DIRECT NUMERICAL SIMULATIONS OF FLUID DRAG FORCES OF NON-SPHERICAL PARTICLE

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1 Eleventh Internatonal Conference on CFD n the Mnerals and Process Industres CSIRO, Melbourne, Australa 7-9 December 2015 DIRECT NUMERICAL SIMULATIONS OF FLUID DRAG FORCES OF NON-SPHERICAL PARTICLE Sathsh K. P. SANJEEVI 1, Johan T. PADDING 1, J. A. M. KUIPERS 1 1 Department of Chemcal Engneerng and Chemstry, Endhoven Unversty of Technology, P.O. Box 513, 5600MB Endhoven, The Netherlands E-mal: s.k.pacha.sanjeev@tue.nl ABSTRACT In the modellng of partcle fludzaton, t s tradtonally assumed that the partcles are sphercal n shape, as ths greatly smplfes the representaton of gas-sold drag and partcle collsons. However, several ndustral processes nvolve partcles whch are hghly non-sphercal n nature. For example, bo-mass gasfcaton process nvolves mlled bo-mass, whch could be approxmated as spherocylndrcal n shape. Ths work nvolves drect numercal smulaton of a sngle non-sphercal partcle. The am of ths study s to characterze the drag coeffcent based on the ynolds number wth partcle orentaton. The smulatons are performed usng the lattce Boltzmann method. The results from these smulatons can be later extended to mult-partcle assembles, whch can be used to derve drag, lft and torque closures for coarse-graned dscrete partcle smulatons. Keywords: LBM, DNS, Non-sphercal partcle. NOMENCLATURE ε Solds volume fracton µ Dynamc vscosty of the flud ν Knematc vscosty ω Proportonalty constant Φ Sphercty φ Incdent angle ρ Macroscopc densty τ laxaton tme C D Drag coeffcent c s Lattce speed of sound D p Equvalent partcle dameter d p Dameter of sphere F Dmensonless force f Dstrbuton functon F f s Force exerted by flud on partcle f (eq) Equlbrum dstrbuton functon h U 0 U Lattce/grd length ynolds number Superfcal velocty latve flud velocty wth respect to partcle INTRODUCTION Practcal applcatons of gas-sold fludzed systems often nvolve partcles whch are non-sphercal, of ether regular or rregular shapes. Therefore, exclusve drag, lft and torque closures must be developed, whch are partcle shape specfc. Several drag closures have been proposed n the past based on experments [Ergun (1952); Wen and Yu (1966)] and smulatons for mono, b- and poly-dsperse spheres [Beetstra et al. (2007); Van der Hoef et al. (2005); Tennet et al. (2011); Yn and Sundaresan (2008, 2009)] as a functon of ynolds number and solds volume fracton φ. One of the earlest and wdely popular drag closures s from Ergun (1952) for randomly packed beds. Hll et al. (2001) n ther work use lattce Boltzmann smulatons to nvestgate the drag force for small to moderate ynolds number flows wth ordered and random array of spheres. The proposed correlaton covers the ynolds number range of and volume fractons n the range Van der Hoef et al. (2005) and Beetstra et al. (2007) proposed drag laws for mono- and b-dsperse arrays of spheres for low and ntermedate ynolds numbers respectvely. cently, Yn and Sundaresan (2008) proposed a drag law for flows at low ynolds number wth b-dsperse spheres wth φ 1 /φ 2 from 1: 1 to 1: 7 and partcle volume fracton from 0.1 to 0.4. Tennet et al. (2011) propose a drag law for mono-dsperse spheres for ynolds number rangng and sold volume fracton 0.1 φ 0.5 usng partcle resolved drect numercal smulatons. The afore-mentoned drag laws are for assembles of sphercal partcles. To the authors knowledge, no drag closures for assembles of non-sphercal partcles have been proposed yet, let alone lft and torque closures. However, there are several works recently reported for non-sphercal partcle fludzaton smulatons usng the dscrete partcle method (DPM) [Zhou et al. (2009, 2011); Zhong et al. (2009); Hlton et al. (2010); n et al. (2012, 2013); Oschmann et al. (2014)]. They use ether emprcal drag cor- Copyrght c 2015 CSIRO Australa 1

2 Models Equatons Partcle orentaton Hader and Levenspel (1989) C D = 24 (1 + A.B ) + C 1+D/ A,B,C, and D are functons of partcle sphercty Φ [ ( ) ] Tran-Cong et al. (2004) C D = 24 d A dn da c dn + [ 0.42(d A /d n ) 2 ( c da dn d A and d n are surface-equvalent sphere and nomnal dameter c s partcle crcularty ) 1.16 ] Hölzer and Sommerfeld (2008) Mandø and Rosendahl (2010) 1 Φ Φ Φ 3/ ( logφ)0.2 1 C D = 8 Φ Φ, Φ, and Φ are regular, lengthwse and crosswse sphercty C D (φ) = C D,φ=0 0 + (C D,φ=90 0 C D,φ=0 0)sn 3 (φ) Table 1: Summary of dfferent drag models for non-sphercal partcles. relatons (parametrzed wth aspect rato and other parameters) proposed for dfferent non-sphercal partcles or use a mult-sphere approach to approxmate complex partcle shapes. Therefore, all these smulatons do not consder the true geometry of the partcles and therefore the reported results may not truly represent actual condtons. Table 1 summarzes the dfferent drag models avalable for non-sphercal partcles. One of the earlest emprcal correlatons s the one from Hader and Levenspel (1989), whch however does not nclude partcle orentaton nformaton and therefore effectvely s not sutable for partcles of arbtrary shape. The correlaton of Tran-Cong et al. (2004), however ncludes orentaton nformaton, but the non-sphercal partcles themselves are created by glung multple spheres together to get the desred partcle shape, whch ntroduces surface roughness. Hölzer and Sommerfeld (2008) proposed a correlaton, whch ncludes two dfferent projected areas to represent partcle orentaton - both lengthwse and crosswse. It s created from a large set of expermental data reported n lterature and also extensve numercal smulatons. The equaton has been tested wth dfferent partcle shapes such as ellpsods, cubods and cylnders. The mean devaton s found to be 14.4% and max. devaton of 29% reported for cubods and cylnders, compared wth expermental results reported n lterature. Snce ths correlaton nvolves extensve data, we compare our smulaton results wth the same and Zastawny et al. (2012) nvolvng the DNS. In ths work, we perform drect numercal smulatons usng lattce Boltzmann method (LBM) to smulate flow around a sngle non-sphercal partcle. We wll show that the LBM predctons are n good agreement wth results obtaned from more tradtonal drect numercal smulaton methods. Ths opens the way to develop closures for lft, drag and torque n mult-partcle assembles. LATTICE BOLTZMANN METHOD The lattce Boltzmann equaton s based on dscretzng the BGK equaton. Ignorng volume forces, t s gven by f ( x+ c t,t + t) = f ( x,t)+ 1 τ ( f (eq) ( x,t) f ( x,t)), (1) where f (eq) ( x, t) s the equlbrum dstrbuton functon and τ s the relaxaton tme. The LBM smulatons are performed n lattce unts and tme s ncremented usng unt tmestep t = 1: f ( x + c,t + 1) = f ( x,t) + 1 τ ( f (eq) ( x,t) f ( x,t)). (2) The above equaton s solved n two separate steps at each tmestep: Collson : f ( x,t + 1) = f ( x,t) + 1 τ ( f (eq) ( x,t) f ( x,t)), Streamng : f ( x + c,t + 1) = f ( x,t + 1). (4) The lattce model used n our smulatons s D3Q19. The equlbrum dstrbuton functon f (eq) ( x, t) derved from the Maxwell-Boltzmann velocty dstrbuton equaton for sothermal condton s gven by ( f (eq) = ρω 1 + c. u c 2 u. u s 2c 2 + ( u. c ) 2 ) s 2c 4 s (3). (5) where ρ s the macroscopc densty, u s the macroscopc velocty, c s s the speed of sound n lattce unts c s = 1 x 3 t and ω s the proportonalty constant. The macroscopc Copyrght c 2015 CSIRO Australa 2

3 varables such as flow densty ρ and velocty u are calculated as, ρ = f (6) u = 1 ρ c f (7) The relaxaton tme τ relates to the lattce vscosty ν by the followng relaton, ( ν = c 2 s τ 1 ) (8) 2 Sphere radus Volume fracton F U Table 3: Sphere radus, volume fracton and the respectve dmensonless drag force F and superfcal velocty U 0 measured. VALIDATION TESTCASES Stokes flow for a smple cubc confguraton Pror to the actual non-sphercal partcles smulaton, a valdaton smulaton wth sphercal partcles s beng performed and compared wth lterature [Krebtzsch (2011)]. In the actual smulaton, a sngle sphercal partcle n smple cubc confguraton wth perodc boundares s smulated at varous volume fractons at Stokes flow ( 0). The flud s subjected to a gravtatonal feld and the partcle s fxed n space. The resultng parameter of nterest s non-dmensonalzed drag force for a sngle partcle F wth respect to the sold volume fracton defned by, F = F f s 3πµU 0 d p (9) where F f s s the force exerted by flud on the partcle, µ s the dynamc vscosty of the flud, U 0 s the superfcal velocty and d p s the partcle dameter. Fgure 1: Dmensonless drag force F as a functon of solds volume fracton for Stokes flow. LBM smulaton parameters Table 2 contans the parameters used n the smulatons n lattce unts. These parameters were mantaned constant throughout all smulatons. Table 3 contans parameters whch were modfed for dfferent smulatons and correspondng dmensonless drag force F measured. The dmensonless drag force F as functon of solds volume fracton s gven n Fg. 1. It can be observed that the results from LBM agree well wth lterature results obtaned from DNS smulatons [Krebtzsch (2011)]. Parameter Value Doman sze Gravtatonal feld for flud forcng Number of teratons laxaton tme 1.0 Knematc vscosty 1/6 Table 2: Fxed parameters for all LBM smulatons n lattce unts. Fgure 2: Face centered cubc (FCC) confguraton. Copyrght c 2015 CSIRO Australa 3

4 soluton free force for a FCC confguraton As another valdaton test case, flow around spheres n a face-centered-cubc (FCC) confguraton (see Fg.2) s smulated at moderate ynolds number = 100 and solds volume fracton, ε = 0.4. The smulaton doman s cubc n shape wth sde lengths rangng from 40 upto 256 and correspondng sphere radus of upto n lattce unts respectvely. The desred s acheved through the combnaton of flud gravtatonal feld and vscosty. The lowest relaxaton parameter used s and the correspondng knematc vscosty s The results are then compared wth Tang et al. (2014). The "resolutonfree" drag force F s computed by fttng the smulaton data obtaned at dfferent resolutons (d p /h) to the form, F = F +C (d p /h) 2, where C s a constant and s plotted n Fg. 3. Present LBM smulatons provde the equaton to be of form, F = (d p /h) 2 (10) compared to the DNS smulatons of Tang et al., F = (d p /h) 2 (11) RESULTS a Fgure 4: The "Ellpsod 1" from Zastawny et al. (2012) wth a/b = 5/2 (and b = c, mplyng prolate ellpsod) wth sphercty Φ = The recent publcaton from Zastawny et al. (2012) contans detaled flow smulaton results of prolate, oblate ellpsod and fbre at dfferent angles of attack. The partcle of our nvestgaton s a prolate ellpsod referenced "Ellpsod 1" n Zastawny et al. (2012) wth rad rato a/b = 5/2 and b = c as n Fg. 4. The sphercty Φ of a non-sphercal partcle s gven by the rato of surface area of volume equvalent sphere as non-sphercal partcle wth respect to surface area of the non-sphercal partcle tself. The partcle under nvestgaton has sphercty, Φ = The LBM flow solver used s hghly scalable, parallelzed wth MPI and has been tested wth lnear scalng up to 262,144 cores as reported n Hartng et al. (2012). b FF LL FF UU φ FF DD Fgure 3: The dmensonless force F obtaned from smulatons at dfferent grd resolutons, as functon of (d p /h) 2. The present work takes the form F = (d p /h) 2 compared to Tang et al. wth F = (d p /h) 2. The present "resoluton-free" force of F = s n close agreement wth the lterature and wthn less than 1% dfference. The dfference n slopes s due to dfferent numercal methods - the present wth LBM and the lterature wth Naver-Stokes based mmersed boundary method DNS. The reason specfcally LBM under-performs s due to frst order explct dscretzaton of the Boltzmann equaton, compared to second order dscretzaton of momentum equatons of NS based DNS. More detals on the NS DNS numercal scheme can be found n Tang et al. (2014). Fgure 5: Forces actng on an nclned non-sphercal partcle. Parameter Value Doman sze Equvalent partcle dameter D p 20 Number of tme steps/teratons Table 4: Varables used n LBM smulatons n lattce unts. The ynolds number s gven by = U D p ν, where D p represents the equvalent partcle dameter of the nonsphercal partcle based on a sphere wth equvalent volume. The range of ynolds numbers smulated s 0 < 100. For a sphere, the flow s stll lamnar for the range of covered here and therefore, we assume the grd resoluton s suffcent for the detaled DNS of the non-sphercal partcle. The smulaton doman s of sze Copyrght c 2015 CSIRO Australa 4

5 10D p 10D p 17.5D p for all. In the actual smulatons, the partcle s moved wth a constant force n the quescent flud. The desred s acheved by varyng forcng and lattce knematc vscosty ν. As the flud doman s perodc, the partcle forcng contnuously adds momentum to the flud. Therefore, the force s balanced by applyng a counteractng force equally dstrbuted n all the flud cells. The maxmum number of processors used n the smulatons s The relevant parameters are summarzed n table 4. As mentoned n Fg. 5, a non-sphercal partcle nclned to the ncdent flow experences both lft and drag force. The ncdent angle nvestgated here s φ = 0,90, where the partcle experences only drag because of symmetry. Drag force C D 10 3 C (Zastawny et al.) D,?= C D,?=90 (Zastawny et al.) C D,?=0 (Holzer and Sommerfeld) C D,?=90 (Holzer and Sommerfeld) C D,?=0 (Present work) C D,?=90 (Present work) Fgure 6: Drag coeffcent C D of the non-sphercal partcle wth respect to ynolds number. It can be observed from Fg. 6 that the smulated results follow the trends of both Zastawny et al. (2012) and Hölzer and Sommerfeld (2008). For ncdent angle φ = 0 at 40, the measured C D les wthn the reported range n the lterature. However at < 40, t can be observed that the measured drag from smulatons s comparatvely larger than correlatons from lterature. Partcularly the devaton s hgher n case of φ = 0 (upto 20% at low ), where the flow ncdent cross secton area s smaller and hence represented by a lower number of lattce cells. Ths ntroduces stronger approxmaton of the boundary as a starcase order leadng to hgher measured drag at low. In case of φ = 90, the agreement s better for all and close to reported results n lterature, as the effectve cross secton s represented by a larger number of lattce cells. CONCLUSION In ths work, we performed drect numercal smulatons of flow around a sngle non-sphercal partcle at dfferent orentaton usng the lattce Boltzmann method. It s observed that there s good agreement between smulated and lterature results for a sngle partcle. Ths opens the way to smulate mult-partcle assembles, generatng closures for lft, drag and torque coeffcents. ACKNOWLEDGEMENT The authors thank the European search Councl for ts fnancal support under ts Consoldator Grant scheme, Contract no (NonSphereFlow). REFERENCES BEETSTRA, R. et al. (2007). Drag force of ntermedate ynolds number flow past mono-and bdsperse arrays of spheres. AIChE Journal, 53(2), ERGUN, S. (1952). Flud flow through packed columns. Chem. Eng. Prog., 48(2), HAIDER, A. and LEVENSPIEL, O. (1989). Drag coeffcent and termnal velocty of sphercal and nonsphercal partcles. Powder technology, 58(1), HARTING, J. et al. (2012). Coupled lattce Boltzmann and molecular dynamcs smulatons on massvely parallel computers. Proceedngs: 25 years HLRZ/NIC, 7-8 February 2012, Jülch, Germany. 2012, 45, 243. HILL, R.J. et al. (2001). Moderate ynolds number flows n ordered and random arrays of spheres. Journal of Flud Mechancs, 448, HILTON, J. et al. (2010). Dynamcs of gas-sold fludsed beds wth non-sphercal partcle geometry. Chemcal Engneerng Scence, 65(5), HÖLZER, A. and SOMMERFELD, M. (2008). New smple correlaton formula for the drag coeffcent of nonsphercal partcles. Powder Technology, 184(3), KRIEBITZSCH, S. (2011). Drect numercal smulaton of dense gas-solds flows. Ph.D. thess, Technsche Unverstet Endhoven. MANDØ, M. and ROSENDAHL, L. (2010). On the moton of non-sphercal partcles at hgh ynolds number. Powder Technology, 202(1), OSCHMANN, T. et al. (2014). Numercal nvestgaton of mxng and orentaton of non-sphercal partcles n a model type fludzed bed. Powder Technology, 258(0), REN, B. et al. (2012). CFD-DEM smulaton of spoutng of corn-shaped partcles. Partcuology, 10(5), REN, B. et al. (2013). Numercal smulaton on the mxng behavor of corn-shaped partcles n a spouted bed. Powder Technology, 234(0), TANG, Y. et al. (2014). A methodology for hghly accurate results of drect numercal smulatons: Drag force n dense gas sold flows at ntermedate ynolds number. Internatonal journal of multphase flow, 62, TENNETI, S. et al. (2011). Drag law for monodsperse gas-sold systems usng partcle-resolved drect numercal smulaton of flow past fxed assembles of spheres. Internatonal journal of multphase flow, 37(9), TRAN-CONG, S. et al. (2004). Drag coeffcents of Copyrght c 2015 CSIRO Australa 5

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