EHed of Curvature on the Temperature Profiles

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1 PROC. OF THE OKLA. ACAD. OF SCI. FOR 1967 EHed f Curvature n the Temperature Prfiles in Cnduding Spines J. E. FRANCIS add R. V. KASER, University f Oklahma, Nrman and GORDON SCOFIELD, University f Missuri, Rna INTRODUCTION The prblem cnsidered in this paper is the determinatin f the effect f surface curvature n the heat flux and temperature distributin in a spine r pin fin. Fr ne-dimensinal, heat-transfer calculatins Schneider (1955) and Jakb (1949) indicate that it is nrmal t assume that the gemetry f the spine is such that the length measured alng the surface is the same as the length measured alng the axis f symmetry. This implies that the slpe f the surface is small. The present wrk includes the effect f a nn-zer slpe and presents the differences btained in predicting heat flux and temperature distributin. Three spines have been cnsidered, each having a circular crss sectin, with the radius varying with psitin alng the spine axis as indicated in Table 1. The psitin alng the spine axis is measured frm the tip, and "a" Is a cnstant. TABLE I. SPINE GEOMETRY ~ Prfile Equatin f Radius radius = ax = y radius = r == y radius = ~~ == y ANALYSIS It 18 assumed that the spine has cnstant thermal cnductivity, and that the cnvective heat transfer cefficient f the fluid surrunding tbt spine 18 a1b cnstant. One-dimena1nal heat transfer is cd8idered SIJ that the length f the spine must be large cmpared t its thickness. Figure 1 illustrates the crdinate system fr this prblem.

2 ENGINEERING SCIENCE 227 y T=T c t y x SYMMETRY Figure 1. Crdinate System The gverning differential equatin is btained by applying an energy balance t a differential vlume as shwn in Figure 1. Cnservatin f energy yields (dztldw) + (dtldx) [(IIA) (daldx)] - (h/k) (PIA) (dsldx) (T - T.. )] = O. (1) where k = thermal cnductivity, h = cnvective cefficient, P = perimeter, and A = crss-sectinal area. Fr a tin f circular crss sectin, a spine, whse surface is described by an equatin f the frm 11 = az", equatin (1) becmes d'tldx') + (2nlx) (dtldx) -(hlk) (2jax ) (1 + (anx 1)2]~ (T Tz ) = O. (2) ~e fllwing substitutin f variables will allw the cnservatin equatin t be written in dimensinless frm: ell = (T - T..IT - T.. ), ~ = xllj fj = hllkj and a = til. The result f this transfrmatin f equatin (2) t dimensinleas frm IS cia" + (2n+'/~) - (2 pia) (IIf') [1 + (a n E.,)'] ~ 4» = O. (3) The bundary cnditins which must be satisfied are as fllws: at z = 1, T = T r fr equatin (3) at ~ = 1,4» = 1, at x = 0, dtldx = 0 r fr equatin (3) at ( = 0, d+ld( = O. Equatin (3) reduces t the generauzed Bessel'.s equatin if the sec- ~ n( term in the radical is assumed small cmpared t unity. Tbls ab Ut :ptin is analgus t assuming db is equivalent t d:e, (see F'1gure 1). 'lsed frm slutin t equatin (3) baa nt been fund, and the

3 228 PROC. OF 'me OKLA. ACAD. OF SCI. FOR 1967 IOlutlns praented here were btained by the applicatin f the furth rder Runge-Kutta methd (McCracken and Dam, 19M). REsuLTS The results f this study are given in three Ulustratins. Table n 8Ummartzes and cmpares the numerical values f dimensinless heat flux btained thrugh the use f the simplifying apprximatin n surface slpe (Bessel's equatin) with the values btained frm the slutin f equatin (8). TABLE n. COMPARATIVE DIMENSIONLESS HEAT FLux Dimensinless Heat Flux = QL/kA(T - T) Prfile Bessel This Paper Figure 2 presents the temperature distributin fr the varius spine prfiles given in Table I. Figure'3 presents the per cent errr between the temperatures btained frm the slutin f equatin (3) and the temperatures btained frm the slutin f Bessel's equatin ,,...---r ~ I~.8 t tl------t-::~~---t n~.6.4 m~.2 F-'~.,_c;._+---~ I~-- t 0.0 ~ / L Figure 2. Dimenslnless Temperature Prfiles

$ENGINEERING SCIENCE 229 5.0 r-----r---...,.---..,...-.--...---- "i : ~ '" 4.0t-----+----+----I----+----I Aa.. A Ita ~ 3.0 1-...-I\r---...,...----t----4-----I-----I.. I ~ LIJ t Z LIJ (,) l&j Q.$

4 ENGINEERING SCIENCE r-----r---...,.---.., "i : ~ '" 4.0t I I Aa.. A Ita ~ I\r---..., t I-----I.. I ~ LIJ t Z LIJ (,) l&j Q. Figure F--...:==:~~...~-+---_+---4_--_ t----t----~~~d_--~~--~ ~ all.l ~ 0.8 Percentage Errr Between Exact and Apprximate Slutins CNCLUSIONS The mre careful analysis presented in this paper des prduce results which are mre accurate than thse f the apprximate methd nnnally used. In the case f temperature distributin the errr invlved is significant thugh nt s great as t preclude the use f the apprximate methd fr sme engineering prblems. In the case f heat flux the tabulated values indicate a difference between the apprximate methd and that f this paper which is large enugh t be f sme cncern, particularly in the case f greater surface curvature. The nature f the Bessel's slutin is ften such that n tabulated values f the apprpriate Bessel's functin are available t implement the equatin, and fr this reasn the numerical slutin indicated in this paper is mre satisfactry fr individuals having access t cmputers. The methds used in this wrk can be extended t handle ther surface shapes as well as straight and circular fins f variable cr.. sectin. Fu~er study is necessary t detennine if the errr intrduced by the nf--dimensina! assumptin is f the same rder f magnitude 88 the en 1r intrduced by neglecting the curvature f the surface. Cmmentary by B. L. Dwty, cotlwztaftt. - I.. believe that the au! hrs have faued t recgnize ne imprtant fact which makes thelr WO k rather meaningless. The assumptin f fin length very much greater Uu 1 tin base, I.e. L >> b, which fa required fr the ne-d1men8lnal 1.0

5 230 PROC. OF THE OKLA. ACAD. OF SCI. FOR 1967 analysis, implies d.b t the surface is apprximated by cu frm: db = (1 + (dy/~)']* ch where dy/dz = 0 (bil). Thus, the "crrectec analysia" the authrs perfo!"dl cannt imprve the results since it des nt relax any f the assumed restrictins. Authr's rep'y.-we wish t thank Mr. E. L. Dwty fr his cmments. The assumptin t ne-dimensinal heat transfer whteh implies that the fin length is much greater than the fin base des nt necessarily imply that db t the surtace is nearly equal t the length cu alng the axis at all values f z. Fr example, in case ill f the paper, near the spine tip ds >> dx. The intent f the present paper was t investigate the temperature prfile and heat flux which are btained when ne assumes that db is nt equal t dx. The differences between this case and the case when db is equal t dx are admittedly small, but in sme applicatins such as thermcuple prbes, they may be significant. In general, fr the spine shape given by y = ax" r y = ax l ''', the effect f curvature shuld becme mre imprtant with larger values f n. REFERENCES Jakb, Max Heat Transfer. Vl. 1, P Jhn Wiley & Sns, New Yrk. McCracken, Daniel D. and William S. Drn Numerical Methds a'lui Frtran Prgramming, p Jhn Wiley & Sns, New Yrk. Schneider, P. J Onductin Heat Transfer, p. 88. Addisn-Wesley Publishing C.

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