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Powder Technology. 28 (1981) 253 - 260 _ 0 Elsevier Sequoia S A., Lausanne -Printed in The Netherlands

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Flow of Particdate Soli& Through TumbIing Milk

S- H R. SWAROOP, A-Z. M_ ABOUZEIDf and D. W. FUEFISTENAU

Department of Materials Science and hfineml Engineering, UniwrsiLy of California. Berkeley. CA 94720 (US A.)

(Reteived May 7.1980; in revised form October 8.1980)

SUMMARY

The fmnsport behavior of particulate solids flowing through tumbling milk depends strongly on the mill opemtingconditions. This paper presents the results of a detailed study to delineate the effect of the important oper- ating mriables on the hold-up, mean residence time and residence time distribution of partic- ulate solids flowing through ball milts and rod mills_ The effectsof feed mte. media load, and mill speed on these transport parameters are d&c-d, emphasizing the fundamental dif- ferences between particulate transport in ball millsand rod milts. Under identrcal dimension- les operating conditions ouer the range of the inuestigation, the mater&z1 hold-up and the

Peclet number of the flow regime in the rod mill were always higher than those in the ball mill. Mechanisttc interpretations of the ob- served transportphenomenaarepresented, and their implica Cons in the con tert of turn bling mill analysis and design pointed out

INTRODUCTION

The processing of granular materials often involves the transport of particulates through rotating cylinders. In developing mathematical models for the design and control of contin- uous particulate processing systems. informa- tion about particulate transport through the device is almost invariably required. Tumbling mills are a typical example of systems that involve transport of matenaI as a sub-process in their operation.

In the recent past, the need to explicitly account for the important sub-processes in

*Presently Associate Professor, Cairo University, Faculty of Engmeering, Dept. of Mining. Giza, Egypt.

continuous grinding systems has been well appreciated, and the material transport sub- process in ball mills has been analyzed to some extent. Kelsall [l] and Kelsall et al [2 - 41 studied the effect of operating conditions on the solids transport m a continuous wet overflow ball mill. In their work, the tracer step and impulse response techniques were employed to obtain the expenmental data which were analyzed in the context of the ideal mixer with delay model. They found that the hold-up of solids in the mill increases with increasing feed rate, ball diameter, and ball load; while the mean residence time decreases with increasing feed rate, ball diameter and increases with ball load. They also concluded that transport behavior in the mill approaches that of a perfect mixer W-X-I increasing hold- up of material in the mill. Mori 151, who studied the residence time distribution of dry solid materials flowing through a ball mill, observed that the dispersion coefficient of solids per unit revolution of the mill is con- stant independent of feed rate and/or mill rotational speed. He also found that the hold- up of material in the mill is proportional to the square root of the feed rate to the drum, i.e., the mean residence time is inversely pro- portional to the square root of the feed rate. The effect of feed rate, ball load and dis- charge opening diameter on the material hold- up and residence time distribution in open- circuit ball mills has been investigated by Keienberg et al. [S] and Mori et al. [7] _

Karra and Fuerstenau Cl53 investigated the effect of discharge opening design on material flow through a rotary drum and found that the residence time distribution tended towards

plug flow at an intermediate discharge open- ing. Abouzeid and Fuerstenau [S] studied the

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effect of mixing aids (Lucite balls of density l-25 g/cm3) on the transport behavior of particulate solids in a constricted-end drum_ They found that the material hold-up in the mill increased linparlv with increasing feed rate, decreased with increasin gballloadtoa _ _ muumum value where it became independent of ball load, and varied in a complicated manner with drum speed. They also found that the variance of the residence time distri- bution increased lincazly with balI load to a limiting value where it became independent of bzll load, in creased nonlinearly with drum speed, and decreased nonlinearly with feed rate. An important conclusion of their work is that in the przseace of balls as mixing aids, there is no segregation in a particulate system with components of widely differing physical properties_ Hogg et ul. 191 fitted a semi- empirical model and predicted the hold-up of material in a laboratory ballmill. Their predic- tions were successful at low feed rates of material to the mill. Mori ef al. [7] and Keienberg et aL [S] proposed using the log- normal distribution function to represent the residence time distribution in ball mills.

Although there have been a reasonable num- bzr of studies of material transport through _ laboratoryscale ball mills, very little has been done to investigate the transport of par&u- lates through rod mills. On the basis of a few experiments in a laboratory wet rod mill, Heyes ef al. [lo] concluded that the contents of a rod mill are extremely well mixed in a manner similar to the contents of a small ball mill under conditions of high hold-up. However, their results are not &equate for describing the &msport.characterisGcs of solids flowing through a rod mill. Detailed investigations concemin g hold-up, mean resi- dence time, and residence time distribution are required for any successful modeling of this type of tumbling milI systems_

This paper deals speci&aIly with material transport through tumbling mill systems, in- cluding both ball mills and rod mills. It pre- sentr the results of a detailed study to delin- eate the effect of operating variables on the hold-up, mean residence time, and residence time distribution of particulate solids flowing through these systems. Comparison between material flow through ball miUs and rod m.ilIs is emphasized. Material transport characteris- ticsofrodmiUswilIbestressedheresincea

relative wealth of information on transport through ball mills exists.

APPARATUS AND EXPERIMENTAL

-The feeder/sampler set-up for this research has been described in another publication [ll] _ The mill used was a s&&zless steel cylin- der 43.8 cm long and 12.7 cm in internal diameter, with eight internal longitudinal lifters of 0.3 cm height. At the feed end of the mill, a 2.5 cm opening permitted introduc- tion of feed t0 the mill. Since the mill was operated in an open-end configuration, the discharge end was completeIy open, except for a wire grid to prevent balls or rods from falling out of the milI. Lucite balls of density 1.25 g/cm3 and hollow lucite rods, both of 1.9 cm diameter, were used to simulate the grinding media without any appreciable grind- ing action. These will be referred to variously as the grinding media, or simply the media, in the rest of this paper_

The material was fed to the mill under specified conditions until steady-state opera- tidn was obtained. At this point a tracer impulse was introduced at the inlet of the mill, and the discharge was sampled at regular inter- vals to obtain the residence time distribution from the response to the tzacer impulse. When all the tracer had discharged from the mill, the mill was stopped and the material hold-up was measure d. In all cases, the bulk material used was 14 X 20 mesh dolomite, a portion of which was colored with a dye so that it could be used as the tracer. However, while investi- gating the effect of mill speed on material trausport through the rod mill, 20 X 28 mesh dolomite was used as the feed material. Based on 0uT earlier studies [ll] , this difference in particle size has no effect on the transport behavior of particulate solids through the miU_ The material hold-up distribution along the mill was measured by replacing the upper half of the mill by a sampler and slowly rotating the mill 180 degrees around its axis. The material hold-up was thus split into six equi- distant samples along the mill axis.

The Peclet number, Pe, and axial dispersion coefficient, D, were obtained from the &men- sionless vanance of tie tracer response at the mill emt in the context of the axial dispersion model for a closed-clod system. Deta& of the method of calculations are discuYed else- where [ll]_

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RESULTS

Hold-up Effect of feed rate on hold-up As the feed rate of material increases, the

hold-up in the mill increases linearly. This is true for both rod and ball mills (see Fig. l)_ In the ball mill, the hold-up is generally lower than that in the rod mill at corresponding feed rates As is clear from Fig_ 1, the maximum feed rate to the ball mill is much less than that for the rod mill. The increase in hold-up with increasing feed rate probably results from the resistance offered by the grinding media (balls or rods) to the flow of particles through the milks. As a result, an increase in feed rate to the system requires a higher head for material to push itself through the grinding media to reach the discharge end [ 81. The reason for the lower hold-up in the ball mill compared with that m the rod mill under identical di- mensionless conditions appears to be due to the difference in the hold-up distribution in the mills. Figure 2 shows the hold-up dislzibu- tion in both rod and ball mills. It can be seen that the hold-up distribution along the mill is nonlinear and that the hold-up proi5les in the two systems have different shapes. In the absence of a reasonably good linear hold-up profile iu both the systems, it is not possible to rigorously explain the difference 111 hold-up between the two systems. Further results on

Fig. 1. Dependenrz of hold-up OP feed rate to the ball mill and rod mill

Fig. 2. Axial hold-up pro?iiles in the ball mill and rod mill under identical dimensionless operating condi- tions

the effect of media on the hold-up distribution along the mill are needed in order to explain the lower hold-up in ball mills compared with rod mills operated under identical chmension- less conditions However, it can be seen horn Fig. 2 that there is a higher build-up of material at the inlet end of the ball mill and this is responsible for a lower limiting feed rate (the feed rate at which material builds up

sufficiently at the inlet end and causes a back- flow) observed in the ball mill.

Effect of load of grinding medi4 on hold-up The hold-up of material in the mill increases

as the media load (balls or rods) increases at the same feed rate. In addition, one can see from the results plotted in Fig. 3 that the hold-up in the ball mill is lower than the

I I I I I

Fig. 3. Dependence of hold-up on media load in the ball and rod mills.

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corresponding hold-up in the rod mdI at all media loads_ The continuous increase in hold- up with media load may be atibuted to the increased resistance to material flow as the media load is increased_ At lower levels of media load, most of the shear zone wdl be ti of the media and the material d ex- perience a relatively smaIl resistance in moving towards the discharge end_ Hence, a smaller driving head is required to push material through the mill, resulting in a lower hold-up at low media loads. As the media load in- creases, an increa~& interaction between the flowing material and the media ensues, result- ing in a higherresistance to material transport. This, in turn, requires a higher material head for the material to discharge at the same rate, resulting in a higher material hold-up_ In contrast, material hold-up decreases with increasingmediaload in a constrict&end mill, the details of which have been discussed else- where [S] _ The reason that the hold-up at any media load is higher in the case of the rod mill is probably related to the differences in the hold-up distribution in the two systems, as already discussed in the foregoing section.

Effect of mill speed on material hold-up Figure 4 shows the material hold-up as a

function OF mill speed for both rod and ball mills. As can be seen from the results given in this figure, the hold-up in the baII mrll at all speeds is lower than the corresponding hold- up in the rod mill under similar operating conditions_ This, again, is perhaps related to the differences in the axial hold-up profiles in

Fig. 4_ Dependence ofhold-up on miU rotational speed intheballmillandrodmill.

the two systems. (This has been ilk&rated in the hold-up distribution in both balI milk and rod miIls in Fig_ 2.) The trend of the hold-up as a function of mill speed is quite similar to the corresponding variation in hold-up with drum speed in rotary drums without any mixmg aids [ 12]_ The hold-up is larger at both low and high m!ll speeds, passing through a minimum at the intermediate speeds. The hold-up is hil:h at low mill speeds because of the high friction to the flow of material as a result of the presence of a thin shear zone [ 131) and is high at high mill speeds because the flowing material cataracts and occupies more volume in the mill [ 141. In both ball and rod mills, the material hold-up is an increasing function of mill speed in the operating range of mill speeds.

Residence time distribution (RTD) The characteristics of the residence time

distribution can be expressed through a single parameter, namely the Peclet number, Pe, which is a m easure of extent ,f dispersion_ Low Peclet numbers indicate a high degree of dispersion and uice-versa. It is important to recognize that, while the Peclet number characterizes the extent of &persion, the intensity of (axial) dispersion is represented by the (axial) dispersion coefficient of materi- al flowing through the gnren system_ Thus, in the context of the axial dispersion model, each of these phenomena must be evaluated through the appropriate parameter_ The independent variables whose effects were investigated in this aspect were the feed rate of material to the mill, media load, and mrll speed_

Effect of material feed rate on RTD Figures 5 and 6 show the Peclet number and

the dispersion coefficient of the material flowing through the rod mrlI and the ball mill as a function of material feed rate. The Peclet number mcreases and the dispersion coeffi- cient decreases with increasing feed rate for both the mills. This behavior for both the mills may be explained by recognizing that, under a fixed media load and mrlI speed, at low feed rates to the milI the level of the material sur- facein themill is low (see the section on hold- up). This implies that the material in the miIl does not completely fill all the voids between the tumbling media. As a result, a large frac-

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Fig. 5. Dependence of Peclet number on the rod 4 and ball mill_

c

feed rate to

Fig_ 6_ Dependence of ax&l dispersion coefficient on feed rate to both the rod 1.41 and ball mill.

tion of the flowing material interacts with an ‘excess’ of tumbling media, resulting in a relativeIy highly dispersed mill discharge. At

higher feed rates, the hold-up of material in the mill increases, filling almost all the voids between the media, and because the material motion is restricted, a low dispersion coeffi- cient results. From Figs. 5 and 6, it can be seen that the values of the axial dispersion coefficient and the Peclet number are about six times higher in the case of the rod mill than in the case of the ball mill under similar conditions. This appears to be due to the axial ‘shoveling action of material by the relatively unconstrained motion of balls in a ball mill, whereas in a rod rnti the motion of rods is restricted in the axial direction, and hence their effect on the axial dispersion of material

would be correspondingly smaller. The im- plications of this signScant difference in the material transport characteristics between dry rod and ball mills over the entire range of admissible feed rates cannot be over- emphasized.

Effec L of media had on RTD The results of this series of experiments are

presented in Fig. 7, which shows the relation between the dispersion coefficient in both the ball and rod mills as a function of media load expressed as a percent of the mill volume, and Fig. 8, which shows the corresponchng varia- tion in Pe. Clearly, the variation of the dlsper- sion coefficient and Peclet number with media load in the two mills chsplay completely opposite trends. This can ‘be explained in terms of the possible nature of particle-media inter- actions in the two systems. As already indi-

Fig.8 Variation of Peclet number with media load in the ball mill and rod mill.

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cated,theaxialco . - t onthe~~otionof rods considerabIy limits their ability to dis- perse the material in the axial direction and this may be the reason the PecIet number is generally high in the presence of rods. In addition, relative to a mill with no tumbling media, the addition of a few rods to the system creates channels of flow between the rods where particleparticle and particlerod interactions tend to become &&ad. In this context, the trajectories of dispersed particles are shortened, and, on the average, the (axial) dispersion of particles with a finite axial com- ponent of motion is reduced. With an increased addition of rods to the mill, the channels for flow through the mill (that is the flow chan- nels between the rods) hecome smaller and the &&ion of material flowing in these re- stricted sections increases. In turn, tbis implies that the axial dispersion coefficient decreases with increasing rod load.

At this stage. it might be appropriate to elaborate on the relationship between the Peclet number and the axial dispersion coef- ficient and its implications in the context of the results and discussion presented in this paper. The Peclet numb-, Pe, end the axial dispersion coef&ient, D, in cm2/sec are related to each other in the form

whereListhe~lenghincm,uisanaverage particulate convective velocity in cm/set, T is the mean residence time, and D is the axial dispersion coefficient. For a mill of given length, the relationship between Pe and D is not necessarily a one-to-one inverse relation- ship because of the possible simultaneous variation of T_ This is clearly demonstrated in Fig_ 9 where the Peclet number and the dis- persion coef5cient are plotted as a function of media load in the rod mill. Thus, while the dispersion coefEcient decreases continuously with increasing rod load, the Peclet number increases with rod load only up to a certain point beyond wluch it remains nearly constant. In the region of constant Pe, the decrease in D is compensated by an identical increase in r, resulting in a constant Pe with increasing rod load. It is our contention that this relatively simple point is easily overlooked

and that in a discussion of the effect of process variables on mixing in tumbling mill systems from a physical standpoint, we may usually only explain these effects on the intensity of dispersion (as characterixed by D) rather than the extent of dispersion in the mill discharge (as characterized by Pe). The effect of process variables on Pe is inevitably coupled with the corresponding effects on the mean residence time in the mill.

In contrast to the effect of media load on dispersion in a rod mill, the addition of balls to a ball mill increases the dwpersion coeffi- cient (Fig. 7) and decreases Pe (Fig. 8) initially. This is because balls, unlike rods, are free to move axially in a random fashion and in- creasing the ball load should then result in increased stirring and axial shoveling of par- ticles back and forth, leading to an increase in the dispersion coefficient Furthermore, as shown schematically in Fig. 10, the axial dispersion of particles re8ected off the media surface would be much greater in the case of balls. The increase in dispersion coefficient with increasing ball load continues until the particulate bed is almost fully impregnated with balls (at about 40% ball load). Balls ad&d to the mill beyond this level probably do not participate very much in particle/ball interaction, but rather tend to interfere with the dispersing action of the other balls in the system. This, it is felt, is responsible for the decrease in the dispersion coefficient observed beyond about 40% ball load_ Once again, the constancy of the Peclet number in this range of ball loads results from the com- pensating influences of D and T (Pig. 8).

Effect of mill speed on RTD The effect of mill rotational speed on the

axial dispersion coefficient and Peclet number for materiel flow in ball and rod mills is presented in Figs. 11 and 12, respectively. In both mills, the dispersion coefficient is low in both the lower and higher range of mill speeds but is bigber in between. The lower dispersion coefficient at low mill speeds may be due to the fewer number of particle- particle and particl~media collisions per unit time under these conditions. In the higher range of mill speeds, the onset of cateracting of the media and the material in the empty space of the mill (as m in the section

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I -0

I I I I I la, ID zo

FLOD LOaD PERCEaT=:F YU “it”“, 50

Fig_ 9. Variation of PecIet number and axial dispersion coefficient with rod load

--LU*l_ DILCTIOm-

PaRTInE -caLL INTERI-

Fig. 10. schematic representation of particle-rod and particle-hall interactions.

Fig. 11. Dependence of arial disperuion coefficient on miIl rok&ianal speed in the ball mill and rod mill-

IX ,

Fig. 12. Variation of Peclet number with mill roa- tional speed in the ball and rod mill.

on hold-up, where this phenomenon is re- flected in higher hold-up) leads to a reduced frequency of particle-particle and particle- media interactions, again resultin~in a decrease in the dispersion coefficient.

In the case of the ball mill, Pe decreases from 11 to 7 as mill speed is increased from 25 to 50% of the critical speed (Fig. 12). Beyond this speed, Pe remains nearly constant, indicating that D X r in the ball mill is nearly constant. This 1s in contrast to the correspond- ing variation in Pe in the rod mill where D X T is decreasing, resulting in a higher Pe at high mill speeds. Thus, in the operatiug range of mill speeds, the Peclet number is almost con- stant in a ball mill whereas it increases with mill speed in a rod mill.

CONCLUSION

In this paper, the results of a detailed study to delineate the effect of operating variables on the hold-up, mean residence time and resi- dence time distribution of particulate solids flowing through tumbling miU systems are presented and compared_ Particulate trsnsport through rod mills has, in the few instances in the past, been assumed to be similar to that through ball mills. The present study has brought out the fundamental differences that exist between particulate flow through these systems. It appears that rod mills operate with

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considerably different material transport char-

acteristics compared wi+h ball mills under - conditions, and that one may not, in

general_ assume material flow through these systems to be similar.

The operating variables investigated include the feed rate to the mill, media load and mill speed_ In both ball and rod mills, the material hold-up is an increasing linear function of the feed rate to the mill_ However, in hall mills the mean residence time for material taansport

is nearly constant with change in the feed rate to the mill, while zt decreases with increasing feedratet.otherodmill.Theseresultsare significant in the context of scaling-up the feed rate to a tumbling mill. For instance_ the variation of the mean residence time with feed rate to a rod mill makes the me-up of feed rate in this case more complia3ted.

Increase in media load was found to lead to an increase in material hoId-up, due to the increased resistance to material flow in both systems_ The effects of media load on the Peclet number in ball and rod mills are very interesting and not intuitively anticipated. While the Peclet number decreased with in- creasing ball charge due to an increased mixing action, it increased with increasing rod charge- This seemingly paradoxical phenomenon can be explained on the bzuis of a postulated mechanism of particlemedia interaction in the two systems_

The variation of material hoid-up with mrll speed in both ball and rod mills was found to be quite complex, although they exhibit similar trends in a broad sense_ Over a range of miB speeds (about 40 - 60% of mill critical speed for rod mills and 50 - 7Q% of mill critical speed for ball mills), the hold-up is approxi- mately constant, but at rotational speeds above or below this range the hold-up increases sharply. The Feclet number of the flow regime in rod mills follows a trend similar to the variation in hoId-up with mill speed, whereas that for flow tbrougb ball mills decreases con- tinuously with in creasingmillspeedupto about 50% of the critical speed, beyond which it remains nearly constant. Thus, in the operating range of mill speeds, the Peclet number is almost constant in a ball mill, whereas it in creaseswithmillspeedinarod

-mill: This important result must be kept in - mind if it is dedred to change the operating

speed of a tumbling mill.

Finally, a very useful general result is that over the entire range of investigation, the material hold-up and Peclet number of the flowregimeint.herodmiJlarebigherthanthe corresponding values in the ball mill under identical dimensionless operating conditions_ The importance of the material hold-up m dete _ - gthe breakage kineticsin a grinding mill is well established. Everything else being eq4 a higher Peclet number for material transport implies a narrower size distribution in the mill discharge. The narrow size distribu- tion observed in industrial rod mill products has generally been related to the breakage kinetics in these mills, but it is clear from the present investigation that material transport in these mills is perhaps an equally important factor, too.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the sup- port of this research by the National Science Foundation and the US_ Bureau of Mines_

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