Natural Convection in a Horizontal Annulus with Oscillating Inner Cylinder Using Lagrangian-Eulerian Kinematics

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1 Natural Cnvectn n a Hrzntal Annulus wth Oscllatng Inner Cylnder Usng Lagrangan-Euleran Knematcs Esam M. Alawadh Kuwat Unversty Mechancal Engneerng Department P. O. Bx # 5969, Safat, 3060 KUWAIT Abstract In ths artcle, natural cnvectn flw n a hrzntal annulus enclsure wth scllatng nner cylnder s studed. An arbtrary Lagrangan-Euleran knematcs descrptn methd s utlzed t smulate flw and temperature felds. Fnte element methd s used t slve the gvernng equatns. The effect f Raylegh number, scllatng eccentrcty, and scllatng frequency f the nner cylnder n the flw structure and heat transfer s presented. The results are valdated wth publshed results fr a statnary nner cylnder case wth dfferent eccentrctes. The results ndcate that the scllatng nner cylnder hghly affect the average Nusselt number at the nner cylnder. Hwever, the scllatng eccentrcty and frequency f the nner cylnder have a negatve effect n the Nusselt number at the uter cylnder. Intrductn The natural cnvectn flw n hrzntal annul has receved ncreased attentn n last decade because f ther mprtance n engneerng. The gemetry f the hrzntal annul s cmmnly fund n slar cllectr-recevers, under grund electrc transmssn cables, vapr cndenser fr water dstllatn, and fd prcesses. On the ther hand, the flw structures n the annulus are nt cmpletely understd, especally fr unsteady flw. Expermental and numercal nvestgatns f the natural cnvectn flw n annul are cnducted n parallel. Kuehn and Gldsten (976, 978) expermentally studed the effect f vertcal eccentrcty f the nner cylnder n the lcal and verall heat transfer ceffcents. Ther expermental data are cmmnly used t valdate mst f the recent numercal studes. The effect f the hrzntal eccentrcty f the nner cylnder was studed by Guj and Stella (995) thrugh expermental and numercal nvestgatns. Ther results ndcated that the average Nusselt number s nearly ndependent t hrzntal eccentrcty. Shahrak (00) demnstrated the effect f temperature dependent physcal prpertes n the streamlnes and temperature dstrbutn n a cncentrc annulus. The varyng vscsty had the strngest effect n the flw felds whle the thermal cnductvty had the strngest effect n temperature prfles. Numercal nvestgatns f three dmensnal natural cnvectn flw nsde a hrzntal cncentrc annulus were accmplshed by Yeh (00). Hs numercal results revealed that hgh temperature znes arund the nner cylnder n the regn near ts tp were ntced. An annulus wth rtatng uter cylnder was nvestgated by Y (998). Natural cnvectn f a mcr-plar flud n an eccentrc annulus enclsure was studed by Char and Lee (998). In cmparsn t a Newtnan flud, they fund that the average Nusselt number was reduced fr a mcr-plar flud. Char and Hsu (998) presented

2 a numercal predctn f turbulent mxed cnvectn n a cncentrc annulus wth rtatng nner cylnder. It was fund that the rtatng nner cylnder caused a sgnfcant reductn n the heat flw n the annulus. Unsteady natural cnvectn n a hrzntal annulus was nvestgated by Mzushma et al. (00). All these expermental r numercal nvestgatns f natural cnvectn flw n annul pertan t flw n fxed r rtatng cylnders. T the best knwledge f the authr, n attentn has been gven t prblem f mxed cnvectn flw wthn a hrzntal annulus wth scllatng nner cylnder. The present study fcuses n the effect f the scllatng nner cylnder n the flw and temperature characterstcs wthn an annulus. Mathematcal Frmulatn: The physcal mdel used n ths wrk s shwn n Fgure. A tw dmensnal hrzntal annulus wth nner radus r and uter radus r s used. The uter cylnder s fxed, whle the nner cylnder s scllatng vertcally wth frequency Ω and maxmum eccentrcty ε max. The nner and uter cylnders are sthermal, wth nner temperature T and uter temperature T, and T > T. The wrkng flud s Ar. Intally, the nner cylnder s at the centre f the annulus and ar s temperature s T n, and T n = T. Buyancy and the scllatns f the nner cylnder effects heat flw frm the nner cylnder t the uter cylnder. T smulate the mtn f the nner cylnder, an arbtrary Lagrangan-Euleran (ALE) knematcs descrptn methd (974) s utlzed t slve the prblem. In ths methd, the cmputatnal mesh can be mved wth flud (Lagrangan), be fxed (Euleran), r be mved n prescrbed way (Huhes, et al., 98). Therefre, the ALE methd s used t numercally nvestgate the effect f scllatns f the nner cylnder upn the flw feld and heat transfer characterstcs. Fnte Element Methd (FEM) s utlzed t slve the gvernng equatns alng wth the bundary cndtns. Ths research fcuses n the effect f the Raylegh, scllatng frequency, and scllatng eccentrcty n the average Nusselt number at the nner and uter cylnders. r T T r Fgure. Physcal mdel and crdnates Gvernng Equatns: Based upn the scales f temperature dfference (T -T ), annulus gap wdth (r -r ), and α/(r -r ), the dmensnless varables are defned as fllws: x X = (a) ( r r )

3 y Y = ( r r ) (b) u U = α ( r r ) (c) v V = α ( r r ) (d) * P P = ρα ( r ) r (e) T T θ = T T (f) The used dmensnless varables t make the gvernng equatns are Raylegh number (Ra), Prandtl number (Pr), and Furer number (F), and they are defnes as, respectvely: Ra ( T T )( r r ) 3 gβ = (a) ν = α να Pr (b) F F t = (c) ( ) α r r t = (d) ( ) α r r where α s the thermal dffusvty, and ν s the knematcs vscsty. The fllwng assumptns are made fr the numercal study: a. The flw s tw-dmensnal, ncmpressble, and lamnar. b. The prpertes f ar are assumed cnstant, except the densty. c. Vscus dsspatn s neglected. d. Bussnesq apprxmatn s vald. e. Radatn effect s nt cnsdered. Based n the abve assumptns and dmensnless varables, the gvernng equatns are expressed as fllws: Mass cnservatn: U V + = 0 Mmentum equatns U U + U F + * ( ˆ U P U U V V ) = + Pr + (3) (4) V V + U F + * ( ˆ V P V V V V ) = + Pr + + Ra Prθ (5)

4 Energy equatn θ θ + U + F Vˆ ( ˆ θ θ V V ) = + θ where s the dmensnless mesh velcty n Y-drectn. The dmensnless ntal and bundary cndtns are descrbed n the fllwng mathematcal frm: F = 0 fr the entre dman: = V = 0 U, θ θ = 0 = n F > 0 alng the nner cylnder: θ = θ, U = 0, V = VmaxCs(π ΩF) alng the uter cylnder: θ = 0 θ, U = V = 0 = where V max s the maxmum velcty f the nner cylnder. The eccentrcty f the nner cylnder ( ε v ) s changng. The eccentrcty s the dstance alng the vertcal axs when the nner cylnder s mved frm ts cncentrc pstn, pstve upward and negatve dwnward. The dmensnless eccentrcty s defned as ε = ε / r r ). The nner cylnder s scllatng accrdng t the relatn: v ( ε = ε Sn(πΩF) (7) max where ε max s the maxmum eccentrcty. The dmensnless heat transfer rate f cnductn mde f heat transfer n an annulus s: Nu cnd = In / (8) ( r r ) The lcal Nusselt number s defned as the actual heat flux at the surface dvded by Nu cnd, as fllws: θ at r = r φ (9a) r ( Nu ) = r / Nucnd θ at r = r φ (9b) r ( Nu ) = r / Nucnd where φ s the angular crdnate alng the nner and uter walls. The average Nusselt number at the nner and uter cylnders are gven, respectvely, by (6) Nu Nu = π = π π 0 π 0 ( Nu ) φ φ d ( Nu ) φ φ d (0a) (0b) Numercal Methd Fnte element methd s used t slve the gvernng equatns f mass, mmentum, and energy equatns. Fur ndes quadrlateral element s used t dscretze the cmputatnal dman. The fnte element mdel cnssts f 3,5 elements and 3,76 ndes. The cmputatnal dman s reduced by half after takng the advantage f symmetry n the prblem. Fgure shws the cmputatn mesh when the nner cylnder at the centre, maxmum eccentrcty, and mnmum eccentrcty. Pre-Cndtnal Generalzed Mnmum Resdual (PGMR) slver was emplyed t slve a set f dscretzed equatns f energy and pressure, whle

5 the Tr-Dagnal Matrx Algrthm (TDMA) slver was emplyed t slve the velcty feld. The advectn terms n the mmentum and energy equatns were frmulated usng Streamlne Upwnd/Petrv Galerkn (SUPG) apprach (Brks et al., 98). The relatve errr f each varable was cmputed at each tme step t examne the cnvergence f the slutn, and the errr s defned as: Π n+ n, j, j Π Π n+, j 0 6 where Π = U, V, P, r θ. The cnvergence mntr represents the sum f the changes f a varable calculated frm the results between the current (n+) th teratn and prevus (n) th teratn dvded by the sum f the current values. Als, the cnservatve f the cntnuty equatn U V Resdual = + () s used t check fr each element at each tme step t ensure the cnservatn f mass s attaned. The resdual f the cntnuty equatn fr each element s less than 0-7. () a) b) c) Fgure. Inner cylnder scllatns and Fnte Element mesh mvement T ensure that the btaned results were mesh ndependent, number f elements n the cmputatnal dman was ncreased by 5% and 50%. The maxmum y-velcty cmpnent was cmputed, and the dfference between the mdels was less than.3%. Therefre, the rgnal mdel was adpted fr the numercal smulatns.. Numercal Valdatn The values f the lcal Nusselt number alng the nner and uter cylnders f the present numercal mdel are cmpared wth expermental results f Kuehn and Gldsten (976, 978). The nner and uter rad are and m, respectvely. The nner and uter cylnder walls are set t temperature f 373 K and 37 K, respectvely. The wrkng flud s ar, and Ra = Fgure 3 shws the prfles f the lcal Nusselt number alng the nner and uter cylnders. The prfles are fr eccentrctes f 0.63, 0, and As shwn, the agreement s excellent fr a steady state analyss.

6 0 9 Present Kuehn and Gldsten Average Nusselt Number Inner Outer a) φ 0 9 Present Kuehn and Gldsten Average Nusselt Number Inner Outer b) φ 0 9 Present Kuehn and Gldsten Average Nusselt Number Inner Outer c) φ Fgure 3. Cmparsn f the lcal Nusselt number dstrbutns wth expermental data [,], fr Ra = and eccentrcty (a) 0, (b) 0.63, and (c) -0.63

7 Results and Dscussn In ths study, the dmensnless radus f the nner and uter cylnders s and.645, respectvely. The examned Raylegh numbers are Ra = 0 4, 5 0 4, and 0 5. These nputs are same as the expermental research accmplshed by Kuehn and Gldsten (976, 978). The nner cylnder scllates wth Ω = 0.5,, and, and eccentrcty ε max = 0.5, 0.5, and Smulatns f nner cylnder scllatns Fgure 4 shws the transent develpment f streamlnes durng ne perdc cycle, F =,.5,.5, and.75. It takes abut F = 0.5 t reach the perdc cndtn. The nner cylnder scllates wth frequency Ω = and eccentrcty ε max = 0.5. The dmensnless stream functn ψ s defned as dψ U =, and V dy dψ = (3) dx a) b) c) d) Fgure 4. Streamlnes fr Ra = 5 0 5, Ω =, and ε max = 0.5 durng ne perdc cycle Fgure 4b shws the nner cylnder at ts maxmum eccentrcty. As the nner cylnder mves upward, the nner cylnder presses ar at the tp regn, and vacant space s nduced at the bttm regn. These tw actns frce ar at the tp regn t flw t the bttm regn. Fgure 4c shws nner cylnder at the centre f the annulus and n ts way t mve dwnward. As the nner cylnder mves dwnward, ar at the bttm regn s pressed, and vacant space s nduced at the tp regn, reversng the drectn f flw. Fgure 4d shws the nner cylnder at ts mnmum eccentrcty. Frced cnvectn due t nner cylnder scllatns affects the natural cnvectn flw and thermal transprt n the annulus. Fgure 5 shws the stherms n the annulus durng ne perdc cycle, fr Ra = 5 0 5, Ω =, and ε max = 0.5. Intally, Fgure 5a ndcates that there s a wde temperature zne at the tp regn. Ths s due t the buyancy effect prmted by the change f densty, and the reverse s ntced at the bttm regn. As the nner cylnder mves upward (Fgure 5b), the heat transfer between the nner and uter cylnder s affected by the mvement f the nner cylnder. The sthermal lnes are clse t each ther and dense at the upper regn f the annulus, shwng hgh heat flw at ths regn. At the bttm regn, heat flw s dmnated by cnductn, makng the Nusselt number lw, as shwn n Fgure 3a. When the nner cylnder mves upward, the cnductn zne at the bttm regn expand, reducng heat flw at the bttm regn. Therefre, at ths nstant, the heat transfer n the annulus s less than the cncentrc annulus case. Fgure 5c

8 shws the nner cylnder at the centre f the annulus and n ts way t the bttm regn f the annulus. When the nner cylnder mves dwnward, Fgure 5d, the heat transfer rate at the bttm regn s greatly prmted by reducng the thermal resstance. Als, the cnvectn cell at the upper regn s expanded by a strng cnvectn, enhancng the heat flw at ths lcatn. Therefre, when the eccentrcty f the nner cylnder s mnmum, the heat transfer frm the nner cylnder t the uter cylnder reaches ts maxmum a) b) c) d) Fgure 5. Istherms fr Ra = 5 0 5, Ω =, and ε max = 0.5 durng ne perdc cycle Average Nusselt number calculatns The effect f the Raylegh number n the average Nusselt number at the nner and uter cylnders s studed. Ra = 0 4, 5 0 4, and 0 5 are selected fr ths study, wth Ω= and ε max = 0.5. Fgures 6a and 6b shw the varatn f Ra wth tme at the nner and uter cylnders, respectvely. The fgures ndcate that the average Nusselt number ncreases frm 0 at F=0 t quckly establshng the perdc scllatns cndtn. The average Nusselt number ncreases wth ncreasng Ra at nner and uter cylnders. At the nner cylnder, Fgure 6a ndcates that the effect f the scllatng nner cylnder n the Nusselt number dmnshes as the Ra decreases. The Ra mstly affects the ampltude f the Nusselt number scllatns and has nsgnfcant effect n the frequency f the Nusselt number. a) b) Fgure 6. Effect f Raylegh number n the average Nusselt number at a) nner, b) uter cylnders fr Ω =, and ε max = 0.5

9 Fr Ra = 0 5, the Nusselt number s scllatng heavly, whle slghtly fr Ra = 0 4. At the uter cylnder, Fgure 6b ndcates that the scllatns f the nner cylnder are hghly affectng the Nusselt number. Als, ncreasng the Ra shfts the Nusselt number prfle upward, wthut sgnfcantly affectng ts scllatng perd r ampltude. At the nner and uter cylnders, the Nusselt reaches ts maxmum value when the eccentrcty f the nner cylnder s mnmum, whle mnmum when the eccentrcty f the nner cylnder s maxmum. When the eccentrcty f nner cylnder reaches ts maxmum value, the cnductn mde f heat transfer zne at the bttm regn s expanded, ncreasng the heat flw resstance n the annulus. On the ther hand, when the eccentrcty f nner cylnder reaches ts mnmum value, the cnvectn mde f heat transfer zne at the tp regn s expanded, and the cnductn dmnated mde zne at the bttm regn s reduced. These tw actns decrease the heat flw resstance n the annulus. The effect f the scllatng eccentrcty f the nner cylnder s examned. Fgures 7a and 7b shw the average Nu fr Ra = at the nner and uter cylnders, respectvely. The nner cylnder scllates wth frequency Ω=, and eccentrcty ε max = 0.5, 0.5, and At the nner cylnder (Fgure 7a), the Nusselt number scllates wth the nner cylnder scllatns, and scllatns ampltude f the Nusselt number sgnfcantly ncreases wth ncreasng the eccentrcty f the nner cylnder. Fr the hghest examned scllatn eccentrcty (ε max = 0.75), the Nusselt number reaches as hgh as 3.88, and as lw as.5. Ths scllatns band s arund the Nusselt number fr the statnary nner cylnder case, where the average Nusselt number s As the eccentrcty f the nner cylnder ncreases, the cnductn-dmnated zne at the bttm regn s expanded mre, greatly reducng the Nusselt number at the nner cylnder. On the ther hand, the cnvectn-dmnated zne at the upper regn s als expanded when the nner cylnder s at the bttm regn, greatly ncreasng the Nusselt number. These tw actns cause these extreme scllatns f the Nusselt number. At the uter surface, ncreasng the scllatng eccentrcty has a negatve effect f the average Nusselt number. The Nu reaches as lw as.73, fr ε max =0.75. Als, as the nner cylnder scllatng eccentrcty ncreases, the Nusselt number decreases. The average Nusselt number at the uter cylnder fr a statnary cncentrc annulus case s 3.5. a) b) Fgure 7. Effect f scllatng eccentrcty f the nner cylnder n the average Nusselt number at a) nner, b) uter cylnders fr Ω =, and Ra = The effect f the scllatng frequency f the nner cylnder n the average Nusselt number at the nner and uter cylnders s studed. Fr ths study, the nner cylnder frequency s Ω = 0.5,, and, wth Ra=4 0 4 and ε max = 0.5. Fgure 8a and 8b shw the varatn f average Nusselt number at the nner and uter cylnders wth tme, respectvely. Fgure 8a ndcates that as the scllatng frequency ncreases, the ampltude f the Nusselt number at the nner cylnder decreases sgnfcantly. Ntce that wth frequency Ω = 0.5, the Nusselt number reaches as hgh as 3.99 and as lw as.5, shwng a sgnfcant effect f the frequency f the nner cylnder. Wth Ω =, the Nusselt number ampltude s hghly reduced. When the

10 nner cylnder at the bttm regn, the heat flw s greatly ncreased, but the heat flw s greatly decreased when the nner cylnder s at the tp regn. A hgh scllatng frequency f the nner cylnder farther decreases the thermal resstance at the tp regn, but father ncreases the resstance at the bttm regn. As a result, the scllatng ampltude f Nusselt number ncreases as the nner cylnder frequency decreases. Fgure 8b shws the average Nusselt number at the uter cylnder. Ths fgure ndcates that as the frequency decreases, the Nusselt number scllatns ampltude decreases. Fr Ω = 0.5, the Nu reaches as hgh as 3.3, and as lw as.9. Fr Ω =, the Nu reaches scllates between 3.5 and 3., shwng nsgnfcant effect f the nner cylnder scllatns. a) b) Fgure 8. Effect f scllatng frequency f the nner cylnder n the average Nusselt number at a) nner, b) uter cylnder fr Ra = 5 0 4, and ε max = 0.5 Cnclusn The heat transfer characterstcs f a heated annulus enclsure wth a transversely scllatng nner cylnder are nvestgated numercally. Fnte element methd s used t slve the gvernng equatns. The scllatns f the nner cylnder are smulated usng An arbtrary Lagrangan-Euleran knematcs methd. The prblem s slved. Sme cnclusns are summarzed as fllws: a. As the Raylegh number decreases n the annulus, the effect f the nner cylnder scllatns n the average Nusselt number dmnshes at the nner cylnder. Hwever, the average Nusselt number at the uter cylnder s hghly affected by the Raylegh fr all examned Raylegh numbers. b. As the scllatng eccentrcty ncreases, the scllatns f average Nusselt number at the nner cylnder s ncreased, but the average Nusselt number at the uter cylnder s decreased. c. The scllatng frequency f the nner cylnder has a sgnfcant effect n the average Nusselt number at the nner and uter cylnders. At the nner cylnder, the scllatng ampltude f the average Nusselt number ncreases as the scllatng frequency ncreases. At the uter cylnder, the average Nusselt number decreases as the scllatng frequency ncreases.

11 Reference ) A. N. Brks, T. J. R. Hughes, Streamlne Upwnd/Petrv-Galerkn Frmulatn fr Cnvectn Dmnated Flws wth Partcular Emphass n the Incmpressble Naver-Stkes Equatn, Cmp. Meth. Appl. Mech. Eng., Vl. 3, pp , 98. ) M. Char and Y. Hsu, Numercal Predcatn f Turbulent Mxed Cnvectn n a Cncentrc Hrzntal Rtatng Annulus wth Lw-Re-Tw-Equatn Mdels, Int. J. Heat Mass Transfer, Vl. 4, N., pp , ) M. Char and G. Lee, Maxmum Densty Effect n Natural Cnvectn f Mcrplar Flud Between Hrzntal Eccentrc Cylnders, Int. J. Engneerng Scence, Vl. 36, pp , ) G. Guj and F. Stella, Natural Cnvectn n Hrzntal Eccentrc Annul: Numercal Study, Numercal Heat Transfer, Part A, Vl. 7, pp , ) C. W. Hrt, A. A. Cks, An Arbtrary Lagrangan-Euleran Cmputng Methd Fr All Flw Speeds, J. Cmput. Phys., Vl. 4, pp. 7-53, ) T. Huhes, W. Lu, T. Zmmermann, Lagrangan-Euleran Fnte Element Frmulatn fr Incmpressble Vscus Flw, Cmput. Meth. Appl. Mech. Eng. Vl. 9, pp , 98. 7) T. H. Kuehn, R. J. Gldsten, An Expermental and Theretcal Study f natural Cnvectn n the annulus between hrzntal cncentrc cylnders, J. Flud Mechancs, vl. 74, pp , ) T. H. Kuehn, R. J. Gldsten, An Expermental Study f natural Cnvectn Heat Transfer n Cncentrc and Eccentrc Hrzntal Cylndrcal Annul, J. Heat Transfer, vl. 00, pp , ) J. Mzushma, S. Hayash, T. Adach, Transtns f Natural Cnvectn n a Hrzntal Annulus, Int. J. Heat Mass Transfer, Vl. 44, pp , 00. 0) F. Shahrak, Mdelng f Buyancy-Drven flw and Heat Transfer fr Ar n a Hrzntal Annulus: Effects f Vertcal Eccentrcty and Temperature-Dependent Prpertes, Numercal Heat Transfer, Part A, Vl. 4, pp , 00. ) C. Yeh, Numercal Investgatn f the Three-Dmensnal Natural Cnvectn Insde Hrzntal Cncentrc Annulus wth Specfc Wall Temperature r Heat Flux, Int. J. Heat Mass Transfer, Vl. 4, pp , 00. ) J. Y, Mxed Cnvectn f Ar Between Tw Hrzntal Cncentrc Cylnders wth a Caled Rtatng Outer Cylnder, Int. J. Heat Mass Transfer, Vl. 4, pp , 998.