Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 54
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The mean plane separation Y , shown in Fig. 4.22, is given approximatelyby the simple power law relation (Antonetti, 1983) −0.097YP= 1.53σHc(4.318)The power law relation shows that Y /σ is a relatively weak function of the relativecontact pressure. Using this relation, the joint conductance may be expressed ashj =kgkg=σ(Y /σ)1.53σ(P /Hc )−0.097(W/m2 · K)(4.319)which shows clearly how the geometric, physical, and thermal parameters influencethe joint conductance. The relation for the specific joint resistance isBOOKCOMP, Inc. — John Wiley & Sons / Page 375 / 2nd Proofs / Heat Transfer Handbook / Bejan[375], (115)376123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES −0.097P1σAa Rj == 1.53hjkgHc(m2 · K/W)(4.320)In general, if the metals work-harden, the relative contact pressure P /Hc is obtained from the relationship1/(1+0.071c2 )PP=Hcc1 (1.62σ/m)c2(4.321)where the coefficients c1 and c2 are obtained from Vickers microhardness tests.
TheVickers microhardness coefficients are related to the Brinell hardness HB for a widerange of metals. The units of σ in the relation above must be micrometers. The unitsof P and c1 must be consistent.The approximation of Hegazy (1985) for microhardness is recommended:Hc = (12.2 − 3.54HB ) σ −0.26m(GPa)(4.322)where Hc , the effective contact microhardness, and HB , the Brinell hardness, are inGPa and the effective surface parameter (σ/m) is in micrometers. If the softer metaldoes not work-harden, Hc ≈ HB . Since HB < Hc , if we set Hc = HB in the specificjoint resistance relationship, this will give a lower bound for the joint resistance or anupper bound for the joint conductance.The simple grease model for joint conductance or specific joint resistance wascompared against the specific joint resistance data reported by Prasher (2001). Thesurface roughness parameters of the bounding copper surfaces and the grease thermalconductivities are given in Table 4.23.
All tests were conducted at an apparent contactpressure of 1 atm and in a vacuum. Prasher (2001) reported his data as specific jointresistance rj = Aa Rj versus the parameter σ/kg , where kg is the thermal conductivityof the grease.TABLE 4.23Surface Roughness and Grease Thermal ConductivityTestRoughness,σ1 = σ2(µm)Conductivity,kg(W/m · K)12345670.1213.513.53.53.53.133.133.130.40.40.250.22Source: Prasher (2001).BOOKCOMP, Inc. — John Wiley & Sons / Page 376 / 2nd Proofs / Heat Transfer Handbook / Bejan[376], (116)Lines: 4368 to 4419———0.86218pt PgVar———Normal PagePgEnds: TEX[376], (116)JOINT CONDUCTANCE ENHANCEMENT METHODS1234567891011121314151617181920212223242526272829303132333435363738394041424344454.17.5377Phase-Change MaterialsPhase-change materials (PCMs) are being used to reduce thermal joint resistance inmicroelectronic systems.
PCMs may consist of a substrate such as an aluminum foilsupporting a PCM such as paraffin. In some applications the paraffin may be filledwith solid particles to increase the effective thermal conductivity of the paraffin.At some temperature Tm above room temperature, the PCM melts, then flowsthrough the microgaps, expels the air, and then fills the voids completely. After thetemperature of the joint falls below Tm , the PCM solidifies. Depending on the levelof surface roughness, out-of-flatness, and thickness of the PCM, a complex joint isformed.
Thermal tests reveal that the specific joint resistance is very small relative tothe bare joint resistance with air occupying the microgaps (Fig. 4.36). Because of thecomplex nature of a joint with a PCM, no simple models are available for the severaltypes of joints that can be formed when a PCM is used.[377], (117)Lines: 4419 to 4431———*37.927pt PgVar———Normal PagePgEnds: TEX[377], (117)Figure 4.36 Specific joint resistance versus σ/k for grease. (From Prasher, 2001.)BOOKCOMP, Inc.
— John Wiley & Sons / Page 377 / 2nd Proofs / Heat Transfer Handbook / Bejan378123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES4.18 THERMAL RESISTANCE AT BOLTED JOINTSBolted joints are frequently found in aerospace systems and less often in microelectronics systems. The bolted joints are complex because of their geometric configurations, the materials used, and the number of bolts and washers used.
The pressuredistributions near the location of the bolts are not uniform, and the region influencedby the bolts is difficult to predict. A number of papers are available to provide information on measured thermal resistances and to provide models to predict the thermalresistance under various conditions.Madhusudana (1996) and Johnson (1985) present material on the thermal andmechanical aspects of bolted joints.
For bolted joints used in satellite thermal design,the publications of Mantelli and Yovanovich (1996, 1998a, b) are recommended.For bolted joints used in microelectronics cooling, the publications of Lee et al.(1993), Madhusudana et al. (1998) and Song et al. (1992b, 1993a) are recommended.Mikic (1970) describes variable contact pressure effects on joint conductance.[378], (118)Lines: 4431 to 4609———NOMENCLATURERoman Letter SymbolsAarea, m2Fourier coefficient, dimensionlessgeometric parameter related to radii of curvature,dimensionlessapparent contact area, m2AaAgeffective gap area, m2Ancoefficient in summation, dimensionlessreal contact area, m2Araradius of source area, mmean contact spot radius, msemimajor diameter of ellipse, mcorrelation coefficient, dimensionlessradius of circle, mstrip half-width, mside dimension of plate, mradius of flat contact, melastic contact radius, maecomposite elastic–plastic contact radius, maepthick-layer limit of contact radius, maLplastic contact radius, mapthin-layer limit of contact radius, mascombination of terms, dimensionlessa∗BOOKCOMP, Inc.
— John Wiley & Sons / Page 378 / 2nd Proofs / Heat Transfer Handbook / Bejan0.20558pt PgVar———Short PagePgEnds: TEX[378], (118)NOMENCLATURE123456789101112131415161718192021222324252627282930313233343536373839404142434445BBiBi(x,y)Bnbb1CCccDddodvEEerf(x)erfc(x)FFiF (k , ψ)F∗FFofepfgf (u)G379Fourier coefficient, dimensionlessgeometric parameter related to radii of curvature,dimensionlessBiot modulus, dimensionlessbeta function of arguments x and y, dimensionlesscoefficient in summation, dimensionlesssemiminor diameter of ellipse, mside dimension of plate, mradius of compound disk, mcorrelation coefficient, dimensionlesschannel half-width, mradius of cylinder, mFourier coefficient, dimensionlesscorrection factor, dimensionlesscontact conductance, dimensionlesscorrelation coefficient, dimensionlesslength dimension, mflux channel half-width, mside dimension of isoflux area, mdiameter of sphere, mdiameter of circular cylinder, mlength dimension, muniform gap thickness, mside dimension of isoflux area, mreference value for average diagonal, mVickers diagonal, mmodulus of elasticity (Young’s modulus), N/m2modulus of elasticity for polymer, N/m2complete elliptic integral of second kind, dimensionlesseffective modulus of elasticity, N/m2error function of argument x, dimensionlesscomplementary error function of argument x, dimensionlessemissivity factor, dimensionlessfactor, dimensionless, i = 1, 2, 3total normal load on a contact, Nincomplete elliptic integral of the first kind of modulus k andamplitude ψ, dimensionlesscombination of terms, dimensionlessload per unit cylinder length, NFourier modulus, dimensionlesselastic–plastic parameter, dimensionlesscombination of terms, dimensionlessaxisymmetric heat flux distribution, dimensionlessparameter, dimensionlessBOOKCOMP, Inc.
— John Wiley & Sons / Page 379 / 2nd Proofs / Heat Transfer Handbook / Bejan[379], (119)Lines: 4609 to 4609———0.00563pt PgVar———Short PagePgEnds: TEX[379], (119)380123456789101112131415161718192021222324252627282930313233343536373839404142434445HHBHB∗HeHepHpHSHVH1H2hhhchghjIgIg,"Ig,pIoIqIγKKnkkgkg,∞ksk1k2LL"LMMgMsmTHERMAL SPREADING AND CONTACT RESISTANCESeffective microhardness, MPaBrinnell hardness, MPabulk microhardness of substrate, MPaadjusted Brinnell hardness, MPaeffective microhardness, MPaelastic–plastic microhardness, MPamicrohardness of softer contacting asperities, MPasubstrate microhardness, MPamicrohardness of softer material, MPaVickers microhardness, MPamicrohardness, layer 1, MPamicrohardness, layer 2, MPaconductance or heat transfer coefficient, W/m2 · Kcoated joint conductance in vacuum, W/m2 · Kcontact conductance, W/m2 · Kgap conductance, W/m2 · Kjoint conductance, W/m2 · Kgap conductance integral, dimensionlessline contact elastogap integral, dimensionlesspoint contact elastogap integral, dimensionlessrelative layer thickness, dimensionlesslayer thickness conductivity parameter, dimensionlessrelative layer thickness conductivity parameter, dimensionlessthermal conductivity parameter, dimensionlesscomplete elliptic integral of first kind, dimensionlessKnudsen number, dimensionlessmodulus related to the ellipticity, dimensionlessthermal conductivity, W/m · Keffective gas thermal conductivity, W/m · Kgrease thermal conductivity, W/m · Kgas thermal conductivity under continuum conditions,W/m · Kharmonic mean thermal conductivity of a joint, W/m · Klayer 1 thermal conductivity, W/m · Klayer 2 thermal conductivity, W/m · Kstrip length, mrelative contact size, dimensionlesspoint in flux tube where flux lines are parallel, mlength scale, mgas rarefaction parameter, mmolecular weight of gas, g-molmolecular weight of solid, g-molcounter, dimensionlessHertz elastic parameter, dimensionlessabsolute asperity slope, dimensionlessBOOKCOMP, Inc.
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