevant as the direction of a granular flow is selected through the frequency of the excitation alone. One can employ such two-way conveyors for example in larger cascading transport systems as control elements to convey the material to different processes as needed. These experiments indicate that the major concepts describing the complex behavior of a vibrated granular system, namely phase transitions, pattern formation, and transport are closely related and yield a rewarding field for future research. Acknowledgements We would like to thank H. El hor, F. Landwehr, S.J. Linz, J. Kreft, T. Schnautz, and S. Strugholtz, for valuable discussions. Support by Deutsche Forschungsgemeinschaft (DFG-Sonderprogramm ‘Verhalten granularer Medien’) is gratefully acknowledged.

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