Stress distribution during a silo filling or a discharging process

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1 9 Naue & Technology Sess disibuion duing a silo filling o a dischaging pocess Ismain BABA AHMED a, Blaise NSOM b,amine M ALLAL a,fouad GHOMARI a a Univesié Aboubek Belkaid,BP 119 Tlemcen, Algéie b Univesié de Beagne Occidenale,Rue Kegoa 931Bes Cedex 3,Fance Absac The exploiaion of he silos equies an accuae knowledge of he poblems elaed o he pocesses of soage and dischage. The physical and mechanical chaaceisics of bulk maeial can be measued and conolled. Meanwhile, a bad managemen of he ineacion beween hese chaaceisics and he soage condiions can give ise o a poo pefomance of he silo (flow ae duing dischaging, aching, ec.), o a failue of he sucue. In ode o invesigae he pincipal disodes which can be obseved duing a filling and a dischage opeaion, he sess ansmission wihin bulk maeials is chaaceized wih he help of discee numeical simulaions. Keywods: Cell hoppe juncion, Disinc Elemen Mehod, Dischaging, Discee simulaions, Filling, Ganula maeial, Silo, Sess ansmission 1. Inoducion Nowadays, diffeen mehods and sysems ae available o silos designes and opeaos which aim a analyzing pessue on silos walls [1], []. Meanwhile, echnological isks such as sucue failue o fomaion of seady aches capable of blocking he opeaion subsis duing a silo empying pocess [3]. These defecs ae caused by he ineacion beween he physical and mechanical chaaceisics of he gains and maes in bulk [4] wih he geomeical and mechanical chaaceisics of he silo and wih he soage condiions [5]-[7].The disibuion of he foces exeing on a pile of dy gains is inhomogeneous. Ceain gains ae submied o high sesses while ohe neighbouing ones suppo weake sesses and may even be fee o move [8]. Noably, an exenal sess applied o such a pile ends o follow pahs involving ouching paicles. Moeove, inceasing he exenal sess applied ses up an enichmen of he sesses pahs involving a lage numbe of gains. In his way, ensiled gains end o ediec he veical sesses o he wall following given pahs []. Fom Guiaa e al. [9], one of he main causes fo failue of gain silos is due o he excessive pessues exeed on he silo wall duing dischage. Pedicing how foces popagae ino ensiled ganula media is hen a eal scienific challenge wih majo indusial focus egading he sockage and he handling of food gains. Coninuous models give saisfacoy esuls fo he sess field wihin he silo a he end of he filling sage. Unfounaely, due o he lage defomaions which he ensiled mae undegoes, hey equie sophisicaed heological laws and numeical mehods as well, wihou meanwhile a eliable pedicion of he kinemaic field [10]. In conas, Disinc Elemen Mehods (DEM) which is based on a mechanical model of individual idealized paicles and no on a coninuum is moe suiable o chaaceize he saisical disibuions of conac foces inside he ensiled mae [11]. The DEM consis in acking he moion of individual paicles and updaing any conac wih neighbouing elemens by using a consiuive conac law [1]. The pogam used in his wok is PFC D (Paicle Flow Code in wo Dimensions) based on he DEM and he validaion of he pogam is obained by compaison wih he esuls obained by Masson and Mainez [1], [13] fo simila configuaions. A fla-boomed and a cylinde-hoppe silos ae consideed, he silo aspec aio and he hoppe slope being he geomeic paamee in he simulaions. While he gains ae polydispese, hei numbe, diamee, densiy, ficion angles (gain-gain and gain-wall), nomal and angenial siffness ae he paicle paamees.. The PFC D pogam The paicles used in he PFC D pogam ae igid disks which ineac only a he conac of anohe paicle o a solid wall. A he conac, he ineacion induces an inenal foce which is esponsible fo an elasic defomaion of he (neveheless igid) paicle. The inenal foces and he paicles displacemens ae obained by acking he moion of each paicle, iself poduced by Revue «Naue e Technologie». n 03/Juin 010. Pages 9 à 36

2 30 Sess disibuion duing a silo filling o a dischaging pocess he popagaion of he peubaions houghou he ganula sysem. In his model, a a conac poin of wo paicles, an ineacion foce is inoduced which is popoional o hei elaive velociy and in opposie diecion. Thus when wo neighbouing paicles ge close he ineacion foce has a epulsive effec; and when hey move away one anohe, i has an aacing effec. This ineacion foce is modelled by a sping. An explici ime-diffeence scheme being applied on he ineacion law, whee he nomal foce ( ) ( ) Fn Fs and he shea foce a a given conac poin a ime ae especively given by: ( Fn ) ( Fn ) + Fn = 1 (1) ( Fs ) ( Fs ) + Fs = 1 () whee ( F n ) 1 and ( F s ) 1 ae especively he nomal and he shea foce a ime 1 and ( F n, F s ) denoes he incemenal nomal and shea foce, iself being given by: F n = kn Un + cnv n (3) Fs = ks U s (4) In hese equaions, k n denoes he sping nomal siffness and c n he nomal damping coefficien, while Un, he elaive nomal displacemen is obained by inegaion of he nomal velociy by: Un= vn (5) Relaions analogous o equaions (3), (4) and (5) ae wien in he angenial diecion. Newon s ( Second Law is hen ) applied o a given paicle wih, ϕ pola coodinaes a ime using he following elaions, o deemine is moion: ϕ + = + & + + = ϕ + ω+ (6) (7) Whee he anslaional and he oaional velociies ae especively given by: & + ω + = & = ω + F i + g m i + In hese equaions, i M I 3i F i zi M i and i (8) (9) especively denoe he summaion of all he foces and momena in he z- diecion (pependicula o he plane of he moion), applied o he paicle a ime, while m is he paicle mass, I is momen of ineia and g he gaviy and is he ime sep. Finally, i should be noiced ha a discepancy exiss in he lieaue beween he values used fo he diffeen model paamees: c n, c s, kn and k s (e.g.: Tsuji and Oikawa, among ohes [14]). 3. Digial Simulaion The ganula flows ae geneally classified in hee disinc modes: he quasi-saic mode, he dense mode and he collision mode. In he quasi-saic mode he movemen of he paicles is elaively slow o non-exising, while he ganula mae is diven as a liquid in he dense mode, and as gas aoms in he collision mode [15]. In he quasi-saic mode of a gain flow inside a silo, he bulk maeial is in consan disubance a each filling o dischage opeaion wih saemen ha he maeial eaches he heshold of upue o each use of he silo, which gives ise o he following fundamenal quesion: how ae disibued he sains and he sesses once his heshold is exceeded? I is especially a poblem of including/undesanding his evoluion a he conac of he mae wih he walls of he silo, since i is in his zone ha he slip occus. Indeed he slip of he gains boh on he ohe gains and on he walls is he main paamee fo he undesanding of such phenomena as he sicking o he sess peaks locaed a he cell-hoppe juncion. In his wok, we will focus on cylindical veical silos, whee wo flow ypes can be idenified: he mass flow, chaaceized by sufficienly seep and smooh walls of he hoppe and he funnel flow when hey ae genly sloping and ough [16]-[19]. The chaaceisics of he simulaed maes ae pesened in Tables 1 &, while he pocedue of he simulaion is

3 Revue «Naue e Technologie». n 03/Juin pesened in Figue 1. In hese simulaions, aboad he equied adiional goals such as he sess field and he sain field a special locaions whee he sudied poo unning phenomena ake place inside he silo, we quanify he paamees specific o he ganula maeials such as he coodinaion numbe, he conac foces, ec. The simulaion of a silo filling and dischaging was pefomed using a DEM sofwae PFC D, suiable fo a D numeical modelling of disconinuous media. Fig. 1. Pocedue used fo he simulaions. Table 1 Inpu daa fo a fla boom silo 1 s case Gain-Gain Gain-Wall Densiy of maeial Angles of Ficion Φ G=45 Φ P=6,5 Nomal & angenial siffness K n=50 000N/m K s= 1 000N/m K n=50 000N/m K s= 1 000N/m Rays of gains 0.09m<<0.11m 0.09m<<0.11m Table Inpu daa fo a conical Hoppe nd case Gain-Gain Gain-Wall Densiy of maeial Angles of Ficion Φ G=5 Φ P=4 Nomal & angenial siffness N/m N/m Rays of gains 0.09m<<0.11m 0.09m<<0.11m

4 3 Sess disibuion duing a silo filling o a dischaging pocess Table 3 Coodinaion numbe fo diffeen gain shape Gain shape Paamees C n Convex Gains Cicula Gains ne squae, equal ay 4 Non convex Gains Cicula Gains ne iangula,equal ay 6 Cicula Gains andom ay Ineganula Vacuum Tiangula Ineganula Vacuum Unspesified 6 Vacuum in no iangula fom wih cuvilinea edge ~ 4 4 Unlimied 4. Resuls of he Simulaion A pile of gains is chaaceized by a newok of conac which ansmis he foces. The andom aspec of he disibuion of he chains of foces was educed o saisical appoaches aking ino accoun he numbe of aveage conacs pe gain, also called coodinaion numbe [0] Coodinaion Numbe If n c denoes he numbe of conacs pe paicle and N he numbe of paicles in he défined measuing space V (close o he desied calculaed zone), he coodinaion numbe Cn is defined as (10) The geomeical shape of hese gains is essenial o chaaceize he sample sudied. Tables 1- pesen he diffeen mechanical and geomeical chaaceisics of he silo and hose of he ganula maeial sudied and Table 3, he coodinaion numbe fo diffeen gain shapes. Figues a-b pesens he evoluion of he coodinaion numbe agains he numbe of cycles fo he age of ou digial simulaions. I is compued a he cell-hoppe juncion, using he Cundall model [11]. Fig. b Coodinaion numbe in a sample space 4. Pessue on he walls duing a filling opeaion The veical and he hoizonal sesses on he silo walls can be calculaed using eqs. (11) - (1). Thei vaiaion along he silo veical wall is shown in Figues 3.a and 3.b, especively, afe a filling opeaion. In each of hese figues, he pessues exeed by he gains on he silo walls using he equaions given above ae called he Janssen equaions [1]. ( z h) ρgr σv = 1 exp Kµ s = ' q σ + ' H γ h z h ni γ ni Kµ s R +1 ( 1 ) n i h z ' (11) (1) In hese equaions, ρ is he mae densiy, g he gaviy, γ =ρg he uni weigh, R he adius of he silo cell, K he Janssen coefficien (equal hee o 0.4 duing a filling Fig. a. Coodinaion numbe in a sample space

Revue «Naue e Technologie». n 03/Juin 010 33 opeaion and beween 0.5 and 0.

5 Revue «Naue e Technologie». n 03/Juin opeaion and beween 0.5 and 0.6 duing a dischage), µ s he gain/wall saic ficion coefficien, h he cell heigh, h' he hoppe heigh, = 1 + gφp ni 3 gθ a paamee whee ΦP denoes he gain/wall ficion angle and θ he hoppe slope; z he σ H q= veical coodinae and finally K a pa of he hoizonal sess. walls (silo hoppe in Figue. 4a) o ficiious and andom walls maeialized by he inesecing lines beween he dead zones and he flow zones inside he mae (flaboom silo, Figue 4.b).The sess peak is a diec consequence of his peubaion. The discepancy which chaaceizes he values of he sess obained in hese cuves can be inepeed. Indeed i clealy shows ha he sudied medium mus no be egaded as a coninuous medium, bu as disconinuous medium. This mae of fac compleely jusifies he use of a DEM []. By wihdawing he lowe wall of he silo, a dischaging akes place and as fom he fis case, we noice he amplificaion of he newok of he conac foces o he aliude of m Figue 5. The vaiaion of he sess along he veical wall is ploed on Figue 6, and i clealy shows he peak of sess on he igh of he op of he dead zone and has as a value of 960 Pa, in hoizonal sess [3], [4]. Fig. 3a Vaiaion of hoizonal sess along silo veical wall Fig. 4a Sess inceasing in swich zone σ v (Pa) Fig. 3b Vaiaion of Veivalal sess along silo veical wall These cuves show a saisfacoy ageemen in he veical pa of he silo (1.5m<z<5.3m), and on he lowe wo hids of he hoppe (0<z<0.8m). Moeove, a disubance of he veical and he hoizonal sess can be noiced on he igh pa of he cell-hoppe juncion (0.8m<z<1.5m). This phenomenon defines he swich zone whee depessue (o ovepessue) ake place eihe in a geomeical silo σ h Fig. 4b Sess deceasing in swich zone

6 34 Sess disibuion duing a silo filling o a dischaging pocess Fig. 5 Ineganula foce newok (silo ; K=0.4) Fig. 7a Evoluion of hoizonal and veical sess duing silo use (case 1) Figue 6: Hoizonal sess on he edge (silo ; K=0.4) Fig. 7b Evoluion of hoizonal and veical sess duing silo use (case ) 5. Paameic sudy The mechanical paamees as measuable quaniies ae impoan fo he ansmission of he sess. Thei inoducion as micomechanical vaiables ino a digial model, using a DEM, can accoun fo he influence which hey have on he macoscopic mechanical behaviou of a ganula medium. The paameic sudy was pefomed wih espec o he wo following mechanical paamees of he ensiled medium: he gain-gain ficion and he gainwall ficion, while he geomeic silo paamee is he slope of is hoppe walls. The ohe paamees such as he maeial densiy, he gain siffness and he silo walls oughness, he silo heigh H, he apeue as well as he gain size ae kep consan. he diffeen esuls obained ae expessed in ems of foce newok, evoluion of hoizonal and veical sess (Figues 7a, 7b, 7c) as well as hei aio, given by equaions (10) e (11)., Fig. 7c Evoluion of hoizonal and veical sess duing silo use (case 3) The sess disibuion can also be shown by dawing he vaiaion (inceasing o deceasing) of he sesses using pecenage bas wih espec o hei values duing he filling opeaion (figs. 8a and 8b). In fig. 8a we can see ha he hoizonal sess inceasing is moe impoan fo

7 Revue «Naue e Technologie». n 03/Juin Fig. 7d Evoluion of hoizonal and veical sess duing silo use (case 4) hoppes wih a slope of 35 and 45 wih low values of φg (0 and 30 ) han fo hoppes wih a slope of 55 whee a deceasing is noiced. The excepion of a hoppe wih a slope of 75 give opposie esuls as he inceasing obained is as impoan as in he hee pevious hoppes (abou 9% fo φp = 10 ). While fo he veical sesses, he inceasing is much moe fo he hoppe wih a slope of 35 whaeve he gain-gain and gain-wal.ficions ae. The inceasing fo θ = 45, 55 and 75 wih φp = 10, θ = 55 and 75 wih φp = 45 ae also negaive. This esul confims he hoizonal ansmission of he sesses. So he geomeic paamee «hoppe slope θ» has a clea influence on he values of he hoizonal and he veical sesses a he swich zone. Slopes lowe han 45 poduce an impoan inceasing of he sesses and convesely, slopes highe han 45 poduce a lowe inceasing of he sesses. These inceasing cause no majo safey isk fo he design of he silos (failue o explosion of he sucue), bu hei bee undesanding allow a bee silo exploiaion (avoidance of dead zones). The angles of ficion paamees φp and φg have an influence on he sess ansmission as hey give o he ensiled mae, pefeed oienaions o he newok of he conac foces. Masson & Mainez [5] obained simila esul wih fla boom silos wih vaiable ficion and siffness. The ensiled mae has a vaiable behaviou accoding o he sae of is use: unde geomeical condiions such as conical silos of vey song slope and slighly huled, i will un ou elaive a ease wihou oo much incease in consain.i will end o have a behavio of coninuous medium.on he ohe hand unde condiions geomeical such as silos fla-boomed and songly slim, he ganula mae will undego a andom sae of sess causing of he nooious dysfuncions, like conainmen, he phenomenon of he vauls, ec. Fig. 8a Evoluion of hoizonal and veical sess duing silo use (case 4) Fig. 8b Evoluion of hoizonal and veical sess duing silo use (case 4) 6. Conclusion This pape consideed he main obsevable phenomena duing a filling and a dischaging opeaion of a silo and hei consequences on he bulk maeial behaviou. In paicula, using diec numeical simulaions, we addessed he pessue on he silo walls. We wee be able o obseve a qui elaionship beween he fom of he pessue cuve of boh hoizonals and veical sess values in he simulaion and he eal meseamen on he silo obained on ohe woks [6], in spie of he smalles values of heses sess value calculed on he simulaion. Then a paameic sudy was pefomed and he ime evoluion of he pessue could be compued. The esuls we obained fo case shown on figue 7b conclude o a nealy linea aio of he hoizonal sess o he veical sess. This means ha in he zone of cell-hoppe juncion, whaeve he sess inceasing could be, he mechanical behaviou of he ganula maeial is simila o ha of a coninuous medium. In opposiion, fo case 4 on figue 7d, he flucuaions in his aio ae so high ha he sess field is anisoopic.

8 36 Sess disibuion duing a silo filling o a dischaging pocess Refeences [1] Reimbe,A. and Reimbe M. Silos, eds Eyolles, Pais (1956) Schulze D (004), Soage of Powdes and Bulk solids in silos, Eds (004) [] Jenike AW: Soage and flow of solids. Bull. N 13 (1964), Eng. Exp. Saion, Univ. Uah, Sal lake Ciy. [3] Bown CJ and Nielsen J: Silos. Fundamenals of heoy, behaviou and design, (1998) Eds. E & FNn SPON, London. [4] Degoue C, Nsom B, Lolive E & Gohens A: Chaaceizaion of Soya, Colza and Rye Seeds, Appl. 186 Rheol 17(3) (006) [5] Teunou E and Fizpaick JJ: Effec of soage ime and consolidaion on food powde flowabiliy, J.Food Eng., 43 (000) [6] Fizpaick JJ, Iqbal T, Delaney C, Twomey T and Keogh MK: J. Food Eng. 64 (004) [7] Fizpaick JJ, Bainge SA & Iqbal T: Flow popey measuemen of food powdes and sensiiviy. [8] Claudin P: La physique des as de sable (1999), EDP Sciences, Les Ulis (Fance) Cundall PA & Sack ODL: A discee numeical model fo ganula assemblies. Geoech, 9 (1979) [9] Guaia M, Couo A & Ayuga F: Numeical Simulaion of Wall Pessue Duing Dischage of Ganula Maeial fom Cylindical Silos wih Eccenic Hoppes. Biosysems Engineeing 85 (003) [10] Masson S, Mainez J, Feelec JF & Paisi D: Simulaion numéique discèe des écoulemens ganulaies confinés. Aces du 16ème Congès Fançais de Mécanique (003), Nice (Fance). [11] P.A. Cundall, O.D.L. Sack, A discee numeical model fo ganula assemblies. Geoechnique (1979), 9:47 6. [1] Masson M & Mainez J: Muliscale simulaions of he mechanical behaviou of an ensiled ganula 09 maeial, Mech. Cohes-Fic. Mae. 5 (000) [13] Goda, T.J. & Ebe, F.: Thee dimensional discee elemen simulaions in hoppes and silos. Powde Technology (005), 158, [14] Tsuji H & Oikawa M: Two Dimensionnel ineacion of inenal Soliay waves in a wo laye Fluid. J.Phys. Soc. Japan, 6 (1993) [15] GDR MiDi: Goupemen De Recheche Milieux Divises, CNRS, GDR181, On dense ganula flow, The Euopean Physical Jounal E, 14 (004) [16] Degoue C, Nsom B, Lolive E & Gohens A: (007). Image Pocessing fo he Measuemen of Flow Rae of Silo Dischage, in Innovaion and Advanced Techniques in Compue and Infomaion. [17] Job, N., Dadenne, A. & Piad, J-P: Silo flow-paen diagnosis using he ace mehod, J. Food Eng., 91(1) (009), [18] Chen, J.F., Roe, J.M., Ooi, J.Y. & Zhong, Z.: Flow paen measuemen in full scale silo conaining ion oe. Chem. Eng. Sci. (005), 60(11), [19] Osendof, M. & Schwedes, J.: Applicaion of paicle image velocimey fo velociy measuemens duing silo dischage. Powde Technology (005), 158, [0] Webe JD: (1966). Recheches concenan les conaines ineganulaies dans les milieux pulvéulens. Bullein de liaison Laboaoie des Pons e chaussées, 337 (1966) 3-1,3-19. [1] Jenike's hoppe design mehodology o he measued values. J. Food Eng. 61 (004) [] Mülle D: Techniques efficaces pou la simulaion de milieux ganulaies pa la méhode des élémens disincs, Thèse de Docoa, N 1545 (1996) EPFL, Swizeland (in Fench), [3] Ismain Baba Ahmed, B. Nsom, M. Allal e F.Ghomai: "Behaviou of Ganula Maeial Duing a Silo Filling o Gischaging", Poceedings of he nd ICMS (Inenaional Confeence on Mechanical Sciences), Oum El Bouaghi, Algeie (008) [4] Ismain Baba Ahmed, B. Nsom, M. Allal e F.Ghomai: " Rheology and Sucue of Ensiled Food Gains", Poceedings of he 5h ISFRS (Inenaional Symposium on Food Rheology and Sucue), Zuich, Suisse (009). [5] Masson M & Mainez J: Effec of paicle popeies and silo geomey on sesses pediced by discee simulaions of bulk maeials. Poceedings EM0, ASCE (00), New Yok. [6] Cason JW & Jenkyn RT: How o exploe peven silo failue wih Rouine Inspecions and Pope 179 Repai, Powde and Bulk Engineeing, Vol. 4 (1990) 18-3.

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body. Descipion of he moion of igid