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Transient Granular Flows in an Inclined Rotating Cylinder: Filling and Emptying David M. Scott,* John F. Davidson, Sim Ee Cheah,† Charmaine Chua, Jeremy G. Gummow,‡ Basil P. M. Lam, and Iryna Reder§ Department of Chemical Engineering, UniVersity of Cambridge, Pembroke Street, Cambridge CB2 3RA, U.K.
This paper reports the results of an experimental and theoretical study into transient granular flows in an inclined, slowly rotating cylinder. The granular materials used were sand, poppy seeds, and glass beads. The following situations have been studied: (i) feeding an initially empty cylinder, (ii) emptying a cylinder which is initially in steady state with equal feed and discharge flow rates, and (iii) the motion of a localized pile of granular material through the cylinder. Axial profiles of bed depth and discharge flow rates have been measured. The mechanistic model used here has previously been used to describe the transient response to large step changes in feed rate, rotation speed and angle of inclination; it has no free parameters. There is good agreement between theory and experiment for sand and poppy seeds, but there are discrepancies for glass beads, which under some conditions flow in a different way from sand and poppy seeds. 1. Introduction Rotary kilns play an important role in the processing of granular materials in the chemical, metallurgical and food industries in which they are used in operations such as mixing, drying, heating, and gas-solid reactions. The kiln is a cylinder without lifters whose axis is inclined to the horizontal and which rotates about its axis. Granular material is fed to the upper end of the kiln and is slowly conveyed along the kiln length as a result of the rotation and gravity. In a typical operation, hot gases or gaseous reagents are passed through the kiln above the bed surface; mass and heat transfer occur between the gas and the bed surface. There is also heat transfer from the gas to the kiln wall and from thence to that part of the granular bed in contact with it; this is regenerative heat transfer from gas to bed via the kiln wall. Transient behavior of the solids motion can occur in a number of industrial situations such as startup, irregularities in the feed rate, and fluctuations in the feed material properties. Perron and Bui1 proposed a nonlinear model to predict the discharge rate following a step change in (i) the feed rate, (ii) the rotation speed, (iii) the discharge dam height, and (iv) the axis of inclination. The model was based on an analysis by Perron and Bui,2 as modified by Sai et al.,3 of the axial velocity of the active layer and the bed as a whole; Perron and Bui2 used a semiempirical model with fitted parameters for solid transport based on dimensional analysis and on the determination of an apparent viscosity characterizing the flow behavior of the bed. The model uses a numerical algorithm to track the development of the bed depth following step changes in operating variables. Perron and Bui2 and Sriram and Sai4 showed that the model could give a good description of experimental transients,3,4 though in some cases the model worked less well in describing initial lags and responses when more than one operating variable was changed. Spurling et al.5 developed unsteady-state equations, derived from published steady state equations incorporated * To whom correspondence should be addressed. Tel.: +44-(0)1223334782. Fax: +44-(0)1223-334796. E-mail:
[email protected]. † Current address: 63F, Two International Finance Centre, 8 Finance Street, Hong Kong. ‡ Current address: Trigonos, Windmill Hill Business Park, Whitehall Way, Swindon, Wiltshire SN5 6PB, U.K. § Current address: BG Group, 100 Thames Valley Park, Reading, Berkshire, RG6 1PT, U.K.
into a partial differential equation, with no adjustable parameters. Extensive experiments for granular flow in a rotating cylinder gave good agreement with theory, for bed depth along the cylinder as well as discharge rate, for step changes in (i) feed rate, (ii) rotation speed, or (iii) axis of inclination. The theory also was in excellent agreement with the data of Sai et al.3 for discharge rate against time, following a step down in rotation speed. A brief summary of the work of Spurling et al.5 is given by Scott and Davidson.6 Later, Descoins et al.7 used a model, essentially the same as that of Spurling et al.,5 to study transients, and conducted experiments which confirmed that the model works well in predicting the response of discharge rate to a step change in rotational speed. Further, they used the model in the novel context of filling, and showed that the model works well in predicting the response of discharge rate to feeding an initially empty cylinder. This paper describes an extension of the work of Spurling et al.5 to the following cases, including comparison with experimental measurements of both bed depths and discharge rates. (i) Feeding a steady flow of granular material to a rotating cylinder which is initially empty, the filling process being predicted by the theory. This aspect of the work is similar to that of Descoins et al.7 (ii) Emptying a rotating cylinder, initially in the steady state with equal feed and discharge rates, the feed being stopped at the start of the experiment. The subsequent emptying is predicted by the theory. (iii) The motion of a localized pile of granular material, put into the inlet of an initially empty rotating cylinder over a short period of time, was observed and the bed depths as a function of time were compared to the predictions of the model. The results should be useful for designers and operators for calculating times required for startup and emptying of kilns, e.g. following shutdown, to answer the questions: (i) How long does it take to empty the kiln? (ii) How long does it take, after startup and initiation of the feed, to get the kiln back online? There is also academic interest in the development of the theoretical model to answer the question: what is the appropriate boundary condition at the front and back of a granular pile moving down a rotating cylinder, the bed depth tending to zero at the front and back of the pile?
10.1021/ie800156r CCC: $40.75 2009 American Chemical Society Published on Web 07/30/2008
160 Ind. Eng. Chem. Res., Vol. 48, No. 1, 2009 Table 1. Physical Properties of Granular Materials
surface texture particle diameter (µm) bulk density (kg/m3) angle of repose (deg)
sand
glass beads
poppy seed
rough 490 1750 36
smooth 1000 1500 26
rough 600 670 37
2. Experiments The experimental apparatus consisted of a Perspex cylinder of length 1 m and radius 0.051 m. The cylinder was lined with a series of circumferential sandpaper strips each of width 23 mm and separated by an axial distance of 2 mm. The sandpaper lining provided a rough surface. The clear slits allowed the bed depth to be measured using a millimeter scale which was marked around the circumference at the edge of each gap. The cylinder was mounted on rollers and driven by an electric motor. The apparatus was mounted on a table hinged at the exit end of the cylinder so that the angle of inclination could be adjusted. The angle of inclination was measured using an electronic spirit level. The experiments reported in this paper were conducted at an angle of inclination of 3.2° and at a Froude number (Fr ) Rω2/g) approximately 0.001-0.002, where R is the radius of the cylinder, ω is rotation speed (rad/s), and g is the acceleration due to gravity. The Froude number is important in that it gives an indicationsbut no more than an indicationsas to whether the bed will be in rolling or avalanching mode (see ref 8 and p 205 of ref 9). The values used here indicate rolling rather than avalanching, confirmed by direct observation. Granular material was fed to the raised end of the cylinder through a gravity feed hopper with a chute. The hopper had various caps with different orifice diameters to allow different flow rates. The flow rates from the hopper were measured, by timing weight change on an electronic balance, and found to be constant. The flow could be stopped with a stopping-cap. The chute at the hopper exit was held at an angle greater than the angle of repose for the granular material being used in order to prevent blockages on the chute. The end of the chute was bent upward so the granules entered the cylinder with little axial momentum. A bucket was placed on a balance at the discharge end of the cylinder to collect and weigh material leaving the cylinder. Bed depths were measured using the “stop-start” technique in which the cylinder rotation and, in the case of filling and steady state experiments, the feed were stopped while measurements were taken and started again on completion of the measurements; for emptying experiments, there was only the kiln rotation to stop, there being no feed. Spurling et al.5 showed that this stop-start process did not affect granular bed development. The arc length of the bed along the cylinder wall was measured, from which bed depths were calculated by assuming that the surface of the bed in a cross section was flat; this assumption has been verified by Lim.9 A lamp was shone into the cylinder from the discharge end to make the bed surface more visible. The “pulse” experiments were performed by pouring 265 g of sand from a cup into the empty cylinder at its elevated end; the progress of the 265 g heap of sand as it moved down the cylinder was monitored using the stop-start technique already described. Physical properties of the granular materials are given in Table 1.
Figure 1. Flow of granular material through a rotating cylinder. (a) Cross section through the cylinder axis, approximately normal to the bed surface. (b) Cross section normal to the cylinder axis. Table 2. Materials, Flowrates, and Rotational Speeds for Filling and Emptying Experiments figure number
material (see Table 1)
2 4 5
sand poppy seed sand
flowrate (g/s)
rotational speed (rps)
5.2 1.8 5.2
0.084 0.093 0.109, 0.093
3. Modeling The cylinder is shown schematically in Figure 1. In steady state operation, the volumetric flow rate of granular material, Q, is uniform along the cylinder and equal to the feed and discharge flow rates. A steady state model for the transport of granular material was derived independently by Saeman10 and Kramers and Croockewit11 for low rotational speeds assuming that the shape of the bed surface in a cross section is locally flat and that the rate of change of bed depth with axial distance is small. As the cylinder rotates, most of the particles move in solid body rotation, each particle describing a circular arc. On reaching the surface, each particle falls down a line of steepest descent, inclined at angle γ to the horizontal. Because the axis of the cylinder is inclined to the horizontal, a particle falling down the free surface has a component of its motion in the axial direction. The model gives tan β dh/dx 4πn [(2R - h)h]3⁄2 + (1) 3 sin γ tan γ where n is rotational speed, β is the angle of inclination of the cylinder, γ is the angle of repose of the granular material, and h is bed depth, a function of x, the distance from discharge; see Figure 1. In the steady state, numerical integration of eq 1 gives the axial profile of bed depth. A boundary condition for the value of h at the discharge end x ) 0, h(0), is required. Putting h(0) ) 0 leads to a singularity in dh/dx at x ) 0, though eq 1 is integrable. However, for numerical integration it is convenient to give h(0) a small value, and here, the value h(0) ) 0.000 01 m has been used. The axial profile of bed depth with this choice of h(0) is little different from the profile resulting from a different choice, for example h(0) ) dp, the diameter of a granule, the minimum possible value for bed depth; for our experiments dp was in the range 0.000 49-0.001 m (see Table 1), compared with h(0) ) 0.000 01 m noted above. The important point is that the theoretical predictions are insensitive to the choice of h(0). Generally, good qualitative agreement, and reasonable quantitative correlation, has been found by previous workers between the experimental measurements and the model calculations. For example, Lebas et al.12 have shown that the model agrees with their experiments on bed depth in a cylinder with no end dam, and Liu and Specht13 agree with the conclusions regarding residence time distributions. For a review, see the work of Spurling.14 Q)
(
)
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Figure 2. Granular bed depth profiles on feeding the initially empty cylinder with sand (Table 1): cylinder length 1 m and radius 0.051 m, angle of inclination 3.2°, rotational speed 0.084 rps; sand flow rate 5.2 g/s. Experimental measurements (data points) and model predictions (solid lines) are shown at different times after starting the feed: (a) 13, (b) 90, (c) 205, (d) 371, (e) 1204 s, and (f) steady state.
In the transient case, there is axial variation of Q. A material balance on an axial element gives1,5 ∂A ∂Q ) (2) ∂t ∂x Here t is time and A is the cross-sectional area of the granular bed, A ) R2[cos-1(1 - y) - (1 - y)√y(2 - y) ] (3) where R is the radius of the cylinder and y ) h/R. Use of eq 1 for Q and eq 3 for A, together with eq 2, leads to a nonlinear diffusion equation for the bed depth, h. The nonlinear diffusion equation must be solved numerically. Here it has been solved in an approximate way by dividing the cylinder of length L into N cells each of length δ ) L/N in each of which the bed depth is assumed to be uniform. A material balance on cell i, i ) 1,..., N, gives dAi ) Qi+1 - Qi (4) dt where Ai, hi, and Qi are the cross-sectional area of the bed in cell i, the height of the bed in cell i, and the volumetric flow rate from cell i, respectively. The feed flow rate is QN+1. For i ) 1,..., N, Qi is given by eq 1 with δ
( dhdx ) ) i
hi - hi-1 δ
(5)
and h0 ) 0.000 01 m. To prevent numerical problems when Q ) 0, the zero flow rate condition was taken to be a feed rate of 10-7 kg/s; this strategy was used by Descoins et al.7 Thus the partial differential equation, eq 2, is replaced by N simultaneous ordinary differential equations. The value N ) 500 was taken for the computations; all results were sufficiently insensitive to changes in N above this value. As a check, it was verified that the eventual steady state achieved on feeding an initially empty cylinder with a constant flow rate was the same as that given directly by eq 1, and that the transient response to a step change in one of the operating variables was the same as that computed by Spurling et al.5 The numerical solutions of the differential equations were found using Mathematica. 4. Results Experimental measurements are reported of the response of the axial profile of bed depth to the introduction of feed at a constant flow rate to an initially empty cylinder and to the cessation of feed to a bed which has reached steady state. For these filling and emptying experiments, flow rates and rotation speeds were chosen to give a steady state maximum bed depth of about half the cylinder radius, typical of operating practice. The values used are shown in Table 2. The flow rates are those from a 6 mm circular orifice. Additional measurements are
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Figure 3. Schematic view of bed of poppy seed, from above, with feed to the empty cylinder before steady state has been established.
reported on the development of the shape of a pile of granular material which was tipped into the empty cylinder. Figure 2 shows the response of an initially empty cylinder to a constant feed flow rate of 5.2 g/s of sand, at rotation speed 0.084 rps. As time progresses the bed depth increases at the feed end and the sand moves into the bed. Eventually the bed depth profile reaches a steady state when the feed and discharge flow rates are equal. The solid lines show the results of the model. Figure 2 shows that the model gives a good prediction of the development of the bed depth profile.
While observing the behavior of the poppy seed during the feeding stage, it was noted that at the beginning the motion of the bed did not correspond to the rolling mode, as was the case for sand. Possibly because of the low density of the poppy seed, the granules could move in the slumping or avalanching mode, which gave rise to oscillations or “snaking” along the cylinder, as shown schematically in Figure 3, and also to oscillations in the discharge flow rate. Nevertheless, the model gave a good prediction of the development of the bed depth profile on feeding an initially empty cylinder with poppy seed at feed flow rate 1.8 g/s and rotation speed 0.084 rps. This is consistent with the expectation that the theoretical model should apply for both the avalanching and rolling modes: in both cases most of the material is in the solid body rotation and the theory assumes that movement of particles down the bed surface is rapid. Whether this movement is sporadic, as for avalanching, or continuous, as for the rolling mode, is immaterial. Figure 4 shows the response of a bed of poppy seed, initially at steady state with feed rate (equal to discharge rate) 1.8 g/s, at rotation speed 0.093 rps, to cessation of feed with no change of rotation speed. As time progresses the bed depth decreases at the feed end, and the poppy seed moves out of the bed. After about 300 s, the upper part of the cylinder is empty; the length of the empty section increases; see Figure 4d-f. Eventually the
Figure 4. Granular bed depth profiles of poppy seed (Table 1) on cessation of feed after the bed has reached a steady state: cylinder length 1 m and radius 0.051 m, angle of inclination 3.2°, rotational speed 0.093 rps. For steady state, the flow rate of poppy seeds is 1.8 g/s and the rotational speed is 0.093 rps. Experimental measurements (data points) and model predictions (solid lines) are shown at different times after cessation of feed: (a) initial steady state, (b) 64, (c) 183, (d) 314, (e) 421, and (f) 646 s.
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Figure 5. Discharge of sand following the cessation of feed. The initial steady state was set up with feed rate ) discharge rate ) 5.2 g/s at a rotation speed 0.109 rps. The discharge took place at 0.093 rps.
cylinder empties. The solid lines show the results of the model. Figure 4 shows that the model gives a good prediction of the development of the bed depth profile. Similar good agreement is found for sand, with initial steady state with feed rate 5.2 g/s and rotation speed 0.109 rps, with emptying at rotation speed 0.093 rps. For this experiment, the mass of discharge after cessation of feed was measured and is compared to the model in Figure 5. Good agreement is found. It is of interest to compare the filling and emptying times with the mean residence time τ for steady state flow. From
Figures 2 and 4, the volume inventory can be calculated from the bed depths, h, at every axial position, x: using the geometry of Figure 1 and integrating along the length of the cylinder gives the volume inventory. This, with the given mass flow rates and bulk densities to get volume flow rates, gives the residence times, which are as follows: Figure 2f (steady state after filling) τ ) 444 s, Figure 4a (steady state before emptying) τ ) 406 s. These times may be compared with approximate filling and emptying times, respectively: Figure 2, 1200 s, Figure 4, 700 s. The ratios are (filling time/τ) ) 2.7, (emptying time/τ) ) 1.7. Note that the time for a heap of granular material to traverse the cylinder is, from Figure 6, about 600 s, comparable with the emptying time. From these very limited data, no general conclusion can be drawn as to any link between mean residence time and filling and emptying times. But, it does look as if the filling and emptying times are appreciably longer than the mean residence times. Figure 6 shows the behavior of a pulse of sand of mass 265 g, which was poured into an initially empty cylinder via the chute, an operation of short duration. The cylinder was rotating at 0.084 rps, as for the filling experiment with sand, Figure 2. The pulse spreads out and moves through the cylinder, and eventually the cylinder empties. The first measurement of the bed depth profile was made 22 s after the pulse of sand was poured into
Figure 6. Granular bed depth profiles for a pulse of 265 g of sand moving through the 1 m cylinder held at an angle of 3.2° to the horizontal with a rotation speed of 0.084 rps. Parts a-e show comparison of experimental results against model predictions for times 22, 147, 297, 397, and 533 s after the injection of the pulse. Part f shows the model predictions for the granular bed progression at times 22, 73, 147, 273, 423, and 533 s. The initial conditions of the model were adjusted to match the experimental results for the first reading (22 s after the injection of the pulse), with the constraint of 265 g of sand entering the cylinder.
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Figure 7. Steady state granular bed depth profiles for glass beads, flow rate of 6.2 g/s and rotation speeds: (a) 0.063, (b) 0.084, (c) 0.093, and (d) 0.109 rps.
the cylinder. The solid lines show the results of the model. In this case, the calculation for the initial condition of the model was set up by postulating a sand flow of 12.04 g/s into the top end of the initially empty cylinder for an injection period of 22 s, thus delivering 265 g of sand. This theoretical injection time, 22 s, was longer than the experimental injection time; it is a coincidence that the numerical value of this time is the same as the time of the first experimental measurement of the bed depth profile. These theoretical values of sand flow rate and injection period produced a bed depth profile which was used as the initial condition, at t ) 0 s, for the model calculation at later times; the values 12.04 g/s and 22 s gave a crude but adequate fit to the first measured sand profile. Figure 6 shows that the model gives a fairly good prediction of the development of the bed depth profile. The differences between the model and the experimental data may be due to uncertainties of measurement: from the data at t ) 22 s, it appears that the inventory, estimated from the data points, would be less than the known quantity of sand, 265 g, originally put in. This difference may be due to the “stopped-kiln” procedure: each depth is measured when the rotation is stopped for a short period. Figure 7 shows steady state bed depth profiles for glass beads at a mass flow rate 6.2 g/s at four rotation speeds. The solid lines show the results of the model, which consistently overestimates bed depth and hold up. An explanation for this discrepancy was found by observing the surface flow of the beads. Some beads, after moving in rigid body rotation close to the cylinder wall, instead of flowing down the free surface as assumed by the model, ran some distance along the cylinder wall, approximately parallel to the cylinder axis. This gives increased mean axial velocity and thus decreased hold up. Experiments with glass beads on feeding and emptying were not conducted, though, surprisingly, the results of an experiment on the behavior of a pulse of beads were well predicted by the model. The fact that the beads slide axially down the cylinder, on moving from the region of solid body rotation into the thin rolling surface region, implies behavior different from that of
rough granular material usually found in industry; the reason may lie in the fact that the glass beads are smooth and almost spherical. 5. Conclusions For granular flow in a sloping rotating cylinder, a theoretical model has been used to predict the processes of filling and emptying. Theoretical predictions were based on the unsteady state equations of Spurling et al.,5 which have no adjustable parameters. The numerical method follows that of Descoins et al.7 in modeling zero flow rate of granular material with a nonzero, but sufficiently small, flow rate, in order to deal the numerical singularity. This is in addition to the numerical singularity at the discharge end x ) 0 which occurs when h f 0. This singularity is removed by imposing h(0) ) h0, and the resulting bed profiles are sufficiently independent of h0 when h0 is sufficiently small. Numerical integrations were performed with h0 ) 0.000 01 m, much less than the minimum possible bed depth of one particle diameter. The new feature of the analysis is a study of bed depth, h, as h f 0, as it must toward the start of filling and toward the conclusion of emptying. The condition h f 0 is also crucial for a finite heap of granular material moving through the cylinder. These theoretical predictions of bed depths, as functions of position and time, agree well with experimental measurements of bed depths for sand and poppy seeds during filling and emptying. The theory also works well for the following situations for unsteady flow of sand and poppy seeds. (i) The rate of discharge following cessation of feed for a cylinder initially in steady state, i.e. discharge rate ) feed rate, is well predicted. (ii) The motion and shape of a slug of sand, injected over a short period at the entrance to a rotating cylinder, is well predicted. In the case of glass beads, the model significantly underestimates bed depth and consequently holdup. This is likely to be
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due to the fact that the glass beads were smooth and spherical and thus able to roll and slide in a way that is uncharacteristic of industrial granular material. The filling and emptying times appear to be appreciably greater that the mean residence time, τ, τ being for steady state when feed rate ) discharge rate. The theory established herein should be useful for industry for answering questions about unsteady operation, for example, (i) How long does it take a rotary kiln to reach steady state, when granular material is supplied to an empty kiln? (ii) How long does it take to empty a rotary kiln after steady state and following shutdown of the feed? Acknowledgment We are grateful to Huntsman Tioxide and the EPSRC for a CASE award for BPML and to Dr. R. Horne of Huntsman Tioxide for helpful advice. Nomenclature A ) cross-sectional area of granular bed (m2) dp ) diameter of a granule (m) Fr ) Rω2/g, Froude number g ) acceleration due to gravity (m/s2) h ) bed depth (m) h0 ) bed depth at the discharge end x ) 0 (m) i ) dummy integer variable L ) length of cylinder (m) n ) rotation speed (rps) N ) number of cells Q ) volumetric flow rate of granular material (m3/s) R ) radius of cylinder (m) x ) axial distance from discharge end of cylinder (m) y ) h/R β ) inclination of cylinder (deg) γ ) angle of repose (deg) τ ) mean residence time in steady state (s)
δ ) axial length of a cell (m) ω ) rotation speed (rad/s)
Literature Cited (1) Perron, J.; Bui, R. T. Fours rotatifs: Mode`le dynamique du mouvement du lit. Can. J. Chem. Eng. 1994, 72, 16. (2) Perron, J.; Bui, R. T. Rotary Cylinders: Solid Transport Prediction by Dimensional and Rheological Analysis. Can. J. Chem. Eng. 1990, 68, 61. (3) Sai, P. S. T.; Surender, G. D.; Damodaran, A. D. Prediction of Axial Velocity Profiles and Solids Hold-Up in a Rotary Kiln. Can. J. Chem. Eng. 1992, 70, 438. (4) Sriram, V.; Sai, P. S. T. Transient Responses of Granular Bed Motion in Rotary Kiln. Can. J. Chem. Eng. 1999, 77, 597. (5) Spurling, R. J.; Davidson, J. F.; Scott, D. M. The transient response of granular flows in an inclined rotating cylinder. Trans. Inst. Chem. Eng. 2001, 79, 51. (6) Scott, D. M.; Davidson, J. F. Dynamics of particles in a rotary kiln. In Granular Materials: Fundamentals and Applications; Anthony, S. J., Hoyle, W., Ding, Y., Eds.; Royal Society of Chemistry: U.K., 2004; p 319. (7) Descoins, N.; Dirion, J.-L.; Howes, T. Solid transport in a pyrolysis pilot-scale rotary kiln: preliminary results - stationary and dynamic results. Chem. Eng. Proc. 2005, 44, 315. (8) Henein, H.; Watkinson, A. P.; Brimacombe, J. K. Experimental study of transverse bed motion in rotary kilns. Met. Trans. 1983, 14B, 191. (9) Lim, S.-Y. Particle Dynamics in rotating cylinders. Ph.D. dissertation, University of Cambridge, U.K., 2002. (10) Saeman, W. C. Passage of solids through rotary kilns: factors affecting time of passage. Chem. Eng. Prog. 1951, 47, 508. (11) Kramers, H.; Croockewit, P. The passage of granular solids through inclined rotary kilns. Chem. Eng. Sci. 1952, 1, 259. (12) Lebas, E.; Hanrot, F.; Ablitzer, D.; Houzelot, J.-L. Experimental study of residence time, particle movement and bed depth profile in rotary kilns. Can. J. Chem. Eng. 1995, 73, 173. (13) Liu, X. Y.; Specht, E. Mean residence time and hold-up of solids in rotary kilns. Chem. Eng. Sci. 2006, 61, 5176. (14) Spurling, R. J. Granular flow in an inclined rotating cylinder. Ph.D. dissertation, University of Cambridge, U.K., 2000.
ReceiVed for reView January 29, 2008 ReVised manuscript receiVed May 23, 2008 Accepted May 27, 2008 IE800156R
