Pseudo Three-Dimensional Modeling of Particle-Fuel Packing using Distinct Element Method

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1 Proceedigs of ICONE5 5th Iteratioal Coferece o Nuclear Egieerig April -6, 007, Nagoya, Japa Pseudo Three-Dimesioal Modelig of Particle-Fuel Packig usig Distict Elemet Method ICONE Daisuke YÛKI Departmet of Sustaiable Eergy ad Eviromet Egieerig, Osaka Uiversity Address: -, Yamada-Oka, Suita, Osaka , Japa Phoe: Fax: yuki_d@qe.see.eg.osaka-u.ac.jp Takashi TAKATA Departmet of Sustaiable Eergy ad Eviromet Egieerig, Osaka Uiversity Address: -, Yamada-Oka, Suita, Osaka , Japa Phoe: Fax: takata_t@see.eg.osaka-u.ac.jp Akira YAMAGUCHI Departmet of Sustaiable Eergy ad Eviromet Egieerig, Osaka Uiversity Address: -, Yamada-Oka, Suita, Osaka , Japa Phoe: Fax: yamaguchi@see.eg.osaka-u.ac.jp KEYWORDS: MOX, Sphere-pac Fuel, Distict Elemet Method, Pseudo Three-Dimesioal Model ABSTRACT Vibratio-based packig of sphere-pac fuel is a key techology i a uclear fuel maufacturig. I the productio process of sphere-pac fuel, a Mixed Oxide (MOX) fuel is formed to spherical form ad is packed i a claddig tube by addig a vibratio force. I the preset study, we have developed a umerical simulatio method to ivestigate the behavior of the particles i a vibrated tube usig the Distict Elemet Method (DEM). I geeral, the DEM requires a sigificat computatioal cost. Therefore we propose a ew approach i which a small particle ca move through the space betwee three larger particles eve i the two-dimesioal simulatio. We take ito accout a equivalet three-dimesioal effect i the equatios of motio. Thus it is amed pseudo three-dimesioal modelig.. INTRODUCTION I the Fast Breeder Reactor (FBR) cycle system, low-decotamiated Mixed Oxide (MOX) fuel that cotais high radioactive rare earth elemets ad Mior Actiides (MA) is to be used. A sphere-pac fuel maufacturig ad vibro-fillig process are suitable for this purpose because of its easy remote hadlig. I the process of sphere-pac fuel productio, a MOX fuel is formed to be spheral ad it is filled i a claddig tube uder a vibratio motio. I the sphere-pac fuel maufacturig, complicated processes are ot required, such as atomizatio of fuel ad a siterig of fie powder to form a fuel pellet. Therefore, oe ca achieve a simplified maufacturig process at low cost. For achievig a high ad uiform packig desity, particles of differet diameters are bleded by addig exteral vibratio. A well-used techique is a ifiltratio method i which the coarse particles are filled first. Subsequetly, the fie particles are filled ito the opeig space of the coarse particles. To optimize the productio method, it is eeded to comprehed the behavior of the particles i a vibratig tube ad to estimate the most effective value of amplitude ad frequecy of the vibratio. For this purpose, we apply the Distict Elemet Method (DEM) to the umerical simulatio of the vibratio-based packig process. The DEM was developed ad applied to rock mechaics aalysis (Cudall, 97). The DEM is a techique to aalyze the dyamic behavior of discotiuous elemet s aggregatio, hece, it is successfully-used to calculate a soil avalache, powder behavior ad so o. It is expected that the DEM is suitable for the aalyses of sphere-pac fuel maufacturig. However, it is recogized that the DEM eeds a sigificat computatioal cost especially i three-dimesioal aalysis (The Society of Powder Techology, Japa, 998). Therefore, two dimesioal aalysis is preferable from a practical viewpoit. The authors performed a two-dimesioal simulatio of particle-fuel packig ad poited out that the ifiltratio of particles with differet diameters caot be expressed i the two-dimesioal aalysis (Yuuki, 006). Therefore, a ew methodology eeds to be developed to simulate a three-dimesioal behavior of particles with practical computatio time. I this study, we propose a pseudo Copyright 007 by JMSE

2 three-dimesioal simulatio method that takes ito accout the three-dimesioal effect i the behavior of particles.. COMPUTATIONAL METHOD I the DEM, goverig equatios are solved for each particle, successively. To calculate a cotact force betwee particles, virtual sprigs ad dashpots i ormal ad tagetial directios are assumed (see Fig.). The virtual sprig expresses a repulsio force ad a frictio force beig proportioal to relative displacemet of the two particles. The virtual dashpot estimate eergy dissipatio due to the relative velocity betwee particles. Cotact forces F ad F s based o the virtual sprigs ad dashpots are writte as: du F = η + Ku, ad () dt dus dψ Fs = ηs( + r ) + Ks( us + rψ ), () dt dt i ormal ad tagetial directios, respectively. u ad u s are ormal ad tagetial compoets of the relative displacemet of the two particles. ψ is relative rotatioal displacemet. η ad η s are dampig coefficiets for ormal ad tagetial directios, K, ad K s are stiffess coefficiets for ormal ad tagetial directio. These two forces are trasformed i horizotal (x), vertical (y) ad rotatioal (ψ ) compoets i the global coordiates. F x, F y ad F ψ are defied as cotact forces i each directio. Two-dimesioal Newto s equatios of motio for particles with cotact force are writte as: dx m xj dt j= dy m = Fyj dt j= dψ I = Fψ j rj dt j= = F, (3) mg, (4). (5) I Eqs.(3)-(5), iteractive forces with all the eighborig particles ( j =,., ) are added. Equatio (3), (4) are the equatios for traslatio motio with two-degree of freedom, ad Eq. (5) is the equatio for rotatio with oe degree of freedom. m is a mass of the particle, I is a iertia momet. Equatios (3)-(5) are discretized with respect to time to obtai the dyamic motios of particles. The Adams-Bashforth method is used for the time itegratio, which is secod-order accurate.. Stiffess ad Dampig The stiffess coefficiets K, K s i Eq. () ad () are determied based o Heltz s theory of elasticity (Timosheko ad Goodier, 970). Let us cosider a situatio show i Fig. where two particles collide. The compressive deformatio is δ ad cotact diameter is b. Accordig to the Heltz s theory of elasticity, δ ad b are writte as Eqs. (6) ad (7) usig Youg s modules (E), Poisso s ratio (ν), the radius of each particles (r, r ) ad the compressive force betwee particles (q): ( ν ) q 4r 4r l l, δ = π E b b 8 rr ν b = q, (7) π r+ r E respectively. The stiffess coefficiet i ormal directio K is described as follows usig two quatities give i Eq. (6) ad Eq. (7). K (6) q π E = = (8) δ 4r 4r ( ν ) l l b b From Eq. (8), it is see that K approaches ifiity whe δ icreases. The compressive force ca become urealistically large, uless sufficietly fie timestep is used. So this method requires a sigificat computatioal cost. The stiffess coefficiet i tagetial directio K s is evaluated by Midli theory (Shiohara, 000) as follows: Tagetial directio K s η s Particle A Frictioal slider Virtual sprig Virtual dashpot K η Particle B 8bsE Ks = (9) ( ν ) where s is the ratio of logitudial stiffess coefficiet to shear coefficiet of rigidity. Dampig coefficiets are obtaied by a coefficiet of reboud of particles (Kawaguchi, 99). Eq. (3) ca be solved with the iitial coditios, i.e. x=0, v=v 0 at 0 secod. The displacemet ad velocity of particles are give by: v x = 0 expαt si βt, β (0) v v = 0 expαt( αsi βt + βcos βt), β () respectively, where α ad β are writte as: Normal directio Fig. The model of cotact force betwee particles η η 4mK α =, β =. () m m Copyright 007 by JMSE

3 r q δ v = v 0 expπαβ. (3) Here we ca estimate the coefficiet of reboud e as the ratio of the collisio velocity ad the departig velocity: v e = = expπαβ. (4) v0 A viscosity coefficiet i ormal directio is obtaied by elimiatig α ad β i Eq.(4) usig Eq. (). b r η = mk γ, δ γ =, δ = l e ( e 0) δ π + γ = ( e = 0) (5) q Fig. Two particles collisio model based o Heltz s theory of elasticity A particle which collides aother oe at t=0 detaches from it at half period i.e. t=π/β. The particle velocity of the particle at the detachig time is give by: 3. PSEUDO THREE-DIMENSIONAL MODEL If three particles are i cotact, a opeig space is formed i the ceter. Let us cosider aother particle with small diameter. If the diameter of the fie particle is small eough, it ca go through the space. I the two-dimesioal model, however, a fie particle caot pass through the opeig space of the coarse particles. Therefore, three-dimesioal represetatio is ecessary for the simulatio of particle-fuel packig. O the other had, the DEM requires a sigificat computatioal load whe it is applied to a three-dimesioal z h r r () () (3) Fig.3 Compariso of () three-dimesioal, () two dimesioal with (3) pseudo three-dimesioal closest packed structure. The upper is top view ad the lower is frot view. 3 Copyright 007 by JMSE

4 cofiguratio. Therefore, for simulatig a ifiltratio of particles ad reductio of a computatioal cost, we propose a ew approach i which the fie particles ca ifiltrate through the cotact of the coarse particles eve i the two-dimesioal computatio. The methodology equivaletly takes the three dimesioal effect ito cosideratio. i.e. pseudo three-dimesioal model. 3. Criterio of cotact of particles of the same diameter Figure 3 shows the compariso of () the three-dimesioal, () two dimesioal ad (3) pseudo three-dimesioal closest packed structure of particles of the same diameter viewed from the top ad the frot. I the two-dimesioal aalysis, there is a problem that a closest packed structure made of particles with the same diameter is icosistet with a actual situatio (Compare Fig.3-() ad ()). For the solutio of this problem, we propose a ew criterio of cotact betwee particles. I the preset method, a virtual diameter is defied to judge whether particles are i cotact each other or ot. The virtual diameter is determied smaller tha the real oe. Resortig to the proposed method, it is able to describe the three-dimesioal closest packed structure i the two-dimesioal simulatio. I Fig. 3-(3), the dotted circles express virtual diameters ad the solid circles describe real diameters. Comparig Fig.3-(3) with Fig.3-(), it is see that the depth of the particles is differet by z, ad the height is smaller by h. I Fig.3-(), the figure made by red lies forms triagular. Therefore, r, h ad z are writte as: r' = r 3 8 h= 3 r 3, (7) z = r. (8) 3 3. Criterio of cotact of particles with differet diameters Next, a cotact betwee particles with differet diameters is discussed. The situatio is show i Fig.4 for () three-dimesioal () two-dimesioal ad (3) pseudo three-dimesioal cotact of the coarse particles ad the fie particle. Whe the ratio of a diameter of the fie particle to a diameter of the coarse oe is less tha oe seveth at least i three-dimesioal situatio, fie particles ca pass through the opeig space betwee coarse particles. However, fie particles caot go through the gap amog the coarse particles i the two-dimesioal aalysis. I Fig. 4-(3), the dotted circles are virtual diameters for criterio of cotact ad solid circles are real diameters. Usig this method, fie particles ca slip through the opeig space betwee coarse particles whe fie particles are at the ceter of coarse particles. The virtual radius is calculated as: 3 3 r' = r. (9) 6 4. ANALYSES OF MAKING PROCESS OF VIBRO-PACKED FUEL, (6) Usig the proposed method of cotact amog particles, the behavior of particle-fuel i the maufacturig process is calculated. Coarse particles are filled first. After coarse z r r () () (3) Fig. 4 Compariso of () three-dimesioal, () two-dimesioal with (3) pseudo three-dimesioal cotact betwee two coarse particles ad a fie particle. The upper is top view ad the lower is frot view. 4 Copyright 007 by JMSE

5 particles come to a stadstill at 0.06 secod, fie particles are poured for 0.06 secod from the top of the cotaier. The cotaier is exterally shaked at the frequecy ad displacemet amplitude show i Table. The vibratio force is applied from 0.5 secod to 0.40 secod. Table shows the iitial value. The aalyses are performed usig two-dimesioal DEM ad pseudo three-dimesioal DEM. 4. Result ad Discussio The result of aalyses is show i Fig. 5. The coarse particles are statioary at 0.06 secod as see i Fig. 5-(). At this momet, fie particles are dropped from up above. At 0.5 secod, some fie particles already ifiltrate the coarse particle bed. It ca be see from Fig. 5-() that other fie particles remais o the coarse particles ad blockage takes place. Exteral vibratio force is applied at 0.5 secod for 0.40 secod duratio. At the ed of vibratio, most of the fie particles ifiltrate ito the coarse particles. Although the aalysis is two-dimesioal, the three-dimesioal pheomea are aalyzed ad the results are reasoable. For evaluatig the effectiveess of the exteral vibratio or the ifiltratio performace, a computatio is made with the same aalytical coditios except the exteral vibratio. I other word, fie particles free-fall to the coarse particles layer. Ad they come to stadstill. Thus the body force actig to the particles is the gravity aloe. Whe we compare Fig. 5-(3) ad Fig. 6, it is foud that the vibratio force cosiderably cotributes to a ifiltratio of fie particles Figure 7 shows the two-dimesioal computatio without the preset model of virtual diameter. Exteral vibratio is added i this computatio with the same Table The iitial coditio of aalyses particles coarse diameter.40e-3 [m] fie diameter.00e-4 [m] desity.0e+4 [kg/m 3 ] Youg s modules.00e+0 [Pa] Poisso s ratio 0.8 [-] coefficiet of frictio 0.5 [-] umber (coarse) 0 umber (fie) 300 cotaier width.00e- [m] height.00e- [m] Youg s modules 3.90E+9 Poisso s ratio 0.5 coefficiet of frictio 0.7 calculatio time step.0e-7 [sec] max step vibratio amplitude.00e-4 [m] frequecy 00 [Hz] coditio as i Fig. 5. I the two- dimesioal aalysis, ifiltratio of particles does ot take place at all as show i Fig. 7. It appears the exteral vibratio force is ot large eough to let the fie particles ifiltrate. () at 0.06 secod 5 Copyright 007 by JMSE

6 () at 0.5 secod (3) at 0.50 secod Fig.5 The sapshot of result, added vibratio Fig.6 The sapshot of result at 0.50 secod, o vibratio Fig.7 The sapshot of result at 0.50 secod, two-dimesioal computatio 6 Copyright 007 by JMSE

7 5. CONCLUSION The packig process of sphere-pac fuel is aalyzed usig a Distict Elemet Method. For simulatig three-dimesioal vibratio-based packig process withi practical computig cost, we have improved the DEM, so that the three-dimesioal effect is expressed i the two-dimesioal computatio. As a result, the behavior i which fie particles ifiltrate i opeig spaces betwee coarse particles ad three-dimesioal closest packig ca be achieved i two-dimesioal aalyses. Therefore, it is cocluded that the dyamic behavior of particles i the maufacturig process of the sphere-pac fuel ca be simulated with the preset methodology based o the two-dimesioal cofiguratio. For the future, we will improve ad validate the preset method. Furthermore, the amplitude ad frequecy of the exteral vibratio are to be optimized by the umerical simulatio to ehace the fuel fillig performace, we will fid the better coditio of fuel fillig. NOMENCLATURE b cotact diameter e reflectio coefficiet E Youg s module F x cotact force for x-directio F y cotact force for y-directio F ψ cotact force for ψ-directio F cotact force for ormal directio F s cotact force for tagetial directio g acceleratio of gravity K stiffess coefficiet for ormal directio K s stiffess coefficiet for tagetial directio m mass of particle q compressive force r radius of particle s the ratio of logitudial stiffess coefficiet to shear coefficiet of rigidity. u relative displacemet v velocity of particle iitial velocity of particle v 0 REFERENCE Cudall PA., 97, A computer model for simulatig progressive large scale movemets i blocky system. I: Muller L, editor. Proc. Symp. It. Soc. Rock Mech.,. Rotterdam: A.A.Balkema. 8. The Society of Powder Techology, Japa, 998 The itroductio of powder simulatio (i Japaese). D. Yuuki, T. Takata ad A. Yamaguchi, 006, Numerical Simulatio of Particle-Fuel Packig Based o Ditict Elemet Method, Aual Meetig of the Atomic Eergy Society of Japa. E49 (i Japaese). Timosheko, S. P. ad J.N. Goodier, 970, Theory of Elasticity McGraw-Hill p.409. K. Shiohara, et al, 000 Movig Bed Reactor, Hokkaido Uiversity Press (i Japaese). T. Kawaguchi, et al, 99, Numerical Simulatio of Fluidized Bed usig the Discrete Elemet Method, Joural of the Japa society of mechaical Egeers, 58, 9-5 (i Japaese). δ η η s ν ψ compressive deformatio dampig coefficiet for ormal directio dampig coefficiet for tagetial directio Poisso s ratio displacemet of relative rotatio 7 Copyright 007 by JMSE