J. Soc. CosmeticChemists9,137-51 (1970) ¸

t97o Societyof Cosmetic Chemiats of Gre,,tBritain

The kineticangleof reposeof powders j. j. KELLY* Presentedat the sy•nposimnon "Powders", organisedby the PharmaceuticalSocietyof Ireland and the Societyof Cosmetic Chemistsof Great Britain, at Dublin, on 17th April 196,9.

Synopsis--The factors which contribute to the KINETIC ANGLE OF REPOSE of POWDERS in the flights of ROTARY DRUMS are experimentally investigated. The results, presented graphically, show the effects of PARTICI,E SIZE, DRUM SPEED and the •naterial type on the kinetic angleof repose.MEASUREMENTS of the STATIC ANGLE OF REPOSE are also taken and correlations with the particle size are statistically derived. Comparisons are made betweenthe static and kinetic anglesof repose. COOLING-DRYING

INTRODUCTION

The final stagesof the manufacturingprocesses for many granular chemical products often involve a drying or cooling stage. In the heavy chemical industry, such as fertiliser manufacture or sugar production, these operationsare commonly carried out in a rotary drum process, whereby the granulesor powdersare brought into contact with either cooling'or dryinggasesfor a fixed periodof time. The drum is mountedat

a slightangle( < 6ønormally)to thehorizontalandhasliftingflightsrunning lengthwiseon the insidesurfaceof the drum circumference. The purposeof the flightsis to improvethe gas/solidcontactby lifting the granulesinto the upperhalf of the drum and cascadingthem downwardsacrossthe flowing gas stream.The granulesto be processedare fed continually to the higherendof the drum,whence,dueto the angleof the drum, togetherwith the lifting action of the flights, the granulesprogressthrough the drum length to the lower end where they exit suitably processed. *University College, Dublin, Ireland. Now at Collegeof Engineering, University of Maryland, College Park 20742, U.S.A. 37

38

JOURNALOF THE SOCIETYOF COSMETICCHEMISTS Research

in this field has been concentrated

in the heat

and mass

transferareasof this processand in the deriving of residencetime relationshipsfor the granularmaterial;very little has beenreportedrelatingto the dynamicsof the granular flow through suchapparatus.The rates of heat and mass transfer are dependenton the effectivenessof the gas/solid contact;the factorswhich dictate the form of the cascadefrom the flights are therefore of fundamental relevanceto the analysisof the operation of this equipment.The powdersare collectedby the flights in the lower half of the drum and cascadecommencesonce the angle of the powder in the flight exceedsits equilibrium value; the top surfaceof the flight holdup maintains a rolling motion towards the flight edgewhere it cascadesinto the gas stream, until the flight is empty.

8=9o o

8=180 ø

Kinetic angle of repose.

Forces actingon a granule.

Figure 1.

Forces acting on a rolling particle in a rotary drum flight.

Earlier designprocedures for the shapeof flightsassumedthat the angle of the material in the flight xvasa constantfor any one material and so it was thereforepossible,knowingthis angle,to calculatethe quantity in the flight at any positionon the top half of the drum circumference and therefore the cascaderate at any drum speed.A closeranalysisof this phenomenon showsthat this assumptionis erroneous.The powdersin the flight are continually subjected to two imposed forces, gravitational and centrifugal, and, for the rolling powderson the surface,a self-generatedfrictional forceopposesthis rollingmotion. Fig. 1 illustratestheseforces.Whilst the gravitational force always acts in the vertical direction, the centrifugal force, acting outwardsfrom the drum's centre, has a varying direction of actiondependenton the positionof the flight on the drum circumference;

Figure •.

Facing page 39

Apparatus for measurement of kinetic angle of repose.

THE KINETIC

ANGLE OF REPOSE OF POWDERS

the kinetic angle of reposeof the powdersis thereforea variable, its value for any drum anglebeingdictatedby the balanceof thesethree forces. Schofieldand Glikin (1), assumingthat the flow of the powdersacross the top surface of the flight holdup representedconditions of dynamic equilibrium, balancedthese forcesand arrived at the following expression for the kinetic angle of repose:

)•=tan-1

!•+ u (Cos0 -- • Sin 0)* ................ (i) 1--u {• Cos 0 q-Sin 0) The range over which this relationshipmay be applied with reasonable accuracy(i.e. suitablefor flight designpurposes)has beeninvestigatedby Kelly and O'Donnell (2), who concludedthat it held generallytrue up to values of u=0.4. The parameter u equalsthe ratio of the centrifugal to gravitational forcesacting on the surfacepowders,and, as a dimensionless numberwhosevalue definesthe drum speed,is the one best suitedto rate the accuracyof the aboveequation.At higherdrum speeds,the assumption of dynamic equilibrium conditionsexisting no longerholds;when u= 1.0, the centrifugaland gravitational forcesare equal, the drum is at its critical speedwhenthe powderswill no longercascade,but will adhereto the inner drum circumferencethroughout the rotation. The only property of the powder containedin the above relationshipis the kinetic coefficientof friction (•); this representsthe value of the coefficientof friction of the slidingpowdersthat allowsfor a balanceof the forceslisted above. The kinetic coefficientof friction, as a property of the powder, is dependenton characteristics suchasthe particlesizeand shape,its moisture content,possiblyits particle density, etc. Much work has been reported on the static angleof reposeand coefficientof friction of powders(3); in particular, the effect of particle size has been investigatedwhere it has been shown that the static angle of reposeand hence the static coefficientof friction, is inverselyrelated to the particle size. MEASUREMENT

OF KINETIC COEFFICIENT

OF FRICTION

The apparatususedin theseexperimentsis shownin Fig. 2. It consists of a horizontalrotary drum which has eight circularflights located at 45ø intervals on the insidecircumference;a variable speeddrive is fitted to the drum. The flights are half filled with the material to be analysed.Photographsare taken of the drum at differentspeeds.Readingsof the anglesof * Table of symbols in page 47.

40

JOURNALOF THE SOCIETYOF COSMETICCHEMISTS

repose(M and the positionalangle of the flight (0) are taken from each photograph.For any one set of conditions(i.e. powder, particle size and drum speed),thesereadingsare analysedin a computerprogrammewhich computesthe value of I• giving the least mean standarddeviation from the theoreticalanglesof reposeaspredictedfrom equation(I) EXPERIMENTAL

REAI)INGS

AND RESULTS

Two separatematerialswereinvestigatedwith sizesasfollows: Pumice

BSS Sieve Size

7+ 14 --14+ 18 --18+ 25 --•25+ 36 --36552 --52+ 72 --72 + 100

Sugar

BSS Sieve Size --18+

25

---25+ --36+ --52+

36 52 72

For the pumice,drum speedsof 8, 16,24 and 32 rev min- l wereinvestigated; for the sugar, 16 and 24 rev min- 1. The resultsof this investigationare tabulatedin Table1. Table

I

Values of IX for pumice and sugar powders rev min- 1

B.S. sieve

size

Ave. particle

s,ze

Pumice 8

Sugar

16

24

32

16

24

--0,94 0.81

--0.91 0.80

0.49 0.44

0.42 0.37

7+ 14 ---14.4. 18 --18+ 25 ---25.4- 36

1 800 1 025 725 510

1.00 0.98 1.00 0.97

0.88 0.97 0.92 0.91

0.89 0.93 0.95 0.93

---36.4- 52 52 -4- 72

360 255

0.96 0.87

0.90 0.86

0.87 0.79

0.94 1.01 0.96 0.95 0.77 0.77

- -72+100

180

0.82

0.79

0.76

0.63

Theseresultsare showngraphicallyin Figs. 3--5.

THE KINETIC

ANGLE OF REPOSE OF POWDERS

41

rev min-t

10--

rev rain-1

24 I6

i 500

i

i

IOOC•

1500

rev rain -I rev rnm-I

I 2000

dp, pm

Figure3, Plot: Particlesizeand kinetic coefficientof frictionfor pumicepowders.

DISCUSSION

From the results,the followingobservations may be immediatelymade with regard to the values of p.

•[fect of particlesize(Figs. 3 and 4) The resultsindicatea falling off in the value of p with a decreasein the averageparticle size. In the caseof pumice,this effectis only true for powderswhoseaverageparticle sizeis lessthan about 500 pm; above this value, and at all drum speeds,p is reasonablyconstant;below this critical

42

JOURNALOF THE SOCIETYOF COSMETICCHEMISTS 10

..16 0.9

rev rain-I

.•. 32 rev min -I

0,8

0,7

0.1

0

I00

200

300

400

500

600

700

800

dp, /4m l:igttre 4.

Plot: Particle size and kinetic coefficientof friction for sugar powders.

value, all the curves show a definite downward trend indicating easier

flowingconditions.With sugarpowders,the trend downwardswith average particle size is more definite and consistentover the limited range investigated.

Effectof drum speed(Fig. ,•) With both materials, very little effect of drum speed on the kinetic angleof reposeis evident; a slight trend of reductionin p with increasein

THE KINETIC

ANGLE OF REPOSE OF POWDERS

43 elO25.P

J'0-

i••----"•' •.._ •"• '• -•---, •"•.-'-"•

/'•, 725.P

----.--'•= 5 I0 'P e•'"'""•'-/ 1800'P

0,9-

0.8--

510

255.P

0.7-

180,P 0,6-

0,5--

"'" •..

.

360.S 255.S

Average particle size O,

= P N

•

Pumice

Sugar

o x• 8I

I

16

I

•-

I

32

N, rev rain'1

Figure 5.

Plot: Drum speedand kinetic coefficientof friction for pumice and sugarpowders.

drum speedis noticeablewhich could possiblybe detectedby a more rigorousstatisticalappraisalof theseresults. Effectof material(CompareFigs. $ and The muchlower valuesof ixobtainedfor the sugarpowdersindicatea definiteeffect. Thismeansthat overthecommonrangeof sizesinvestigated,

44

JOURNALOF THE SOCIETYOF COSMETICCHEMISTS

sugar powdersflow much more easily than pumicepowdersof the same size.This would suggestadditionaleffectsdue to propertiesof the material suchas the particle shapeand/or particle density. MEASUREMENT

OF •,

In the readingof the anglesof reposefrom the photographicslidesfor the finer gradesof both pumice and sugar powders,a questionarosein decidingon the correctangle for measurement.Fig. 6 illustrates this problem; the pumice powder in these flights lies within the BSS sieve size range --52+ 100. It is noted that the angle of reposevaries within each flight; particularlyin the upperhalf of the drum;at approximatelyhalf way alongthe powdersurface,the angiechangesnoticeably.In readingoff the angleL for the evaluationof I•, it was decidedto take the lower angle (left hand in Fig. 6); it is arguedthat this is the more correctone as it is the angleat the terminationof the particles'"roll downwards''; it is alsothe angle of cascadein rotary drum flights. The powderhas had sometime to reach this loweranglewhereasthe upperor right hand angle,whichis alwaysthe biggerof the two, is being formed by particlescommencingtheir roll. The particlesat the lowerpoint have reachedtheir 'terminalfrictionalvelocity' and therefore, at this point, the assumptionof dynamic equilibrium is justified. The highervalue of •, occursas the frictional forcehas not built up to its equilibriumvalue and this conditionallowsfor a greater, though unstableand temporary, angleof repose.The phenomenonwaslessevident with the larger sizedpowders,which,it is presumed,reachedtheir 'terminal frictional velocity' almost immediately after the commencementof their roll. STATIC ANGLE OF REPOSE

The results of measurementsof the static angles of repose for the pumicepowdersare givenin TableII; Figure 7 plotsthesefigures.As with the kinetic readings,the curve changesits character at the 500 I•m size. Over the-range of particle sizes500-1 800 [tm, the static angleof repose decreaseswith particle size; below this range, the static angle of repose risessharplywith decreasein particle size.Theseexperimentswere carried out accordingto the method as describedby Train (6). For each size, the •nean value of four separately taken values was obtained; the repeatability was good, the mean averagedeviation from the mean values given

-.

Figure 6.

330

Photograph of apparatus withpumice powder of B.S.sievesizerange--52 d-100 in closed flights.

Facing page 44

THE KINETIC

ANGLE OF REPOSE OF POWDERS

45

33'5 F

32.5

30,!

29.•

..,...."'•" /

=Regression lines

28'5 0

200

400

soo

eoo

•000

•200

•400

•600

•eoo

2000

Particlesize, •m

Figure7. Staticangleof reposereadingsfor pumicepowders.

in TableII, neverexceeded 0.5ø.A leastsquares regression analysison the

topfourpointsandagainon the lowerfourpointsprovidethe following equations:-

kst (dp>500 1 000 tun may produce an error in measured angle of several degrees. You do not state which method of Trains you have used. 'rue LEC•rURER:It is a poured angle of repose and exactly as described by Pilpel (3). Four readings were taken of each angle of repose and the repeatability was excellent, the mean average deviation never exceeding 0.$ ø. For the mean angles of repose the same sample was used each time for each of the four readings in each size range; I have no doubts about this curve. DR. T. M. ]o•s: I was wondering about the differences;excluding that 180 gm point, all your differencesare of the order of •--1 ø. This is very small when you consider a 2 000 gm particle. I am wondering whether that shape is possibly an important variable in Fig. 77

Tu• L•CTURER:It may be, but 1 do not think so. We separated the pumice by a normal sieve analysistaking samplesout of each sieve;we did not analyse the particle shape.

MR.J. E. B•,•C•URN: Couldweestablish the sizeof the problem'caused by the difference between taking into account the static angle of reposeand the kinetic angle of repose,i.e. what is the error by assuminga static angle in the whole of the drier?

T• L•CtURER: The formula for the drum holdup is given elsewhere{2). It is directly related to the flight holdup, and it makes allowance for the variations in the kinetic angle of repose.The kinetic angle of repose of, say, pumice, is of the order of 48ø or higher, at the commencementof cascade.The static angle of repose is much lower. The relevant area for this differencecalculation would be that bounded by these two angles in the flight. The static angles are of the order of $0ø•so it is quite a sizeable

area.To take 30øinsteadof the current45øwouldmeansomething neara $$«%error in the designholdup calculation. A MEMBER OF •'mZ AUmENCE: I would like to question the wall effects of your

apparatus; the ratio of the flight to particle size seemsquite high. T}•E LEC•U•: In fact, it is not very large, about 20:1. In packed distillation columns, etc. 8:1 or 12:1 are accepted to miniraise wall effects. TUE s• M•BER OF T• AUDieNCE: You can not really compare packed distillation columns with moving particles in flights.

50

JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS

THE LECZURER:Agreed, of course,but it is important to realise that the particles are not moving in the flight until they reach the top surface, i.e. until the granules staffc to roll.

THE SaMEMEMBERor THE AUDIENCE:I disagreevery profoundly. The movement of the particles will go down a large number of layers.

THE LECTURER:It may go down a little of course,but the bulk of the particleswill be stationary. We have a film somewhere to support me on this point. Equation I is based on this assumption and whilst there is a certain movement in the particles belowthe top rolling layersto a depth of perhapsone or two particles, the bulk of the particles do not move until they reach the top. THE SaMEMEMBEROFZHE AUDIENCE:This is preciselymy point becausethe wall affects the particles in the same way that they would on a much larger scale. ANOTHER MEMBER OF THE AUDIENCE: I would like to comment on Dr. Jones's

contribution. I think it is very dangerousto compareresultswith two different kinds of material, especiallywhen one is magnesium.What is this magnesium?Our laboratory found that magnesium can differ considerably from manufacturer to manufacturer, althoughhaving the sameparticle size.It dependson the previoushistory of the surface, on moisture, etc. What are the interparticulate forces xvhich Dr. Jones talked about? Is this due to moisture capillary attraction? I would suggestthat the differencesin characteristicsbetween particles of these sizesare remarkably small and the effectsare mainly due to moisture and to the fact that the particles hook together. Thefollowingwrittencommunication wassubsequently received from ProfessorH. E. Rose:

This paper is particularly interesting in that it suggestsa means by which the dynamical friction characteristicsof powdersmay be investigated;a field of work in which but few simple experimental techniques are available. As stated in the synopsis,the investigationrelates to the determination of the kinetic angle of reposeof materials in the flights of rotary drums and this aim is achieved. In this connection it should be noted that the kinetic angle of repose is a function of the frictional propertiesof the powder (which propertiescan probably, at least approximately, be characterisedby the static angle of friction) and of the dynamicsof the system--such as the size and speedof rotation of the drum. Thus, whether the kinetic angle of friction so obtained is applicable to casessuch as the flow from hoppersis questionable,and attempts to usethe data in sucha way would imply a recklessdisregardfor the laws of mechanics. With referenceto .Fig. 5, would it not be expectedthat for zero speedof rotation {N=0) the kinetic angle of friction would correspondto the static angle?This does not appear to be the caseand it would be interestingif the reasonfor this difference could be explained..Fig. 7 is as expected. For particle size below about 500 gm the cohesiveforcesincreaserapidly with decreasingparticle size and so the static angle of repose increasesrapidly with decreasingsize. This can be very important because,for example, with Portland

THE KINETIC ANGLE OF REPOSE OF POWDERS

cement the angle of reposeapproximates to 90ø and a tunnel, extending from the bottom to the top of a storage silo, can form. For particles with a diameter >0.Sram the static angle of friction is roughly constant, since it is largely controlled by the geometry of the particles. The static angleof friction can, for thesesizes,increaseor decreasewith increasingparticle size, dependingon changesof geometry, surface characteristics,etc. with size; but the changeswill be small, as in Fig. 7. In page 44 it is stated that the angie of the slopeof the surfaceof the material in the flightschangesabout halfway along the flight. A similar phenomenonoccurswith the changein a rotating ball mill and a treatment by Sullivan and Rose (7) shows that the surfaceis an equi-angularspiral. The lecturermade thefollowing written comments:

The kinetic angle of reposeis fundamentally different from the static angie in that a third force, due to the centrifugal action, is present in the equilibrium balance of forces which determine the value of the angle. As the direction of this centrifugal force on the moving plane of powdersvaries over the 360ø rotation, the value of the kinetic angle follows a sinusoidalcurve pattern as shownin Fig. 1 (2). At low drum speeds,this curve flattens out at about 45øfor powderswith a kinetic coefficientof 1.0. This is in agreementwith equationI where, with l)-•0, •,=tan- • (1.0). On the other hand, as pointed out by Professor Rose, the static angle of these particlesis about 30ø (Fig. 7). The lowestdrum speedshownin Fig. 5 is 8 rev rain-1 and at this speed, a continually moving powder surface was maintained, i.e. the dyna,nic equilibrium assumptionwas justified and the result of the centrifugal force was to yield a greater angle of reposeat the turning points on the sinusoidal curve

(i.e. 90ø and 270ø where [t=tan-• •.}•). If the drum speedis loweredto, say, 2 rev rain-•, the conditions of dynamic equilibrium break down and a 'stop and start' motion of the powder in the flights result, cascadestarting at about 55ø, leaving an angie of about 35ø, the plane of powders remain static until the drum motion has

broughtits angleback to 55ø. It still averagesout at 45ø. In summary, the answer to ProfessorRose amounts to the fact that in the formation of the static angles, no opposingcentrifugal force is present, and the angle is therefore

of a somewhat

lower

value.

(7) Sullivan, R. M. E. and Rose, H. E. Treatise on internal mechanismsof ball, tube and rod mills (1958) Constable, London.