Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 52
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— John Wiley & Sons / Page 364 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 4060 to 4103———-1.19986pt PgVar———Normal Page* PgEnds: Eject(4.302)To implement the procedure (Antonetti and Yovanovich, 1983, 1985) for finding H from the three correlation equations requires an iterative method.To initiate the iterative method, the first guess is based on the arithmetic averageof the substrate and layer microhardness values:H1 =[364], (104)(4.303)[364], (104)JOINT CONDUCTANCE ENHANCEMENT METHODS123456789101112131415161718192021222324252627282930313233343536373839404142434445365[365], (105)Lines: 4103 to 4109———*22.25099pt PgVar———Normal PagePgEnds: TEX[365], (105)Figure 4.30 Vickers microhardness of a silver layer on a nickel substrate. (From Antonettiand Yovanovich, 1985.)BOOKCOMP, Inc.
— John Wiley & Sons / Page 365 / 2nd Proofs / Heat Transfer Handbook / Bejan366123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESThe dimensionless spreading–constriction resistance parameter is defined asΨ( , φn ) = 4k2 a Rc(4.304)where k2 is the thermal conductivity of the substrate that is coated, a is the contact spot radius for the layer on the substrate, and Rc is the spreading–constrictionresistance of the contact spot.The spreading–constriction resistance parameter with a layer on the substrate is(Antonetti and Yovanovich, 1983, 1985)∞16 J12 (δn )Ψ( , φn ) = φ γ ρπ n=1 (δn )3 J02 (δn ) n n n(4.305)The first of these, φn , accounts for the effect of the layer though its thickness andthermal conductivity; the second, γn , accounts for the contact temperature basis usedto determine the spreading–constriction resistance; and the third, ρn , accounts for thecontact spot heat flux distribution.
For contacting surfaces it is usual to assume thatthe contact spots are isothermal. The modification factors in this case are γn = 1.0 and φn = K(1 + K) + (1 − K)e−2δn τ (1 + K) − (1 − K)e−2δn τ(4.306)where K is the ratio of the substrate-to-layer thermal conductivity, τ = t/a is thelayer thickness-to-contact spot radius ratio, andsin δn ρn =2J1 (δn )Lines: 4109 to 4155———4.24411pt PgVar———Normal PagePgEnds: TEX[366], (106)(4.307)The parameter δn are the eigenvalues, which are roots of J1 (δn ) = 0.Tabulated values of C were reported by Antonetti (1983) for a wide range of theparameters K and τ .
Details of the thermomechanical model development are givenin Antonetti (1983) and Antonetti and Yovanovich (1983, 1985).The thermomechanical model of Antonetti and Yovanovich (1983, 1985) has beenverified by extensive tests. First the bare joint was tested to validate that part of themodel. Figure 4.31 shows the dimensionless joint conductance data and theory plottedversus the relative contact pressure for three joints having three levels of surfaceroughness. The two surfaces were flat; one was lapped and the other was glass beadblasted. All tests were conducted in a vacuum.
The agreement between the modelgiven by the correlation equation and all data is very good over the entire range ofrelative contact pressure.The bare surface tests were followed by three sets of tests for joints having threelevels of surface roughness. Figure 4.32 shows the effect of the vapor-deposited silverlayer thickness on the measured joint conductance plotted against the contact pressure. For these tests the average values of the combined surface roughness parameterswere σ = 4.27 µm and m = 0.236 rad.
For the contact pressure range the substrateBOOKCOMP, Inc. — John Wiley & Sons / Page 366 / 2nd Proofs / Heat Transfer Handbook / Bejan[366], (106)JOINT CONDUCTANCE ENHANCEMENT METHODSSpecimens 08/09Specimens 10/11Specimens 26/27Specimens 34/35Dimensionless Conductance h/mk ⫻ 104123456789101112131415161718192021222324252627282930313233343536373839404142434445367[367], (107)101( (h = 1.25 PmkH0.95Lines: 4155 to 4157———3.42099pt PgVar———Normal PagePgEnds: TEX[367], (107)100100101Relative Pressure P ⫻ 104HFigure 4.31 Dimensionless contact conductance versus relative contact pressure for bare Ni200 surfaces in a vacuum. (From Antonetti and Yovanovich, 1985.)microhardness was estimated to be HS = 2.97 GPa.
The layer thickness was between0.81 and 39.5 µm. The lowest set of data and the theoretical curve correspond to thebare surface tests. Agreement between data and model is very good. The highest setof data for layer thickness of t = 39.5 µm corresponds to the infinitely thick layerwhere thermal spreading occurs in the layer only and the layer microhardness controls the formation of the microcontacts. Again, the agreement between experimentand theory is good.BOOKCOMP, Inc. — John Wiley & Sons / Page 367 / 2nd Proofs / Heat Transfer Handbook / Bejan368102Specimens08/0910/11CoatingNone”18/190.81 m22/2312/1314/1516/171.2 m1.4 m5.1 m39.5 mUpper boundInfinite coating39.5 m5.1 m1.4 m1.2 mContact Conductance (kW/m2 .
K)123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES[368], (108)0.81 m101Lines: 4157 to 4180———0.097pt PgVarLower boundNo coating———Normal PagePgEnds: TEX[368], (108)100100103Pressure (kN/m2)Figure 4.32 Effect of layer thickness and contact pressure on joint conductance: vacuum dataand theory.
(From Antonetti and Yovanovich, 1985.)The difference between the highest and lowest joint conductance values is approximately a factor of 10. The enhancement is clearly significant. The agreementbetween the measured values of joint conductance and the theoretical curves for thelayer thicknesses: t = 0.81, 1.2, 1.4, and 5.1 µm is also very good, as shown inFig. 4.32. All the test points for bare and coated surfaces are plotted in Fig. 4.33as dimensionless joint conductance versus relative contact pressure. The agreementbetween experiment and theory is very good for all points.BOOKCOMP, Inc.
— John Wiley & Sons / Page 368 / 2nd Proofs / Heat Transfer Handbook / BejanJOINT CONDUCTANCE ENHANCEMENT METHODS102Series A = 4.27 mSeries B = 1.28 mSeries C = 8.32 mDimensionless Conductance h⬘ ⫻ 104mk⬘123456789101112131415161718192021222324252627282930313233343536373839404142434445369[369], (109)( (h⬘ = 1.25 P 0.95mk⬘H⬘101Lines: 4180 to 4180———4.097pt PgVar———Normal PagePgEnds: TEX[369], (109)100100101Relative Pressure P ⫻ 104H⬘102Figure 4.33 Dimensionless joint conductance for a bare and silver layer on Ni 200 substratesversus relative contact pressure. (From Antonetti and Yovanovich, 1985.)A parametric study was conducted to calculate the enhancement that can beachieved when different metal types are used. The theory outlined earlier will nowbe applied to a common problem in electronics packaging: heat transfer across analuminum joint.
What is required is a parametric study showing the variation in jointconductance as a function of metallic coating type and thickness for fixed surfaceBOOKCOMP, Inc. — John Wiley & Sons / Page 369 / 2nd Proofs / Heat Transfer Handbook / Bejan370123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESTABLE 4.20LeadTinSilverAluminumAssumed Nominal Property Values of Four Coatingsk (W/m · K)H (kg/mm2 )32.458.4406.0190.03.08.540.085.0roughness and contact pressure. The thermophysical properties of the coatings andthe aluminum substrate material are presented in Table 4.20.Figure 4.34 shows the effect of the metallic layers on joint conductance.
As shownin this figure, except for a very thin layer (about 1 µm), the performance curves arearranged according to layer microhardness. Lead with the lowest microhardness has[370], (110)Lines: 4180 to 4188———-0.83296pt PgVar———Normal PagePgEnds: TEX[370], (110)Figure 4.34 Effect of layer thickness for four metallic layers. (From Antonetti and Yovanovich, 1983.)BOOKCOMP, Inc. — John Wiley & Sons / Page 370 / 2nd Proofs / Heat Transfer Handbook / BejanJOINT CONDUCTANCE ENHANCEMENT METHODS123456789101112131415161718192021222324252627282930313233343536373839404142434445371the highest contact conductance, and silver with the highest microhardness has thelowest contact conductance.
The thermal conductivity of the coating appears to playa secondary role.The unusual shape of the curves is attributable to the fact that the assumed effectivehardness curve shown in Fig. 4.30 has three distinct zones. Moreover, because themicrohardness of silver is much closer to aluminum than are the microhardness oflead and tin, respectively, the transition from one region to the next is not abruptin the silver-on-aluminum effective microhardness curve, and this is reflected in thesmoother contact conductance plot for the silver layer shown in Fig. 4.34. It shouldbe noted that in the model that has been used, the load is assumed to be uniformlyapplied over the apparent contact area.4.17.2Ranking Metallic Coating Performance[371], (111)In his research on the effects of soft metallic foils on joint conductance, Yovanovich(1972) proposed that the performance of various foil materials may be ranked according to the parameter k/H , using the properties of the foil material.
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