Thermal-Mechanical Models for Non-Conforming Surface Contacts

Transcription

1 Thermal-Mechaical Models for No-Coformi Surface Cotacts M. M. Yovaovich Departmet of Mechaical Eieeri Waterloo, Otario, Caada Fudametals ad Applicatios of Coductio May 5, 000

2 Outlie Itroductio types of cotacti surfaces types of o-flat surface cotacts Ojectives Model developmet spreadi resistace ap resistace Sample calculatio of joit resistace Summary

3 Itroductio: Prolem Descriptio Steady heat trasfer at iterface formed y cotact etwee silico chip ad metallic heat sik Mechaical cotact etwee two o-coformi, rouh surfaces Heat Sik Iterface Material Silico Chip Ceramic Sustrate

4 Itroductio: Types of Cotacts 3

5 Itroductio: No-Flat Surfaces 4 Cotacti surfaces ca e cocave or covex Three possile o-flat surface comiatios: Lare variatios i cotact areas, aps ad thermal joit resistace models

6 Ojectives 5 Develop aalytical joit resistace model for metallic heat sik o silico chip o-coformi (covex/covex), smooth surfaces elastic cotact ap filled with as or liquid Assumptios: heat sik modeled as aseplate with effective uiform heat trasfer coefficiet all edes adiaatic eliile radiative heat trasfer across iterface

7 Model Developmet 6 Joit resistace j s + - spreadi/costrictio resistace ectaular cross sectios trasformed to equivalet circular disks: L L π adii of curvature for covex cotacti surfaces: ρ d L L π d s L - ap resistace L d

8 Model Developmet 7 Cotact radius from Hertz elastic cotact model: a 3 F 4 ρ ρ ρ + ρ 3 where: υ E + υ E E υ, F E, υ -load - You s modulus - Poisso s ratios

9 Spreadi esistace 8 Total spreadi (costrictio) resistace: s ψ ψ s, + s, + where: 4k a 4k a Dimesioless spreadi resistace: ψ ( ε, τ Bi) f, ψ ψ 4k 4k a a s, s, ε a τ t Bi h k h A iterface - relative cotact size - relative thickess - Biot umer Q heat sik T heat sik

10 Spreadi esistace 9 Fuctio of oudary coditio at cotact area: where: isoflux equivalet isothermal φ δ β tah + ψ 6 π ε 8 ψ π ε δ 3 J ( δ ε) ( δ ) J 0 φ ( δ ε) ( δ ε) 3 δ J ( δ ) J si ( δ τ) + δ Bi ( δ Bi) tah( δ τ) β β β 5 β 70 β π ( 4 +) 0 φ

11 Spreadi esistace -Calculatios 0 Three calculatio alteratives for spreadi resistace: solve eeral equatios with computer alera system, such as Maple or Mathematica We-ased caculatio tools availale at: Correlatio for ifiitely lo flux tue limit, τ > 0. 7 ψ C C ε + C3 ε + C5 ε + C7 ε

12 Gap esistace Elemetal flux tue: d Q ( r) k T π r dr t + M Local ap thickess for small deformatios: M - as rarefactio parameter k - ap thermal coductivity r r r t t + t + ( d + d ), 0 r ρ ρ Gas parameter: a r M αβ Λ α β - accommodatio parameter -as parameter Λ - molecular mea-free-path t t t

13 Gap esistace Total ap heat trasfer: Q Averae ap coductace: h Local ap temperature drop: T () r d Q π k t + A A Q T j d k + ( r) d T 0 f ( r), T j a M r dr () r f r dr r + M + a a r ( d d )

14 Gap esistace - Solutios 3 For liht cotact loads where a <<, assumi ives simple, closed form solutio: f ( r) πk, K l + a For liquid or rease i ap: ad: K M 0 πk d d + d l + d + ( a) K K M d + d

15 Computatio Procedure 4 Start equivalet flux tue radii ad radii of curvature ρ,, ρ, iterface effective physical parameter ad radius of cotact,a relative cotact radius, relative thickess, ad Biot umer ε, ε, τ,, Bi τ spreadi resistace i solid ψ, s, spreadi resistace i solid ψ, s, total spreadi resistace + s s, s, rarefactio parameter, ap parameter M, K ap resistace joit resistace j s +

16 Summary 5 Models developed for thermal joit resistace for o-coformi, smooth hemispherical surfaces Joit resistace cosists of spreadi ad ap resistaces coected i parallel Gap resistace model developed for various ases, liquids or reases Example provided of cotact etwee a silico chip ad a alumium heat sik

17 Ackowledemets 6 The author ackowledes the cotiued fiacial support of the Natural Scieces ad Eieeri esearch Coucil of Caada The assistace of M. Stevaovic ad P. Teertstra i the preparatio of the paper ad presetatio material is reatly appreciated