Fluid Flow and Heat Transfer from a Cylinder Between Parallel Planes

Transcription

1 JOURNAL OF THERMOPHYSICS AN HEAT TRANSFER Vol. 18, No. 3, July September 4 Flui Flow an Heat Transfer from a Cyliner Between Parallel Planes W. A. Khan, J. R. Culham, an M. M. Yovanovich University of Waterloo, Waterloo, Ontario NL 3G1, Canaa An integral approach is employe to investigate the effects of blockage on flui flow an heat transfer from a circular cyliner confine between parallel planes. The integral form of the bounary-layer momentum equation is solve using the moifie von Kármán Pohlhausen metho, which uses a fourth-orer velocity profile insie the hyroynamic bounary layer. The potential flow velocity, outsie the bounary layer, is obtaine by the metho of images. A thir-orer temperature profile is use in the thermal bounary layer to solve the energy integral equation for isothermal an isoflux bounary conitions. Close-form solutions are obtaine for the flui flow an heat transfer from the cyliner with blockage ratio an Reynols an Prantl numbers as parameters. It is shown that the blockage ratio controls the flui flow an the transfer of heat from the cyliner an elays the separation. The results for both thermal bounary conitions are foun to be in a goo agreement with experimental/numerical ata for a single circular cyliner in a channel. Nomenclature b = blockage ratio, ientical to /S T C = total rag coefficient C f = friction rag coefficient C p = pressure rag coefficient C f = skin-friction coefficient C p = pressure coefficient c p = specific heat of flui, J/kg K = cyliner iameter, m f = shear force in flow irection, N p = pressure force in flow irection, N h = average heat transfer coefficient, W/m K j = number of cyliners in transverse irection k = thermal conuctivity, W/m K Nu = average Nusselt number base on iameter of cyliner, ientical to h /k f Pr = Prantl number, ientical to ν/α p = pressure, N/m q = heat flux, W/m Re = Reynols number, ientical to U /ν r,θ = polar coorinates S T = vertical istance between parallel planes s = istance along curve surface of circular cyliner measure from forwar stagnation point, m T = temperature, C U(s) = potential flow velocity just outsie bounary layer, m/s u,v = velocity components in bounary layer, m/s w(z) = complex potential in Cartesian coorinates, m /s x, y = Cartesian coorinates Y = istance normal to an measure from surface of circular cyliner, m z = complex variable in Cartesian coorinates, m Receive 31 October 3; presente as Paper at the 4n Aerospace Sciences Meeting an Exhibit, Reno, NV, 5 8 January 4; revision receive 3 January 4; accepte for publication 31 January 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 rive, anvers, MA 193; inclue the coe /4 $1. in corresponence with the CCC. Grauate Research Assistant, epartment of Mechanical Engineering, Microelectronics Heat Transfer Laboratory. irector, Microelectronics Heat Transfer Laboratory. istinguishe Professor Emeritus, epartment of Mechanical Engineering, Microelectronics Heat Transfer Laboratory. Fellow AIAA. 395 α = thermal iffusivity, m /s δ = hyroynamic bounary-layer thickness, m δ T = thermal bounary-layer thickness, m δ 1 = isplacement thickness, m δ = momentum thickness, m ζ = ummy variable η = imensionless normal istance θ = angle measure, ra or eg λ = pressure graient parameter µ = absolute viscosity of flui, kg/m s ν = kinematic viscosity of flui, m /s ρ = ensity of the flui, kg/m 3 τ = shear stress, N/m ψ, φ = stream an potential functions, m /s Subscripts f = flui or friction p = pressure r,θ = plane polar coorinates s = separation T = temperature w = wall = freestream conitions Introuction THE main objective of this stuy is to investigate analytically the effects of blockage ratio b = /S T on the flui flow an heat transfer from a cyliner uner ifferent thermal bounary conitions. This parameter plays an important role in etermining the flui flow an heat transfer from a cyliner confine between parallel planes. In practice, a cyliner is place in flows restricte by walls. This configuration is foun in many applications, such as crossflow heat exchangers, shroue heat sinks, an electric heating elements in boilers. It has been observe by Žukauskas 1 an others that, as the blockage ratio increases, the velocity aroun the circular cyliner outsie the bounary layer increases an the pressure an velocity istributions insie the bounary layer are change accoringly. Potential flow velocity outsie the bounary layer can be obtaine by the metho of images, which is a technique use in potential flow moeling to represent the presence of a channel wall by creating an image of the cyliner. The flui flow an heat transfer from a cyliner (infinite flow conitions) have been stuie analytically by Khan et al., an experimentally/numerically by many researchers incluing Žukauskas, 1 Roshko, 3 Achenbach, 4 Schlichting, 5 Wieselsberger, 6 Churchill, 7 Sucker an Brauer, 8 Žukauskas an Žiugža, 9 Eckert

2 396 KHAN, CULHAM, AN YOVANOVICH an Soehngen, 1 Churchill an Bernstein, 11 Morgan, 1 Hilpert, 13 Refai Ahme an Yovanovich, 14 Krall an Eckert, 15 Giet, 16 an Sarma an Sukhatme. 17 However, the problem of flui flow an heat transfer from a cyliner place insie a channel has not been stuie analytically so far. A review of existing literature reveals that few experimental/ numerical stuies exist regaring the configuration of the present problem. Perkins an Leppert 18 investigate local heat transfer coefficients from a uniformly heate cyliner with water in crossflow. They use both potential flow an experimental pressure istributions to investigate an correlate the effects of blockage on the velocity an heat transfer istributions. Žukauskas 1 analyze the work of Akilba yev et al. 19 about the influence of channel blockage on the flow an heat transfer of a tube in a restricte channel. Accoring to Žukauskas, 1 they showe that increasing channel blockage ratio from to.8 cause the minimum pressure point to be isplace from θ = 7 to 9 eg, an the separation point to move ownstream to θ = 1 eg. Their theoretical calculations, by the metho of Merk using the potential flow velocity istribution, showe that the heat transfer on the front portion of the tube increases with an increase in blockage ratio. Žukauskas an Žiugža 9 performe a series of experiments with ifferent freestream geometries to investigate the effects of channel blockage. They expresse the velocity istribution in the outer bounary layer in terms of channel blockage an use it to estimate the heat transfer behavior of a cyliner. Vaitiekünas et al. 1 investigate numerically the effects of the channel blockage on the imensionless shear stress, the location of U max, point of bounary-layer etachment, an the local heat transfer coefficients. They approximate the velocity istribution outsie the bounary layer by the moifie Hiemenz polynomial in which the coefficients are functions of channel blockage. These functions were base on the analysis of the experimental ata of Žukauskas an Žiugža. 9 They 1 foun satisfactory agreement with the experimental results of Žukauskas an Žiugža. Hattori an Takahashi performe experiments on force convection heat transfer from a single row of circular cyliners in crossflow. They measure local an average Nusselt numbers for a cyliner in the Reynols number range from 8 to an gave a correlation for the average Nusselt number. Later Yamamoto an Hattori 3 verifie numerically their heat transfer values for the same arrangement. They foun goo agreement with those obtaine from experiments in water by Hattori an Takahashi. It is obvious from the literature survey that no analytical stuy exists to give a close-form solution for the flui flow an heat transfer from a circular cyliner in a channel for a wie range of blockage ratios, Reynols numbers, an Prantl numbers. In this stuy, a circular cyliner in a channel is consiere in crossflow with a Newtonian flui (Pr.71) to investigate the effects of the blockage ratio on the flui flow an heat transfer from the cyliner for a wie range of parameters, incluing blockage ratio, Reynols numbers, an Prantl numbers. Close-form solutions are obtaine for the rag coefficients an Nusselt numbers uner ifferent thermal bounary conitions, which can be use for a wie range of parameters. U an the ambient temperature is assume to be T. The surface temperature of the cyliner wall is T w in the case of the isothermal cyliner an the heat flux is q for the isoflux bounary conition. The flow is assume to be laminar, steay, an two imensional. Using an orer-of-magnitue analysis, the reuce equations of continuity, momentum, an energy in the plane polar coorinates for an incompressible flui can be written as follows: the continuity, u r r + u r r + 1 u θ r θ = (1) the θ momentum, u θ u r r + u θ u θ r θ = 1 ( p rρ θ + ν u θ r the r momentum, the energy, T u r r + u ( θ T r θ = α T r + 1 r u θ r u ) θ r () p r = (3) + 1 r ) T r These equations can be rewritten by aopting a curvilinear system of coorinates in which s enotes istance along the curve surface of the circular cyliner measure from the forwar stagnation point A an Y is the istance normal to an measure from the surface as shown in Fig.. In this system of coorinates, the velocity components u θ an u r are replace by u an v in the local s an Y irections, whereas r θ an r are replace by s an Y, respectively. The potential flow velocity just outsie the bounary layer is enote by U(s), which will be etermine by the metho of images. Therefore, the governing equations in this curvilinear system will be as follows: the continuity, u s + v Y = (5) the s momentum, u u s + v u Y = 1 p ρ s + ν u (6) Y the Y momentum, p Y = (7) (4) Analysis Consier a uniform flow of a Newtonian flui (Pr.71) past a fixe circular cyliner of iameter, confine between parallel planes, as shown in Fig. 1. The approaching velocity of the flui is Fig. 1 Physical moel an coorinate system. Fig. Flow over a circular cyliner.

3 KHAN, CULHAM, AN YOVANOVICH 397 the energy, u T s + v T Y = α T (8) Y the Bernoulli equation, 1 ρ p U(s) = U(s) s s (9) Hyroynamic Bounary Conitions At the cyliner surface, that is, at Y =, u u =, Y = 1 p µ s At the ege of the bounary layer, that is, at Y = δ(s), u = U(s), u Y =, (1) u Y = (11) Thermal Bounary Conitions The bounary conitions for the uniform wall temperature (UWT) an uniform wall flux (UWF) are T = T w for UWT Y =, T Y = q for UWF k f (1) Y =, Y = δ T, T = T, T Y = (13) T Y = (14) Velocity istribution Insie Bounary Layer Assuming a thin bounary layer aroun the cyliner, the velocity istribution in the bounary layer can be approximate by a fourthorer polynomial as suggeste by Pohlhausen, 4 u/u(s) = (η η 3 + η 4 ) + (λ/6)(η 3η + 3η 3 η 4 ) (15) where η = Y/δ(s) 1 an λ is the pressure graient parameter given by λ = δ U(s) (16) ν s With the help of velocity profiles, Schlichting 5 showe that the parameter λ is restricte to the range 1 λ 1. Velocity istribution Outsie Bounary Layer When the metho of images is use, a cyliner confine between parallel planes (Fig. 1) can be moele as a system of infinite transverse row of oublets superimpose on a uniform flowfiel (Fig. 3). Kochin et al. 5 an Perkins an Leppert 18 pointe out this case. For this case, the complex potential can be written as w(z) = U z + j = µ π(z ijs T ) = U z + µ ( ) πz coth S T S T (17) where j is the number of oublets or cyliners. Therefore, the complex velocity will be w (z) = U ( µπ / S T )[ 1 / sinh (πz/s T ) ] (18) At the stagnation points, z =±/ an w (±/) =. Therefore, Eq. (18) gives µ/s T = (U S T /π) sinh (π /S T ) (19) Fig. 3 Transverse row of oublets or circular cyliners. When this value is substitute into Eq. (17), the require complex potential will be w(z) = φ + iψ = U [z + C coth(πz/s T )] () where φ an ψ are the potential an stream functions an C is a constant given by C = (S T /π) sinh (π /S T ) (1) The stream function ψ in polar coorinates (r,θ) can be obtaine from Eq. () as follows: ψ = U r sin θ [ ] sin(πr sin θ/s T ) C U () cosh(πr cos θ/s T ) cos(πr sin θ/s T ) The raial an transverse components of velocity at the surface of the cyliner can be written as u r = 1 ψ, u θ = ψ (3) r θ r r = / r = / which gives u r =, u θ = U f (θ) (4) where f (θ) = sin θ ( )[ πc cos(π cos θ/s T ) sin θ S T cosh(π cos θ/s T ) cos(π sin θ/s T ) ( ) π sin θ + sin S T sinh(π cos θ/s ] T ) cos θ + sin θ sin(π sin θ/s T ) [cosh(π cos θ/s T ) cos(π sin θ/s T )] Setting C 1 = π /S T an substituting the value of C, weget f (θ) = sin θ sinh ( C1 )[ cos(c 1 cos θ)sin θ cosh(c 1 cos θ) cos(c 1 sin θ) (5) + sin(c 1 sin θ) sinh(c ] 1 cos θ)cos θ + sin θ sin(c 1 sin θ) [cosh(c 1 cos θ) cos(c 1 sin θ)] (6)

4 398 KHAN, CULHAM, AN YOVANOVICH The resultant potential flow velocity is U = U f (θ) (7) Temperature istribution Assuming a thin thermal bounary layer aroun the cyliner, the temperature istribution in the thermal bounary layer can be approximate by a thir-orer polynomial, (T T )/(T w T ) = A + Bη T + Cη T + η3 T (8) where η T = Y/δ T (s). When the aforementione thermal bounary conitions are use, the temperature istribution will be (T T )/(T w T ) = 1 3 η T + 1 η3 T (9) for the isothermal bounary conition an T T = (qδ T /3k f ) ( 1 3 η ) T + 1 η3 T for the isoflux bounary conition. (3) Bounary-Layer Parameters In imensionless form, the momentum integral equation can be written as where Uδ ν ( δ s + + δ ) 1 δ U δ ν s = δ U δ 1 = δ δ = δ 1 1 [ 1 u U ] u Y Y = (31) η (3) [ u 1 u ] η (33) U U When velocity istribution from Eq. (15) is use, Eqs. (3) an (33) can be written as assuming δ 1 = (δ/1)(3 λ/1) (34) δ = (δ/63)(37/5 λ/15 λ /144) (35) Z = δ ν, K = Z U s Equation (31) can be reuce to a nonlinear ifferential equation of the first orer for Z, which is given by Z s = H(K ) (36) U where H(K ) = f (K ) K [ + f 1 (K )] isauniversal function an is approximate by Walz 6 using a straight line, with f 1 (K ) = f (K ) = 1 63 K = H(K ) =.47 6K (37) 63(3 λ/1) 1(37/5 λ/15 λ /144) ( + λ 6 )( 37 5 λ 15 λ 144 ) (38) (39) λ ( λ ) 15 λ (4) 144 When Eq. (36) is solve with Eq. (37), the local imensionless momentum thickness can be written as δ =.485 Re θ 1 f 6 (θ) f 5 (ζ ) ζ (41) The local imensionless bounary-layer thickness can be obtaine from Eq. (16), δ/ = λ/ Re g(θ) (4) where g(θ) is the first erivative of f (θ) with respect to θ obtaine from Eq. (6). When Eqs. (35) an (4) are solve an the results are compare with Eq. (41), the values of the pressure graient parameter λ are obtaine corresponing to each position along the cyliner surface. These values are positive from θ θ 1 = π/, for example, region 1, an negative from θ 1 = π/ θ θ s, for example, region, as shown in Fig.. Thus, the whole range of interest θ θ s can be ivie into two regions, an the λ values can be fitte separately for each blockage ratio by the least-squares metho into two polynomials. These polynomials can be use to etermine the rag an the local heat transfer coefficients in both regions. Flui Flow The first parameter of interest is flui friction, which manifests itself in the form of the rag force F, where F is the sum of the skin-friction rag f an pressure rag p. Skin-friction rag is ue to viscous shear forces prouce at the cyliner surface preominantely in those regions where the bounary layer is attache. The component of shear force in the flow irection is given by f = τ w sin θ θ (43) A where τ w is the shear stress along the cyliner wall, which can be etermine from Newton s law of viscosity, τ w = µ u (44) y y = In imensionless form, it can be written as C f = τ w / 1 ρu = 1 3[ (λ + 1) / Re ] f (θ) g(θ)/λ (45) The angle of separation epens only on the velocity istribution outsie the bounary layer. Khan et al. have shown that, for infinite flow conitions, separation is calculate to occur at θ s = eg. From Eq. (45) it can be shown that the angle of separation epens on the blockage ratio. The friction rag coefficient is efine as C f = C f sin θ θ = C f sin θ θ + C f sin θ θ (46) θ s Because no shear stress acts on the cyliner surface after bounarylayer separation, the secon integral will be zero an the friction rag coefficient can be written as C f = C f sin θ θ [ θ1 1 g(θ) = 3 (λ 1 + 1) f (θ) sin θ θ Re λ 1 + θ 1 ] g(θ) (λ + 1) f (θ) sin θ θ λ (47)

5 KHAN, CULHAM, AN YOVANOVICH 399 The rag coefficient C f is calculate for ifferent blockage ratios an correlate into a single expression: C f = [ exp(.95b3.44 )] (48) Re 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 force in the flow irection is given by p = p cos θ θ (49) In imensionless form, it can be written as C p = A C p cos θ θ (5) where C p is the pressure coefficient an is efine as C p = p / 1 ρu (51) The pressure ifference p is obtaine by integrating Eq. () with respect to θ. Then the pressure rag coefficient is calculate, using Eq. (5), for ifferent blockage ratios an correlate into a single expression: C p ={ exp(.76b.63 )} [ ] exp( 5.81b.15 ) + (5) Re The total rag coefficient C can be written as the sum of both rag coefficients: C = [ exp(.95b3.44 )] Re + [ exp(.76b.63 )] + [ exp( 5.81b.15 )] Re (53) Heat Transfer The secon parameter of interest in this stuy is the imensionless average heat transfer coefficient Nu for large Prantl numbers. This parameter is etermine by integrating Eq. (8) from the cyliner surface to the thermal bounary-layer ege. When the presence of a thin thermal bounary layer δ T along the cyliner surface is assume, the energy integral equation for the isothermal bounary conition can be written as s δt (T T )u Y = α T Y Y = (54) When velocity an temperature profiles are use, an ζ = δ T / δ<1isassume, Eq. (54) can be simplifie to δ T s [U(s)δ T ζ(λ+ 1)] = 9α (55) Because s = (/)θ s = (/) θ an U(s) = U f (θ), therefore, δ T θ [ f (θ)δ T ζ(λ+ 1)] = 45 (56) PrRe This equation can be rewritten separately for the two regions mentione earlier, that is, δ T θ [ f (θ)δ T ζ(λ 1 + 1)] = 45 (57) PrRe for region 1, an δ T θ [ f (θ)δ T ζ(λ + 1)] = 45 (58) PrRe for region. These two equations can be solve separately in the two regions for the local thermal bounary layer thicknesses, δ T1 = 1 Re 1 δ T = 1 Re f 1 (θ) (λ 1 + 1) f (θ) 3 9 f 3 (θ) λ f (θ) g(θ) where functions f 1 (θ) an f 3 (θ) are given by f 1 (θ) = f (θ) = θ θ λ1 g(θ) (59) (6) f (θ)(λ 1 + 1) θ (61) θ 1 f (θ)(λ + 1) θ (6) f 3 (θ) = f 1(θ) λ f (θ) (63) λ + 1 For the isothermal bounary conition, the local heat transfer coefficient is efine as follows: h(θ) = k f ( T / Y ) Y = T w T = 3k f δ T (64) Thus, local heat transfer coefficients for both regions are written as h 1 (θ) = 3k f / δt1, h (θ) = 3k f / δt (65) which give the local Nusselt numbers for isothermal bounary conition: Nu 1 (θ) = 3 3 (λ 1 + 1) f (θ) g(θ) Re 1 9 f 1 (θ) λ (66) 1 Nu (θ) = 3 3 f (θ) g(θ) Re 1 9 f 3 (θ) λ (67) The average heat transfer coefficient is efine as h = 1 π h(θ) θ = 1 [ h(θ) θ + π θ s h(θ) θ ] (68) It has been observe experimentally by many researchers that, at low Reynols numbers (up to Re = 5, accoring to Žukauskas an Žiugža 9 ), there is no appreciable increase in the local heat transfer after the separation point. However, at high Reynols numbers, the local heat transfer increases from the separation point to the rear stagnation point. Hence, the average heat transfer coefficient can be written as h = 1 π h(θ) θ = 1 [ θ1 h 1 (θ) θ + π θ 1 h (θ) θ ] (69)

6 4 KHAN, CULHAM, AN YOVANOVICH When Eqs. (59 65) are use, Eq. (69) can be solve for the average heat transfer coefficient that gives the average Nusselt number for an isothermal cyliner: Nu isothermal = [ exp(.65b.5 )] Re 1 (7) For the isoflux bounary conition, the energy integral equation can be written as s δt (T T )u Y = q ρc p (71) For constant heat flux an thermophysical properties, Eq. (71) can be simplifie to [ f (θ)δ T θ ζ(λ+ 1)] = 45 (7) PrRe Rewriting Eq. (7) for the two regions in the same way as Eq. (56), one obtains the imensionless local thermal bounary layer thicknesses δ T1 / an δ T / uner isoflux bounary conition, δ T1 = 1 Re 1 δ T = 1 Re θ (λ 1 + 1) f (θ) 3 λ1 g(θ) 45 f 4 (θ) λ f (θ) g(θ) (73) (74) Results an iscussion Flow Characteristics The effects of the blockage ratio b on the velocity istribution outsie the bounary layer are shown in Fig. 4. It shows that as the blockage ratio ecreases the velocity outsie the bounary layer ecreases. These results are compare with the experimental ata of Akilba yev et al. 19 (reporte by Žukauskas 1 ) for two blockage ratios. A goo agreement between potential theory an experiment is observe for the front part of the cyliner where laminar bounary layer exists. The imensionless local shear stress, C f (Re ),isplotte in Fig. 5 for b =.5. It shows that C f is zero at the stagnation point an reaches a maximum at θ 6 eg. 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, where bounary-layer separation occurs. Beyon this point, C f remains close to zero up to the rear stagnation point. These results are compare with the numerical results of Vaitiekünas et al. 1 for the same blockage ratio. The results are again in goo agreement for the front part of the cyliner. In Fig. 6, it can be seen that the angle of separation epens on the blockage ratio. As the blockage ratio increases, the location of the bounary-layer separation moves backwar. This movement is ue to the change in the velocity istribution outsie the bounary layer. In Fig. 7, the effects of blockage ratio on the rag coefficient can be seen for ifferent Reynols numbers. It is clear that the rag coefficient ecreases with the blockage with f 4 (θ) = θ/(λ 1 + 1) + (θ π/)/(λ + 1) (75) The local surface temperatures for the two regions is obtaine from Eq. (3), T 1 (θ) = qδ T1 / 3k f (76) T (θ) = qδ T / 3k f (77) The local heat transfer coefficient is obtaine from its efinition as h 1 (θ) = q/ T 1 (θ), h (θ) = q/ T (θ) (78) which will give local Nusselt numbers for isoflux bounary conition as follows: Nu 1 (θ) = 3 3 (λ 1 + 1) f (θ) g(θ) Re 1 45θ λ (79) 1 Nu (θ) = 3 3 f (θ) g(θ) Re 1 45 f 4 (θ) λ (8) Following the same proceure for the average heat transfer coefficient as escribe earlier, one obtains the average Nusselt number for an isoflux cyliner as Fig. 4 Effect of blockage ratio on velocity istribution. Nu isoflux = [ exp( 1.54b.77 )]Re 1 (81) Combining the results for both thermal bounary conitions, we have Nu Re 1 = { exp(.65b.5 ) for UWT exp( 1.54b.77 ) for UWF (8) Fig. 5 ratio. istribution of imensionless shear stress for given blockage

7 KHAN, CULHAM, AN YOVANOVICH 41 Fig. 8 rag coefficient as a function of Reynols number Re for ifferent blockage ratios. Fig. 6 Effect of blockage ratio on angle of separation. Fig. 9 Local Nusselt number for two thermal bounary conitions. Fig. 7 Effect of blockage ratio an Reynols number on rag coefficient. ratio. The variation of the rag coefficient C with Reynols number Re for ifferent blockage ratios is shown in Fig. 8. In this case, the rag coefficient ecreases with the increase in Reynols number Re for a specific blockage ratio. The present results are compare with the experimental results of Wieselsberger 6 for infinite flow conitions. They are in goo agreement except at Re = 1 3, where a ownwar eviation (3.75%) in the experimental results was notice. No physical explanation coul be foun in the literature for this eviation. Heat Transfer Characteristics The comparison of local Nusselt numbers for the isothermal an isoflux bounary conitions for a given blockage ratio is shown in Fig. 9. The isoflux bounary conition gives a higher heat transfer coefficient over the larger part of the circumference. On the front part of the cyliner (up to θ 4 eg), there is no appreciable effect of bounary conition. Higher heat transfer coefficients have also been observe experimentally by Perkins an Leppert 18 for a blockage ratio of.41 with the isoflux bounary conition. The results for average heat transfer from a single isoflux cyliner are shown in Fig. 1 for a given blockage ratio an Prantl number, where they are compare with the experimental an numerical ata Fig. 1 Average Nusselt number for isoflux cyliner. of Hattori an Takahashi an Yamamoto an Hattori. 3 It is clear that the present results are in very goo agreement with the previous work for a given range of Reynols numbers. It is clear from Fig. 11 that the heat transfer values are higher for the smaller blockage ratios, they ecrease as the blockage ratio ecreases, an finally they approach to the values for an infinite cyliner. The heat transfer values increase with the Reynols number. The effects of the blockage ratios on the heat transfer

8 4 KHAN, CULHAM, AN YOVANOVICH parameter Nu /Re 1/ Pr1/3 for the two thermal bounary conitions are shown in Fig. 1. Figure 1 shows that the heat transfer rates are higher for the isoflux bounary conition. Heat transfer rates, for both thermal bounary conitions, ecrease first with the blockage ratio an then become constant. The average Nusselt numbers for the isothermal cyliner for a given blockage ratio are compare in Fig. 13 with the experimental results of Niggeschmit 7 (reporte by Hausen 8 ) an Hausen. 8 The average Nusselt number Nu values are foun to be in a goo agreement with both results. However, both previous results are foun to be higher at high Reynols number ue to freestream turbulence. Fig. 11 Effect of blockage ratio an Reynols number on average Nusselt numbers. Fig. 1 Effect of blockage ratio an thermal bounary conition on heat transfer. Fig. 13 Average Nusselt number for isothermal bounary conition. Summary The influence of blockage ratio on the flui flow an heat transfer from a circular cyliner, place between two parallel planes, has been investigate. The metho of images is use to obtain the velocity istribution outsie the bounary layer. Three correlations are obtaine, Eq. (53) for total rag coefficient, Eq. (7) for heat transfer from an isothermal cyliner, an Eq. (81) for heat transfer from a cyliner uner the isoflux bounary conition. These correlations can be use to etermine the rag coefficient an the imensionless heat transfer coefficient from a cyliner confine in a channel with ifferent blockage ratios. The present results inicate goo agreement with the experimental/numerical results for a wie range of blockage ratio, Reynols numbers, an Prantl numbers. Acknowlegments The authors gratefully acknowlege the financial support of Natural Sciences an Engineering Research Council of Canaa an the Centre for Microelectronics Assembly an Packaging. References 1 Žukauskas, A., Avances in Heat Transfer, Acaemic Press, New York, 197, pp Khan, W. A., Culham, J. R., an Yovanovich, M. M., Flui Flow an Heat Transfer from a Pin-Fin: Analytical Approach, AIAA Paper 3-163, Jan Roshko, A., Experiments on the Flow Past Circular Cyliners at Very High Reynols Number, Journal of Flui Mechanics, Vol. 1, No. 3, 1961, pp Achenbach, E., Total an Local Heat Transfer From a Smooth Circular Cyliner in Cross Flow at High Reynols Number, International Journal of Heat an Mass Transfer, Vol. 18, No. 1, 1975, pp Schlichting, H., Bounary Layer Theory, 7th e., McGraw Hill, New York, 1979, Chap Wieselsberger, C., New ata on The Laws of Flui Resistance, NACA TN 84, March Churchill, S. W., Viscous Flows: The Practical Use of Theory, Butterworths Ser. in Chemical Engineering, Butterworths, Boston, 1988, pp Sucker,., an Brauer, H., Investigation of the Flow Aroun Transverse Cyliners, Wärme- un Stoffübertragung, Vol. 8, No. 3, 1975, pp Žukauskas, A., an Žiugža, J., Heat Transfer of a Cyliner in Crossflow, Hemisphere, New York, Eckert, E. R. G., an Soehngen, E., istribution of Heat-Transfer Coefficients Aroun Circular Cyliners in Cross Flow at Reynols Numbers from to 5, Transactions of the ASME, Vol. 74, April 195, pp Churchill, S. W., an Bernstein, M., A Correlating Equation for Force Convection from Gases an Liquis to a Circular Cyliner in Cross Flow, Journal of Heat Transfer, Vol. 99, No., 1977, pp Morgan, V. T., The Overall Convective Heat Transfer from Smooth Circular Cyliners, Avances in Heat Transfer, Vol. 11, Acaemic Press, New York, 1975, pp Hilpert, R., Ẅ armeabgabe von geheizten rahten un Rohren, Forsch. Geb. Ingenieurwes, Vol. 4, 1933, pp Refai-Ahme, G., an Yovanovich, M. M., Analytical Metho for Force Convection from Flat Plates, Circular Cyliners, an Spheres, Journal of Thermophysics an Heat Transfer, Vol. 9, No. 3, 1995, pp Krall, K. M., an Eckert, E. R. G., Heat Transfer to a Transverse Circular Cyliner at Low Reynols Number Incluing Refraction Effects, Heat Transfer, Vol. 3, 197, pp. 5 3.

9 KHAN, CULHAM, AN YOVANOVICH Giet, W. H., Investigation of Variation of Point Unit Heat-Transfer Coefficient Aroun a Cyliner Normal to an Air Stream, Transactions of the ASME, Vol. 71, May 1949, pp Sarma, T. S., an Sukhatme, S. P., Local Heat Transfer from a Horizontal Cyliner to Air in Cross Flow: Influence of Free Convection an Free Stream Turbulence, International Journal of Heat an Mass Transfer, Vol., No. 1, 1977, pp Perkins, H. C., an Leppert, G., Local Heat Transfer Coefficients on a Uniformly Heate Cyliner, International Journal of Heat an Mass Transfer, Vol. 7, No., 1964, pp Akilba yev, Z. S., Isata yev, S. I., Krashtalev, P. A., an Masle yeva, N. V., The Effect of Channel Blockage on the Local Heat Transfer Coefficient of a Uniformly Heate Cyliner, Problemy Teploenergetyki i Priklanoi Teplofiziki, Vol. 3, 1966, pp Merk, H. J., Rapi Calculations for Bounary Layer Transfer Using Wege Solutions an Asymptotic Expansions, Journal of Flui Mechanics, Vol. 5, 1959, pp Vaitiekünas, P. P., Bulota, A. J., an J. J., Analysis of the Effect of uct Blocking on Crossflow an Heat Transfer of a Cyliner, Heat Transfer Soviet Research, Vol. 17, No. 4, 1985, pp Hattori, N., an Takahashi, T., Heat Transfer from a Single Row of Circular Cyliners Place in the Transverse irection of Water Flow, Transactions of the Japan Society of Mechanical Engineers, Pt. B, Vol. 59, No. 568, 1993, pp Yamamoto, H., an Hattori, N., Flow an Heat Transfer Aroun a Single Row of Circular Cyliners, Heat Transfer Japanese Research, Vol. 5, No. 3, 1996, pp Pohlhausen, K., Zur Näherungsweise Integration er ifferential Gleichung er Laminaren Reibungschicht, Zeitschrift für angewante Mathematic un Mechanic, Vol. 1, 191, pp Kochin, N. E., Kibel, I. A., an Roze, N. V., Theoretical Hyromechanics (translate from 5th Russian eition), Interscience, New York, 1964, Chap Walz, A., Ein neuer Ansatz für as Greschwinligkeitsprofil er laminaren Reibungsschicht, Lilienthal-Bericht, Vol. 141, 1941, p Niggeschmit, W., ruckverlust un Warmeubergang bei fluchtenen, versetzten un teilversetzten querangestromten Rohrbuneln, issertation, armstat, Germany, Hausen, H., Heat Transfer in Counterflow, Parallel Flow an Cross Flow, McGraw Hill, New York, 1983, Chap..