Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 31
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The boundary condition over the contact area can be modeled as (1) uniform heat flux or (2) isothermal. The complete solution for these two boundary conditionshas been reported by Yovanovich et al. [138]. The general solution for the dimensionless constriction p a r a m e t e r 4klaRc depends on several dimensionless geometric and thermophysicalparameters: x = t/b, "r, 1 -- tl/b, "c2 = h/b, ~ =a/b, ~ = k~/k2, Bi = hb/k2, IX. The p a r a m e t e r Ix describesFIGURE 3.17 Two-dimensional flux channel witharea change.g=-1/2g=0qt = t~ + t 2Bi = h b/k~e=a/bx = t/bRc=4k~aRoq'Ifa.Ib..... I.- ID..rR ~ = 4k~aR~D~: = k~/k 2t1 R,D.
~-4{ ex,+Ke(x-x,)+--~ZFIGURE 3.18 Compound circular disk with conductance.t3.44CHAPTER THREEthe heat flux distribution over the contact area. When g = O, the heat flux is uniform, andwhen B =-1/2, this heat flux distribution is called the equivalent isothermal distributionbecause it produces an almost isothermal contact area provided a/b < 0.6.The general solution is given asJl(~n()4klaRs = 8(g + 1________2_~) A,,(n, e)B,,(n, 'r,,'1:1)5n----"~TC~.(3.141),,= lThe coefficients A, are functions of the heat flux parameter B.
They become, for p =-1,4:A,=and for g = 0:A.-2e sin 8,e82 j2(8,)=-2dl(5,e)2 25.Jo(8.)The function B,, is defined asB, =@. tanh (~n~l) -- q)n(3.142)1-Gand the two functions that appear in the above relationship are defined asK-1G =~cosh (8,xl)[cosh (8,'q) - % sinh (8,1h)]Kand8, + Bi tanh (8,x)% = 8, tanh (8,x) + Bi(3.143)(3.144)The eigenvalues 8, are the positive roots of Jl(-) = 0.Characteristics of %. This function accounts for the effects of the parameters 8n, "¢, andBi. For limiting values of the parameter Bi it reduces toand% = tanh (8,x)Bi ~% = coth (8,x)Bi --) 0For all 0 < Bi < oo and for all values x > 0.72, tanh (8,x) = 1 for all n _> 1.
Therefore (1)n= 1 forn>l.Characteristics of Bn. When x~ > 0.72, tanh (8,x) - 1, G - 1 for all 0 < Bi < 0% thereforeB, - 1 for n > 1. These characteristics lead to the previously discussed flux tube solutions.The general solution for the compound disk can be used to obtain the spreading resistances for the several cases shown in Figs. 3.19 and 3.20.Spreading Resistance Within Isotropic Finite Disks With Conductance.The dimensionlessconstriction resistance for isotropic (~: = 1) finite disks (Xl < 0.72) with negligible thermal resistance at the heat sink interface (Bi = oo) is given by the following solutions.For g = _1/~:4kaR, = 8 ~ Jl(8.e) sin (8.e)For pt = 0:4kaR, = - 16 ~ .
j2(8.e)3 2tanh (8.x)n¢.:~ 8,,Jo(8,,)n--~.:83 J20(5.)tanh (8.x)(3.145)(3.146)If the external resistance is negligible (Bi --) ¢~), the temperature at the lower face of the diskis assumed to be isothermal. The solutions for isoflux g = 0 heat source and isothermal basetemperature were given by Kennedy [42] for (1) the centroid temperature and (2) the areaaverage contact area temperature.,_a(a) 0 < 1 < < o o(c) e--> O, y--) oo, 0 < K < oo__.at___~ a,T=Oi(b) Bi ~ o o , ) , ~ oo, 0 < n < ooFIGURE3.19(d) e --> O, a ---> oo, 0 < ~: < ooS p e c i a l c a s e s o f t h e c o m p o u n d d i s k w i t h 1<, 1.aIi(c) ~:= 1,~,-~ oo(a) K: = 1!a=attftfiiT=0(b) ~¢ = 1, Bi ~ ooFIGURE3.20•m(d) e --> O, ~: = 1, y--) ooS p e c i a l c a s e s o f t h e c o m p o u n d d i s k w i t h 1< = 1.3.453.46CHAPTER THREECorrelation Equation for Spreading Resistance Within Finite Disk With Conductance.The solution for the isoflux boundary condition and with external thermal resistance wasrecently reexamined by Song et al.
[156] and Lee et al. [157]. These researchers nondimensionalized the constriction resistance based on the centroid and area-average temperaturesusing the square root of the contact area as recommended by Chow and Yovanovich [15] andYovanovich [132, 137, 144-146, 150], and compared the analytical results against the numerical results reported by Nelson and Sayers [158] over the full range of the independent parameters Bi, e, and x. Nelson and Sayers [158] also chose the square root of the contact area toreport their numerical results.
The analytical and numerical results were reported to be inexcellent agreement.Lee et al. [157] recommended a simple closed-form expression for the dimensionless constriction resistance based on the area-average and centroid temperatures. They defined thedimensionless spreading resistance parameter as ~ = X/nkaRc and recommended the following approximations.1/2(1-iE)3/2(DcFor the area-average temperature:I[/ave =For the centroid temperature:1Vmax= - ~(3.147)(1 - e)q~cwithBi tanh (ScX) + 5c~Pc= Bi + 8c tanh (ScX)and8c = n +(3.148)The above approximations are within +10 percent of the analytical results [156, 157] andthe numerical results [158]. They do not, however, indicate where the maximum errorsoccur.Circular Contact Area on Single Layer (Coating) on Half-SpaceEquivalent Isothermal Circular Contact.
Dryden [16] obtained the solution for an equivalent isothermal circular contact of radius a that is in perfect contact with an isotropic layer ofthermal conductivity kl and thickness tl that is also in perfect thermal contact with a substrateof thermal conductivity k2. This system is shown in Fig. 3.21.Dryden based his solution on the axisymmetric flux distribution:aQq(r) = 2ha(a2 - r2)mYTTYYwhere Q is the heat transfer rate through the contact area.The spreading resistance, based on the area-average temperature, is obtained from the integral:1 I~[~2 exp(~tl/a)+~l exp(-~tl/a)]Jl(~)sin~Rs = nklaL2 exp(~tl/a) - ~,1exp(-~tl/a)~2d~(3.149)F I G U R E 3.21 Circular contact area on single layer(coating) on half-space.with ~,~ = (1 - kz/kl)/2 and ~,2 = (1 + kz/kl)/2.
The parameteris a dummy variable of integration. The constriction resistance depends on the thermal conductivity ratio kl/k2 andCONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.47the relative layer thickness tl/a. Dryden gave simple asymptotes for the constrictions for thinlayers (h/a < 0.1) and thick layers (h/a > 10). These asymptotes were also presented as dimensionless spreading resistances defined as 4k2aR,. They are as follows.Thin-Layer Asymptotekl](4k2aR,)mi=n1 + \[41[t,l[k2~- ]\ a J L ~ -(3.150)Thick-Layer Asymptote(2)(4k2aRs)thick=-~l-- ~-)(~)(-~-1) In l+kJk2,311,These asymptotes provide results that are within 1 percent of the full solution for relativelayer thickness tl/a < 0.5 and tl/a > 2.The dimensionless constriction is based on the substrate thermal conductivity k2. The preceding general solution is valid for conductive layers where kl/k2 > 1 as well as for resistivelayers where kl/k2 < 1.
The infinite integral can be evaluated numerically by means of computer algebra systems, which provide accurate results.Isoflux Circular Contact.Hui and Tan [39] gave the solution for a circular contact of radiusa that is subjected to a uniform and steady heat flux q and that is in perfect contact with anisotropic layer of thermal conductivity kl and thickness tl that is also in perfect thermal contact with a substrate of thermal conductivity k2. This system is shown in Fig.
3.21.Hui and Tan [39] reported the following integral solution for the dimensionless spreadingresistance:4k2aRs=-~2 \-~(] +--re 1 - \-~(] j £- [1 + (k,/k2)tanh (~tl/a)]~ 2T,/,,,/////,,~~ ri a ",'////////'2,.,.1kl2ka3k3~t1z,(3.152)which depends on the thermal conductivity ratio kl/k2 andthe relative layer thickness tl/a. The dimensionless constriction is based on the substrate thermal conductivity k2. Theabove general solution is valid for conductive layers wherekl/k2 > 1 as well as for resistive layers where kl/k2 < 1.t2z2ZFIGURE 3.22 Flux tube with two layers.Circular Contact Area on Multiple Layerson Circular Flux TubeThe effect of single and multiple isotropic layers or coatingson the end of a circular flux tube has been determinedby Antonetti [2] and Sridhar et al. [107].
The heat enters theend of the circular flux tube of radius b and thermal conductivity k3 through a coaxial, circular contact area that is inperfect thermal contact with an isotropic layer of thermalconductivity kl and thickness h. This layer is in perfect contact with a second layer of thermal conductivity k2 and thickness t2 that is in perfect contact with the flux tube havingthermal conductivity k3 (Fig.
3.22). The lateral boundary ofthe flux tube is adiabatic and the contact plane outside thecontact area is also adiabatic. The boundary condition overthe contact area may be (1) isoflux or (2) isothermal. Thedimensionless constriction resistance ~2 layers = 4k3aRc is defined with respect to the thermal conductivity of the flux3.48CHAPTERTHREEtube, which is often referred to as the substrate.
This constriction resistance depends on several dimensionless parameters: relative contact size e = a/b where 0 < e < 1; two conductivityratios, 1(21= k2/kl and 1(32= k3/k2; two relative layer thicknesses, Xl = tl/a and x2 = t2/a; and theboundary condition over the contact area. The solution for two layers is given as~1/2 layers - -where16 1~E n = 1~n,~1(211(32~(3.153)¢"'~= 8,J0(8,)32P,,~(3.154)and the boundary condition parameter is defined assin (8,e)9,,,~ = 2J1(8,~)isothermal contact1(3.155)isoflux contactThe thermal conductivity ratios are defined above. The layer parameters d + and O- come fromthe following general function:O± = (1 + 1(21)(1 + 1(32)+--(1 -- 1(2,)(1 + 1((32)exp(-ZS.eXl)+ (1 - 1(2,)(1 - K32) exp(-28.e'c2) + (1 + 1(2,)(1 + 1(32) exp(-28,,e(a:1 + I:2))The eigenvalues 8. that appear in the solution are the roots of Jl(') -O.The two-layer solution can be used to obtain the solution for a single layer of thermal conductivity k 1 and thickness t, in perfect contact with a flux tube of thermal conductivity k2.
Inthis case the dimensionless constriction resistance ~1 ~ay~rdepends on the relative contact sizee, the conductivity ratio 1(2,, and the relative layer thickness "c1:~1layer - -16 _1~E n = 1(3.156)~n,~1(21~O-and the general layer function reduces to0-+= 2[(1 + 1(21)(1 - 1(2,)] exp(-28,ex,)Transient Spreading ResistanceIntroduction.Transient spreading resistance occurs during startup and is important in certain micro-electronic systems. The spreading resistance can be defined with respect to thearea-average temperature as a single point temperature such as the centroid. Solutions havebeen reported for isoflux contact areas on half-spaces, circular contact areas on circular fluxtubes, and strips on channels.Spreading Resistance of hoflux Circular Contact Area on Half-Space.Beck [6] reportedthe following integral solution for a circular area of radius a that is subjected to a uniform andconstant flux q for t > 0:d~4kaRs =~-8 f f j2(O erf (~V~o) ;2(3.157)where the dimensionless time is defined as Fo = ~/a 2.
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