Numerical simulation of turbulent forced convection in a venturi channel with fully developed flow at the inlet

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1 Avalabl onln at Advancs n Appld Scnc Rsarch, 2014, 5(2): ISSN: CODEN (USA): AASRFC Numrcal smulaton of turbulnt forcd convcton n a vntur channl wth fully dvlopd flow at th nlt Srg Wndsda Igo 1, Kokou N wutcha 2, K. Palm 1, Lucan Mhascu 3 and D. J. Bathébo 4 1 Insttut d Rchrch n Scncs Applqués t Tchnologs (IRSAT/CNRST). Départmnt Enrg, 03 BP 7047 Ouagadougou 03, Burkna Faso 2 GPTE-LES Départmnt d Physqu, Unvrsté d Lomé, BP 1515, Togo 3 Polytchnc Unvrsty of Bucharst (UPB), Dpartmnt of Mchancal Engnrng, Romana 4 Laborator d Enrgs Thrmqus Rnouvlabls (LETRE), Unvrsté d Ouagadougou, 03 BP 7021 Ouagadougou 03, Burkna Faso ABSTRACT In ths work, a 2D turbulnt forcd convcton ar flow n a vntur channl has bn numrcally smulatd usng flunt. Transfrs quatons ar dscrbd by th Rynolds Avragd Navr-Stoks quatons (RANS) and th vcous modl s th turbulnt standard k-ε modl. A fully-dvlopd powr-law profl s assumd for th nlt stram ws vlocty. Th rsults ar prsntd as vlocty fld, statc and dynamc prssurs flds, stramlns fld, vortcty and turbulnt kntc nrgy flds. Thy show that th rgon of maxmum turbulnt kntc nrgy s th dffusr wall rgon xtndng to th channl xt and th turbulnc shfts away from th dffusr wall n th flow drcton. Ky words: vntur channl, turbulnt flow, fully dvlopd flow, CFD mthods, flunt INTRODUCTION Vntur channls ar wdly usd as scrubbrs for partcls and gasous collcton from ndustral xhaust [1] or to mtr gas flows [2]. Ths dvcs consst of channl wth thr parts : a convrgnt scton, a throat and a dvrgnt scton or dffusr. Numrcal turbulnt studs n vntur channls ar scarc bcaus of th on hand, th complxty of th gomtrcal confguraton and th othr hand, th dffculty to modl th turbulnt flow along th vntur. Among th turbulnc modls n gas flow, th standard k-ε modl [3] s th most popular. In many numrcal studs n vntur channls, th vlocty gas fld s calculatd usng ths modl [4,5]. Howvr, t s wll know that nar th walls occurs mportant vcous forcs, so, th standard k-ε modl whch s sutabl for larg Rynolds numbrs s no longr applcabl. To solv ths problm, two solutons can b usd. Th frst soluton consst to us th standard k-ε modl wth wall functons n ordr to forc th frst nod of th computatonal grd to b n th sub-layr. Th scond soluton consst to us drctly th k-ε LRN (Low Rynolds Numbr) modl [6]. Th gas ntry condtons ar on of th smulaton problms. It s rar n practc to ncountr an unform nlt flow nto a vntur channl. Igo t al. studd th turbulnt gas clanng n a rctangular vntur scrubbr usng th k-ε LRN modl wth fully dvlopd flow at th nlt and show that th turbulnc s dvlopng from th nlt to th outlt of th vntur channl [7]. Thr CFD mthod was basd on a mathmatcal transformaton of th vntur wall and th fnt volum mthod whch has bn mplmntd by Thomas and Gauss algorthms. Usng a concal dffusr (Azad dffusr) whch has a total dvrgnc angl 8 and a fully dvlopd flow at th nlt, Okwuob and Azad [8] showd that th pak of th turbulnt fluctuatons shfts away from th wall along th stramws drcton of th flow. Thy also show that th rat of turbulnt nrgy producton rachs a maxmum valu at th dg of th wall layr. To faclty th turbulnt modlng and smulaton n gas flow, CFD tools whch nclud pr-procssors, solvrs and post-procssors ar mor and mor usd. Among ths tools, th solvr post-procssor flunt [9] and th pr-procssor gambt [10] ar wdly usd. Many studs basd on ths tools ar wdly bn prformd [11,12,13]. Rcntly, Igo t al. [14] studd th turbulnt gas flow n a vntur channl usng flunt and gambt. Thy showd 359

2 that th vntur ffct s wll prdctd by th flunt cod. Thrfor, th obctv of th prsnt study s to xtnd ths work and provd mor rsults of turbulnt flow n vntur channls. PROBLEM FORMULATION Th vntur channl s composd of two ron plats of sctons lngths (L 1, L 2, L 3 ). Th nlt damtr s R o and th throat damtr s R th. Th wall plats ar subctd to a constant tmpratur T w. A fully-dvlopd turbulnt ar flow wth avrag vlocty U ntrs at th vntur channl wth an unform tmpratur T. Y R o Ar O R th x L 1 L 2 L 3 Fg 1. Schmatc rprsntaton of th studd systm n th (O,X,Y) rfrntal It s assumd that th flow s ncomprssbl and th transfrs ar two-dmnsonal, axsymtrc and stady stat. Th RANS quatons modlng th turbulnt ncomprssbl ar flow through th vntur channl ar: -Contnuty quaton U = 0 -Momntum quaton P U U ( ) ρu U µ = + + ( ρu u ) x (1) (2) U, P ar rspctvly th avrag vlocty and prssur, u s th vlocty fluctuaton. Th Rynolds strsss ( ρ u u ) s dfnd by th Boussnsq approxmaton: U U 2 ρ uu + = µ t + ρkδ (3) 3 µ t s th turbulnt vscosty, k s th turbulnt kntc nrgy and δ s th Kronckr symbol. To clos ths quatons, th standard k-ε modl quatons ar: µ t k ( ρu k) = µ + Q ρε x x + σ k µ t ε ε ( ρu ε) = µ + + ( C Q ρc ε) k σ ε ε1 ε2 (4) (5) 360

3 2 k µ t = ρcµ ε (6) Q s th trm of producton and s dfnd by: Q U U U = µ t + (7) C µ = 0.09; σ k = 1; σ ɛ = 1.3; C ɛ1 = 1.44; C ɛ2 = 1.92 ar th constants of th k-ε modl. NUMERICAL SIMULATION Msh Th msh s gnratd usng th Gambt 2.2 softwar. Du to symmtry, only th half doman nds to b consdrd. Th msh sz s vry clos nar th walls to tak account th turbulnt boundary layr. Fg 2. Channl msh Smulaton Transfr quatons ar solvd usng flunt 6.2, a computatonal flud dynamcs softwar whch nabls accurat smulaton of flow n channls. Th Standard Wall Functon s usd for th tratmnt of th turbulnt boundary layr. To us ths functon, ach wall-adacnt cll s cntrod should b locatd wthn th log-law layr (30 < y + < 300). In ordr to prformd th smulaton, th followng boundars condtons ar consdrd: -At th nlt A fully-dvlopd powr-law profl (n=7) s assumd for th stramws vlocty. As w consdr th lowr half channl, th nlt vlocty profl s: U = Uo 1 + y R o 1/n U o s th cntrln vlocty. U o = U ( 2n + 1)( n + 1) 2n 2 U s th avrag nlt vlocty whch s corrlatd to th nlt Rynolds numbr R. UDh R = υ Dh s th hydraulc damtr and υ th ar nlt cnmatc vscosty. Th computng of th nlt vlocty profl s prformd by usng th flunt 2D UDF cod. Th nlt turbulnt kntc nrgy k and hs rat of dsspaton ε can b xprssd as: 361

4 3 k = 2 ε = C ( I U ) 2 3/4 µ 3/2 ( k ) /l I s th nlt turbulnc rat and l th turbulnc scal: I 1/ 8 ( R) = 0.16 for ntrnal fully dvlopd flows. l=0.07dh -At th walls : T=T w and th no-slp condtons ar appld for th flow. -At th symmtry axs : symmtry boundary condton s mposd. -At th outlt : outflow boundary condton s mposd. RESULTS AND DISCUSSION In th prsnt study, smulatons wr prformd for R o =0.3m, R th =0.15m, L 1 =0.48m, L 2 =0.1m, L 3 =0.72m, U =13 ms -1, T = T w = 300K. Bfor prnt th rsults, w vrfy that th msh s conform to th standard wall functon law. Fg 3. Wall Yplus voluton along th vntur channl Wall Yplus (fgur 3) s btwn 30 and 60. Th msh s conform to th standard wall functon law. Fg 4. Inlt vlocty 362

5 Fgur 4 shows th nlt vlocty computd usng th flunt 2D UDF cod. As sn, th nlt vlocty profl s turbulnt fully-dvlopd and s an agrmnt wth our hypothss. Fg 5. Contours of vlocty magntud (m/s) Fg 6. Contours of statc prssur (Pascal) Fg 7. Contours of dynamc prssur (Pascal) Fgurs 5-6 show rspctvly th vlocty and th statc prssur flds n th vntur. W not that th flow vlocty ncrass as th flow movs n th convrgng scton to rach a maxmum valu at th vntur throat. In fact, n th convrgng scton, thr s a contnuous ncras of th prssur drop du to th acclraton of th ar vlocty rsultng of th convrson of th ar potntal nrgy nto kntc on. Accordng to th flow rat consrvaton law 363

6 and consdrng th vntur damtr rato (R th /R o ) qual to 0.5, th vlocty must doubl n th throat and w can not t on Fgur 5. In th dvrgng scton, th dcras of th vlocty s du to th xpanson of th channl damtr. Th voluton of th statc prssur along th vntur s th oppost of th vlocty on. Th lowst prssurs ar obsrvd n th throat. Th larg dprssons sn at th throat cornrs ar du to th chang of th flow drcton. Th flow n th vntur channl and s thn an agrmnt wth th Brnoull law : ths s vntur ffct. Fgur 7 shows th dynamc prssur fld. Th dynamc prssur s by dfnton proportonal to th squar of th vlocty. W ndd notc that th dynamc prssur and th vlocty hav approxmatly th sam voluton n th channl. Th dynamc prssur ncrass from th nlt to th throat and dcrass from th throat to th outlt. Th hghst valus ar obsrvd n th vntur throat. Fg 8. Contours of stram functon (Kg/s) Fg 9. Channl vlocty vctors colord by vlocty magntud (m/s) Fgurs llustrat th flow structur, rspctvly th stramlns fld, th channl vlocty vctors fld and th dffusr vlocty vctors fld. W not that stramlns (fgur 8) ar vry clos n th convrgnt scton and thy ar mrgd n th throat du to th rducton of th channl scton and th augmntaton of th vlocty. In th dvrgnt scton, th stramlns ar dtachd from th wall partcularly at th channl xt. Th stramlns voluton n th channl shows that th flow s non-sparatd bcaus th channl dvrgnc angl (about 6 ) and th dffusr aspct rato (0.72/0.15=4.8) ar low. Accordng to th dagram of typcal flow rgons [15], ths gomtrcal paramtrs cannot crat a sparatd flow n th channl. Th obsrvaton of th vlocty vctors (fgur 9) shows that th flow movs from th channl nlt to th channl outlt, but w can s th chang of vlocty vctors (fgur 10) whos movmnts bcom vry random n th dffusr wall rgon. Ths stuaton s typcal n th dffusrs and s du to a rtardd flow (advrs prssur gradnt) whch appars n ths knd of channl. And t s wll know that ths phnomnon contrbut to amplfy th vlocty fluctuatons. 364

7 Fg 10. Dffusr vlocty vctors colord by vlocty magntud (m/s) Fg 11. Contours of vortcty magntud (1/s) fgur 11 shows th contours of th vortcty magntud. W not that som low ampltud vortcs appar n th dffusr wall rgon xtndng to th channl xt. Ths rsult corroborat th prvous on. In fact, t s wll know that a vortcs zon s an mportant rgon of vlocty fluctuatons. Fg 12. Contours of turbulnt kntc nrgy (k) (m 2 /s 2 ) 365

8 Fg 13. Contours of turbulnt ntnsty (%) Fgurs show th contours of th turbulnc ntnsty and th turbulnt kntc nrgy rspctvly. W not that th turbulnt kntc nrgy and th turbulnc ntnsty hav approxmatly th sam voluton and thr maxmum valus ar obsrvd n th dffusr wall rgon xtndng to th channl xt. Ths rsult s th consqunc of th two prvous rsults. In a channl, th turbulnc ntnsty s of cours mprovd n th zons whr th vlocty fluctuatons ar mportant, and ths zon for th studd channl s th dffusr wall rgon xtndng to th channl xt. W obsrv also that th turbulnc shfts away from th wall n th flow drcton. CONCLUSION A forcd turbulnt convcton ar flow n a 2D vntur channl was numrcally nvstgatd n th prsnt study usng flunt. A fully-dvlopd powr-law profl s assumd for th nlt stramws vlocty. Transfrs quatons ar dscrbd by th Rynolds Avragd Navr-Stoks quatons and th vcous modl s th turbulnt standard k-ε modl. Th maor rsults ar: -Th vntur ffct s wll prdctd. -Th hgh vlocty fluctuatons zon s th dffusr wall rgon xtndng to th channl xt. -Th turbulnc s vry mportant n th dffusr wall rgon xtndng to th channl xt and shfts away from th dffusr wall n th flow drcton. Acknowldgmnts Th authors thank th AUF (Agnc Unvrstar d la Francophon) for hs fnancal support. Thy also thank th Polytchnc Unvrsty of Bucharst (UPB) for hs tchncal support. REFERENCES [1] R.H. Boll, Industral and Engnrng Chmstry Fundamntals,1973,12, [2] W. Jtschn, M. Ronzhmr, S. Khodabakhsh, Vacuum, 1999, 53, [3] B.E. Laundr, D.B. Spaldng; Mathmatcal modls of turbulnc, acadmc prss, London, [4] S.I. Pak, K.S. Chang, Journal of Hazardous Matral, 2006, 138, [5] F. Ahmadvand, M.R. Tala, Chmcal Engnrng Journal, 2010, 160, [6] B.E. Laundr, B.I. Sharma, Lttrs n hat and mass transfr, 1974, 1, [7] S.W. Igo, PhD thss, Ouagadougou Unvrsty (Ouagadougou, BF, 2011) [8] Flunt Inc; Flunt 6.2 tutoral Gud, Lbanon, NH, [9] Flunt Inc; Gambt 2.2 tutoral Gud, Lbanon, NH, [10] A. Mad, Q.Y. Chang, N.S. Zhong, J.W. Jan, R. Athar, Appld Mchancs and Matrals, 2012, [11] G.G. Vádla, B. Rodrgo, A.S.G. José, R.C. José, Industral and Engnrng Chmstry Rsarch, 2012, 51, [12] A. Mad, Y. Changq, S. Zhongnng, W. Janun, G. Hafng, Nuclar Engnrng and Dsgn, 2013, 256, [13] P.A.C. Okwuob, R.S. Azad, ournal of fluds mchancs, 1973, 57, [14] S.W. Igo, M. Lucan, P. Tudor, 2nd Intrnatonal confrnc of thrmal qupmnt, rnwabl nrgy and rural 366

9 dvlopmnt, Jun 2013, Bal Olanst Romana (TE-RE-RD 2013, Romana) [15] F.M Wht; Fluds mchanc,3rd dton, McGraw-Hll, Inc., Nw York,