Finite Element Analysis of Mixed Convection Heat Transfer through a Vertical Wavy Isothermal Channel

Transcription

1 Proceedigs of the World Cogress o Egieerig 00 Vol II WCE 00, Je 30 - Jly, 00, Lodo, U.K. Fiite Elemet Aalysis of Mixed Covectio Heat Trasfer throgh a Vertical Wavy Isothermal Chael H.Shokohmad, S.M.A.Noori Rahim Abadi Abstract I this aer, mixed covectio heat trasfer throgh a vertical wavy isothermal chael is ivestigated merically. I reset stdy, the hot sisoidal vertical walls are at costat temeratre ad the cold flow eters the chael at the bottom side. The merical model is based o a D Navier-Stokes icomressible flow ad eergy eqatio solver o strctred grid. The goverig eqatios cosist of cotiity, mometm ad eergy eqatios are solved merically by fiite elemet method sig Characteristic Based Slit (CBS algorithm. The effect of Reyolds, Pradtl ad Grashof mbers o flow ad thermal fields are ivestigated. The variatios of local Nsselt mber alog the vertical walls are also reseted. Keywords Mixed Covectio, Fiite Elemet Method, Wavy Chael. M I. INTRODUCTION ixed covectio ivolves featres from both forced ad atral flow coditios. I mixed covectio flows, the forced covectio ad free covectio effects are comarable i magitdes. Ths, mixed covectio occrs if the effect of boyacy forces o a forced flow or the effect of forced flow o a boyat flow is sigificat. The goverig odimesioal arameters for the descritio of mixed covectio flows are Grashof mber (Gr, Reyolds mber (Re ad Pradtl mber (Pr. The ratio Gr/Re is also amed Richardso mber (Ri that idicate the stregth of the atral ad forced covectio flow effects The limitig case Ri 0 ad Ri corresod to the forced ad atral covectio flows, resectively. It is ecessary to stdy the heat ad mass trasfer from a irreglar srface becase irreglar srfaces are ofte reset i may alicatios sch as micro-electroic devices, flat-late solar collectors ad flat-late codesers i refrigerators [], ad geohysical alicatios (e.g., flows i the earth s crst [], dergrod cable systems, electric machiery, coolig system of micro-electroic devices, etc. I additio, rogheed srfaces cold be sed i the coolig of electrical ad clear comoets where the wall heat flx is kow. Oe of the reasos why a rogheed srface is more efficiet i heat trasfer is its caability to romote flid motio ear the srface; i this way a comlex wavy srface, a sm of two or H.Shokohmad is rofessor of the School of Mechaical Egieerig, College of Egieerig, Uiversity of Tehra, Tehra, Ira ( hshokoh@ t.ac.ir. S.M.A.Noori is M.S stdet of the School of Mechaical Egieerig, College of Egieerig, Uiversity of Tehra, Tehra, Ira (corresodig athor to rovide hoe: ; a.oori364@ gmail.com. more sisoidal srfaces, is exected to romote a larger heattrasfer rate tha a sigle sisoidal srface. This comlex geometry will romote a corresodigly comlicated motio i the flid ear the srface; this motio is described by the oliear bodary-layer eqatios. This exectatio is the basis of the crret stdy eve thogh oly lamiar mixed covectio is stdied. A vast amot of literatre abot covectio alog a sisoidal wavy srface is available for differet heatig coditios ad varios kids of flids [3 7]. Recetly Ashjaee et al. [8] have ivestigated the roblem of free covectio alog a vertical wavy srface exerimetally ad merically. The ivestigatio was carried ot for three differet amlitde wavelegth ratios ad Rayleigh mber based o the legth of the wavy srface ragig from.90 5 to Reslts idicate that the freqecy of the local heat trasfer rate is the same as that of the wavy srface ad the average heat trasfer coefficiet decreases as the as the amlitde- wavelegth ratio icreases. The atral covectio heat trasfer from a isothermal vertical wavy srface was first stdied by Yao [9 ] ad sig a exteded Pratdl s trasositio theorem ad a fiite-differece scheme. He roosed a simle trasformatio to stdy the atral covectio heat trasfer from isothermal vertical wavy srfaces, sch as sisoidal srface. Chi ad Cho [] stdied the atral covectio heat trasfer alog a vertical wavy srface i microolar flids. Che ad Wag [3,4] aalyzed trasiet forced ad free covectio alog a wavy srface i microflids. Cheg [5,6] has ivestigated coled heat ad mass trasfer by atral covectio flow alog a wavy coical srface ad vertical wavy srface i a oros medim. The aim of this stdy is to ivestigate the effects of arameters sch as Grashof mber, Reyolds mber ad Pradtl mber o flow ad thermal fields throgh the chael. The local Nsselt mber of vertical walls alog the chael at the wide rage of goverig arameters (Re, Pr, Gr mbers are reseted. The goverig eqatios icldig cotiity, Navier Stokes ad eergy eqatios are solved merically by Galerki fiite elemet method based o the characteristic based slit (CBS algorithm. II. GOVERNING EQUATIONS A two-dimesioal vertical wavy chael ad related dimesioless bodary coditios are illstrated i Fig.. The shaes of the side wavy srfaces rofile are as the followig atter: x D ± Asi( πyk ( ISBN: ISSN: (Prit; ISSN: (Olie WCE 00

2 Proceedigs of the World Cogress o Egieerig 00 Vol II WCE 00, Je 30 - Jly, 00, Lodo, U.K. Where A is the dimesioless amlitde of the wavy srfaces ad k is the mber of dlatio (z5. D ad L (L6D are the mea diameter ad the legth of the chael, resectively. The cold flid eters the chael with the coditios, TT c, U 0 ad v0. The sisoidal vertical walls of the chael are at costat temeratre TT h. The flow is assmed to be lamiar ad the flid is assmed to be icomressible, with costat hysical roerties excet for the desity variatio which is take ito accot throgh the Bossiesq aroximatio. Also viscos dissiatio ad ressre work are cosidered egligible. + 0 U mometm eqatio: + + v + t V mometm eqatio: + + v + ( t Re Eergy eqatio: ( Re + x y (3 (4 v v + + Gr / Re. T x y (5 t + + v Re.Pr T ( x T + y (6 III. NUMERICAL METHOD The goverig eqatios are solved by CBS fiite elemet method. The CBS algorithm for the soltio of the Navier Stokes ad eergy eqatio eqatios ca be smmarized by the followig stes[9]:.soltio of the mometm eqatio withot the ressre term..calclatio of the ressre sig the Poisso eqatio. 3.Correctio of velocities. 4.Calclatio of eergy eqatio or ay other scalar eqatio. By alyig the CBS method, the goverig eqatios become as follows: Ste: Fig, hysical model Based o the characteristics scales of D, U 0, T h ad T c, the dimesioless variables are defied as follows: x Gr x D ρu 0 y T y D 3 gbd ( T h T c ν T Tc T T h c Pr U 0 ν α v v U 0 U 0 D Re ν ( Therefore the o-dimesioal goverig eqatios are (the stars were omitted for simlicity: Cotiity eqatio: ISBN: ISSN: (Prit; ISSN: (Olie t[( C ~ + k ~ f t( k ~ + f ] ~ M τ (7 Ste: ( M ~ + t θθ H [( ~ ~ t G + θ G tθ H ~ Ste3: ~ ~ ~ M Ste4: t[ G T ( ~ f ~ t + θ ~ + P ] (8 ] (9 ~ ~ ~ E M [ ~ E t CE E + C + KTT + f ] ~ t( K ~ ee + K + fes (0 WCE 00 s

3 Proceedigs of the World Cogress o Egieerig 00 Vol II WCE 00, Je 30 - Jly, 00, Lodo, U.K. Where overlie arameters rereset the odal qatities. I the above eqatios, ~ ad ~ are itermediate velocities, C, G, H ad k are discrete covectio, gradiet, τ Lalacia ad viscos oerators ad M is the mass matrix ad also θ is the coefficiet of stability arameter ad θ is a coefficiet for switchig betwee exlicit θ 0 ad imlicit ( 0 < θ < scheme of solvig the eqatios. I additio we have the followig relatio betwee the remaiig coefficiet matrices: C CE C, K T H Kτ, K Ke K, Pr, M E M ( M M β The terms, f s, K, P, K e ad K are de to discretizatio alog the characteristics ad f, f ad f e cotai the bodary coditios. The term f es cotais sorce terms. The overlied arameters rereset the odal qatities. The o-real time ste, t, (sedo_time ste accelerates soltio to steady state as fast as ossible. The sedo time ste is locally calclated ad sbjected to stability coditio. a: ( ( (3 h t ( + β Where h is the elemet size, β is the artificial comressibility arameter [7] ad is the velocity. IV. RESULT AND DISCUSSION The mai arameters of Rayleigh, Pradtl ad Reyolds mbers o variatios of mea Nsselt mbers of left, right ad bottom ad thermal ad flow fields are examied. The local Nsselt mber is calclated by the followig eqatio: N (3 Where deotes the ormal directio o a lae. The dimesioless stream fctio ψ is defied as: ψ ψ, v (4 b: ( ( (3 Fig. Isothermal lies(a ad stream fctios(b at Gr0 5, Pr0.7 for differet vales of Reyolds mber; (.Re0, (.Re00, (3.Re500 I this stdy a strctred liear triaglar mesh corresodig 300 odes is tilized for all cases. Nmerical soltios are obtaied for varios vales of Gr , Pr0.0 0 ad Re ISBN: ISSN: (Prit; ISSN: (Olie WCE 00

4 Proceedigs of the World Cogress o Egieerig 00 Vol II WCE 00, Je 30 - Jly, 00, Lodo, U.K. eterig flid immediately reaches to T h de to low velocity ad mometm. With icreasig Reyolds mber the isotherms sreads all over the chael. It is also aaret that the comressio of isotherms is icrease de to crvatre of side walls. a: ( ( (3 a: ( ( (3 b: ( ( (3 Fig. 3 Isothermal lies(a ad stream fctios(b at Re0, Pr0.7 for differet vales of Grashof mber; (,Gr0 4, (.Gr0 5, (3.Gr0 6 Effect of Reyolds mber: Fig. shows the isothermal lies ad stream fctios at Gr0 5 ad Pr0.7 for differet vales of Reyolds mber. For Re0, two vertices rodce de to iform velocity of flid at the ilet of the chael. Also lots for stream fctios show that icremet of Reyolds mber reslts i rodctio of small vertices at valleys. For Re0 the temeratre of cold ISBN: ISSN: (Prit; ISSN: (Olie b: ( ( (3 Fig. 4 Isothermal lies(a ad stream fctios(b at Re0, Gr0 5 for differet vales of Pratdl mber; (.Pr0.0, (.Pr0.7, (3.Pr0 WCE 00

5 Proceedigs of the World Cogress o Egieerig 00 Vol II WCE 00, Je 30 - Jly, 00, Lodo, U.K. Effect of Grashof mber: Fig.3 shows the isothermal lies ad streamlies at Re00 ad Pr0.7 for differet vales of Grashof mber. With icremet of Grashof mber effect of boyacy forces icreases ad excels the effect of forced covectio which leads to decremet of eaks of isothermal lies at the middle of the chael. Icremet of Grashof mber also reslts i comressio of isotherms at the ilet of the chael. Also at higher vales of Grashof mber the temeratre of cold eterig flids will reach to T h sooer tha that of lower Grashof mber. Stream fctios for at higher vales of Grashof mber become more codese at the regios ear the side walls de to stroger covectio effects at the middle of the chael. Effect of Pratdl mber: Fig.4 shows the isothermal lies ad streamlies at Re00 ad Gr0 5 for differet vales of ratdl mber. At low vales of Pratdl mber, the temeratre of the flid raidly reaches to the temeratre of hot walls de to large vale of thermal diffsivity. With icreasig Pratdl mber, icremet of temeratre of flid will hae slowly, as a reslt the isotherms are more comressed alog the chael ear the hot walls. At Pr0.0, a circlatio will be rodce at the valleys of the hot walls. With icremet of Pradtl mber theses vertices will disaear gradally. mber will become relative maximm at the eaks of hot walls de to crvatre of walls which reslts i icremet of comressio of isotherms. Fig. 6. Variatios of local Nsselt mber alog the wavy wall of the chael for differet vales of Reyolds mber; Gr0 5, Pr0.7. Local Nsselt mber: Fig.5 shows the variatios of local Nsselt mber alog the hot wall for differet vales of Grashof mber for Re0 ad Pr0.7. At the ilet of the chael the local Nsselt mber is very large de to miimm thickess of bodary layer. With icreasig y, the local Nsselt mber will decrease de to decremet of temeratre differece ad Fig. 7. Variatios of local Nsselt mber alog the wavy wall of the chael for differet vales of Pratdl mber; Re0, Gr0 5. Fig. 5. Variatios of local Nsselt mber alog the wavy wall of the chael for differet vales of Grashof mber; Re0, Pr0.7. icremet of thickess of bodary layer, this tred will occrs more raid with icremet of Grashof mber. Local Nsselt Fig.6 shows the lots for differet vales of Reyolds mber for Gr0 5 ad Pr0.7. At Re0 the temeratre of the flid reaches to T h raidly, therefore the local Nsselt mber become zero. Icremet of vales of Reyolds mber will reslts i icreasig of local Nsselt mber. Similar to figre 5 the variatios of local Nsselt mber has relative maximm de to crvatre of hot walls. Also with icreasig y, the local Nsselt mber will decrease de to decremet of temeratre differece ad icremet of thickess of bodary layer. ISBN: ISSN: (Prit; ISSN: (Olie WCE 00

6 Proceedigs of the World Cogress o Egieerig 00 Vol II WCE 00, Je 30 - Jly, 00, Lodo, U.K. Fig.7 shows the variatios of local Nsselt mber alog the wavy wall of chael for differet vales of Pratdl mber for Gr0 5 ad Re0. For Pr0.7 local Nsselt mber raidly become zero becase the temeratre of the flid reaches the temeratre of hot walls. With icreasig the vales of Pratdl mber local Nsselt mber of hot wall will icrease de to ehaced thermal mixig. Similar to the other lots the local Nsselt mber will decrease with icreasig y throgh the chael de to decreasig the temeratre differece. V. CONCLUSION I this work, mixed covectio heat trasfer throgh a wavy vertical chael has bee ivestigated merically by Galerki fiite elemet method based o the characteristic based slit (CBS algorithm. The effects of arameters sch as Grashof mber, Reyolds mber ad Pradtl mber o flow ad thermal fields throgh the chael. Reslts showed that with icreasig Reyolds mber ad decreasig Pratdl mber a secodary flow is rodced at the valleys of hot walls. At higher vales of Reyolds ad Pratdl mbers the temeratre of the flid reaches to the vale T h later. Local Nsselt mber of hot walls will decrease gradally throgh the chael ad has relative maximm vale at the eak of hot wavy wall. Also icreasig the vales of Reyolds ad Pratdl mber reslt i icreasig the local Nsselt mber of hot wall. [] C.P. Chi, H.M. Cho, Trasiet aalysis of atral covectio alog a vertical wavy srface i microolar flids, It. J. Eg. Sci. 3 ( J.-H. Jag et al. / Iteratioal Joral of Heat admass Trasfer 46 ( [3] C.K. Che, C.C. Wag, Trasiet aalysis of force covectio alog a wavy srface i microolar flids, AIAA J. Thermohys. Heat Trasfer 4 ( [4] C.C. Wag, C.K. Che, Trasiet force ad free covectio alog a vertical wavy srface i microolar flids, It. J. Heat Mass Trasfer 44 ( [5] C.Y. Cheg, Natral covectio heat ad Mass trasfer ear a wavy coe with costat wall temeratre ad cocetratio i a oros medim, Mech. Res. Comm.7 ( [6] C.Y. Cheg, Natral covectio heat ad mass trasfer ear a vertical wavy srface with costat wall temeratre ad cocetratio i a oros medim, It. Comm. Heat Mass Trasfer 7 ( [7] P. Nithiaras, C.-B. Li. A artificial comressibility based characteristic based slit (CBS scheme for steady ad steady trblet icomressible flows Comt. Methods Al. Mech. Egrg. 95 ( ACKNOWLEDGEMENT The athors are leased to ackowledge the sort of this stdy by Uiversity of Tehra, Tehra, Ira. Athors also ackowledge hel of Miss Moayedi for her valable sggestios. REFERENCES []J.-H. Jag, W.-M. Ya, Mixed covectio heat ad mass trasfer alog a vertical wavy srface, It. J. Heat Mass Trasfer 47 ( []P.K. Das, S. Mahmd, Nmerical ivestigatio of atral covectio iside a wavy eclosre, It. J. Therm. Sci. 4 ( [3]C.P. Chi, H.M. Cho, Trasiet aalysis of atral covectio alog a vertical wavy srface i micro olar flids, It. J. Eg. Sci. 3 ( [4]D.A.S. Rees, I. Po, A ote o free covectio alog a vertical wavy srface i a oros medim, ASME J. Heat Trasfer 6 ( [5] D.A.S. Rees, I. Po, Free covectio idced by a vertical wavy srface with iform heat flx i a oros medim, ASME J. Heat Trasfer 7 ( [6] Y.T. Yag, C.K. Che, M.T. Li, Natral covectio of o-ewtoia flids alog a wavy vertical late icldig the magetic field effect, It. J. Heat Mass Trasfer 39 ( [7] L.S. Yao, Natral covectio alog a vertical comlex wavy srface, It. J. Heat Mass Trasfer 49 ( [8] M. Ashjaee, M. Amiri, J. Rostami, A correlatio for free covectio heat trasfer from vertical wavy srfaces, Heat Mass Trasfer 44 ( [9]L.S. Yao, Natral covectio alog a wavy srface, ASME J. Heat Trasfer 05 ( [0]L.S. Yao, A ote o Pradtl_s trasositio theorem, ASME J. Heat Trasfer 0 ( [] S.G. Molic, L.S. Yao, Mixed covectio alog a wavy srface, ASME J. Heat Trasfer ( ISBN: ISSN: (Prit; ISSN: (Olie WCE 00