NUMERICAL SIMULATION OF SUDDEN-EXPANSION PARTICLE-LADEN FLOWS USING THE EULERIAN-LAGRANGIAN APPROACH

THERMAL SCIENCE: Year 2012, Vol. 16, No. 4,. 1005-1012 1005 NUMERICAL SIMULATION OF SUDDEN-EXPANSION PARTICLE-LADEN FLOWS USING THE EULERIAN-LAGRANGIAN APPROACH by Mohamed Al MERGHENI a, b*, Jean-Charles SAUTET b, Hmaed BEN TICHA c, and Sass BEN NASRALLAH c a Centre de Recherches et des Technologes de l Energe, LMEEVED, Hammam-Lf, Tuns b CORIA UMR 6614 CNRS Unversté et INSA de ROUEN, Sant Etenne du Rouvray, France c LESTE Ecole Natonale d Ingéneurs de Monastr, Monastr, Tuns Orgnal scentfc aer DOI: 10.2298/TSCI110128036M A Lagrangan-Euleran model for the dserson of sold artcles n sudden-exanson flows s reorted and valdated. The flud was calculated based on the Euleran aroach by solvng the Naver-Stokes equatons. A Lagrangan model s also aled, usng a Runge-Kutta method to obtan the artcle trajectores. The effect of flud turbulence uon artcle dserson s taken nto consderaton through a statstcal model. The redcted axal mean velocty and turbulent knetc energy of both hases agree well wth exermental data reorted by Sommerfeld. Key words: Euleran-Lagrangan model, gas-artcles, sudden-exanson Introducton Two-hase flows can be found n several ndustral rocesses nvolvng e. g. transort conveyng, searaton of sold artcles, and ulverzed-coal combuston. Nowadays, two categorcal aroaches for redctng tow-hase flows are Euleran and Lagrangan. In the Euleran aroach two hases are consdered to be searate nterenetratng contnua, and searate equatons of moton are solved for each hase. Recent models of ths tye are those of Elghobash [1], Lun [2], and Smonn [3]. Ths method may be referably used for dense two-hase flows, for examle, n fludzed beds or two-hase flows wth hase transton, e. g. from bubbly flow to mst flow. In the Lagrangan aroach the dsersed hase s treated by solvng Lagrangan equatons for the trajectores of a statstcally sgnfcant samle of ndvdual artcles, whle the lqud hase s treated as a contnuum n the Euleran aroach. The two-way coulng between both hases s also accounted, Berlemeont et al. [4, 5], Lan et al. [6, 7]. The Euleran-Lagrangan aroach allows an easy mlementaton of hyscal effects occurrng on the scale of the artcle sze as, for examle, artcle-artcle nteractons and artcle-wall collsons. The key element of the Euleran-Lagrangan aroach s how t takes account of the effects of turbulent fluctuatons on artcle, as well as the effects of artcles on turbulence roertes of the lqud hase. * Corresondng author; e-mal: med_alleste@yahoo.fr

1006 THERMAL SCIENCE: Year 2012, Vol. 16, No. 4,. 1005-1012 Hstorcally, turbulent two-hase free jets have been the subject of many studes but, untl very recently t has been dffcult to fnd n the lterature a well-documented numercal study of a two-hase turbulent sudden exanson jet. For an mroved understandng of the characterstcs of sudden exanson artcle-laden flows are necessary. A Euleran-Lagrangan model was used to solve the governng equatons of artcle and lqud hase. The Euleran framework was used for the lqud hase, whereas the Lagrangan aroach was used for the artcle hase. The steady-state equatons of conservaton of mass and momentum were used for the lqud hase, and the effect of turbulence on the flow-feld was ncluded va the standard k-e model. The artcle equaton of moton ncluded the drag force. Turbulence dserson effect on the artcles was smulated by statstcal model. The effects of artcles on the flow were modelled by arorate source terms n the momentum equaton. Ths aer s organzed as follows. In next secton, the mathematcal model s dscussed. In the secton Numercal rocedure and boundary condtons, the numercal rocedure and boundary condtons are resented. Fnally, the numercal results whch the redcted axal mean velocty and turbulent knetc energy of both hases are comared wth exermental data of Sommerfeld [8]. Mathematcal model Ths secton descrbes the mathematcal model for turbulent lqud-artcle flows assumng that the artculate hase s dluted, so that nter-artcle effects are neglected. It s also assumed that the mean flow s steady and the materal roertes of the hases are constant. The lqud hase has been calculated based on the Euler aroach. The artcle hase was treated followng the Lagrangan aroach, whch means that the arameters of every artcle are functons of tme. For the coulng of hases, the PSI-CELL method of Crowe et al. [9] was chosen. By ths method all nfluences of the dsersed hases on the contnuous hase are accounted for through source terms n momentum equaton. Gas-hase flow model The moton of the flud s descrbed by the contnuty and the Naver-Stokes equatons. Turbulence s modelled by the standard model Patanakar [10]. Accordng to the exerment, the sudden-exanson artcle-laden flows are consdered to be steady axsymmetrcal turbulent lqud-artcle flows, wth these assumtons, the corresondng governng equatons for the lqud hase then are: contnuty ( ru ) 0 (1) Naver-Stokes j ruu j U m j j P ( ruu j) S j In these equatons U denotes the Cartesan velocty comonents, P the flud ressure, and r and m are the densty and vscosty of the flud, resectvely. u (2)

THERMAL SCIENCE: Year 2012, Vol. 16, No. 4,. 1005-1012 1007 The turbulence was reresented here by the k-e model, whch mles the need to solve two addtonal equatons, namely: turbulent knetc energy m k eff ruj k Pk re j s k j (3) turbulent dssaton m e eff e ruj e ( k CP 1 k C 2re) j se j U U j U meff mmt, Pk m eff j j where m t s the turbulent vscosty gven by (4) m t = C m rk 2 /e. For the k-e model, the followng standard Table 1. Turbulence model constants coeffcents are used, tab. 1. The nteracton between artcles and the lqud hase yelds source terms n the governng equaton for conservaton of momentum s consdered. The standard exresson Constant Value C m 0.09 C 1 1.44 C 2 1.92 s k 1.0 s e 1.3 for the momentum equaton source terms due to the artcles has been used. It s obtaned by tme and ensemble averagng for each control volume n the form: S u 1 mn u u t out n r ( ) g 1 D (5) Vcv n r Here V cv s the control volume, m the mass of an ndvdual artcle, and n the number of real artcles. Partcle equaton of moton The equaton of artcle moton than can be wrtten as follows: d m u, 3rCDm ( u u, ) u u, mg r 1 (6) dt 4Dr r dx, u, (7) dt Here x, are the co-ordnates of the artcle oston, u, the velocty comonents, D s the artcle dameter and r the artcle densty. The term on the rght-hand sde of eq. (6) s the drag force. Drag force s always resent and s generally the domnatng force for artcle moton n most regons of the flow. Here C D s the drag coeffcent, whch vares wth artcle Reynolds number, the drag coeffcent for the sold artcles s gven as: 24 ( 1 015. Re0. 687 Re 1000 C D Re (8) 0. 44 Re 1000 The artcle Reynolds number s defned as:

1008 THERMAL SCIENCE: Year 2012, Vol. 16, No. 4,. 1005-1012 D u u, Re (9) m The effect of turbulence on the artcle moton s modelled n the resent work by a statstcal model. The nstantaneous lqud velocty along the artcle trajectory s samled from a Gaussan velocty dstrbuton, wth the equal RMS value u = v = 2/3k n all two Cartesan co-ordnate drectons to smulate sotroc turbulence. The nstantaneous flud velocty s assumed to nfluence the artcle moton durng a gven tme erod, called the nteracton tme, before a new fluctuaton comonent s samled from the Gaussan dstrbuton functon. In the resent model, the smulaton of the nteracton tme of a artcle wth the ndvdual turbulent eddes s lmted by the lfetme of the turbulent eddy, T e, or the transt tme, T r, for the artcle to traverse the eddy,. e. D t = mn.(t e,t r ) (10) where T e = 0.2k/e and the eddy length scale l e =C 075. m k 1.5 /e. The transt tme T r can be determned from the lnearzed form of the artcle momentum equaton, gven by Gosman et al. [11]: l e Tr tp ln1 (11) t u u, where t P s the artcle relaxaton tme, defned as: t 4 3 r D rc u u D The Lagrangan aroach can follow only a moderate number of artcles. The real number of artcles n the flow doman, however, s very large. For ths reason, the secal term arcel of artcles was defned. Parcels reresent a large bulk of artcles wth the same sze, mass, velocty and oston. The model s arranged such that every arcel reresents the same mass of a dserse hase. Numercal rocedure and boundary condtons The boundary condtons and flow arameter of the lqud hase and artcle hase, as well as for velocty and turbulent knetc energy at the nlet are gven n tab. 2 and tab. 3. Partcles are ntroduced nto the flow at 25 dfferent nlet ostons between r = 0 m and r = 12.75 10 3 m. The resultng set of equatons s solved by usng a fnte volume dscretsaton scheme and alyng an teratve soluton Table 2. Flow arameters Lqud hase Partcle hase rocedure based on the SIMPLE algorthm. Ths code was extended by ntroducng the addtonal Densty r = 830 kg/m 3 source terms to account for the Dameter D = 450 µm resence of artcles. Knetc vscosty n = 5.205 cst * Densty r = 2500 kg/m 3 A soluton for the lqud feld assumng no artcles s ntally ob- Inlet velocty U n = 6.021 m/s Mass flux f = 0.213 kg/s Mass flux f =2.29 kg/s taned, and the redcton of the artcle moton s carred out. Therefore eqs. (6) and (7) are solved by usng a * 1 cst = 10 2 m 2 /s r, (12)

THERMAL SCIENCE: Year 2012, Vol. 16, No. 4,. 1005-1012 1009 standard 4 th order Runge-Kutta scheme. The source terms are redcted smultaneously durng trajectory calculaton. The lqud feld s then re-comuted wth the contrbuton of artcle source terms. The detal of numercal scheme sees Merghen et al. [12, 13]. Results and dscusson Table 3. Boundary condtons for the lqud hase u n = 6.021; v n =0; k 4 At the nlet k n = 0.006(u 2 + v 2 m ); en n C 3 D 2 No-sl condtons for velocty and the At the wall wall-functon aroxmatons for near-wall grd nodes Along the axs v = 0 and F 0, F uk,,,e In ths secton, ths smulatons result for lqud and artcle veloctes and turbulence knetc energy are resented. Ths smulatons were erformed for a 2-D downward fully develoed sudden-exanson, whch s D1 = 25.5 10 3 m dameter, D2 = = 51 10 3 m dameter large and 1.0 m long, fg. 1. In order to obtan a better defnton and, ossbly a deeer hyscal understandng of the flow feld of a sudden-exanson jet confguraton, At the ext Full develoed flow condtons the radal rofles at var- ous axal statons (x = 9, 50, 100, 150, Fgure 1. The jet flow confguraton of Sommerfeld [8] 200, and 300 mm) are resented one at the nlet of the sudden exanson geometry, three wthn the recrculaton zone and two n the re-develoment zone. Fgures 2(a) and 3(a) show the resence of recalculatng flow regons n the frst two calculaton sectons for both hases. Comarng the exermental and numercal results at x = 9mm of the mean veloctes of flud and artcle, t can be observed the mean veloctes are not well redcted. But fgs. 2(b) and 3(b) show that the smulated mean artcle and flud veloctes are n favourable agreement wth the exermental data. For the turbulent knetc energy of both hases s shown n fgs. 4 and 5. For the near ext regon fgs. 4(a) and 5(a) the turbulent knetc ncrease radally towards the shear layer at r = Fgure 2. Axal mean velocty of the lqud hase

1010 THERMAL SCIENCE: Year 2012, Vol. 16, No. 4,. 1005-1012 Fgure 3. Axal mean velocty of the artcle hase = 0.125 10 3 m, whereas the other statons (x = 100, 150, and 200 mm) n the centre of the ntal regon the gas turbulence ncreases wth ncreasng dstances to the nozzle for the flud and artcles. The reason for ncreased turbulence s hgh radal dffusve turbulence from the shear layer nto the centre of the flow. In general, the numercal redctons of both hases are n good agreement wth Sommerfeld [8] measurements esecally for axal mean veloctes of both hases. The dffer- Fgure 4. Turbulent knetc energy of the lqud hase Fgure 5. Turbulent knetc energy of the artcle hase

THERMAL SCIENCE: Year 2012, Vol. 16, No. 4,. 1005-1012 1011 ence between the measurement and redcton of radal velocty s caused by the turbulent model of lqud hase. Also, the small magntude of the radal velocty n the exermental results may carry some error. There could be two reasons for the dfference between measurements and redcton of turbulent knetc energy of both hases. One s due to the assumed nlet condtons, as there are no exermental data avalable. The other s due to the redcton ablty of the k-e model for lqud hase n sudden exanson flow. Fgures 4 and 5 show that, deste these two factors, numercal results of turbulent knetc energy of both hases are n good agreement wth measurements n the downstream regon where the effect of the nlet condtons become weak and the k-e s sutable for a channel flow. Concluson Usng the Euleran-Lagrangan model, redcted results of velocty and turbulent knetc energy of flud and sold artcles of sudden-exanson n a downward fully develoed channel flow s reresented. A Euleran-Lagrangan model was used to solve the couled governng equatons of artcle-laden flows. The steady-state equatons of conservaton of mass and momentum were used for the flud, and the effect of turbulence on the flow-feld was ncluded va the standard k-e model. The artcle equaton of moton ncluded the drag force. Turbulence dserson effect on the artcles was smulated as a contnuous Gaussan random feld. The effect of artcles on the flud was consdered by ncluson of arorate source terms n the momentum equaton. The redcted axal mean velocty and turbulent knetc energy of both hases agree well wth exermental of Sommerfeld [8]. It s concluded that the Euleran-Lagrangan model has been successfully aled n redctng sudden-exanson artcles-laden flows. Nomenclature C m,c 1,C 2 coeffcents of the turbulence model C D drag coeffcent D center jet dameter, [mm] D P artcle dameter, [µm] g acceleraton due to gravty, [ms 1 ] l e eddy sze, [m] m artcle mass r radal co-ordnate, [m] Re artcle Reynolds number S source term T r resdence tme of the artcle, [s] T e turbulent eddy lfetme, [s] Dt nteracton tme of artcle, [s] u axal mean velocty, [ms 1 ] x axal co-ordnate, [m] V control volume, [m 3 ] Greek symbols e knetc energy dssaton rate, [m 2 s 3 ] k knetc energy of turbulence, [m 2 s 2 ] m dynamc vscosty, [kgm 1 s 1 ] r gas densty, [kgm 3 ] r artcle densty, [kgm 3 ] t artcle tme, [s] F artcle-loadng rato References [1] Elghobash, S., Partcle-Laden Turbulent Flows: Drect Smulaton and Closure Models, Al. Sc. Res., 48 (1991), 3-4,. 301-314 [2] Lun, C. K. K., Numercal Smulaton of Dlute Turbulent Gas-Sold Flows, Internatonal Journal of Multhase Flow 23 (1997), 3,. 575-605 [3] Smonn, O., Predcton of the Dsersed Phase Turbulence n Partcle-Laden Jets, 4 th Internatonal Symosum on Gas-Sold Flows, Portland, Ore., USA, 1991,. 23-26

1012 THERMAL SCIENCE: Year 2012, Vol. 16, No. 4,. 1005-1012 [4] Berlemont, A., Desjonqueres, P., Gouesbet, G., Partcle Lagrangan Smulaton n Turbulent Flows, Internatonal Journal of Multhase Flow, 16 (1990), 1,. 19-34 [5] Berlemont, A., Grancher, M. S., Gouesbet, G., Heat and Mass Transfer Coulng Between Vaorzng Drolets and Trubulence Usng a Lagrangan Aroach, Internatonal Journal of Heat Mass Transfer, 38 (1995), 16,. 3023-3034 [6] Lan, S., et al., Modellng Hydrodynamcs and Turbulence n a Bubble Column Usng the Euler-Lagrange Procedure, Internatonal Journal of Multhase Flow, 28 (2002), 8,. 1381-1407 [7] Lan S., Sommerfeld, M., Turbulence Modulaton n Dsersed Two-Phase Flow Laden wth Solds from a Lagrangan Perseectve, Internatonal Journal of Multhase Flow, 24 (2003), 4,. 616-625 [8] Sommerfeld, M., Partcle Dserson n a Plane Shear Layer, 6 th Worksho on Two-Phase Flow Predctons, Erlangen, Germany, 1992 [9] Crowe, C. T., Sharma, M. P., Stock, D. E., The Partcle-Source-In Cell(PSI-CELL) Model for Gas-Drolet Flows, Journal of Fluds Engneerng, Transactons of the ASME, 99 (1977), 2,. 325-332 [10] Patankar, S. V., Numercal Heat Transfert and Flud Flow, Hemshere Publshng Co., New Yourk, USA, 1980 [11] Gosman, A. D., Ioanndes, E., Asects of Comuter Smulaton of Lqud-Fuelled Combustors, AIAA aer 81-0323, Journal of Energy, 7 (1983), Nov.-Dec.,. 482-490 [12] Merghen, M. A., et al., Interacton Partcle-Turbulence n Dsersed Two-Phase Flows Usng the Euleran-Lagrangan Aroach, Internatonal Journal of Flud Mechancs Research, 3 (2008), 35,. 273-286 [13] Merghen, M. A., et al., Measurement of Phase Interacton n Dsersed Gas-Partcle Two-Phase Flow by Phase Doler Anemometry, Thermal Scence, 2 (2008), 12,. 59-68 Paer submtted: January 28, 2011 Paer revsed: Arl 25, 2011 Paer acceted: May 7, 2011