Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 32
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The spreading resistance is based on thearea average temperature. Steady state is obtained when Fo ---) oo and the solution goes to4kaRs = 32/(3r~2).CONDUCTION AND THERMALCONTACTRESISTANCES(CONDUCTANCES)3.49Beck gave approximate solutions for short and long times. For short times (Fo < 0.6),4kaRs =--rtS[~F°rt -~+F°71;~FO2 + ~FO3 +15F°4]5i2,t(3.158)and for long times (Fo > 0.6),4kaRs322 [= 3re---f - r t 3 / 2 ~111 ]1 - 3(4Fo) ~ 6(4Fo) 2 - ------------S12(4Fo)(3.159)The maximum errors of about 0.18 percent and 0.07 percent occur at Fo = 0.6 for the shortand long-time expressions, respectively.Transient Spreading Resistance of Isoflux HyperellipseContact Area on Half-SpaceThe hyperellipse is defined by (x/a)" + (y/b)" = 1 with b < a, where n is the shape parameterand a and b are the axes along the x and y axes, respectively, as shown in Fig.
3.14. The hyperellipse reduces to many special cases by setting the values of n and the aspect ratio parameter 7 = b/a, which lies in the range 0 < y _< 1. Therefore the solution developed for thehyperellipse can be used to obtain solutions for many other geometries, such as ellipses andcircles, rectangles and squares, diamondlike geometries, etc. The transient dimensionlesscentroid constriction resistance kV~Ro where R0 = To/Q is given by the double-integralsolution [152]:gV'Aerfc2V/-~~Ov ~dr din(3.160)where the area of the hyperellipse is given by A = (4y/n)B(1 + 1/n, 1/n) and B(x, y) is the betafunction. The dimensionless time is defined as Fov~ = o~t/A.
The upper limit of the radius isgiven by r0 = 7/[(sin m)~ + 7~(cos m)']~" and the aspect ratio parameter 7 = b/a. The precedingsolution has the following characteristics: (1) for small dimensionless times, Fov~ __ 4 x10-2 and kX/-ARo = (2/V'~)~/-F-ov~ for all values of n and y; (2) for long dimensionless timesFo,,/-x _>103. The results are within 1 percent of the steady-state values, which are given by thesingle integral:27kV"-ARo = rtV~£~,2dm[(sin (It))n + 7n(cos (o)n] lln(3.161)which depends on the aspect ratio T and the shape parameter n. The dimensionless spreading resistance depends on the three parameters Fov~, 7, and n in the transition region: 4 x10-2 ___Fox/-x _< 103 in some complicated manner that can be deduced from the solution forthe circular area.
For this axisymmetric shape we put 7 = 1, n = 2 into the hyperellipse double integral, which yields the following closed-form result valid for all dimensionless time[152]:[ 111(1)]kV/-ARo = x/-F--ov7 v ~ - v ~ exp(-1/(4rt Fov~)) + 2v~x/-F--Ov~ erfc 2v~x/-F-ov7(3.162)where the dimensionless time for the circle of radius a is Fov~ = o~//(7ca2).3.50CHAPTER T H R E ETransient Spreading Resistance of Isoflux Regular PolygonalContact Area on Half-SpaceFor regular polygons having sides N > 3 as depicted in Fig.
3.14, the area is A = Nraitan rr/N,where ri is the radius of the inscribed circle. The dimensionless constriction resistance basedon the centroid temperature kV~Ro is given by the following double integral [152]:kV~Ro=2~/Ntan(r~/U~fortlNfo1,....erfc(r)2V'N tan (r~N) ~/-F--ov~dr do)(3.163)where the polygonal area is expressed in terms of the number of sides N and for conveniencethe inscribed radius has been set to unity. This double-integral solution has identical characteristics to those of the double-integral solution given above for the hyperellipse, i.e., for smalldimensionless time, Fov~ < 4 × 10-1 and kX/--ARo= (2/V/-~)V'FO~A for all polygons N > 3; forlong dimensionless time, Fov~ > 103. The results are within 1 percent of the steady-state values given by the following closed-form expression [152]:kV,-~R0= 1 ~N[ 1 + sin (rr/N) ]~- tan (n/N) Incos (n/N)(3.164)The dimensionless spreading resistance kX/-ARo depends on the parameters Fov~ and N inthe transition region 4 × 10 -2 <_ F o v ~ <_ 10 3 in some complicated manner that, as describedabove, can be deduced from the solution for the circular area.
The steady-state solution yields~0 values of 0.5617, 0.5611, and 0.5642 for the equilateral triangle (N = 3), the square (N = 4),and the circle (N ~ oo). The difference between the values for the triangle and the circle isapproximately 2.2 percent, whereas the difference between the values for the square and thecircle is less than 0.6 percent. The following procedure is proposed for computation of thecentroid-based transient spreading resistance for the range 4 x 10 -2 _< Fov~ _<10 6.
The closedform solution for the circle is the basis of the proposed method. For any planar, singly connected contact area subjected to a uniform heat flux take, ~ 0 - kX/ARo.~ff0[1(~0(Fov~ ~ oo) = 2x/-F--ov~ 1 - exp(-1/(4rt Fov~)) + 2x/-F--o-------~Aerfc1)]2X/-~V ~ v ~(3.165)The right side of the above equation can be considered to be a universal dimensionless timefunction that accounts for the transition from small times to near steady state. The proposedprocedure should provide quite accurate results for any planar, singly connected area.
A simpler expression based on the Greene approximation [25] of the complementary error functionis proposed [152]:~I/01v0(Fov~ ~ oo)= zN/-~ [1-e-Z2+alV~ze -~2(z+13)2](3.166)where z = 1/(2~/-~v'-F--ov~) and the three correlation coefficients are al = 1.5577, a2 = 0.7182,0.7856. This approximation provides values of ~0 with maximum errors of less than 0.5percent for Fov~ > 4 × 10-2.a3 =Transient Spreading Resistance Within Semi-InfiniteFlux Tubes and ChannelsIsoflux Circular Contact Area on Circular Flux Tube.
Turyk and Yovanovich [118] reported the analytical solutions for transient spreading resistance within semi-infinite circularCONDUCTIONAND THERMALCONTACT RESISTANCES (CONDUCTANCES)3.51flux tubes and two-dimensional channels. The circular contact and the rectangular strip aresubjected to uniform and constant heat fluxes. The dimensionless spreading resistance for theflux tube is given by the series solution16 1 ~ J2(6.e) erf (~5.ex/-F-do)4kaRs - ~ e .=~~53J~(8,)(3.167)where e = a/b < 1, Fo - o~t]a2, and 5n are the roots of Jl(X) = 0. The series solution approachesthe steady-state solution presented in an earlier section when the dimensionless time satisfiesthe criterion Fo > 1/62 or when the real time satisfies the criterion t > a2/(o~2).Isoflux Strip on Two-Dimensional ChannelThe dimensionless spreading resistance withina two-dimensional channel of width 2b and thermal conductivity k was reported askRs = 1_~ ~~36 m = 1sin 2 (mne) erf (mneV~o)(3.168)m3where e = a/b < 1 is the relative size of the contact strip, and the dimensionless time is definedas Fo = t~t/a2.
There is no half-space solution for the two-dimensional channel. The transientsolution is within 1 percent of the steady-state solution when the dimensionless time satisfiesthe criterion Fo _>1.46/~2.CONTACT, GAP, AND JOINT RESISTANCESAND CONTACT CONDUCTANCESPoint and Line Contact ModelsThe thermal resistance models for steady-state conduction through contact areas and gapsformed when nonconforming smooth surfaces are placed in contact are based on the Hertzelastic deformation model [37, 41,117, 121] and the thermal spreading (constriction) resultspresented previously. The general elastoconstriction models for point and line contacts arereviewed by Yovanovich [143].
In the general case the contact area is elliptical and its dimensions are much smaller than the dimensions of the contacting bodies. The gap that is formedis a function of the shape of the contacting bodies, and in general the local gap thickness isdescribed by complex integrals and special functions calledelliptical integrals [8, 10]. Two important special cases areIconsidered in the following sections: sphere-fiat and circularcylinder-flat contacts. The review of Yovanovich [143] canSpherebe consulted for the general case.elGap£2Io IElastoconstriction Resistance of Sphere-Flat Contacts.The contact resistance of the sphere-fiat contact shown inFig.
3.23 is discussed in this section. The thermal conductivities of the sphere and flux tube a r e k l and k2, respectively.The total contact resistance is the sum of the constrictionresistance in the sphere and the spreading resistance withinthe flux tube. The contact radius a is much smaller than thesphere diameter D and the tube diameter. Assuming isothermal contact area, the general elastoconstriction resistancemodel [143] becomes:IFIGURE 3.23 Sphere-flatcontact with gap.Rc -12ksa(3.169)3.52CHAPTER THREEwhere ks = 2klk2/(kl + k2) is the harmonic mean thermal conductivity of the contact, and thecontact radius is obtained from the Hertz elastic model [117]:2a [3FA] '/3O-[Dz ](3.170)where F is the mechanical load at the contact, and the physical parameter is defined as1E(3.171)(1 - v 2) +A = ~-E1E2where Vl and v2 are the Poisson's ratio and E1 and E2 are the elastic modulus of the sphere andflat contacts, respectively.Elastoconstriction Resistance of Cylinder-Flat Contacts.The thermal contact resistancemodel for the contact formed by a smooth circular cylinder of diameter D and thermal conductivity kl, and a smooth flat of thermal conductivity k2, was reported by McGee et al.
[63]to beksRc- 2n kl In- 2k---~--2---n- k--TIn (4nF*)(3.172)where ks is the harmonic mean thermal conductivity and A is the Hertz physical parameterdefined above. The dimensionless mechanical load is defined as F* - FA/(DL), where Fis thetotal load at the contact strip and L is the length of the cylinder and the flat.Gap Resistance Model of Sphere-Flat Contacts. The general elastogap resistance modelfor point contacts [143] reduces for the sphere-flat contact to1(o)R g - --L kg.olg.p(3.173)where L = D/(2a) is the relative contact size defined previously.
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