Granular Motion in a Rotary Kiln: the Transition from Avalanching to Rolling

We report measurements of flow transitions, from avalanching to rolling, for granular material in rotary kilns. In the avalanching mode, the surface slips periodically; in the intervals between avalanches, all particles rotate with the kiln. In the rolling mode, the surface particles slide down continuously; the material underneath the surface rotates with the kiln. Our measurements give Froude numbers (Rw/g) for transitions, which are significantly different for sand and Ti02 powder. For the avalanching mode, we measured cycle times and deduced t12, the avalanche time; t12 was also measured directly by video photography. For kilns of diameters 0.2-0.5 m, both methods give t12, of order 1-2 sec and it appears to be proportional to vl, l being the chord length of the granular bed, the maximum distance of fall for avalanche material. Simple theory, assuming the avalanche particles slide down a frictional surface, gives fair estimates oft12 and may be a basis/or predicting avalanche-to-rolling transitions in large industrial kilns.


Introduction
A rotary kiln for processing granular material is usually a cylinder rotating slowly about its axis, inclined at a few degrees to the horizontal. Granular material is fed continuously at the top end of the cylinder. Within the cylinder, the material forms a bed which may occupy about one third of the cylinder volume. The bed surface is inclined to the horizontal at about the angle of repose for the granular material. A full-size industrial kiln is typically about 3 m diameter and 50 m long and rotates once in 5-10 minutes. At these slow rotation speeds, the flow regime of the granular material may be in one of two alternative modes, as follows.
(1) The bed may be in "avalanching" mode, otherwise described as slumping: for most of the time, the whole bed rotates at the same angular velocity, w, as the kiln, thus undergoing "solid body" rotation. Then there is an avalanche, starting when the bed surface reaches the static angle of repose y 5 : during the avalanche, material from the upper part of the bed surface slides rapidly down; when the avalanche stops, the bed surface is inclined at a lesser angle yd, the dynamic angle of repose. Solid body rotation of the whole bed ensues, until the surface inclination again reaches angle y 5 , when avalanching occurs and the cycle is repeated.
(2) At higher kiln speeds, the motion of near-surface particles is continuous, the "rolling" mode. Most of the bed rotates with the kiln in solid body rotation, but the surface and near-surface particles fall continuously down the slope, whose inclination is approximately Yrl· This paper is concerned with the transition from avalanching to rolling. In all cases centrifugal accelerations are small, i.e., the Froude number Fr<< 1, where Fr=Rw 2 /g, R being the kiln radius and g the acceleration of gravity. The surface of the bed is essentially flat. The avalanching/rolling transition was addressed by Henein eta! [1,2] who proposed the Froude number as a suitable parameter to define the transition from avalanching to rolling. Their experiments showed that very low values of Fr, less than 10-5 , are consistent with avalanching. Then there is a transition region where 10-s < Fr < 10-4; values of Fr above 10-4 are consistent with the rolling mode.
The use of Froude number is important for scaleup. If the transition from avalanching/slumping to rolling were uniquely described by Fr, then large kilns would have the transition at lower speeds than small kilns.
The work described here was a study of the transition from avalanching/slumping to rolling in drums of diameters 194, 288 and 500 mm, to test the effect of scale-up, albeit over a small range of R. Experiments were done with two granular materials, namely (i) sand in the size range 300-500 11m and (ii) titanium dioxide powder as discharged from a rotary kiln. The sand was chosen as a free-flowing granular material. The titanium dioxide powder represented the powder near the kiln discharge: it had a wide range of particle sizes, with a mean particle size of about 250 11m and a very wide size distribution with particle diameters up to about 10 mm; the powder was somewhat cohesive.
The results from observations of avalanching to rolling transitions for the three drum diameters and the two powders are presented in terms of Froude number.
There were also observations of cycle time for the avalanching mode, this being the time interval between one avalanche and the next, the form of the powder bed being identical at the beginning and end of each cycle: for example a cycle could be assumed to begin when the inclination of the powder bed is a maximum. Analysis of the relation between cycle time and rotation speed, w, casts light on the mechanism of avalanching, particularly the time for an avalanche, which is finite. While this does not yet give a new criterion for scale-up, it suggests how an improved criterion might be developed.

Apparatus
The experiments used a cylindrical steel drum of internal diameter 500 mm, length 300 mm, with a horizontal axis. The drum was flanged at one end: mounted on the flange was a flat vertical transparent Perspex plate, diameter 600 mm, so that the granular bed in the drum could be viewed from the front. This plate contained a small opening port, with a Perspex lid forming a flush internal face; the port was used to load powder into the drum and to unload. The other end of the drum was vertical steel plate, welded to the cylindrical part and connected to a shaft at the back: this shaft supported the drum and was driven by a variable speed motor with speed control and interchangeable gearbox. Thus the drum could be rotated at speeds from 0.1 to 6 revolutions per minute. The cylindrical part of the drum was lined with sandpaper.
To study the effects of varying drum diameter, two cylindrical Perspex inserts were made, of diameters 194 and 288 mm. Each was closed at the front end by a 600 mm diameter Perspex plate that replaced the above mentioned 600 mm diameter plate on the drum. The two cylindrical Perspex inserts were each closed at the back end to give a (length/diameter) ratio=3/5, as for the 500 mm drum, and the cylindrical parts were lined with sandpaper. When the drum operated with an inserted cylinder, the granular bed was contained entirely within the insert.
In this way, there were three alternative rotary drums of diameters 194, 288 and 500 mm with the capability of viewing the slumping or rolling granular bed through a transparent end plate.

Experiments: slumping/rolling observations
A number of experiments were performed to test the effect of the nature of the granular material and the percentage fill of the drum on the bed behaviour i.e., whether it was (i) slumping, (ii) transition or (iii) rolling. Sand of size range 300-500 11m and Ti0 2 particles discharged from a kiln were used. These Ti0 2 particles were in two forms namely (i) raw calciner discharge with a wide size range, up to 10 mm, or (ii) with the larger particles sieved out, so that there was still a wide size range, but up to 1 mm.
The following experiments were made with the objective of plotting bed behaviour diagrams, such as Figures 1 and 2, which were suggested by Henein et a!. [1]. The steps in the experiments were as follows: (1) An amount of granular material was introduced into the cylinder, which was then rotated to give a bed with a flat surface, albeit inclined. With the drum rotation stopped, the chord length of the surface was measured and from thence the percentage fill calculated; the percentage fill is the ratio (area of the crescent-shaped cross section of the bed) /nR 2 .
(2) With the drum rotating steadily, the angular speed was measured by timing one revolution. The bed behaviour was classified as slumping, transitional or rolling, using the following definitions: (i) In the slumping or avalanching mode, there was a clearly cyclical motion: over a distinct period, the whole bed rotated with the drum, in solid body rotation; then there was an avalanche. The times for these separate parts of the cycle were the subject of further measurements, see below. • 288 6 ...
. Bed behaviour diagram: transition from slumping (avalanching) to rolling for sand main part of the bed moved with the drum, in solid body rotation. (iii) There was a rather ill-defined transition region, in which some parts of the surface were falling continuously: other parts were subject to cyclical motion. As will be evident from the results, the transitional region included an appreciable range of drum speeds.

Experiments: cycle time in the slumping mode
With the drum speed slow enough to give welldefined slumping, the cycle time was measured for a range of drum speeds, for each of the three drum diameters. Two methods of observation were used, as follows.
(i) The time for ten slump cycles was recorded. This measurement was repeated nine times, so there were ten recordings of the time for ten slump cycles. The drum rotation speed was measured by simply timing a revolution. These observations gave the total cycle time t 13 . From analysis of these results, given below, the duration of the avalanche, t 12 , could be deduced. (ii) Direct measurements of the avalanche duration, t 12 , were obtained from video pictures of the cyclical motion. The video camera was set up to observe the granular bed through the transparent front wall of the drum. The video picture included a digital stopwatch. The video was played back at a low framing rate: the tape was stopped at the beginning and end of an avalanche; from the stopwatch readings, the time t 12 for the avalanche was obtained. The kiln speed and the chord length of the particle bed were measured as before, giving the percentage fill. For each speed/percentage fill combination, the video photography and . . ....
. analysis were repeated five times, to give an average value of t12· 3. Results and discussion 3.1 Bed behaviour diagrams: Froude number as a scaling parameter Henein et al. [1] used the bed behaviour diagram, plotting either bed depth or percentage fill against Froude number, Fr, and delineating areas of the diagram as slumping, transition or rolling. Figures 1 and  2 show our data plotted in this form as percentage fill against Fr, the plotting points indicating transitions, measured in the way described above. Figures 1 and 2 show bed behaviour diagrams for our data, respectively for sand and raw Ti0 2 calciner discharge. For sand, Figure 1, it appears that the slumping/transition boundary is in the region 0.00002 < Fr < 0.00004; the transition to rolling is less well defined, in the region 0.00006 < Fr < 0.0001. Figure 2 shows the results for raw Ti0 2 calciner discharge: here the slumping/transition boundary is in the region 0.0001 < Fr < 0.0002; the transition to rolling is in the region 0.0006 < Fr < 0.0015.
From these results, the following conclusions may be drawn: (1) The Froude number gives a rough guide to transitions in the flow regimes. But the transition Froude numbers are markedly different between sand and raw Ti0 2 • Thus the boundary for slumping is about five times higher for Ti0 2 as compared with sand; the boundary for transition to rolling is about ten times higher for Ti0 2 as compared with sand.
(2) With regard to the effect of drum diameter, the data in Figure 1 show that the Froude numbers for the two transitions tend to increase as the drum diameter decreases; the same effect is approximately true of the data in Figure 2.
The overall conclusion is that although the Froude number is a very rough guide to behaviour, there is certainly not a unique Froude number for each of the two observed transitions. There must therefore be grave doubts about the use of these transition Froude numbers for industrial sized rotary kilns.
With this in mind, the measurements of cycle times in the slumping mode were undertaken, with the objective of understanding the mechanism of the slumping/avalanching mode. Figure 3 shows the sequence observed at slow rotation speeds with clearly defined avalanching or slumping giving a sequence of bed states 1, 2 and 3 shown in the diagram. An avalanche starts at state 1, when the bed surface is at maximum angle Ys (degrees), the static angle of repose. Then a finite quantity of granular material slides down the surface. At time t 12 , the avalanching material comes to rest, relative to the drum, when the inclination of the bed surface is Yct (degrees), the dynamic angle of repose; this is condition 2 in Figure 3. Between conditions 2 and 3, the granular bed rotates with the drum in solid body motion at angular velocity ffi. At condition 3, the bed is in the same state as at 1. Subsequently the cycle repeats itself, with a cycle time t 13 , measured in the experiments.

Cycle time results and analysis
As the angular speed, ffi, varies, it is assumed that the avalanche time t 12 is constant, plausible because t 12 is usually much less than t 13 , so the drum rotation during time t 12 is small. During the 2-3 sequence, there is solid body rotation, so it follows that Ys-Yct=ffi(t13-td180/rc and hence  Table 1. This analysis neglects the change in path length of the avalanche, TB in Figure 3, as  the percentage fill changes. The effect of path length, TB, on avalanche time t 12 is considered further in Section 3.3 below. Table 1 also shows the slope of the best fit line in each diagram, Figures 4, 5 and 6. From equation (1) the slope should be (ys-Yd)n/180, the difference between static and dynamic angles of repose. There appears to be a size effect, (ys-Yd) diminishing as drum size increases. The bed inclinations (i) just before an avalanche and (ii) just after an avalanche, were measured for sand by Powell and Ramsay [5,6], using photography. Their results suggested Ys=33o and yd=30o: thus Ys-Yd=3o, in good agreement with Table 1. Figure 7 shows the corresponding data for Ti0 2 , both the raw calciner discharge (sizes up to 10 mm) and the calciner discharge with large particles sieved out, leaving sizes up to 1 mm. The data are much more scattered than for sand, but t 13 is still roughly linear against 1/ro, and the deduced values of avalanche time t 12 = 1-2 sec, are of the same order as with sand. The slopes of the two fitted lines are: (i) 7.2° for raw Ti0 2 and (ii) 5.0° for sieved Ti0 2 . These angles, representing (y 5 -yd) the difference between static and dynamic angles of repose, are greater than the values for sand, given in Table 1. The higher values of Ys-Yd for Ti0 2 are entirely plausible, the Ti0 2 powder being somewhat cohesive.
Henein et al. [1] report experimental measurements of slumping frequency N (min-1 ) against kiln rotation speed, n, in revolutions per minute (rpm) for two drum sizes and a variety of granular materials. Their most complete data set, for limestone particles, is shown in Figure 8. Now l/t 13 is the slumping frequency, and ro=2nn/60 so equation (1) can be transformed, with appropriate conversion of units, to give (2) Curve fitting of Equation (2) to the data in Figure 8 gives values of (y 5 -yd) and t 12 : the value of (ys-Yd) is obtained from the slope at the origin, because dN/dn=360/(y 5 -yd) at n=O and t12 is obtained by curve fitting at finite n. This procedure gives Ys-Yd=6 degrees for both kiln diameters, and t 12 =1.13 sec for

Avalanche times t 12
Bird [3] and Herbert [4] plotted their data for sand as in Figures 4-6. For each drum diameter, they fitted a straight line to the data; by extrapolation to 1/ro=O, they obtained values of avalanche time t12. plotted in Figure 9 as the open points. The solid points are from their direct measurements of avalanche times, using video photography as described above. The chord length, t, is the length TB shown in Figure 3, the distance from the top to the bottom of the particle bed. Also shown in Figure 9 are data points from fitting Equation (2) to the data of Henein et at. [1] in Figure 8.  Figure 9. Bearing in mind the differences of granular material, Henein et at's [1] results agree remarkably well with ours.
The reason for plotting t 12 against '>}tis for comparison with the time taken for a single particle to slide down a smooth surface of length t. The slope being y, it is readily shown that with no friction, (3) Figure 9 shows a plot of Equation (3), with y=34o, from which it is clear that the avalanche times are much greater than the time for free fall down a slope of length t.
An alternative is to consider a single particle sliding down a slope with surface friction. Assume that the inclination of the slope is Ys: the normal force between the slope and the particle, per unit mass, is g cosy 5 • We assume the friction coefficient, between the particle and the slope, to be tanyd; the dynamic friction is relevant because the particle is in motion. The frictional force on the particle, parallel to the slope, is thus g tanyd cosy 5 : this acts against the gravity component g siny" so the net force causing acceleration is g(siny 5 -tanydcosy 5 ) in place of gsiny in deriving Equation (3). It follows that the flight time, with friction, is t 12 = [2t/g(siny 5 -tanydcosys)P 12 (4) Bird [3] and Herbert [4] used, for sand, Ys=36o and Yct=34°, giving the line plotted in Figure 9, in fair agreement with most of the data. However, Powell [5] and Ramsay [6] give Ys=33o, Yct=30o from video measurements, so Ys-Yct=3°, in fair agreement with the results in Table 1. Figure 9 shows a plot of Equation (4), using Powell and Ramsay's values of Ys and Yct· In spite of the fair agreement between Equation (4) and the data in Figure 9, there must be reservations about its validity for scale-up, as follows: (1) The range of values of '>}t in Figure 9 is small. Data are needed for much larger cylinders.
(2) There is the unexplained difference between the data in Figure   For the avalanching mode, the granular material moves cyclically: the bed rotates with the kiln in solid body motion till its surface reaches a maximum inclination, the static angle of repose Ys· Then there is an avalanche, of duration t 12 : when the avalanche terminates, the inclination is Yct. the dynamic angle of repose. The avalanche time was measured in two ways.
(i) Measuring the total cycle time t 13 and plotting it against 1/ (rotation speed ro) gave, by extrapolation to 1/ro=O, values of t 12 ; these are of order 1-2 sec for our cylinders, 198-500 mm dia. (iii) Video photography gave direct measurements of ti2· The values of ti 2 from method (i) are somewhat lower than from method (ii). 4. The avalanche time ti 2 is approximately proportional to\}[; here lis the chord length of the sloping granular bed, the maximum length of fall for avalanche material. The times ti 2 are much greater than the time for a particle to fall down a smooth slope of length !. But ti 2 is reasonably well predicted by theory for a particle falling down a frictional slope of inclination Ys, the static angle of repose: between the falling particle and the slope, the coefficient of friction is tanyd, Yd being the dynamic angle of repose.