Review of Thermal Conductance Models for Joints ...

1 JOURNALOFTHERMOPHYSICS ANDHEATTRANSFERV ol. 17, No. 1, January March 2003 Review of Thermal Conductance Models for JointsIncorporating enhancement MaterialsI. Savija, J. R. Culham, and M. M. Y ovanovich University of W aterloo, W aterloo, Ontario N2L 3G1, CanadaandE. E. Marotta IBM Corporation, Inc., P oughkeepsie, New Y ork 12601A comprehensive Review of analytical and empirical Models for calculating the Thermal Conductance acrossmechanically formed Joints is presented. A historical perspective of modeling procedures for a range of interfacecon gurations is presented, including bare contacting surfaces for conforming rough surfaces as well as interfacialsurfaces augmented with enhancement materials such as greases, metallic foils, polymeric compliant materials, lms, and coatings.

2 Given the wide range of interface materials available and their associated thermophysical andsurface properties, the Models presented provide an effective procedure for determining the signi cance of theseproperties in the prediction of contact , gap, and overall joint area,m2a= contact spot radius, mb= radius of heat ux channel, mbt= surface roughness parameterC= constriction parameter correction factorc1,c2= coef cients for Vickers microhardnessd= equivalent V ickers interaction depth, mE= Y oung s modulus, MPaF= load,Nfg= correction factorg1,g2= temperature jump distances, mH= hardness, MPah= Conductance , W/m2 KIg= gap integralKn= Knudsen number, identical to3= k= Thermal conductivity, W/ m Kkm= mean harmonic Thermal conductivity, identical to2k1 k2=.

3 K1Ck2/M= gas parameter, mm= mean absolute asperity slopeP= apparent contact pressure, MPaPr= Prandtl numberQ= heat ow rate, WR= resistance , C/WT= temperature, Kt= thickness, mY= mean plane separation, m , = accommodation coef cients = ratio of speci c heats = effective gap thickness, m3= molecular mean free path, mReceived 21 February 2002; revision received 25 July 2002; accepted forpublication 23 August 2002. Copyrightc 2002 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved. Copies of this papermay be made for personal or internal use, on condition that the copier paythe $ per-copy fee to the Copyright Clearance Center, Inc.

4 , 222 Rose-wood Drive, Danvers, MA 01923; include the code 0887-8722/03 $ incorrespondence with the CCC. Candidate, Department of Mechanical Engineering. Associate Professor, Director, Microelectronics Heat TransferLaboratory. Distinguished Professor Emeritus, Department of Mechanical Engineer-ing. Fellow AIAA. Senior Engineer, Development, Product Packaging, Power and CoolingGroup. Senior Member AIAA. = relative gap thickness = Poisson s ratio = mean rms roughness, mSubscriptsB= Brinellbare= barebulk= bulkc= contacte= elasticf= foilg= gap, gasj= joint `= layermin = minimump= plasticpol = polymers= solid , = indices for contacting surfaces0= reference condition1, 2 = indices for contacting surfacesSuperscriptsC= dimensionless quantity0= effectiveIntroductionHEA T ow across a mechanical joint results in a temperaturedrop, which depends on the Thermal resistance of the contact -ing interface.

5 Thermal joint resistance is a function of several geo-metric, physical, and Thermal parameters such as surface roughnessand waviness; surface microhardness; Thermal conductivity of thecontacting solids, including layers, coatings, and lms; propertiesof any interstitial materials; and the contact substances, such as gases, greases, oils, liquids, etc.,which completely ll the gaps formed between contacting asperitiescan perfectly wet interfacial surfaces, producing interfaces whichhave relatively high joint conductances. For instance, helium, whichhas a higher Thermal conductivity than air, enhances the gap con-ductance, providing a higher overall joint Conductance . Greases,such as Dow Corning DC-340, when used at low contact pressurescan enhance joint Conductance , whereas oils, which have higherthermal conductivities than both greases and gases, can be used4344SA VIJA ET further increase the Thermal joint Conductance .

6 Composites, in-cluding greases doped with particulates such as silver combine thewettability of greases with the high conductivity of metals to furtherenhance joint conductive layers, in the range of 1 50 m thickness, whenvapor deposited on contacting surfaces, can increase joint conduc-tance by at least an order of magnitude. The most effective materialsare those which combine high Thermal conductivity with a hard-ness that is lower than the contacting asperities. The hardness ofthe deposited materials is typically more important than the thermalconductivity. As an alternative to deposited layers, interstitial metal-lic foils made of aluminum, copper, indium, lead, tin, etc., can beplaced between contacting rough surfaces to increase signi cantlythe joint less conventional materials, such as nonmetallics, includ-ing rubber or soft plastics, can be effective when inserted betweencontacting surfaces especially when contact pressures are very lowand the hardness of the material is much lower than that of thecontacting surfaces.

7 Phase change materials that ow at elevatedtemperatures have recently been used to create Joints with very highjoint conductances when contact pressures are very objective of this paper is to present a thorough Review ofjoint Conductance Models that deal with interstitial substances, lms,and coatings for the enhancement of Thermal joint conductances ofconforming, rough surfaces. Whenever possible, the Models will becompared against experimental of Thermal contact , Gap,and joint ConductancesWhen two real (rough) surfaces are placed in mechanical contact ,an interface is formed that consists of numerous discrete, microcon-tact spots and a gap that separates the two surfaces as shown inFig.

8 1 and in more detail in the microscopic view shown in Fig. the surfaces are nominally at, that is, they have negligible sur-face waviness (out-of- atness), if the surface asperity heights havea Gaussian distribution with respect to the mean plane, and if theyare randomly distributed in the contact plane, then the discrete mi-crocontacts are assumed to be randomly distributed throughout theapparent contact real contact area, which is much smaller than the apparent(nominal) contact area, depends on the contact spot density and themean contact spot area. If the contact spots are modeled as circular,Fig. 1 Typical interface geometry for conforming rough 2 Microscopic view of a contact interface in a conforming real contact area is said to depend on the contact spot densityand mean contact spot contacting asperities, depending on their microgeometry andphysical properties and the apparent contact pressure, can undergoelastic, plastic, or elastoplastic and mechanical Models are available for prediction ofthe ratio of real to apparent contact areaAr=Aa 1, the contactspot densityn, and the mean contact spot radiusawith respectto the relative mean plane separation DY= given the surfaceroughnesses 1and 2and the mean asperity slopesm1andm2.

9 Therelative mean plane separation depends on the mode of contactingasperity heat transfer across the interface shown in Fig. 2 is givenby the relationQjDQcCQgCQr(1)whereQcis the conduction via the microcontacts,Qgconductionthrough the interstitial substance, andQrheat transfer by radiation ifthe interstitial substance is transparent to radiation, for example, dryair. If the interstitial substance is opaque (absorbing gases, liquids,and solids), thenQrD0 andQjDQcCQg(2)If the contact is made in a vacuum with no interstitial substancein the gaps, thenQjDQc(3)In all of the preceding cases, the temperature drop across theinterface is given by the relation1 TjDQjRj(4)whereRjrepresents the joint resistance which is related to threeresistances if the interstitial gap substance is transparent to radiation,1=RjD1=RcC1=RgC1=Rr(5)or two resistances if the interstitial substance is opaque,1=RjD1=RcC1=Rg(6)and only the contact resistance if the contact is made in a vacuumand there is no interstitial substance in the gaps,RjDRc(7)

10 If conductances are used to model heat transfer across the joint ,QjDhjAa1Tj(8)then the corresponding Conductance relationships arehjDhcChgChr(9)hjDhcChg(10)hjDhc(11)SA VIJA ET interface temperaturesTj<600 C, radiation heat transfer canbe assumed negligible, that is,Rr a material (metallic or nonmetallic) is inserted betweencontacting surfaces, as shown in Fig. 1, and the material thicknesstis much greater than the surface roughnesses 1and 2, that is,t Dp. 1C 2/, then a more complicated mechanical and ther-mal joint is obtained. The joint now consists of two interfaces, eachformed between the contacting surfaces and the inserted two interfaces, in the general case, are different because theirgaps may be occupied by different substances, their surface rough-nesses may be different, and the contacting surface asperity defor-mation may be the general case where interstitial substances are present in thetwo gaps, the overall joint resistance depends on ve resistances, ;Rg1;Rc2;Rg2;Rl/(12)and the overall joint Conductance depends on ve conductances, ;hg1;hc2;hg2;hl/(13)where the subscripts denote interfaces 1 and 2.