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Avalanching of granular material in a horizontal slowly rotating cylinder: PEPT studies
Powder Technology 138 (2003) 25 – 30 www.elsevier.com/locate/powtec
Avalanching of granular material in a horizontal slowly rotating cylinder: PEPT studies S.-Y. Lim a, J.F. Davidson a, R.N. Forster b, D.J. Parker c, D.M. Scott a,*, J.P.K. Seville d a
Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA, UK b Cemex, 103 Great Valley Parkway, Malvern, PA 19355, USA c Positron Imaging Centre, University of Birmingham, Birmingham, B15 2TT, UK d Department of Chemical Engineering, University of Birmingham, Birmingham, B15 2TT, UK
Abstract A detailed analysis of the motion af a single particle in a slowly rotating horizontal cylinder in the avalanching mode has been carried out using the positron emission particle – tracking technique to follow the motion of a tagged particle. Cylinders of diameters 240, 390, and 488 mm, and sand of mean diameter 500 Am were used; the tracer was an irradiated sand particle. It was found that a tracer particle rotates with the bed in rigid body motion and then traverses the active surface region in a series of discrete avalanches before being absorbed once again into the rigid body rotation. The mean number of avalanches required to traverse the active region decreased from 3.5 to 4 at low rotational speed to 1 at higher rotational speed at the transition to the rolling mode. D 2003 Elsevier B.V. All rights reserved. Keywords: Rotating cylinders; Avalanching; PEPT studies
1. Introduction Rotating cylinders are commonly used for processing granular materials in the mineral, ceramic, cement, metallurgical, chemical, pharmaceutical, food and waste industries, and are relatively simple in design and operation. They are suited to processes that require high temperatures at near-atmospheric pressure. There are considerable economic incentives to develop a more fundamental understanding of the motion of granular solids in such cylinders. Some rotating cylinders are slightly inclined, with the granular material entering at the upper end and flowing by gravity to the lower end; other cylinders are horizontal. Some cylinders are without internals; others contain lifting flights on the inside surface of the wall. In cylinders without flights, the granular material forms a bed in the lower part of the cylinder. The motion of the particles in the transverse plane (perpendicular to the cylinder axis), as it moves from one end of the cylinder to the other, can be characterised as one of several possible behaviours, depending on the operating conditions (rotational speed and degree of fill), the flow characteristics of the particles and the friction * Corresponding author. E-mail address: [email protected] (D.M. Scott). 0032-5910/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2003.08.038
coefficient with the wall of the cylinder wall [1]. The current research deals with slowly rotating cylinder without flights, for which the important forms of bed behaviour are listed below. Slumping and rolling occur at low cylinder speeds when inertial effects are negligible [1]. Cylinders with rough walls are considered so there is no slipping between the granular bed and the wall. Slumping occurs at very low rotational speeds. The particles that form the bed at the bottom of the cylinder ride up one side of the wall, due to the rotation, forming a bed whose surface is inclined to the horizontal at about the angle of repose. The particles on the sloping surface periodically slump or avalanche from the upper half of the surface to the lower half. Rolling behaviour occurs at higher cylinder speeds and is characterised by an approximately flat upper surface. The angle of the upper surface to the horizontal plane is known as the dynamic angle of repose of the granular material. Slipping behaviour occurs with low particle-to-wall friction: the bed slides up and down the wall of the rotating cylinder, with very little relative motion between the particles. Radial mixing of the granular bed is poor in the slipping mode: to ensure good radial mixing, and consequently good heat and mass transfer, rotary kilns are generally operated in the slumping or rolling regimes.
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S.-Y. Lim et al. / Powder Technology 138 (2003) 25–30
This study is confined to cohesionless particles of approximately uniform size, in cylinders without flights. This paper presents a detailed analysis of the motion of a single particle in a slowly rotating horizontal cylinder in the avalanching mode using the positron emission particle tracking (PEPT) technique to follow the motion of a tagged particle.
used, which was typical of the bulk sand material, which had a mean particle size of 500 Am and bulk density of 1600 kg/m3.
3. Results 3.1. PEPT measurement of avalanche
2. Experimental 2.1. PEPT technique Positron emission particle tracking is a noninvasive technique for following the motion of a single radioactively labelled tracer particle. The technique is based on positron emission tomography, now widely used in medicine, and was developed at Birmingham University by Hawkesworth et al. [2] and Parker et al. [3,4]. The tracer particle is produced by irradiating a particle of glass or sand with a beam of 3He ions from a cyclotron so that some of the oxygen atoms in the solid are converted to the radionuclide 18F. Alternatively, the radionuclide can be applied by adsorption of irradiated water onto the surface of the tracer particle. This radionuclide decays with emission of a positron (a positive electron) and a neutrino. This positron rapidly annihilates with an electron, producing a pair of collinear back-to-back 511-keV g-rays which carry equal and opposite momenta. The g-rays are able to penetrate considerable thicknesses of material allowing the tracer to be detected through the opaque casing of engineering equipment and within three-dimensional granular flows. The pair of g-rays is detected using a ‘‘positron camera’’ [5] consisting of a pair of large rectangular detectors, operating in coincidence so that an event is recorded only if g-rays are detected in both detectors within a resolving time of 12 ns. For each event the tracer particle must lie close to a line connecting the points on each detector where the pair of back-to-back g-rays were detected. Therefore, the particle position can be determined by triangulation from a small number of g-ray pairs. The data were processed [6] to give tracer position as a function of time, and tracer velocity as a function of time and of position. 2.2. Experimental apparatus The experimental apparatus was designed, constructed and operated at the Department of Chemical Engineering, University of Birmingham. It is as described by Spurling [7]. The apparatus consisted of a Perspex cylinder lined on the inner wall with sandpaper and fitted with an aluminium end plate at one end and a thin aluminium orifice dam at the other. Three sizes of cylinder were studied; the diameters were 488, 390 and 240 mm with respective lengths of 1.3, 1 and 1 m. All experiments were conducted with sand as bed material and the cylinder horizontal. A sand tracer was
Fig. 1 is a sample vertical displacement diagram for a tracer particle, showing its vertical movement. A cycle starts from location T1 near the top of the bed (T in Fig. 2). Fig. 3 shows dots representing the position of the tracer particle measured every 20 ms. When the particle is at T1, the bed surface is inclined at the static angle of repose, cs, shown in Fig. 2. A small further rotation of the cylinder makes the bed unstable: an avalanche occurs, terminating when the bed angle is ct (position 2 in Fig. 2; see Ref. [8]). The inclination, ct, corresponds to location S1 on the displacement diagrams (Figs. 1 and 3); this is because the avalanche moves the tracer particle by a distance only about one third of the chord length BT in Fig. 2 (see also Fig. 3). Subsequently, the bed moves in solid body rotation, with the cylinder, from position 2 to position 3 in Fig. 2. A second avalanche then begins, but with the tracer starting in position S1 (Figs. 1 and 3): this avalanche moves the tracer particle from S1 to S2. Subsequent solid body rotation, followed by a third avalanche, moves the tracer from S2 to B1 (Figs. 1 and 3). Note that T1 is not necessarily right at the top of the bed, nor is B1 quite at the bottom. At B1, the tracer particle is taken into the body of the bed, where it is unaffected by avalanches: it thus gets carried round in a circular arc, concentric with the cylinder wall, until the tracer particle reaches a point near the top of the bed (position T2, Fig. 1). The whole cycle then begins again. This solid body rotation appears as curve B1T2 on Fig. 1: the position T2 may not be exactly at the top, nor in exactly the position T1 (Fig. 1). To summarise, the complete cycle for the tracer particle motion is as follows. (1) Three successive avalanches, T1S1, S1S2, S1B1 (Figs. 1 and 3), carry the tracer from the top region to the bottom region of the bed surface. (2) Between avalanches, the particle rests on the bed surface at S1 or S2. During these rest periods, the particle moves a little, due to the solid body rotation of a few degrees from angle ct to angle cs, i.e., from position 2 to position 3 (Fig. 2). Consequently, some vertical movement occurs at positions S1 and S2. Fig. 1 shows short lapses of time between avalanches; these are the rest periods at S1 and S2 (Fig. 1). (3) The final part of the cycle is the transport of the tracer particle in an arc of a circle concentric with the cylinder, from B1 to T2 (Fig. 1). The angle of rotation of the radius vector, from the particle to the centre of the cylinder, is
S.-Y. Lim et al. / Powder Technology 138 (2003) 25–30
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Fig. 1. Vertical displacement of the tracer relative to the centre of the horizontal cylinder with respect to time. The time interval between individual points is about 20 ms.
of order 90j. Therefore, the duration of this part of the cycle is much longer than for the avalanches (1) and the rest periods (2). In Figs. 1 and 3, each dot represents the position of the tracer particle; thus the velocity can be estimated from the fact that the time interval between dots is 20 ms. At each rest position, S1 and S2, there should be a succession of dots forming a line, on account of the solid body rotation. However, the noise in the signal gives a blob of points at S1 and at S2 (Figs. 2 and 3), so the movement due to the small solid body rotation cannot be estimated. Fig. 4 shows the velocity of a tracer particle at the bed surface, calculated from the gaps between dots in Figs. 1 and 3, for the periods when the tracer particle was moving down in an avalanche. The velocity is in the direction TB (Fig. 2), i.e., down the slope of the bed. Note that the avalanche accelerates progressively and then decelerates
progressively; the process of coming to rest is not instantaneous. Fig. 4 also shows the rest periods at S1 and S2 between avalanches. 3.2. Transition from avalanching to rolling mode As was observed in Fig. 3, each particle requires a discrete number of avalanches to convey it from the top to the bottom of the granular bed. The mean number of avalanches to convey a particle from the top to the bottom varied between 1 and 4.25 within the range of operating conditions of the experiments. The total number of avalanches that occurred in at least 20 passes was summed and divided by the actual number of passes to give a mean number of avalanches. It was found that the mean number of avalanches (n) for any experiment, if calculated from 20 or more of passes, remained approximately constant with a standard deviation of roughly 0.3 –0.4.
Fig. 2. Schematic representation of the cyclical avalanching process where cs>ct. There is an avalanche between positions 1 and 2; then there is solid body rotation from position 2 to position 3.
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Fig. 3. Cross-sectional view of a tracer avalanching down the surface of the granular bed. Individual points are locations of the tracer on the bed surface as detected by the PEPT detector every 20 ms.
Fig. 3 shows the occurrence of three avalanches during one active circuit, i.e., during the time the tracer was near or on the surface of the bed. Fig. 5 shows the mean number of avalanches against Froude number (Fr = x2R/g) for different experiments: x is the angular speed of the cylinder (rad/s); R is its radius; g is the gravitational acceleration. The Froude number has been found to be an appropriate dimensionless group for characterising flow in rotating cylinders [9,10]. The error bars in Fig. 5 are the standard deviations of the mean number of avalanches.
Fig. 5 shows that n decreases approximately linearly with increasing Fr; because of the large spread, it is not possible to comment on the effect of cylinder size and degree of fill. As the rotational speed of the cylinder increases, the average number of avalanches to convey a particle from the top to the bottom of the bed decreases, giving transition to a rolling bed. It would seem reasonable to suppose that when n = 1 the motion changes from avalanching to rolling. An experiment was carried out with the bed clearly in the rolling mode; it was found that n = 1 (see Fig. 5). A linear fit to the date of Fig. 5
Fig. 4. Sample velocity profile of a tracer moving down the bed surface. Surface velocities deduced from the data shown in Figs. 1 and 3. The origin is taken to be the centre of the chord (TB in Fig. 2).
S.-Y. Lim et al. / Powder Technology 138 (2003) 25–30
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Fig. 5. The mean number of avalanches needed for a tracer particle to traverse the active layer versus Froude number.
gives n = 1 at Fr = 0.000085; this value is similar in magnitude to the value measured by Henein et al. [9] for transition to rolling mode for sand. Ding et al. [11] discuss the slumping-rolling transition using different criteria. 3.3. Distance travelled in an avalanche The length of the chord is defined as the distance between the ends of the bed in a cross section, i.e., the length BT in Fig.
2. Distances travelled in an avalanche, i.e., distances between two consecutive points (i.e., lengths T1S1, S1S2, S2B1) where the tracer on the bed surface was stationary relative to that surface, were measured for at least 20 passes down the free surface, and divided by the chord length, for each rotational speed and filling. Results are shown in Fig. 6. The bars indicate standard deviations from the mean; note that the spread is large. It can be seen that the mean fraction of the chord length traversed per avalanche is approximately 1/3;
Fig. 6. Mean fraction of chord length traversed per avalanche.
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S.-Y. Lim et al. / Powder Technology 138 (2003) 25–30
Fig. 7. Representation of the slumping mode geometry. The centroid of the wedge of material forming the avalanche moves on average about 1/3 of a chord down the bed surface.
the data suggest that this quantity increases slowly with Froude number. A mechanism for avalanching in which a particle traverses 1/3 of the chord length per avalanche is indicated in Fig. 7. The shaded wedge avalanches from angle cs to the position of the dotted wedge at angle ct. The centroid of the avalanching material moves 1/3 of the chord length, and so on average a particle moves 1/3 of the chord length. This conjecture is consistent with the experimental finding that the tracer moves about 1/3 of the chord length per avalanche and needs an average three avalanches to move across the free surface. Note that a particle would not be expected to travel a distance of the chord length along the free surface.
4. Conclusions This paper presents a PEPT study of the avalanching phenomenon for granular material in a slowly rotating horizontal cylinder. The PEPT data give displacement diagrams that depict the trajectory of a particle: it is clear that a tracer particle does not necessarily slump to the bottom of the bed in a single avalanche but rather in a series of discrete steps. A particle rotates with the bed in rigid body rotation, then traverses an active surface region in a series of avalanches, and is then absorbed once more into the rigid body rotation. The mean number of avalanches (n) required for a tracer particle to traverse the active region decreased from 3.5 – 4.0 at low Fr to around 1.5 at a Fr c 0.00006. The mean fraction of the chord length traversed in a single avalanche increased with Fr from around 0.25 at low Fr to around 0.4 at Fr c 0.00006. There was wide scatter for both quantities. In the rolling mode, particles traverse the free surface without intermediate stops, and thus n = 1 for all particles. The data provide information about the transition from the avalanching to the rolling mode and throw new light on the transition.
Acknowledgements S.-Y. Lim expresses his gratitude to the Cambridge Commonwealth Trust, Her Majesty’s Government, Foreign and Commonwealth Office, and Huntsman Tioxide for their financial support. We are grateful to the EPSRC (UK) for funding the PEPT work.
References [1] H. Henein, A.P. Watkinson, J.K. Brimacombe, Experimental study of transverse bed motion in rotary kilns, Metall. Trans. B 14B (1983) 191. [2] M.R. Hawkesworth, G. Jonkers, N.L. Jeffries, J.F. Crilly, P. Fowles, D.J. Parker, Non-medical applications of a positron camera, Nucl. Instrum. Meth. A 310 (1991) 423. [3] D.J. Parker, P.A. McNeil, M.R. Hawkesworth, P. Fowles, C.J. Broadbent, Positron emission particle tracking—a technique for studying flow within engineering equipment, Nucl. Instrum. Meth. A 236 (1993) 592. [4] D.J. Parker, P.A. McNeil, T.D. Fryer, P. Fowles, C.J. Broadbent, M.R. Hawkesworth, Industrial positron-based imaging: principles and application, Nucl. Instrum. Meth. A 348 (1994) 583. [5] D.J. Parker, R.N. Forster, P. Fowles, P.S. Takhar, Positron emission particle tracking using the new Birmingham positron camera, Nucl. Instrum. Meth. A 477 (2002) 540. [6] R. Stewart, Convection and mixing of granular material stirred by a flat blade. PhD thesis, University of Cambridge, 2001. [7] R. Spurling, Granular flow in an inclined rotating cylinder: steady state and transients. PhD thesis, University of Cambridge, 2001. [8] P. Evesque, Analysis of the statistics of sandpile avalanches using soil-mechanics results and concepts, Phys. Rev. A 43 (6) (1991) 2720. [9] H. Henein, A.P. Watkinson, J.K. Brimacombe, The modelling of transverse solids motion in rotary kilns, Metall. Trans. B 14B (1983) 207. [10] J.F. Davidson, D.M. Scott, P.A. Bird, O. Herbert, A.A. Powell, H.V.M. Ramsay, Granular motion in a rotary kiln: the transition from avalanching to rolling, KONA, Powder Part. (18) (2000) 149. [11] Y.L. Ding, R. Forster, J.P.K. Seville, D.J. Parker, Granular motion in a rotating drum: bed turnover time and slumping-rolling transition, Powder Tech. 124 (2002) 18.
Avalanching of granular material in a horizontal slowly rotating cylinder: PEPT studies S.-Y. Lim a, J.F. Davidson a, R.N. Forster b, D.J. Parker c, D.M. Scott a,*, J.P.K. Seville d a
Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA, UK b Cemex, 103 Great Valley Parkway, Malvern, PA 19355, USA c Positron Imaging Centre, University of Birmingham, Birmingham, B15 2TT, UK d Department of Chemical Engineering, University of Birmingham, Birmingham, B15 2TT, UK
Abstract A detailed analysis of the motion af a single particle in a slowly rotating horizontal cylinder in the avalanching mode has been carried out using the positron emission particle – tracking technique to follow the motion of a tagged particle. Cylinders of diameters 240, 390, and 488 mm, and sand of mean diameter 500 Am were used; the tracer was an irradiated sand particle. It was found that a tracer particle rotates with the bed in rigid body motion and then traverses the active surface region in a series of discrete avalanches before being absorbed once again into the rigid body rotation. The mean number of avalanches required to traverse the active region decreased from 3.5 to 4 at low rotational speed to 1 at higher rotational speed at the transition to the rolling mode. D 2003 Elsevier B.V. All rights reserved. Keywords: Rotating cylinders; Avalanching; PEPT studies
1. Introduction Rotating cylinders are commonly used for processing granular materials in the mineral, ceramic, cement, metallurgical, chemical, pharmaceutical, food and waste industries, and are relatively simple in design and operation. They are suited to processes that require high temperatures at near-atmospheric pressure. There are considerable economic incentives to develop a more fundamental understanding of the motion of granular solids in such cylinders. Some rotating cylinders are slightly inclined, with the granular material entering at the upper end and flowing by gravity to the lower end; other cylinders are horizontal. Some cylinders are without internals; others contain lifting flights on the inside surface of the wall. In cylinders without flights, the granular material forms a bed in the lower part of the cylinder. The motion of the particles in the transverse plane (perpendicular to the cylinder axis), as it moves from one end of the cylinder to the other, can be characterised as one of several possible behaviours, depending on the operating conditions (rotational speed and degree of fill), the flow characteristics of the particles and the friction * Corresponding author. E-mail address: [email protected] (D.M. Scott). 0032-5910/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2003.08.038
coefficient with the wall of the cylinder wall [1]. The current research deals with slowly rotating cylinder without flights, for which the important forms of bed behaviour are listed below. Slumping and rolling occur at low cylinder speeds when inertial effects are negligible [1]. Cylinders with rough walls are considered so there is no slipping between the granular bed and the wall. Slumping occurs at very low rotational speeds. The particles that form the bed at the bottom of the cylinder ride up one side of the wall, due to the rotation, forming a bed whose surface is inclined to the horizontal at about the angle of repose. The particles on the sloping surface periodically slump or avalanche from the upper half of the surface to the lower half. Rolling behaviour occurs at higher cylinder speeds and is characterised by an approximately flat upper surface. The angle of the upper surface to the horizontal plane is known as the dynamic angle of repose of the granular material. Slipping behaviour occurs with low particle-to-wall friction: the bed slides up and down the wall of the rotating cylinder, with very little relative motion between the particles. Radial mixing of the granular bed is poor in the slipping mode: to ensure good radial mixing, and consequently good heat and mass transfer, rotary kilns are generally operated in the slumping or rolling regimes.
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S.-Y. Lim et al. / Powder Technology 138 (2003) 25–30
This study is confined to cohesionless particles of approximately uniform size, in cylinders without flights. This paper presents a detailed analysis of the motion of a single particle in a slowly rotating horizontal cylinder in the avalanching mode using the positron emission particle tracking (PEPT) technique to follow the motion of a tagged particle.
used, which was typical of the bulk sand material, which had a mean particle size of 500 Am and bulk density of 1600 kg/m3.
3. Results 3.1. PEPT measurement of avalanche
2. Experimental 2.1. PEPT technique Positron emission particle tracking is a noninvasive technique for following the motion of a single radioactively labelled tracer particle. The technique is based on positron emission tomography, now widely used in medicine, and was developed at Birmingham University by Hawkesworth et al. [2] and Parker et al. [3,4]. The tracer particle is produced by irradiating a particle of glass or sand with a beam of 3He ions from a cyclotron so that some of the oxygen atoms in the solid are converted to the radionuclide 18F. Alternatively, the radionuclide can be applied by adsorption of irradiated water onto the surface of the tracer particle. This radionuclide decays with emission of a positron (a positive electron) and a neutrino. This positron rapidly annihilates with an electron, producing a pair of collinear back-to-back 511-keV g-rays which carry equal and opposite momenta. The g-rays are able to penetrate considerable thicknesses of material allowing the tracer to be detected through the opaque casing of engineering equipment and within three-dimensional granular flows. The pair of g-rays is detected using a ‘‘positron camera’’ [5] consisting of a pair of large rectangular detectors, operating in coincidence so that an event is recorded only if g-rays are detected in both detectors within a resolving time of 12 ns. For each event the tracer particle must lie close to a line connecting the points on each detector where the pair of back-to-back g-rays were detected. Therefore, the particle position can be determined by triangulation from a small number of g-ray pairs. The data were processed [6] to give tracer position as a function of time, and tracer velocity as a function of time and of position. 2.2. Experimental apparatus The experimental apparatus was designed, constructed and operated at the Department of Chemical Engineering, University of Birmingham. It is as described by Spurling [7]. The apparatus consisted of a Perspex cylinder lined on the inner wall with sandpaper and fitted with an aluminium end plate at one end and a thin aluminium orifice dam at the other. Three sizes of cylinder were studied; the diameters were 488, 390 and 240 mm with respective lengths of 1.3, 1 and 1 m. All experiments were conducted with sand as bed material and the cylinder horizontal. A sand tracer was
Fig. 1 is a sample vertical displacement diagram for a tracer particle, showing its vertical movement. A cycle starts from location T1 near the top of the bed (T in Fig. 2). Fig. 3 shows dots representing the position of the tracer particle measured every 20 ms. When the particle is at T1, the bed surface is inclined at the static angle of repose, cs, shown in Fig. 2. A small further rotation of the cylinder makes the bed unstable: an avalanche occurs, terminating when the bed angle is ct (position 2 in Fig. 2; see Ref. [8]). The inclination, ct, corresponds to location S1 on the displacement diagrams (Figs. 1 and 3); this is because the avalanche moves the tracer particle by a distance only about one third of the chord length BT in Fig. 2 (see also Fig. 3). Subsequently, the bed moves in solid body rotation, with the cylinder, from position 2 to position 3 in Fig. 2. A second avalanche then begins, but with the tracer starting in position S1 (Figs. 1 and 3): this avalanche moves the tracer particle from S1 to S2. Subsequent solid body rotation, followed by a third avalanche, moves the tracer from S2 to B1 (Figs. 1 and 3). Note that T1 is not necessarily right at the top of the bed, nor is B1 quite at the bottom. At B1, the tracer particle is taken into the body of the bed, where it is unaffected by avalanches: it thus gets carried round in a circular arc, concentric with the cylinder wall, until the tracer particle reaches a point near the top of the bed (position T2, Fig. 1). The whole cycle then begins again. This solid body rotation appears as curve B1T2 on Fig. 1: the position T2 may not be exactly at the top, nor in exactly the position T1 (Fig. 1). To summarise, the complete cycle for the tracer particle motion is as follows. (1) Three successive avalanches, T1S1, S1S2, S1B1 (Figs. 1 and 3), carry the tracer from the top region to the bottom region of the bed surface. (2) Between avalanches, the particle rests on the bed surface at S1 or S2. During these rest periods, the particle moves a little, due to the solid body rotation of a few degrees from angle ct to angle cs, i.e., from position 2 to position 3 (Fig. 2). Consequently, some vertical movement occurs at positions S1 and S2. Fig. 1 shows short lapses of time between avalanches; these are the rest periods at S1 and S2 (Fig. 1). (3) The final part of the cycle is the transport of the tracer particle in an arc of a circle concentric with the cylinder, from B1 to T2 (Fig. 1). The angle of rotation of the radius vector, from the particle to the centre of the cylinder, is
S.-Y. Lim et al. / Powder Technology 138 (2003) 25–30
27
Fig. 1. Vertical displacement of the tracer relative to the centre of the horizontal cylinder with respect to time. The time interval between individual points is about 20 ms.
of order 90j. Therefore, the duration of this part of the cycle is much longer than for the avalanches (1) and the rest periods (2). In Figs. 1 and 3, each dot represents the position of the tracer particle; thus the velocity can be estimated from the fact that the time interval between dots is 20 ms. At each rest position, S1 and S2, there should be a succession of dots forming a line, on account of the solid body rotation. However, the noise in the signal gives a blob of points at S1 and at S2 (Figs. 2 and 3), so the movement due to the small solid body rotation cannot be estimated. Fig. 4 shows the velocity of a tracer particle at the bed surface, calculated from the gaps between dots in Figs. 1 and 3, for the periods when the tracer particle was moving down in an avalanche. The velocity is in the direction TB (Fig. 2), i.e., down the slope of the bed. Note that the avalanche accelerates progressively and then decelerates
progressively; the process of coming to rest is not instantaneous. Fig. 4 also shows the rest periods at S1 and S2 between avalanches. 3.2. Transition from avalanching to rolling mode As was observed in Fig. 3, each particle requires a discrete number of avalanches to convey it from the top to the bottom of the granular bed. The mean number of avalanches to convey a particle from the top to the bottom varied between 1 and 4.25 within the range of operating conditions of the experiments. The total number of avalanches that occurred in at least 20 passes was summed and divided by the actual number of passes to give a mean number of avalanches. It was found that the mean number of avalanches (n) for any experiment, if calculated from 20 or more of passes, remained approximately constant with a standard deviation of roughly 0.3 –0.4.
Fig. 2. Schematic representation of the cyclical avalanching process where cs>ct. There is an avalanche between positions 1 and 2; then there is solid body rotation from position 2 to position 3.
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Fig. 3. Cross-sectional view of a tracer avalanching down the surface of the granular bed. Individual points are locations of the tracer on the bed surface as detected by the PEPT detector every 20 ms.
Fig. 3 shows the occurrence of three avalanches during one active circuit, i.e., during the time the tracer was near or on the surface of the bed. Fig. 5 shows the mean number of avalanches against Froude number (Fr = x2R/g) for different experiments: x is the angular speed of the cylinder (rad/s); R is its radius; g is the gravitational acceleration. The Froude number has been found to be an appropriate dimensionless group for characterising flow in rotating cylinders [9,10]. The error bars in Fig. 5 are the standard deviations of the mean number of avalanches.
Fig. 5 shows that n decreases approximately linearly with increasing Fr; because of the large spread, it is not possible to comment on the effect of cylinder size and degree of fill. As the rotational speed of the cylinder increases, the average number of avalanches to convey a particle from the top to the bottom of the bed decreases, giving transition to a rolling bed. It would seem reasonable to suppose that when n = 1 the motion changes from avalanching to rolling. An experiment was carried out with the bed clearly in the rolling mode; it was found that n = 1 (see Fig. 5). A linear fit to the date of Fig. 5
Fig. 4. Sample velocity profile of a tracer moving down the bed surface. Surface velocities deduced from the data shown in Figs. 1 and 3. The origin is taken to be the centre of the chord (TB in Fig. 2).
S.-Y. Lim et al. / Powder Technology 138 (2003) 25–30
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Fig. 5. The mean number of avalanches needed for a tracer particle to traverse the active layer versus Froude number.
gives n = 1 at Fr = 0.000085; this value is similar in magnitude to the value measured by Henein et al. [9] for transition to rolling mode for sand. Ding et al. [11] discuss the slumping-rolling transition using different criteria. 3.3. Distance travelled in an avalanche The length of the chord is defined as the distance between the ends of the bed in a cross section, i.e., the length BT in Fig.
2. Distances travelled in an avalanche, i.e., distances between two consecutive points (i.e., lengths T1S1, S1S2, S2B1) where the tracer on the bed surface was stationary relative to that surface, were measured for at least 20 passes down the free surface, and divided by the chord length, for each rotational speed and filling. Results are shown in Fig. 6. The bars indicate standard deviations from the mean; note that the spread is large. It can be seen that the mean fraction of the chord length traversed per avalanche is approximately 1/3;
Fig. 6. Mean fraction of chord length traversed per avalanche.
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Fig. 7. Representation of the slumping mode geometry. The centroid of the wedge of material forming the avalanche moves on average about 1/3 of a chord down the bed surface.
the data suggest that this quantity increases slowly with Froude number. A mechanism for avalanching in which a particle traverses 1/3 of the chord length per avalanche is indicated in Fig. 7. The shaded wedge avalanches from angle cs to the position of the dotted wedge at angle ct. The centroid of the avalanching material moves 1/3 of the chord length, and so on average a particle moves 1/3 of the chord length. This conjecture is consistent with the experimental finding that the tracer moves about 1/3 of the chord length per avalanche and needs an average three avalanches to move across the free surface. Note that a particle would not be expected to travel a distance of the chord length along the free surface.
4. Conclusions This paper presents a PEPT study of the avalanching phenomenon for granular material in a slowly rotating horizontal cylinder. The PEPT data give displacement diagrams that depict the trajectory of a particle: it is clear that a tracer particle does not necessarily slump to the bottom of the bed in a single avalanche but rather in a series of discrete steps. A particle rotates with the bed in rigid body rotation, then traverses an active surface region in a series of avalanches, and is then absorbed once more into the rigid body rotation. The mean number of avalanches (n) required for a tracer particle to traverse the active region decreased from 3.5 – 4.0 at low Fr to around 1.5 at a Fr c 0.00006. The mean fraction of the chord length traversed in a single avalanche increased with Fr from around 0.25 at low Fr to around 0.4 at Fr c 0.00006. There was wide scatter for both quantities. In the rolling mode, particles traverse the free surface without intermediate stops, and thus n = 1 for all particles. The data provide information about the transition from the avalanching to the rolling mode and throw new light on the transition.
Acknowledgements S.-Y. Lim expresses his gratitude to the Cambridge Commonwealth Trust, Her Majesty’s Government, Foreign and Commonwealth Office, and Huntsman Tioxide for their financial support. We are grateful to the EPSRC (UK) for funding the PEPT work.
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