Fluid Flow Around and Heat Transfer from Elliptical Cylinders: Analytical Approach

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1 JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER Vol. 19, No., April June 5 Flui Flow Aroun an Heat Transfer from Elliptical Cyliners: Analytical Approach W. A. Khan, J. R. Culham, an M. M. Yovanovich University of Waterloo, Waterloo, Ontario NL G1, Canaa An integral metho of bounary-layer analysis is employe to erive close-form expressions for the calculation of total rag an average heat transfer for flow across an elliptical cyliner uner isothermal an isoflux thermal bounary conitions. The Von Kármán Pohlhausen integral metho is use to solve the momentum an energy equations for both thermal bounary conitions. A fourth-orer velocity profile in the hyroynamic bounary layer an a thir-orer temperature profile in the thermal bounary layer are use. The present results are in goo agreement with existing experimental/numerical ata an, in the limiting cases, can be use for circular cyliners an finite plates. Nomenclature a semimajor axis of elliptical cyliner, m b semiminor axis of elliptical cyliner, m C D total rag coefficient C D f friction rag coefficient C Dp pressure rag coefficient C f skin friction coefficient, τ w /ρuapp C p pressure coefficient, p/ρuapp c imensional focal istance, m c p flui specific heat, kj/kg K D iameter of circular cyliner, m E(e) complete elliptic integral of secon kin e eccentricity, (1 ɛ ) g 11, g metric coefficients h average heat-transfer coefficient, W/m K k thermal conuctivity, W/m K L length of finite plate, m L characteristic length a, m Nu L Nusselt number base on characteristic length L h/k f P pressure, N/m Pr Prantl number ν/α q heat flux, W/m Re L Reynol number base on characteristic length LU app /ν r raius of curvature of surface, m s istance along curve surface of elliptical cyliner measure from forwar stagnation point, m T temperature, C U app approach velocity, m/s U(s) velocity in invisci region just outsie bounary layer, m/s u s-component of velocity in bounary layer, m/s v η-component of velocity in bounary layer, m/s x, y Cartesian coorinates Receive 8 April 4; presente as Paper 4-7 at the AIAA 7th Thermophysics Conference, Portlan, OR, 8 June 1 July 4; revision receive 5 July 4; accepte for publication 8 July 4. Copyright c 4 by the American Institute of Aeronautics an Astronautics, Inc. All rights reserve. Copies of this paper may be mae for personal or internal use, on conition that the copier pay the $1. per-copy fee to the Copyright Clearance Center, Inc., Rosewoo Drive, Danvers, MA 19; inclue the coe /5 $1. in corresponence with the CCC. Postoctoral Fellow, Department of Mechanical Engineering. Member AIAA. Associate Professor an Director, Microelectronics Heat Transfer Laboratory, Department of Mechanical Engineering. Distinguishe Professor Emeritus, Department of Mechanical Engineering. Fellow AIAA. δ hyroynamic bounary-layer thickness, m δ 1 isplacement thickness, m δ momentum thickness, m δ T thermal bounary layer thickness, m ɛ axis ratio b/a η istance normal to an measure from surface of elliptical cyliner, m θ angle measure from stagnation point, raians λ pressure graient parameter µ absolute viscosity of flui, kg/ms ν kinematic viscosity of flui, m /s ξ, θ moifie polar coorinates ρ flui ensity, kg/m τ shear stress, N/m Subscripts a ambient f flui p pressure s separation T thermal w wall Introuction MANY inustrial applications, where heat loa are substantial an space is limite, require the use of tubular heat exchangers for the cooling of electronic equipment. In these applications, elliptical geometries outperform circular geometries. Elliptical cyliners offer less flow resistance an higher heat-transfer rates than circular cyliners. They provie more general geometrical configurations than circular cyliners. In the limiting cases, they represent a finite-length plate when the axis ratio ɛ an a circular cyliner when the axis ratio ɛ 1. Thus, a systematic analytical investigation of elliptical geometries can provie flow an heat-transfer characteristics not only from elliptical cyliners of ifferent axis ratios but also from circular cyliners an finite-length flat plates. Because of space limitations, existing literature is reviewe for the elliptical cyliners only. There have been few experimental/numerical stuies of flui flow an heat transfer from elliptical cyliners. Schubauer 1, conucte experiments to etermine velocity istributions in the laminar bounary layer on the surface of an elliptic cyliner with axis ratio 1:. He foun that the velocity istributions in the bounary layer, its thickness, an its tenency to separate from the surface of the boy are governe almost entirely by the velocity istribution in the region of potential flow outsie the bounary layer. He got goo agreement with the approximate metho, evelope by Pohlhausen, for the forwar part of the cyliner. The same approximate metho was use by Schlichting an Ulrich 4 to calculate 178

2 KHAN, CULHAM, AND YOVANOVICH 179 bounary-layer parameters for elliptical cyliners of ifferent axis ratios, mentione by Schlichting. 5 Ota et al. 6,7 experimentally stuie heat transfer an flow aroun elliptical cyliners of axis ratios 1: an 1:. Their results show that the heat-transfer coefficient of an elliptical cyliner is higher than that of a circular one with equal circumference an the pressure rag coefficients of the former are much lower than those of the later. Žukauskas an Žiugža 8 experimentally stuie flui flow an heat transfer from an elliptical cyliner with 1: ratio between minor an major axes an with air flow parallel to either axis. They foun higher heat-transfer coefficients for elliptical cyliners. Moi et al. 9 stuie experimentally the aeroynamics of a set of two-imensional, stationary elliptic cyliners with ifferent axis ratios in the subcritical Reynol number range They presente extensive results on static pressure istribution, Strouhal number, an near-wake geometry as functions of the angle of attack an Reynol number. They also etermine the separation points using the analytical Gortler series solution approach. Konjoyan an Dauin 1 measure experimentally the effect of freestream turbulence intensity on heat an mass transfer at the surfaces of a circular cyliner an an elliptical cyliner with axis ratio 1:4. They foun that the effect of turbulence intensity appeare to be as important as the influence of velocity an seeme to be inepenent of the pressure graient an of the egree of turbulence isotropy. Jackson, 11 D Allessio an Dennis, 1,1 an D Allessio 14 numerically stuie the flow of a viscous incompressible flui past an incline elliptic cyliner. They obtaine solutions for Reynol numbers up to 1 an for various inclinations. Goo agreement was foun with the existing results. Li et al. 15 experimentally showe that the heat-transfer rate with elliptical pin fins is higher than that with circular pin fins, whereas the flow resistance of the former is much lower than that of the latter in the Reynol-number range from 1 to 1 4. The results of these stuies are applicable only over a fixe range of conitions. Furthermore, no analytical stuies exist that provie close-form solutions for flui flow an heat transfer from elliptical cyliners for a wie range of axis ratio an Reynol an Prantl numbers. The following stuy will be use to erive close-form expressions for the rag an heat-transfer coefficients from elliptical cyliners of arbitrary axis ratio. For the limiting cases, these expressions will be use for circular cyliners an finite plates. The basic parameters involve in this problem are the minor major axis ratio, ɛ b/a, an the Reynol an Prantl numbers. Analysis Consier uniform flow of a Newtonian flui past a fixe elliptical cyliner with major axis a an minor axis b. The cyliner is oriente so that the major axis is parallel to the irection of the net flow in the main stream, thus making one en of the major axis a point of stagnation A (Fig. 1). The flow is assume to be laminar, steay, an two-imensional. The approach velocity of the air is U app an the ambient temperature of the air is T a. The surface temperature of the wall is T w in the isothermal case an the heat flux is q for the isoflux case. The raius of curvature of the surface is r. For a comparison of total rag an heat transfer from an elliptical cyliner with those from a circular cyliner an a flat plate, a characteristic length is use in both the Reynol an the Nusselt numbers. This characteristic length L is the equivalent iameter of a circular cyliner whose perimeter is the same as that of the elliptical cyliner an that of the flat plate. In this case, the istance traverse by the flow will be the same along the surfaces of the three objects. This length is L 4aE(e)/π (1) where E(e) is the complete elliptic integral of the secon kin. In the limiting cases, when ɛ 1, this characteristic length gives the iameter D of a circular cyliner, an when ɛ, it represents the length L of a flat plate. Coorinate System an Governing Equations Following D Allessio, 14 the mapping between Cartesian (x, y) an moifie polar (ξ,θ) coorinates is x c cosh ξ cos θ, y c sinh ξ sin θ () where c is the imensional focal istance c a 1 ɛ with ɛ b/a The surface of an ellipse is efine by ξ ξ,sothat tanh ξ ɛ () The elliptic coorinate system gives the metric coefficients g 11 g c sinh ξ + sin θ (4) The equations of continuity, momentum, an energy for steay-state force convection of a Newtonian constant-property flui with no heat generation can be expresse as V (5) V V (1/ρ) P + ν V (6) V T α T (7) where V is the velocity vector of the flowfiel, P is the pressure graient for force convection, an T is the temperature. For analyzing flui flow aroun an heat transfer from an elliptical cyliner, it is convenient to use a curvilinear system of coorinates in which s enotes istance along the curve surface of the elliptical cyliner measure from the forwar stagnation point an η is the istance normal to an measure from the surface (Fig. 1). In this system of coorinates, the velocity components in the local s- an η-irections are enote by u an v. The potential flow velocity just outsie the bounary layer is enote by U(s). Using an orer-of-magnitue analysis, one can obtain the following simplifie bounary layer equations corresponing to Eqs. (5 7): Continuity u s + v η (8) s-momentum u u s + v u η 1 P ρ + ν u (9) η η-momentum P η (1) Energy Fig. 1 Laminar flow over an elliptical cyliner. u T s + v T η α T η (11)

3 18 KHAN, CULHAM, AND YOVANOVICH Bernoulli equation 1 ρ P Hyroynamic Bounary Conitions At the cyliner surface, η, U(s) U(s) At the ege of the bounary layer, η δ(s), u U(s), (1) u, v (1) u η, u η (14) Thermal Bounary Conitions The bounary conitions for the isothermal an isoflux cyliners are T T w for uniform wall temperature (UWT) η, T η q for uniform wall flux (UWF) k f (15) η δ T, T T a, T η (16) Velocity Distribution Assuming a thin hyroynamic bounary layer aroun the cyliner, the velocity istribution in the bounary layer can be approximate by the fourth-orer polynomial suggeste by Pohlhausen as u/u(s) ( η H η H + ( H) η4 + (λ/6) ηh η H + η H ) η4 H (17) where η H η/δ(s). This satisfies the bounary conitions (1) an (14). Outsie the bounary layer, the velocity istribution may be approximate by the potential-flow solution 4 U(s) U app(1 + ɛ)sin θ 1 e cos θ (18) Temperature Distribution Assuming a thin thermal bounary layer aroun the cyliner, the temperature istribution in the thermal bounary layer can be approximate by the thir-orer polynomial (T T a )/(T w T a ) A + Bη T + Cη T + Dη T (19) where η T η/δ T (s). Using the thermal bounary conitions (15) an (16), the temperature istribution is (T T a )/(T w T a ) 1 η T + 1 η T () for the isothermal bounary conition an T T a (qδ T /k f ) ( 1 η ) T + 1 η T for the isoflux bounary conition. (1) Bounary-Layer Parameters In imensionless form, the momentum integral equation is written as U(s)δ ν ( δ + + δ ) 1 δ U δ ν δ U(s) u η η () where δ 1 δ δ δ 1 1 [ 1 u ] η H () U(s) [ u 1 u ] η H (4) U(s) U(s) Using the velocity istribution from Eq. (17), Eqs. () an (4) can be written as Assuming that δ 1 (δ/1)( λ/1) (5) δ (δ/6)(7/5 λ/15 λ /144) (6) Z δ ν, K Z U Equation () can be reuce to the nonlinear ifferential equation of the first orer for Z, Z H(K ) (7) U(s) where H(K ) f (K ) K [ + f 1 (K )] isauniversal function an is approximate by Walz 16 by a straight line, with f 1 (K ) f (K ) 1 6 K H(K ).47 6K 6( λ/1) 1(7/5 λ/15 λ /144) ( + λ 6 )( 7 5 λ 15 λ 144 λ ( λ ) 15 λ (8) 144 Equation (7) can be solve for the imensionless local momentum thickness, δ L.61 (1 e cos θ) θ ReL E(e) sin 6 θ ) sin 5 θ θ (1 e cos θ) (9) Using the potential-flow velocity, Eq. (18), the imensionless local bounary-layer thickness can be written as δ L.886 (1 e cos θ) λ () ReL ɛ (1 + ɛ)e(e) cos θ By solving Eqs. (6) an () an comparing the results with Eq. (9), the values of the pressure-graient parameter λ corresponing to each position along the cyliner surface are obtaine. These values are foun to be positive from θ θ 1 9 eg (region I, boune by ACA) an negative from θ 1 θ θ s (region II, boune by CDC), as shown in Fig. 1. Thus, the entire range of interest θ θ s can be ivie into two regions an the λ values can be fitte separately by the least-squares metho into two polynomials for each axis ratio. Flui Flow The first topic of interest is flui friction, which manifests itself in the form of the rag force F D, which is the sum of the skin-friction rag D f an pressure rag D p. Skin-friction rag is ue to viscous shear forces prouce at the cyliner surface, preominantly in those regions to which the bounary layer is attache. The component of

4 KHAN, CULHAM, AND YOVANOVICH 181 shear force in the flow irection is given by D f τ w sin θ A s (1) s where A s per unit length of cyliner. In the case of the elliptical geometry, the raial istance r to a point on the ellipse surface ξ ξ is r a 1 e cos θ () Therefore, the length is a 1 e cos θ θ () The shear stress on the cyliner wall τ w can be obtaine from Newton s law of viscosity. In imensionless form, it can be written as C f.8 ɛ (1 + ɛ) E(e) (λ + 1) sin θ cos θ (4) ReL (1 e cos θ) λ The friction rag coefficient is efine as C D f π C f sin θ θ θs C f sin θ θ + π θ s C f sin θ θ (5) Because no shear stress acts on the cyliner surface after the bounary-layer separation, the secon integral will be zero an the friction rag coefficient is written as C D f θs C f sin θ θ (6) The analytical efinition for the bounary-layer separation shows that the separation point is characterize by zero transverse velocity graient at the wall. The angle of separation θ s calculate in this stuy epen upon the velocity istribution, Eq. (17), chosen insie the bounary layer. The angle of separation, for ifferent axis ratios, is given in Table 1. It is important to note that these results follow the tren of Schlichting an Ulrich 4 an Schubauer 1 but o not confirm the Moi et al. 9 tren. The friction-rag coefficients are calculate for ifferent axis ratios an a general correlation is euce in terms of arbitrary axis ratio an Reynol number: C D f ɛ1.5 ReL (7) In the limiting cases when ɛ 1, this gives 5.78/ Re D for a circular cyliner, an when ɛ, it gives 1.5/ Re L for a finite flat plate. The pressure rag is ue to the unbalance pressures that exist between the relatively high pressures on the upstream surfaces an the lower pressures on the ownstream surfaces. The component of pressure rag in the flow irection is given by D p P cos θ A s (8) s Table 1 Angle of separation for ifferent axis ratios of elliptical cyliner ɛ θ s which can be rewritten in terms of the pressure-rag coefficient as C Dp π C p cos θ 1 e cos θ θ (9) The unbalance pressure, P P 1 P, between upstream an ownstream surfaces can be obtaine by integrating the θ-component of Eq. (6) from P 1 to P. Using an orer-of-magnitue analysis, the simplifie θ-component of Eq. (6) in elliptic cyliner coorinates (ξ,θ) can be written as u ξ u θ g 11 ξ + u θ u θ g 11 θ 1 P g 11 ρ θ + ν g11 { } u θ θ (4) where g 11 are the metric coefficients an are given by Eq. (4), u ξ an u θ are the velocity components in the ξ- an θ-irections an are given by u ξ 1 φ g 11 ξ, u θ 1 φ (41) g 11 θ where φ is the potential function for a steay flow past an elliptical cyliner an is obtaine from the complex potential φ U app a (1 + ɛ)/(1 ɛ)cosh ξ cos θ(1 ɛ tanh ξ) (4) Calculating the velocity components an their erivatives on the surface of the elliptic cyliner, one can fin, from Eq. (8), the pressure coefficients for ifferent axis ratios. These coefficients are then use, in Eq. (9), to etermine the pressure-rag coefficients for ifferent axis ratios. Using these coefficients, a general correlation in terms of the axis ratio an Reynol number is foun to be C Dp ( /Re L )ɛ.95 (4) The total rag coefficient C D can be written as the sum of both rag coefficients, ( ) ɛ1.5 C D ReL Re L ɛ.95 (44) In the limiting cases when ɛ 1 C D / ReD /Re D (45) for a circular cyliner an, when ɛ, it gives for a finite flat plate. C D 1.5 / ReL (46) Heat Transfer For the isothermal bounary conition, the energy integral equation can be written as δt (T T a )u η α T η η (47) Using the velocity an temperature profiles, an assuming that ζ δ T /δ < 1, Eq. (47) can be simplifie to δ T [U(s)δ T ζ(λ+ 1)] 9α (48) This equation can be rewritten separately as δ T [U(s)δ T ζ(λ 1 + 1)] 9α (49) for region I an δ T [U(s)δ T ζ(λ + 1)] 9α (5)

5 18 KHAN, CULHAM, AND YOVANOVICH for region II. Multiplying both equations by U(s)ζ an integrating separately in the two regions, one can solve these two equations for the local thermal bounary-layer thicknesses to fin ( ) δt (θ) L L Pr 1.6 (1 + ɛ)e(e) (1 e cos θ) f (θ) ɛ(λ 1 + 1) sin θ (1 e cos θ) f 5 (θ) ɛ sin θ where with f (θ) f 5 (θ) θ λ 1 cos θ λ cos θ for region I for region II (51) sin θ(λ 1 + 1) θ (5) f (θ) λ f 4(θ) λ + 1 θs η (5) f 4 (θ) sin θ(λ + 1) θ (54) θ 1 For the isothermal bounary conition, the local heat-transfer coefficient is / T h(θ) k f (T w T a ) k f (55) η δ T Thus, the local Nusselt numbers for both regions can be written as Nu L (θ) isothermal (1 + ɛ)e(e) L Pr 1. ɛ(λ 1 + 1) sin θ cos θ for region I (1 e cos θ) f (θ) λ 1 ɛ sin θ cos θ for region II (56) (1 e cos θ) f 4 (θ) λ The average heat-transfer coefficient is efine as h 1 π h(θ) θ 1 [ θs π ] h(θ) θ + h(θ) θ (57) π π θ s It has been observe experimentally by many researchers that, at low Reynol numbers, there is no appreciable increase in the local heat transfer after the separation point. However, at high Reynol numbers, the local heat transfer increases from the separation point to the rear stagnation point. Hence, the average heat transfer coefficient is h 1 θs h(θ) θ 1 [ θ1 θs ] h 1 (θ) θ + h (θ) θ (58) π π Using Eqs. (51 54), Eq. (58) can be solve for the average heattransfer coefficient, which gives the average Nusselt number for an isothermal elliptical cyliner of arbitrary axis ratio ɛ as ( ) Nu D isothermal exp (59) L Pr 1 ɛ.1 For the isoflux bounary conition, the energy integral equation can be written as x δt (T T a )u η q ρc p (6) θ 1 For constant heat flux an thermophysical properties, Eq. (6) can be simplifie to [ U(s)δ T ζ(λ+ 1)] 9 ν (61) Pr Rewriting Eq. (61) for the two regions in the same way as Eq. (48), one can obtain the local thermal bounary layer thicknesses δ T1 an δ T uner isoflux bounary conition. The local surface temperatures for the two regions can then be obtaine from Eq. (1) as T 1 (θ) qδ T1 / k f (6) T (θ) qδ T / k f (6) The local heat transfer coefficient can now be obtaine from its efinition as h 1 (θ) q/ T 1 (θ), h (θ) q/ T (θ) (64) Following the same proceure for the average heat-transfer coefficient, one can obtain the average Nusselt number for an isoflux cyliner as ( ) Nu L isoflux exp (65) L Pr 1 ɛ 1.79 This Nusselt number is 6% greater than the average Nusselt number for an isothermal circular cyliner. Combining the results for both thermal bounary conitions, we have ( ) exp UWT Nu L ɛ.1 ( ) L Pr exp UWF (66) ɛ 1.79 which gives the imensionless Nusselt numbers for elliptical cyliners of arbitrary axis ratio uner isothermal or isoflux bounary conitions. In the limiting cases, when ɛ 1, it represents the average Nusselt numbers for a circular cyliner, { Nu D.59 UWT D Pr 1.61 UWF (67) An, when ɛ, it represents the average Nusselt number for a finite flat plate as { Nu L.75 UWT L Pr 1.91 UWF (68) where L is the length of the plate. Results an Discussion The imensionless shear stress, C f ReL,atthe surfaces of cyliners of ifferent axis ratios is shown in Fig.. This shows that C f is zero at the stagnation point for each case an reaches a maximum at a certain angle, which ecreases with the axis ratio. The increase in shear stress is cause by the eformation of the velocity profiles in the bounary layer, a higher velocity graient at the wall, an a thicker bounary layer. In the region of ecreasing C f preceing the separation point, the pressure graient ecreases further an finally C f falls to zero at the separation angle which increases with the axis ratio. These angles in raians are presente in Table 1. Beyon the separation point, C f remains close to zero up to the rear stagnation point. The results for the circular cyliner are also shown in the same figure for comparison. The total rag coefficients versus axis ratio are plotte in Fig. for ifferent Reynol numbers. It is clear that the total rag coefficient ecreases from the circular cyliner to the

6 KHAN, CULHAM, AND YOVANOVICH 18 Fig. Distribution of shear stress on surface of cyliners in air. Fig. 4 Variation of total rag coefficient with Reynol number. Fig. Variation of total rag coefficient with axis ratio. finite flat plate. For larger axis ratios, these coefficients epen upon the Reynol numbers, but for smaller axis ratios, the coefficients are inepenent of Reynol numbers. The total rag coefficients versus Reynol numbers are shown in Fig. 4 for ifferent axis ratios. As expecte, the rag coefficients are highest for the circular cyliner an lowest for the flat plate. The rag coefficients of the elliptical cyliner lie between these two limits an ecrease with the Reynol numbers. The present results for circular cyliner an finite flat plate are compare with the experimental ata of Wieselsberger 17 an Janour, 18 respectively. It is clear that the present results are in goo agreement with the experimental ata. The friction rag coefficients of the finite flat plate, obtaine from the present moel, are also compare with the experimental 18 an numerical 19 ata in Fig. 5. The theoretical correlations of Van Dyke an Kuo 1 further confirm the present flui flow moel. The heat-transfer parameter Nu L /Re 1/ L Pr1/ is plotte vs the axis ratio ɛ in Fig. 6. The curve shows that the heat-transfer parameter increases very slowly from the circular cyliner (ɛ 1) to a certain axis ratio (ɛ.5) an then increases rapily up to (ɛ.1), an after that it becomes constant for the finite flat plate. The average Nusselt numbers of isothermal elliptic cyliners versus Reynol numbers are presente in Fig. 7. It is clear that Nu L increases linearly with Re L on a log log plot for all cases. Nusselt numbers for the circular cyliner are foun to be lower than for any geometry consiere in this stuy. The present results for the circular cyliner are compare with the empirical correlation of Churchill an Bernstein an the finite flat-plate results are compare with the Fig. 5 Comparison of friction rag coefficients of a finite flat plate. Fig. 6 Variation of heat transfer parameter with axis ratio.

7 184 KHAN, CULHAM, AND YOVANOVICH Fig. 7 Variation of average Nusselt number with Reynol number. Fig. 1 Variation of average Nusselt number with axis ratio for isoflux elliptic cyliner. Yovanovich Teertstra moel. It is clear that the present results are in goo agreement with the existing ata. The results of heat transfer from a single infinite isoflux elliptical cyliner of axis ratio 1: are shown in Fig. 8, where they are compare with the experimental results of Rieher, 4 Ota et al., 6 an Žukauskas an Žiugža. 8 Although the Rieher configuration was claime as obscure by Ota et al., 6 it shows goo agreement with the present results. For the same isoflux bounary conition, the average Nusselt numbers of elliptical cyliners with axis ratios 1:4 are compare with the experimental ata of Konjoyan an Dauin 1 in Fig. 9. It can be seen that Nu L increases linearly with Re L on a log log plot. The present results for elliptical cyliners are in goo agreement with the existing ata. The variation of average Nusselt numbers of isoflux elliptical cyliners with axis ratio is shown in Fig. 1. The present results are compare with the experimental ata of Ota et al. 6,7 for two ifferent Reynol numbers. The results show goo agreement with the experimental ata. Fig. 8 Variation of average Nusselt number with Reynol number for elliptic cyliner of axis ratio 1:. Conclusions Three general correlations, one for rag coefficient an two for heat transfer, have been etermine. The rag coefficients are lower, whereas the average heat-transfer coefficients are higher for elliptical cyliners than for circular ones. The effects of the axis ratio of the elliptical cyliner upon rag an the average heat-transfer coefficients are also observe an compare for the two extremes with experimental/numerical values obtaine from the open literature. The rag an the average heat-transfer coefficients epen on the Reynol number as well as the axis ratio. The present results are in goo agreement with the experimental results for the whole laminar range of Reynol numbers in the absence of freestream turbulence an blockage effects. The correlations obtaine in this stuy can be use to etermine the imensionless rag an heat-transfer coefficients from an elliptical pin fin, where the Reynol number epen on the characteristic length. Acknowlegments The authors gratefully acknowlege the financial support of Natural Sciences an Engineering Research Council of Canaa an the Center for Microelectronics Assembly an Packaging. Fig. 9 Variation of average Nusselt number with Reynol number for isoflux elliptic cyliner of axis ratio 1:4. References 1 Schubauer, G. B., Air Flow in a Separating Laminar Bounary Layer, NACA TR-57, Dec Schubauer, G. B., Air Flow in the Bounary Layer of an Elliptic Cyliner, NACA TR-65, Aug. 199.

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