Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 29
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The centroid temperature is2qa[1To = rt --k sinh-1 --• + • sinh-1 •1(3.110)where the aspect ratio parameter is • = a/b > 1. The area-average temperature is-~=2 qa(1 1•[1 (1~3/2]1--~ sinh -1 --• + --• sinh -~ • + ~ 1 + -~- - 1 +-~]jj(3.111)The two expressions for the rectangular area give the results kaRs = 0.2806 and kaR, = 0.2366for the spreading resistance of a square contact area based on the centroid and area-averagetemperatures, respectively. The ratio of centroid temperature to area-average temperature isTo/T = 1.1857.
The differences between the results for the circle and those for the square arevery small when the spreading resistances are based on ~ = V/-A. The dimensionless spreading resistances become kVARo = 0.5611 and kX/-ARo = 0.5642 for the square and circle,respectively, and kX/AR = 0.4728 and k ~= 0.4787 for the square and circle, respectively,where R is based on the area average temperature.Regular Polygonal Areas. The equilateral triangle, square, pentagon, and hexagon and thecircle are members of the regular polygonal contact family as shown in Fig.
3.14. The dimensionless spreading resistance based on the centroid temperature is obtained from the integral[132].kX/-~Ro= l ~nNt a n (n/N) In[l+sin(n/N)]cos (n/N) ,N >3(3.112)The above expression gives kX/ARo = 0.5516 for the triangle (N = 3), which is approximately2.3 percent smaller than the value for the circle where (N = oo). The corresponding~aluefor the area-average basis was reported by Yovanovich and Burde [134] to be kVA-R =0.4600, which is approximately 4 percent smaller than the value for the circle.3.36CHAPTERTHREE( )N=3TriangleN=6HexagonN-5PentagonN=4SquareN ---~ o oCircleY_[bTY~"0Semi-CircleRectangleEllipsea_>ba_bHyper-Ellipsea>b(x}n+0<_n< ooFIGURE 3.14 Singlyconnected planar areas.H y p e r e l l i p s e C o n t a c t Areas.
The family of contact areas defined by the hyperellipse(x/a) n + (y/b)" = 1 with b _<a is shown in Fig. 3.14. Yovanovich et al. [137] used a numericaltechnique to determine the spreading resistances for several values of the shape parametern = 0.5, 1, 2, 4, oo and for a range of the aspect ratio 0.1 _<tx = b/a <_1.The centroid temperature-based spreading resistance is obtained from the integral [137]kX/'-ARo= 1 ~? Son[sin"(3.113)( 1 ) " } - a n COS n (.0] TMwhere {x - b/a is the aspect ratio parameter and B is the beta function [1], which depends onthe shape parameter n:.
.(1+1nN o n s y m m e t r i c C o n t a c t Areas. Yovanovich and Burde [134] reported centroidal and areaaverage spreading resistances of several nonsymmetric contact areas such as semicircles,isosceles triangles of different aspect ratios, and squares with corners removed. For these contact areas Yovanovich and Burde Leportedrthat all numerical results lie in the ranges 0.4424 <kX/~ R < 0.4733 and 0.5197 < k V A R o < 0.5614, and that -T/To = 0.84 + 1.7 percent.Circular Annular Contact Areas on Half-SpaceIsoflux Annulus.Yovanovich and Schneider [135] reported the temperature distributionwithin a circular annular contact a < r < b subject to a uniform flux q:T(r)=2qb_r E a+_a 2K a(3.114)CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.37where k is the thermal conductivity of the half-space.
The special functions that appear in thepreceding expression, K(x) and E(x), are complete elliptical integrals of the first and secondkinds, respectively [1, 10]. The heat sink t e m p e r a t u r e was set to zero for convenience. Thethermal spreading resistance is based on the area-average temperature:1 a2 ) f: T(r)2rtr drT= rt(b 2 _The dimensionless thermal spreading resistance with e = a/b < 1 is given by3214kbR,- 3rc2 (1 - e2)2 [1 + e 3 - (1 + e2)E(e) + (1 - e2)K(e)](3.115)The above result reduces to the case of a circular contact area e = 0 on a half-space where4kbR, = 32/(31t2).Isothermal Annulus.Smythe [95] obtained the solution for the capacitance of a thin circular annulus.
The solutions were recast [135] into the following dimensionless thermal spreading resistance expressions:4 [In 16 + In [(1 + e)/(1 - e)]](1 + e)4kbR, = ~t](3.116)for 1.000 < 1/e < 1.10, andrff2_4kbR,- [cos_l e "~"V'I -- e 2 tanh -~ e][1 + 0.0143e -1 tan 3 (1.28e)](3.117)for 1.1 < l i t < oo.Doubly Connected Isoflux Contact Areas on Half-SpaceThe numerical data of Martin [154] for the dimensionless thermal spreading resistancekX/-Ac R, for three doubly connected regular polygons (Fig.
3.15)---equilateral triangle,square, and circle--are functions of e = X/Ai/Ao, where Ai and Ao are the inner and outer projected areas of the polygons (Fig. 3.15). The contact area is Ac = Ao - Ai. Accurate correlationequations with a maximum relative error of 0.6 percent were given for the range 0 _<e < 0.995:\all JAo! Jr--2bFIGURE 3.15 Doubly connected regular polygonal areas.2b3.38CHAPTER THREEand for the range 0.995 _<• _<0.9999:kPoRs = a5 In [ (l&)a4- 1 ](3.119)where Po is the outer perimeter of the polygons, and the correlation coefficients a0 through a5are given in Table 3.12.Coefficients for DoublyConnected PolygonsTABLE 3.12a0a~a2a3a4a5CircleSquareTriangle0.47890.999571.50560.3593139.660.316040.47320.999801.51500.3730268.590.315380.46021.000101.51010.38637115.910.31529The correlation coefficient a0 represents the dimensionless spreading resistance of the fullcontact area, in agreement with results presented previously.
Since the results for the squareand the circle are very close for all values of the parameter •, the correlation equations for thesquare or the circle may be used for other doubly connected regular polygons such as pentagons, hexagons, etc.Effect of Contact Conductance on Spreading ResistanceMartin et al. [61] used a numerical technique to determine the effect of a uniform contact conductance h on the spreading resistance of square and circular contact areas. The dimensionless spreading resistance values were correlated with an accuracy of 0.1 percent by thefollowing expression:kX/ARs = cl -c2tanh (c3 In Bi- c4) ,0< Bi < oo(3.120)with Bi = hV/-A/k.
The correlation coefficients cl through Ca are presented in Table 3.13.TABLE 3.13clc2c3CaCoefficients for Square and CircleCircleSquare0.461590.0174990.439001.16240.457330.0164630.470351.1311When Bi _<0.1, the values predicted by the expression approach the values correspondingto the isoflux boundary condition; and when Bi >_100, the predicted values are within 0.1 percent of the values obtained for the isothermal boundary condition. The transition from theisoflux values to the isothermal values occurs in the range 0.1 _<Bi __ 100.CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.39Spreading Resistance in Flux Tubes and ChannelsGeneral Expression for Circular Contact Area With Arbitrary Flux on Circular Flux Tube.The general expression for the dimensionless spreading (constriction) resistance 4kaRs for acircular contact subjected to an arbitrary axisymmetric flux distribution f(u) [131-133] isobtained from the series4kaRs =(8/rt)J~(8,,~)foI uf(u) au "= ~.Jo(5.)2 2fo'uf(u)Jo(Sn~-U)du(3.121)where 8.
are the roots of Jl(') = 0 and e = a/b is the relative size of the contact area.General Expression for Flux Distributions of the Form (1 - n2)". Yovanovich [131-133]reported the following general solution for axisymmetric flux distributions of the form f(u) =(1 - u2)", where the parameter gt accounts for the shape of the flux distribution.
The abovegeneral expression reduces to the following general expression:)J~ + l(~n E)4kaR, :--~16 (bt + 1)2ur(g + 1) 1' . -2~ Jl~( ~)neJ0~,)(~,~(3.122)where F(x) is the gamma function [1] and Jv(x) is the Bessel function of arbitrary order v.The above general expression can be used to obtain specific solutions for various values ofthe flux distribution parameter g. Three particular solutions are considered next.Equivalent Isothermal Contact Area; IX = - ½ . The isothermal contact area requires thesolution of a difficult mathematical problem that has received much attention from numerousresearchers [24, 42, 65, 70, 71, 84, 132, 133].Mikic and Rohsenow [65] proposed the use of the flux distribution corresponding tog = - ½ to approximate an isothermal contact area for small relative contact areas: 0 < ¢ < 0.5.The general expression becomes4kaRs =----81 ~1if, ~..=Jl(~ne)3sin2 (8,~)(3.123)~)nJo(~)n)An accurate correlation equation of this series solution is given later.Isoflux Contact Area: Ix = 0.
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