NUMERICAL INVESTIGATION OF PILING AND HOPPER FLOW OF ELLIPSOIDAL PARTICLES

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1 NUMERICAL INVESTIGATION OF PILING AND HOPPER FLOW OF ELLIPSOIDAL PARTICLES Z.Y. Zhou 1, S.D. Lu 1, R.P. Zou 1, A.B. Yu 1, D. Pnson 2 and P. Zull 2 1 Laboratory for Smulaton and Modellng of Partculate Systems School of Materals Scence and Engneerng The Unversty of New South Wales Sydney, NSW 2052, Australa z.zhou@unsw.edu.au 2 BlueScope Steel Research, P.O. Box 202, Port Kembla NSW 2505, Australa ABSTRACT Shape s one of the most mportant propertes of granular materals that sgnfcantly affects the packng structure and flow n a partculate system. In ths work, the effect of partcle shape on the angle of repose of granular ples, and on partcle flow n a hopper, s nvestgated by the dscrete element method (DEM usng ellpsodal partcles (varyng from dsk to cylndrcal. It s shown that partcle shape sgnfcantly affects the angle of repose and hopper dscharge rate. An equvalency between the aspect rato and sldng/rollng frcton coeffcents s establshed. It s shown that by ncreasng the sldng and rollng frcton coeffcents, spheres can generate qualtatvely consstent results wth ellpsods, but only n a small rage of aspect rato varyng from 0.8 to 1.2. The proposed method s useful n reducng the computatonal tme for ndustral scale smulaton by DEM when a large number of partcles and partcle shape are both consdered. INTRODUCTION Partcle shape s one of the mportant propertes of granular materals, and t affects the packng/flow structures whch are crtcal to transport propertes such as permeablty (related to pore connecton and thermal conductvty (related to partcle connecton. For example, n the packng of non-sphercal partcles, studes reveal that porosty (1-packng fracton decreases to a mnmum and then ncreases when partcles become more non-sphercal (Zou and Yu, 1996; Donev et al., 2004; Guses et al., For the formaton of ples of non-sphercal partcles, the angle of repose ncreases wth ncreasng non-sphercty (Yong and Warkentn, 1975; Matutts et al., 2000; Fredman and Robnson, 2002; Zhou and Oo, These studes are manly based on experments and modellng technques such as Monte Carlo algorthms wth ether sequental addton or collectve rearrangement. The dscrete element method (DEM (Cundall and Strack, 1979, whch can take nto account all dynamc factors related to not only geometry but also forces, has been wdely used to study packng and plng processes, as revewed by Zhu et al. (2008. However, so far such studes manly consder sphercal partcles. Recently, DEM has been extended to study the granular flow of non-sphercal partcles (for example, Vu-Quoc et al., 2000; Cleary and Sawley, 2002; Langston et al., 2004; La et al., 2004; Frage et al., The shapes consdered can be regular or rregular. However, due to the 1

2 dffculty and complexty n representng the rregular shapes and the assocated computatonal requrements, to date, such studes are largely lmted to regular shapes. In partcular, ellpsodal partcles are commonly used because of ther ablty to represent partcle shapes varyng from oblate to prolate. Recently, attempts have been made to use DEM to model the packng and fludzaton of ellpsodal partcles (for example, Zhou et al., 2009, 2001; Hlton et al., 2010; Delaney et al., DEM s also used to study the formaton of granular ples, but manly n two-dmensons (for example, Zurguel and Mulln, So far, few attempts have been made to study the effect of partcle shape on granular ples and hopper flow systematcally (e.g. ellpsods wth aspect rato varyng n a large range. In ths work, DEM s used to examne the effect of partcle shape on the angle of repose of a formed ple. An equvalency between the aspect rato and sldng/rollng frcton coeffcents s establshed, and ts applcaton s demonstrated n ple formaton and partcle dscharge rate from a hopper. MODEL DESCRIPTION As orgnally proposed by Cundall and Strack (1979, a partcle can have two types of moton: translatonal and rotatonal. The governng equatons for the translatonal and rotatonal moton of partcle wth mass m and moment of nerta I can be wrtten as: k = j =1 c, j d, j m dv / dt ( f + f + m g (1 I dω k / dt = = ( M j + M j + t n, M j 1, r, j where v and ω are the translatonal and angular veloctes of the partcle, respectvely, and k s the number of partcles nteractng wth the partcle. The forces nvolved are: the gravtatonal force m g, and nter-partcle forces whch nclude elastc force f c,j and vscous dampng force f d,j. The torques actng on partcle by partcle j nclude: M t,j generated by the tangental force, M r,j commonly known as the rollng frcton torque, and also the torque M n,j generated by the normal force when the normal force does not pass through the partcle centre, whch s appled to the ellpsodal partcles. Equatons used to calculate the nteracton forces and torques between two spheres have been well-establshed and wdely used n the lterature (Zhu et al., In ths work, the non-lnear model for spheres s extended to ellpsods. As noted above, an addtonal torque s ntroduced, whch s caused by the normal force. Ths s because the normal drecton of the contact force does not necessarly pass through the centre of an ellpsod, whch, together wth the tangental forces and rollng torque, governs the rotatonal moton of the partcle. Another parameter s the reduced radus R*, R*=1/2(A B -0.5 where A and B are related to the rad of the partcle shape curvature at a contact pont as dscussed by (Dzugys and Peters, The explct tme ntegraton method s wdely used to solve the translatonal and rotatonal motons of a system of dscrete partcles n DEM smulatons (Cundall and Strack, Although t s establshed for spheres, such a method can be extended to ellpsods. The dffcultes assocated wth the extenson manly le n two aspects: partcle-partcle detecton and partcle orentaton. The methods have been well documented elsewhere (Ln and Ng, 1995; Goldsten, 1980; Dzugys and Peters, 2001; Zhou et al., (2

3 SIMULATION CONDITIONS A ple can be formed n varous ways such as the njectng, dschargng, and tltng methods, whch have been used by dfferent researchers for dfferent purposes (Carrgy, 1970; Grassell & Herrmann, In the present work, the so-called dschargng method s used, n connecton wth the prevous study (Zhou et al., 1999, Thus, the smulatons are carred out n a rectangular contaner wth a fxed mddle plate and two sde outlets, as shown n Fgure 1(a. A packng s formed by pourng partcles from a certan heght nto the contaner. A small velocty wth random drecton and a random orentaton are assgned to each ellpsod to be dropped. Durng the packng process, partcles may collde wth neghborng partcles, or bounce back and forth. Ths dynamc process proceeds untl all partcles reach ther stable postons wth an essentally zero velocty as a result of the dampng effect for energy dsspaton. Then, the nstantaneous openng of the outlets starts a dschargng process n whch partcles drop nto the bottom of the contaner under gravty. Some partcles reman on the mddle plate after the dscharge, formng a stable ple. Then the angle of repose can be determned from the surface profle of the ple. For the sold flow n a hopper wth a flat bottom, the geometry used s shown n Fgure 1(b wth a thckness of 6d v. The orfce dameter at the centre of the bottom s 10d v. The perodc boundary condtons n the front and rear drecton are used for both cases. (a Fgure 1: Geometry of the contaner used n the smulaton: (a formaton of granular ple; and (b hopper flow wth a flat bottom. Spherods wth aspect ratos varyng from 0.25 to 2.5 are used n ths work to produce shapes from oblate to prolate. The defntons of partcle sze, aspect rato and sphercty are shown n Table 1. In the present work, b=c for spherods. d v s the equvalent dameter of a sphere wth the same volume as an ellpsod, and set to 10 mm n ths work. All the partcles wth dfferent aspect ratos have the same volume. The table also shows other physcal propertes of partcles and parameters n the DEM smulaton. These propertes may affect the plng process of ellpsods, as s the case for spheres (Zhou et al., However, as the present work s focused on the effect of aspect rato these parameters are fxed n ths study. (b 3

4 Table 1 Partcle parameters used n the DEM smulaton Parameters Values Partcle number 5,000 (ples, 20,000 (hopper Partcle sze (2a, 2b=2c, d v d v =2(abc 1/3 =10 mm Partcle aspect rato (η=a/b 0.25~2.5 Partcle densty 2,500 kg/m 3 Sldng frcton coeffcent, µ s,pp 0.4 Sldng frcton coeffcent, µ s,pw 0.4 Rollng frcton coeffcent, µ r,pp 0.05 mm Rollng frcton coeffcent, µ r,pw 0.1 mm Normal dampng coeffcent, c n 0.3 Tangental dampng coeffcent, c t 0.3 Young s modulus, E Pa Posson rato, ν 0.3 Tme step, t s NB: a s the prncpal radus n the polar drecton, b and c are prncpal rad n the equatoral plane. For a prolate spherod, aspect rato η>1; for a sphere, η=1; and for an oblate spherod, η<1. RESULTS AND DISCUSSION Effect of aspect rato on the angle of repose of sandples For valdaton purposes, a comparson has been made between the measured and the ples smulated. The materals used n the experments are two knds of Chocolate Candes, and ther physcal propertes are: a=6.53 mm, b=c=14.67 mm, aspect rato=0.445, and densty= 1313 kg/m 3 for large partcles; and a=12.44 mm, b=c=8.78 mm, aspect rato=0.706, and densty=1415 kg/m 3 for small partcles. It should be noted that to be comparable wth experment, wall boundary condtons are used here. Fgure 2 shows the snapshots llustratng the formaton process of ples. Fgure 3 presents a comparson of formed ples between the experments and DEM smulatons. It can be observed that the otbaned profles of the ple are very comparale, llustratng that the proposed DEM model can produce the relable results. Fgure 2: Snapshots showng the formaton of ples for regular Chocolate Candes. 4

5 (a (b Fgure 3: Comparson of formed ple between the observed and the smulated: (a large partcles wth model thckness dp; and (b small partcles wth model thckness dp. The varaton of angle of repose wth aspect rato under perodc boundary condtons s shown n Fgure 4(a. The nset fgures are the sandples produced for dfferent aspect ratos. The relatonshp between the angle of repose and aspect rato reveals that spheres have the lowest angle of repose. For oblate spherods, wth aspect rato decreasng from 1.0, the angle of repose frst ncreases then reaches a maxmum at aspect rato 0.5, then decreases. For prolate spherods, the angle of repose ncreases then almost reaches a constant after aspect rato 1.8. The relatonshp s smlar to the varaton of coordnaton number wth aspect rato for partcle packng (Donev et al., 2004; Zhou et al., It demonstrates that prolate and oblate partcles follow dfferent trends of angle of repose wth aspect rato. 2.2 Angle of repose (θ/θ Aspect rato (η (a (b Fgure 4: (a Angle of repose for dfferent aspect ratos (the nsets are stable ples formed for dfferent aspect ratos; and (b dmensonless angle of repose wth aspect rato. Fgure 4(b shows the varaton of dmensonless angle of repose wth aspect rato. The angle of repose of ellpsodal partcles wth aspect rato η can be wrtten as: θ = θ f ( (3 0 η 5

6 where θ 0 s the angle of repose for spheres. f(η represents the effect functon of partcle shape on the angle of repose by aspect rato. Accordng to the data obtaned shown n Fgure 4(b, two correlatons are obtaned for f(η for the oblate and prolate spherods, respectvely: f ( η = e 2.9η(1 η ( η 1 η f ( η = ( η > 1 Zhou et al. (1999 proposed a correlaton to calculate θ 0, the angle of repose for sphercal glass beads. θ 0 s a functon of partcle propertes and wrtten as (Zhou et al., 1999: θ0 = µ µ µ µ d s, pp s, pw r, pp r, pw (6 where µ s,pp and µ s,pw are sldng frcton coeffcents between partcle and partcle, and between partcle and wall, respectvely; and µ r,pp and µ r,pw are the rollng frcton coeffcents between partcle and partcle, and between partcle and wall, respectvely. d s the partcle sze. It should be noted that Eq. (6 s obtaned under the condton that the effects of front and rear walls have been elmnated, consstent wth the condtons used n ths work. Thus, Eqs. (3-6 provde a method to estmate the angle of repose of ellpsodal partcles wth dfferent aspect ratos. Effect of partcle shape on the dscharge rate from a hopper The dscharge rate of a hopper has been nvestgated for a long perod and most of the researchers have focused on sphercal/crcular partcles. The Beverloo equaton (Beverloo et al., 1961 has been well establshed to estmate the dscharge rate for spheres. The effect of partcle shape on the flow rate has been examned by some nvestgators (for example, Cleary and Sawley, 2002; L et al., For the further test of the proposed method, the dscharge rate from a 2D hopper (Fgure 1(b s examned. Fgure 5 shows the snapshots of partcle flow patterns for dfferent aspect ratos. It can be observed that the general flow patterns are smlar, whch conssts of plug flow zone, convergng flow zone, and stagnant zone. However, for spheres, wall effect s relatvely sgnfcant, causng mxed regons on the wall. Fgure 6 shows the partcle dscharge rate for dfferent aspect ratos. It can be seen that sphercal partcles have the hghest dscharge rate. As the partcles becomng more platy or elongated, the dscharge rate decreases. (4 (5 (a (b (c Fgure 5: Snapshots of flow patterns n a 2D slot hopper for dfferent aspect ratos: (a 0.5; (b 1.0; and (c

7 Dscharge rate (kg/s Aspect rato Fgure 6: Effect of aspect rato on the dscharge rate from a hopper. A comparson wth spheres In DEM smulatons of ndustral problems (for example, Pnson and Wrght, 2007, even for spheres, the number of partcles used n the system s lmted by the current computng capacty. If the effect of partcle shape needs to be consdered (e.g. usng the DEM-ellpsods code, the computaton becomes at least ten tmes slower than for sphercal partcles. Thus, n practce, one soluton to tackle ths problem s to ncrease the sldng frcton and/or rollng frcton coeffcents of spheres to represent the effect of partcle shape to some degree. However, such a method s not well quantfed n the lterature. An attempt s made n ths work as gven below. For a gven materal and shape (e.g. ellpsods, ts angle of repose can be theoretcally estmated by Eqs. (3-6. If spheres are used to represent ellpsodal partcles, the materal propertes of spheres, e.g. sldng/rollng frcton coeffcents, should be adjusted. If let µ s =a s µ s and µ r =a r µ r where a s and a r are adjustng coeffcents, and assume partcle and wall have the same propertes, then Eq. (6 can be wrtten as θ = ( a µ d s s, pp ( arµ r, pp Comparng the angle of repose by Eqs. (3-6 wth the one by Eq. (7 gves f 0.49 ( η = ( a s ( ar where f(η s a functon of aspect rato, and determned by Eqs. (4-5. a s and a r are two unknown parameters to be determned. One assumpton can be made by a s = a r. Thus, as = ar = 1.5 f (η Physcally speakng, the sldng frcton coeffcent s ndependent of partcle shape. Thus, the parameter that can be adjusted s the rollng frcton coeffcent only. Ths condton gves a = 1 and a f ( η s r = 5.5 (7 (8 (9 (10 7

8 Such methods represented by Eqs. (9 and (10 are tested n the case for granular ples and hopper flow as gven n Fgure 7. Fgure 7(a shows a comparson of the angle of repose obtaned from dfferent methods. The square symbols represent the angle of repose obtaned usng ellpsods, the crcular symbols represent the data obtaned usng spheres by adjustng the sldng and rollng frcton coeffcents accordng to Eq. (9, and the delta symbols for the results generated by adjustng the rollng frcton coeffcent accordng to Eq. (10. It s llustrated that the results are consstent only n a small range of aspect ratos, e.g. from 0.8 to 1.2. Fgure 7(b shows the partcle dscharge rate from dfferent methods n a hopper. Clearly, comparng wth the data by ellpsods, the methods accordng to Eqs. (9 and (10 generate low dscharge rate, even n the small range of aspect rato close to 1.0. The method by Eq. (10 s much worse, although t s more seasonable theoretcally. Also, the varaton trends produced are consstent only when aspect rato s larger than 0.6. The tests shown n Fgure 7 ndcate that by adjustng the materal propertes, e.g. sldng frcton and rollng frcton coeffcents, the qualtatvely consstent results can be generated. It should be noted that those results are on the bass of Eqs. (9 and (10. To establsh the relatonshp of equvalency between sldng/rollng frcton coeffcents and aspect rato, more smulatons should be carred out by combng the dfferent values of sldng/rollng frcton coeffcents. Angle of repose (degree Ellpsods Spheres by Eq. (9 Spheres by Eq. ( Aspect rato Dscharge rate (kg/s Ellpsods 3.5 Spheres by Eq. (9 Spheres by Eq. ( Aspect rato (a (b Fgure 7: Varaton of (a the angle of repose of granular ples, and (b partcle dscharge rate n a hopper for dfferent aspect ratos of ellpsods. CONCLUSIONS Ellpsodal partcles are used n ths work to examne the effect of partcle shape on the angle of repose of granular ples and the dscharge rate n a hopper usng DEM technque. It s shown that when the partcle becomes more non-sphercal, the angle of repose of ples generally ncreases. A correlaton s establshed between the angle of repose and aspect rato of ellpsods. The partcle dscharge rate n a hopper s also affected sgnfcantly by aspect rato, and decreases wth aspect rato devatng from 1.0. An equvalency between the aspect rato and adjustng sldng/rollng frcton coeffcents s establshed. It ndcates that spheres can be used to 8

9 qualtatvely estmate the effect of partcle shape on propertes, but only n a small range of aspect rato varyng from 0.8 to 1.2. ACKNOWLEDGEMENTS The authors are grateful to Australan Research Councl (ARC and BlueScope Steel Research for the fnancal support of ths work, and the NCI Natonal Faclty for the support n computaton. REFERENCES Beverloo, W.A., Lenger, H.A. and van de Velde, J., The flow of granular solds through orfces, Chemcal Engneerng Scence, 115, Carrgy, M.A., Experments on angles of repose of granular materals. Sedmentology, 14, Cleary, P.W. and Sawley, M.L., DEM modellng of ndustral granular flows: 3D case studes and the effect of partcle shape on hopper dscharge. Appled Mathematcal Modellng, 26, Cundall, P.A. and Strack, O.D.L., Dscrete numercal model for granular assembles. Geotechnque, 29, Pnson, D. and Wrght, B. Industral applcaton of dscrete partcle smulaton at BlueScope Steel. Dscrete Element Methods (DEM 07 August 27-29, 2007 Brsbane Australa. Delaney, G. W., Hlton, J. E. and Cleary, P. W., Defnng random loose packng for nonsphercal grans. Physcal Revew E, 83, Donev, A., Csse, I., Sachs, D., Varano, E., Stllnger, F. H., Connelly, R., Torquato, S. and Chakn, P.M., Improvng the densty of jammed dsordered packngs usng ellpsods. Scence, 303, Dzugys, A. and Peters, B., An approach to smulate the moton of sphercal and non-sphercal fuel partcles n combuston chambers. Granular Matter, 3, Frage, F.Y., Langston, P.A. and Chen, G.Z., Dstnct element modellng of cubc partcle packng and flow. Powder Technology, 186, Fredman, S. P., and Robnson, D.A., Partcle shape characterzaton usng angle of repose measurements for predctng the effectve permttvty and electrcal conductvty of saturated granular meda. Water Resources Research, 38, Grassell, Y. and Herrmann, H.J., On the angles of dry granular heaps. Physca A 246, Goldsten, H., Classcal Mechancs. Addson-Wesley Publshng Company. Guses, R., Xang, J.S., Latham, J.P. and Munjza, A., Granular packng: numercal smulaton and the charactersaton of the effect of partcle shape. Granular Matter, 11, Hlton, J.E., Mason, L.R. and Cleary, P.W., Dynamcs of gas-sold fludsed beds wth nonsphercal partcle geometry. Chemcal Engneerng Scence, 65, Langston, P.A., Al-Awamleh, M.A., Frage, F.Y. and Asmar, B.N., Dstnct element modellng of non-sphercal frctonless partcle flow. Chemcal Engneerng Scence, 59, L, J.T., Langston, P.A., Webb, C. and Dyakowsk, T., Flow of Sphero-dsk partcles n rectangular hoppers a DEM and expermental comparson n 3D. Chemcal Engneerng Scence, 59, La, J.T., Langston, P.A., Webb, C. and Dyakowsk, T., Flow of sphero-dsc partcles n rectangular hoppers - a DEM and expermental comparson n 3D. Chemcal Engneerng Scence, 59, Ln, X. and Ng, T.T., A three-dmensonal dscrete element model usng arrays of ellpsods. Geotechnque, 47,

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