Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 30
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The preceding general solution with g = 0 yields the isofluxsolution reported by Mikic and Rohsenow [65]:4kaRs16 1 ~ ,J](8,~)(3.124)An accurate correlation equation of this series solution is given later.Parabolic Flux Distribution: IX = ½. Yovanovich [131-133] reported the solution for theparabolic flux distribution corresponding to gt = 1/2.4kaRs-- 2__441 2~l~ e n=lJl(~n¢)3sin2 (8,e)8,,J0(8,){1(8,~) z18,e}(3.125)t a n (SnE)Effect of Flux Distribution on Circular Contact Area on Half-SpaceThe three series solutions just given converge very slowly as e --->0, which corresponds to thecase of a circular contact area on a half-space. The corresponding half-space results are givenby Strong et al. [112]: 4kaRs (g =-½) = 1, 4kaRs (g = 0) = 32/(3rt2), 4kaR, (g = ½) = 1.1252.3.40CHAPTER THREESimple Correlation Equations of Spreading Resistancefor Circular Contact AreaYovanovich [131-133] r e c o m m e n d e d the following simple correlations for the three flux distributions:4kaRs = al(1 - a2~)(3.126)in the range 0 < ~ < 0.1 with a m a x i m u m error of 0.1 percent and4kaRs = a~(1 - ~.)a3(3.127)in the range 0 < ~ < 0.3 with a m a x i m u m error of 1 percent.
The correlation coefficients for thethree flux distributions are given in Table 3.14.TABLE 3.14 Correlation Coefficientsfor l.t = -1/2, 0, 1/2~.--1/201~ala211.41971.501.08081.41111.351.12521.40981.30a3Accurate Correlation Equations for Various Combinations of Contact Area,Flux Tubes, and Boundary ConditionGeneral Accurate Correlation Equation. Solutions are also available for various combinations of contact areas and flux tubes such as circle/circle and circle/square for the uniformflux, true isothermal, and equivalent isothermal b o u n d a r y conditions [71].Numerical results were correlated using the polynomial4kaRs = Co + C1~ + C3¢3 + C5¢5 + C7E7(3.128)The dimensionless spreading (constriction) resistance coefficient Co is the half-spacevalue, and the correlation coefficients C1 through C7 are given in Table 3.15.General Approximate Correlation Equation for Applications.
For microelectronic applications, an accurate engineering approximation [74] that is valid for a circular contact on a circular or square flux tube or a square contact on a square flux tube iskV~cRs = 0.475 - 0.62~ + 0.13~ 3(3.129)where ~ = V/Ac/At, and Ao At are the contact and flux tube areas, respectively. The m a x i m u merror with respect to the exact solution is less than 2 percent for 0 < ~ <_0.5 and less than 4 percent for 0 < ~ _<0.7.TABLE 3.15Coefficients for Correlations of Dimensionless Spreading Resistance 4kaRsFlux tube geometry andcontact boundary conditionCircle/circle, uniform fluxCircle/circle, true isothermalCircle/square, uniform fluxCircle/square, equivalentisothermal fluxCoC1C36"5C71.080761.000001.08076-1.41042-1.40978-1.241100.266040.344060.18210-0.000160.043050.008250.0582660.022710.0389161.00000-1.241420.209880.027150.02768CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.41Square Contact Area on Square Flux Tube.Mikic and Rohsenow [65] reported the solutionfor an isoflux square contact area on the end of a semi-infinite square flux tube.
The solutionwas recast [74] to give the dimensionless spreading resistance:k ~ c R s = ~3---~2[mZ=1 sin2" ~-S(mTt¢) + _~-L31¢--27' _.2nm=:] 1,~= sin2=m2n2%/m(m/re) sin22+(n/re)n2(3.130)Circular Contact Area on Square Flux Tube.
Sadhal [155] reported the general solution foran isoflux or equivalent isothermal elliptical contact area on a rectangular flux tube. His general solution gives the dimensionless spreading resistance for an isoflux square contact areaon a circular flux tube that has the formkX/~Rs- 2g3---~ ,,=j2(2nV'-~) + ~J](2V'-~V"m 2 + n 2)n3m=, ,=(m 2 + n2)3/2(3.131)with the relative size para _meter defined as • = V/A~/A, = (V/-~/2)a/b. The dimensionless isofluxspreading resistance kVAcR, has the half-space values 0.47890 and 0.47320 for the circularand square contact areas, respectively, as • ~ 0.
Negus et al. [74] reported accurate correlations for the circle/circle, circle/square, and square/square combinations corresponding to thepreceding series solutions given by Eq. 3.128.General Spreading Resistance Expression for Circular Annular Areaon Circular Flux TubeGeneral Expression for Arbitrary Flux. The spreading resistance of a circular annulus ofinner and outer radii a and b, respectively, on one end of a semi-infinite circular flux tube ofradius c and thermal conductivity k is considered here.The general expression for the dimensionless spreading resistance 4kbR, for a circularannular contact subjected to an arbitrary axisymmetric flux distribution f(u) is given by [131]4kbR,=(1 -e2)(8/~)~,uf(u) du2 Jl(~n){1-~'[Jl(~'~n)]/[Jl(~n)]} I 1n=l~2J2(~n)uf(U)Jo(~nU) du (3.132)Spreading Resistance of Isoflux Annular Contact on Circular Flux Tube.
The previousgeneral expression reduces for the isoflux case that corresponds to f(u) = 1 to the followingseries [128, 131-133]:_4kbRs-~(116e2)2J] ([38,)3 21- •(3.133)with the parameters 0 < [3 = b/c < 1 and 0 < e < 1. The eigenvalues 8n are the roots of Jl(') = 0.This solution reduces to the solution for the isoflux circular contact area on the end of a circular flux tube when [3= 0. For small values of 13and 0 < e < 1 one can use the closed-form solution reported previously for a circular annular contact on a half-space.Spreading Resistance Within Two-Dimensional ChannelsGeneral Expression for Arbitrary Flux.
The steady-state spreading resistance due to conduction through a strip of width 2a on one end of an infinitely long two-dimensional channelof width 2b and thermal conductivity k (Fig. 3.16) has solutions reported by Smythe [98],Mikic and Rohsenow [65], Veziroglu and Chandra [119], and Sexl and Burkhard [90]. Solu-3.42CHAPTERTHREE//tions have been obtained for the isothermal strip, the isofluxstrip, and a general solution developed for arbitrary flux distribution over the strip. Yovanovich [130] reported the following general solution:////////////kR=-;11~2fo f(u) c l uS/,,2b(1) ~sin (nrte) ~~n2f(u) COS (nrceu) du1°--S,//k,-/(3.134)where f(u) with 0 < u = x/a < 1 represents the arbitrary fluxdistribution over the contact strip, and e = a/b < 1 is the relative size of the contact strip.//General Expression f o r Flux Distribution of the Form,FIGURE 3.16 Two-dimensionalflux channel.1 ((1 - uZ)~.
Yovanovich [130] chose the general flux functionf(u) = (1 - u2y with parameter ~, which gives (1) the isofluxcontact when kt = 0, (2) the equivalent isothermal strip whenkt = - ½ , and (3) the parabolic flux distribution when kt = 1/5 todevelop another general solution:3)(1)~sin(nrre)[~]kR= = -~ V kt + -~,,=n2.+¢1/2)J~,+(1/2)(n/I;~)(3.135)Setting ~ = -1/2 and kt = 0 in the above general solution gives the two solutions reported byMikic and Rohsenow [65].Equivalent Isothermal Contact Area~" Ix = -~k.For ~t = -1/21 ( 1 ) ~ 1 sin (nr~e)Jo(n~e)kR= = --~,, =n-------------5~Isoflux Contact A r e s IX= 0.(3.136)For la = 01 (1)2 ~1 sin 2 (nrre)kR= = --~.=rt3(3.137)Parabolic Flux Distribution: IX= ~&.
The parabolic flux distribution kt = 1/2 gives2 /l~2x-~ sin (nr~e)kR= = --~ ~ } 3_~,,__ n S ~ Jl(n/I;e)(3.138)True Isothermal Contact A r e ~ Veziroglu and Chandra [119] used a conformal mappingtechnique to obtain the closed-form solution for the true isothermal strip:kRslln{1r~sin [(rd2)e](3.139)The true isothermal strip solution and the equivalent isothermal flux solution predict valuesof the dimensionless spreading resistance that are in close agreement provided e < 0.4.
Theparabolic flux distribution gives the greatest values of the spreading resistance, followed bythe isoflux values, which are greater than the values for the isothermal strip.CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.43Effect of Area Change.Smythe [98] reported the analytical solution for the spreading resistance for the case of twocoaxial channels of widths 2a and 2b where a < b are in perfect contact of width 2a (Fig. 3.17)"/kR, = ~2ar/Z///////__~+In1-e+ 2 In(3.140)4eEffect of Single and Multiple Layers (Coatings)on Spreading Resistance/i///////2b[The influence of single and multiple layers of different thermal conductivities and thicknesses is of great interest. Thesolutions given below can be used to determine the effect ofoxide layers and coatings on the spreading resistance. Solutions are available for semi-infinite flux tubes and finite systems such as circular disks.=_iSpreading Resistance Within Compound Disks With Conductance. The compound disk is shown in Fig.
3.18. The~~-disk consists of two isotropic materials of thicknesses h, t2and thermal conductivities kl, k2 that are in perfect contact.The radius of the compound disk is denoted b and its thickness is denoted t = tl + t2. The lateral boundary r = b is adiabatic; the face at z = t is either cooled by a fluid through the film conductance h or in contactwith a heat sink through a contact conductance h. In either case h is assumed to be uniform.The face at z - 0 consists of the heat source area of radius a, and the remainder of that facea < r < b is adiabatic.
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