Analytical Modeling of Natural Convection in Horizontal Annuli

Transcription

1 Analytcal Mdelng f Natural Cnvectn n Hrzntal Annul Peter Teertstra, M. Mchael Yvanvch, J. Rchard Culham Mcrelectrncs Heat Transfer Labratry Department f Mechancal Engneerng Unversty f Waterl Waterl, Ontar, Canada 12 January, rd Aerspace Scences Meetng and Exhbt, Ren NV

2 Outlne Intrductn and prblem descrptn Lterature revew and bjectves Mdel develpment Valdatn Summary and cnclusns

3 Prblem Defntn 2D hrzntal annulus Steady state, natural cnvectn Cncentrc nner and uter cylnders Isthermal bundary cndtns, T > T Gemetry: Relatve bundary sze Effectve gap spacng Shape, rentatn δ e P / P d d ( d d ) 2 (spheres) (spheres)

4 Parameter Defntns Ttal heat transfer rate nn-dmensnalzed by sselt number ( T T ) selected as characterstc length: P k Fr P / P lmt, scale length related t nner bdy dmensns nly Smlar results fr smlar bdy shapes, rentatns P P k Q Q l ( T T )

5 Parameter Defntns Dmensnless cnductn shape factr Ra << Ra Effectve cnductvty Raylegh number k eff k RaP lm cr P SP gβ P SP keff, 1 k ( T T )( P ) υ α 3

6 Lterature Revew - Data Expermental and mercal Studes Cncentrc sphercal enclsures Over 20 publcatns wth average heat transfer data Mst expermental data fr hgh Raylegh number, bundary layer flw All ther data frm numercal smulatns Other enclsure gemetres mercal data fr crcular, plygnal, rhmbc, ellptcal cylnders Crrelatns f expermental, numercal data Vald fr lmted ranges f Raylegh number Gemetry-dependent

7 Lterature Revew - Mdels Analytcal mdels avalable fr cncentrc, eccentrc crcular annulus Rathby & Hllands 28 Kuehn and Gldsten 29 Byd 30 presents general crrelatn prcedure fr 2D annulus wth arbtrarly-shaped bundares Requres crrelatn ceffcent values frm emprcal data Dffcult t mplement fr nn-standard bundary shapes

8 Objectves Analytcal mdelng f natural cnvectn n hrzntal annulus Ra P Full range f frm cnductn t cnvectn Applcable t wde range f gemetres Inner and uter bundary shapes and rentatn Relatve bundary szes Physcally-based analyss Valdate mdel usng expermental, numercal data frm the lterature Crcular annulus Annul wth dfferent nner, uter bundary shapes

9 Mdel Develpment Assume lnear superpstn f dffusve, cnvectve lmts * S P P + cnv Kuehn and Gldsten 8,9 data fr crcular annulus d d 2.6 cnv S * P P

10 Mdel Develpment General mdel based n Churchll and Usag 33 cmpste slutn technque n n 1 n 1 1 P SP + + tr bl Cmbnatn f three asympttc slutns SP tr bl cnductn shape factr transtn flw cnvectn lamnar bundary layer cnvectn Cmbnatn parameter n determned frm valdatn wth numercal, expermental data

11 Cnductn Shape Factr Crrelatns, mdels, frm handbks mercal smulatns ( δ e ) Apprxmate methd frm equvalent crcular annulus 2π Effectve dameter rat SP d d ( d d ) e Dmensnless cnductn shape factr SP e ln ln 2 4π Inner permeter Enclsed area π ( A P 2 ) + 1 d P π 2 4A P d + 2 π π

12 Bundary Layer Cnvectn Assumptns Lamnar flw T b unfrm Nn-ntersectng bundary layers Seres cmbnatn f resstances R R + R R T T b Q Nn-dmensnalze usng sselt number T Tb Tb T R R bl Q Q φ T T b T T b R R R T b T Q 1 k ( R + R ) 1+ 1 φ

13 Bundary Layer Cnvectn Cnvectn mdeled usng Yvanvch 31 and Jafarpur 36 A F( 14 Pr)G A Ra A Lamnar bundary layer cnvectn asymptte bl bl ( Pr) F G Ra 1+ 1 φ + φ 1 4 P P ( Pr) ( 1 1 ) 5 4 [ ( ) ( ) ] P P G G 5 4 bl F G P Ra P 1 4 P P ( ) ( ) F Pr RaP [ ( ) ] d d 5 4 (crcular annulus) 4

14 Transtn Flw Bundary layers merge when Mdel as equvalent crcular annulus Three dstnct regns are frmed Central regn Radal cnductn Buyancy nduced flw Ra < Ra cr Fr narrw gap spacng, δ, e << r temperature, velcty n central regn u T T b ge β 12υ y δ 2 e ( T T ) ( T T ) b δ e 2 2, T b T y δ e T 2 y δ e 2

15 Transtn Flw Enthalpy balance n tp-end and bttm-end regns Q, ρ c β ( T T ) 720ν Transtn flw asymptte tr tr p g e 1 90π π 2 3 δ 3 ( δ e P ) RaP ( 1+ P P ) 3 ( d d 1) RaP ( 1+ d d ) (crcular annulus) e

16 Valdatn: Crcular Annulus

17 Crcular Annulus

18 Crcular Annulus

19 Cncentrc Ellptc Cylnders vertcal hrzntal

20 Square and Damnd n Crcle

21 Hexagn n Crcular Cylnder

22 Crcle n Square Cylnder

23 Summary Analytcal study f natural cnvectn heat transfer fr sthermal, hrzntal annul Mdel develped based n cmbnatn f analytc, asympttc relatnshps Dffusve lmt Lamnar bundary layer cnvectn Transtn flw cnvectn Valdated usng prevus data fr smlar, dfferent nner and uter cylnder shapes 6 9% RMS dfference between mdel and data