A PHENOMENOLOGICAL DESCRIPTION OF MIXING- SEGREGATION EFFECTS DURING SHEAR

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1 A PHENOMENOLOGICAL DESCRIPTION OF MIXING-SEGREGATION EFFECTS DURING SHEAR DEFORMATION OF PARTICULATE SOLIDS Poceeding of Euopean Conge of Chemical Engineeing (ECCE-6) Copenhagen, 6- Septembe 7 A PHENOMENOLOGICAL DESCRIPTION OF MIXING- SEGREGATION EFFECTS DURING SHEAR DEFORMATION OF PARTICULATE SOLIDS V. Dolgunin, a V. Bochev b, A. Klimov a, R. Shubin b a Depatment of Technological Eqipment and Food Technology, State Technical Univeity, 6 Sovetkaya t., 39, Tambov, Ruia b Depatment of Mahine and Appaatu of Chemical Technology, State Technical Univeity, 6 Sovetkaya t., 39, Tambov, Ruia Abtact The mixing egegation kinetic in the coue of hea flow of paticulate olid at low hea ate i tudied analytically and expeimentally. The expeimental eeach wa caied out by the ue of the conveyo hea cell developed peviouly []. The phenomenological deciption of the egegation mixing dynamic duing hea flow of nonelatic coheionle ough pheical paticle i uggeted. Only one expeimental contant (egegation coefficient) i ued to foecat the egegation mixing dynamic. The method of the egegation coefficient detemination i developed. Keywod: paticulate olid, hea defomation, mixing, egegation. Intoduction In ou peviou pape [] we have uggeted the expeimental unit and method of egegation and mixing exploation duing hea defomation of paticulate olid. Then a mathematical deciption of the mixing effect duing hea flow of paticulate olid wa caied out. The kinetic paamete of the mixing flux wee detemined on the bai of the analyi of chaotic tanveal movement of pheical nonelatic coheionle paticle (tanveal ma tanfe) in a eticted envionment. Thee chaotic movement ae fomally analogical to the quaidiffuional mixing of paticle. That i why the mixing flux of paticle i expeed a follow [] jm Ddif b c y,b ut c y, b b d in / 4 ( d ) d u y c y () whee D dif i the coefficient of quaidiffuional mixing, c i the concentation of tet paticle, b i the bulk denity of paticulate olid, u t du dybd i the mean elative velocity between paticle of neighbouing flow laye, d i the mean paticle

2 V. Dolgunin et al. diamete, b i the geometical paamete, b b d i the mean ditance between paticle, b i the geometical paamete b, calculated at,9. Howeve, the eal paticulate olid i not unifom in complex of phyical and mechanical popetie and then the chaotic movement ae accompanied by ytematic diplacement of paticle becaue of egegation effect. Although egegation ha long been known and ha been ued ove centuie in human activitie, e.g., in gain and ceeal cleaning, gold dut mining, etc., the cientific cognition of thi phyical phenomenon i till in it infancy [3]. Thi i becaue the egegation phenomenon, albeit imple in appeaance, ha extemely complex and divee phyical mechanim. Fo example, only fo the fomation of heap, Entad [4] evealed poible mechanim of egegation. Fo thi eaon, to decibe egegation, the tool of mathematical tatitic and pobability theoy ae widely ued, []. The coeponding tatitical imulation model do not epeent the phyical eence of the poce and have limited pedictive powe, but can be ueful in ome pecific cae of decibing the egegation in ytem unde complex hydodynamic condition. A highe pedictive powe i chaacteitic of model contucted by deteminitic analyi [, 6 8]. Howeve, many of thee model wee developed fo diffeent pecific cae and ae often contadict one anothe. Fo example, model popoed by Bagnold [6] and Stephen and Bidgwate [7] pedicted oppoite diection of the diplacement of lage paticle, depending on the hea velocity gadient diection. Pactice howed [, 8] that both of the two appoache to modeling the egegation kinetic ae mutually excluive even within the bulk of a ingle hea flow. In the context of the above, it i vey impotant to pefom invetigation to eveal mechanim and kinetic law of egegation fo the mot geneal and pactically impotant cae of flow of ganula media, e.g., hea defomation flow.. Segegation kinetic duing hea defomation of paticulate olid The egegation flux j i detemined on the bai of the mechanim of hydomechanical egegation []. Theeby the appoache ae ued which wee developed ealie in [] fo the mixing-egegation pediction of paticle duing thei apid hea flow, when momentum i caied by the inetia of paticle and exchanged in the coue of intepaticle colliion. In the peent pape thee appoache ae developed in tem of "low" hea flow of paticulate olid when polonged intepaticle contact take place. Thi mechanim peuppoe that egegation take place due to geneation of the exce moment of foce acting on the nonunifom paticle. Thi mechanim i adapted hee fo the condition of the low hea ate, when the colliion foce between paticle ae negligible. In accodance with the mechanim of hydomechanical egegation and taking into account the fact that the egegation kinetic at low hea ate doen't depend pactically on the hea te [9] we fomulated the egegation diving foce a the elative exce moment of foce acting on a tet paticle in a nonunifom medium:

3 A PHENOMENOLOGICAL DESCRIPTION OF MIXING-SEGREGATION EFFECTS DURING SHEAR DEFORMATION OF PARTICULATE SOLIDS 3 M M M M () whee M M G M F i the um moment of gavity M G fiction M F foce acting on a tet paticle of the mixtue, M i analogou moment acting on an aveage paticle of the mixtue. The gavity foce moment i calculated a ealie in [, ]. Auming that the fictional foce value i popotional to the hea te value we detemined M a follow d 4 co in bd tg d M F in (3) whee i the hea te, i the inclination angle of the flow to the hoizontal, i the mean value of the angle between the hea uface and adiu of a tet paticle in the contact point of inteacting paticle. Taking into account that the egegation intenity i popotional to the hea ate [9] we expeed the egegation flux in the following way js Kc bum Kcbbdu dy M (4) whee K i the egegation coefficient, u i the mean elative hea velocity between inteacting paticle. Theeby the egegation coefficient acquie the phyical ene of the elation between tanveal and tangent velocitie of tet paticle and M paamete i adequate to the epaation facto which detemine the inclination of mixtue paticle to egegation. Taking into conideation thee cicumtance we uggeted a method fo the egegation coefficient detemination a the elative tanveal velocity of the tet paticle in the conveyo hea cell []. The coefficient i detemined a follow K ut / M bddu / dy () whee u t i the mean tanveal velocity of a tet paticle detemined expeimentally in the conveyo hea cell [] uing unifom bulk paticle. The expeimental eult hown on Fig. and allow to conclude that the egegation coefficient i the expeimental contant, which doen t depend on hea ate and ize of lage and mall pheical coheionle nonelatic tet paticle fo the invetigated ange of the paticulate atio and hea ate K,, 3 3, 4 4, d b а) d t 3, m 6 K b) 3 3 3, 3,4 3,6 3,8 4 F du dy, Fig. Segegation coefficient dependence on the tet paticle diamete (a) and the hea ate (b) fo gla bead (d b = m) 4 3 3

4 V. Dolgunin et al. K a) K b),6,6, d b, d t 3, m 4 du dy 3,, Fig. Segegation coefficient dependence on the tet paticle diamete (a) and the hea ate (b) fo ceamic ganule (d b = m) 3. Mathematical imulation of egegation mixing effect The mathematical deciption of the joint egegation mixing effect in the coue of hea defomation of paticulate olid i baed on the geneal ma tanfe equation taking into account the fluxe of convection, egegation and mixing of paticle, diffeing in ize and denity. Neglecting the mixing flux towad hea diection we fomulated thi equation adapted to the teady two-dimenional hea flow a follow c b ucb x jm j y (6) whee u i the mean paticle velocity towad hea diection x, ( x, y ) Cateian coodinate, i the time. The bounday condition fo Eq. (6) ae fomulated in the abence of tanvee mateial flow at the uppe and lowe boundaie of the moving laye of paticle. The initial condition ha the fom c x.4, y, cx.4, y, The kinetic paamete D dif, and M of thi equation ae calculated, depending on the flow chaacteitic, uing coeponding phyicomechanical chaacteitic of a ganula mateial, and the kinetic contant K i found by the popoed expeimental and analytical method (Fig and ). The flow chaacteitic, uch a u y and y, wee gotten expeimentally by mean of the method uing the conveyo hea cell []. The compaion of the calculated and expeimental eult hown on Fig. 3 and 4 eveal thei adequacy. The tandad deviation between the named eult of egegation mixing dynamic modeling i 6 pecentage. Thu the Eq. (, 4, 6) ha athe high pedictive powe, which i confimed by the eult of tudying the egegation kinetic and dynamic. Futhemoe, the tudy howed that not only can the mathematical model of the hea flow epaation mechanim be ued to decibe the epaation of a mixtue of paticle, but alo thi model can be applied to pedict the velocitie of both mall and lage ingle pheical coheionle paticle duing hea defomation uing a ingle kinetic coefficient. 4

5 A PHENOMENOLOGICAL DESCRIPTION OF MIXING-SEGREGATION EFFECTS DURING SHEAR DEFORMATION OF PARTICULATE SOLIDS y 3, m с, kg kg y 3, m y 3, m с, kg kg - с, kg kg - y 3, m 4 с, kg kg x 3, m Fig. 3 Segegation dynamic duing hea defomation of ceamic ganule (d t = m, d b = m) in the conveyo hea cell at the mean hea ate.47 - expeimental; calculated y, 3, m y, 3, m 4 c, kg kg - y, 3, m 48 c, kg kg - c, kg kg - 3 y, 3, m c, kg kg x, 3, m Fig. 4 Segegation dynamic duing hea defomation of gla bead (d t = m, d b = m) in the conveyo hea cell at the mean hea ate.76 - expeimental; calculated

6 V. Dolgunin et al. Refeence. Dolgunin V., Bochev V. and Klimov A. The conveyo hea cell fo detemination of paticle tendency to egegation and mixing. 4 th Euopean conge of chemical engineeing, full text of pape in CD-ROM; Ganada, Spain, 3.. Dolgunin V., Ukolov A. Segegation modeling of paticle apid gavity flow // Powde Technology. 99. N 83. P Savage, S.B., Intepaticle Pecolation and Segegation in Ganula Mateial: A Review, in Development in Engineeing Mechanic, Selvaduai, A.P.S., Ed., Amtedam: Elevie Science, 987, p Entad, G.G., Segegation of Powde and It Minimization, in Poc. of the nd Iael Conf. fo Conveying and Handling of Paticulate Solid, Kalman, H., Ed., Jeualem, Iael, May 6--8, 997, p.... Pehin, V.F., Modeling of Mixing of a Ganula Mateial in a Co Section of a Rotating Dum, Teo. On. Khim. Tekhnol., 986, vol., no. 4, p Bagnold, R.A., Expeiment on a Gavity-Fee Dipeion of Lage Solid Sphee in a Newtonian Fluid unde Shea, Poc. Roy. Soc. London, Se. A, 94, vol., p Stephen, D.J. and Bidgwate, J., The Mixing and Segegation of Coheion le Paticulate Mateial: Pat I. Failue Zone Fomation; Pat II. Micocopic Mechanim fo Paticle Diffeing in Size, Powde Technol., 978, vol., p Dolgunin, V.N., Kudy, A.N., and Ukolov, A.A., Development of the Model of Segegation of Paticle Undegoing Ganula Flow Down on Inclined Chute, Powde Technol., 998, vol. 6, p.. 9 Bidgwate, J., Cooke, M.H., and Scoott, A.M., Intepaticle Pecolation: Equipment Development and Mean Pecolation Velocitie, Tan. I Chem. E, 978, vol. 6, p. 7.. Dolgunin, V.N., Ukolov, A.A., and Klaen, P.V., Model of the Segegation Mechanim in a Rapid Gavity Flow of Paticle, Teo. On. Khim. Tekhnol., 99, vol. 6, no., p.. 6