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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [Fi r s [26 1 Lin e 5.7 6 —— No r m PgE [26 1 CHAPTER 4 Thermal Spreading and Contact Resistances M. M.YOVANOVICH Department of Mechanical Engineering University of Waterloo Waterloo, Ontario, Canada E. E. MAROTTA Thermal Technologies Group IBM Corporation Poughkeepsie, New York 4.1 Introduction 4.1.1 Types of joints or interfaces 4.1.2 Conforming rough solids 4.1.3 Nonconforming smooth solids 4.1.4 Nonconforming rough solids 4.1.5 Single layer between two conforming rough solids 4.1.6 Parameters influencing contact resistance or conductance 4.1.7 Assumptions for resistance and conductance model development 4.2 De±nitions of spreading and constriction resistances 4.2.1 Spreading and constriction resistances in a half-space 4.2.2 Spreading and constriction resistances in flux tubes and channels 4.3 Spreading and constriction resistances in an isotropic half-space 4.3.1 Introduction 4.3.2 Circular area on a half-space Isothermal circular source Isoflux circular source 4.3.3 Spreading resistance of an isothermal elliptical source area on a half-space 4.3.4 Dimensionless spreading resistance of an isothermal elliptical area 4.3.5 Approximations for dimensionless spreading resistance 4.3.6 Flux distribution over an isothermal elliptical area 4.4 Spreading resistance of rectangular source areas 4.4.1 Isoflux rectangular area 4.4.2 Isothermal rectangular area 4.4.3 Isoflux regular polygonal area 4.4.4 Arbitrary singly connected area 4.4.5 Circular annular area Isoflux circular annulus Isothermal circular annulus 261
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262 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [262 Lin e 1.0 p —— Lon g PgE n [262 4.4.6 Other doubly connected areas on a half-space Effect of contact conductance on spreading resistance 4.5 Transient spreading resistance in an isotropic half-space 4.5.1 Isoflux circular area 4.5.2 Isoflux hyperellipse 4.5.3 Isoflux regular polygons 4.6 Spreading resistance within a compound disk with conductance 4.6.1 Special cases of the compound disk solution 4.6.2 Half-space problems 4.6.3 Semi-inFnite flux tube problems 4.6.4 Isotropic Fnite disk with conductance 4.7 Spreading resistance of isotropic Fnite disks with conductance 4.7.1 Correlation equations 4.7.2 Circular area on a single layer (coating) on a half-space Equivalent isothermal circular contact 4.7.3 Isoflux circular contact 4.7.4 Isoflux, equivalent isothermal, and isothermal solutions Isoflux contact area Equivalent isothermal contact area Isothermal contact area 4.8 Circular area on a semi-inFnite flux tube 4.8.1 General expression for a circular contact area with arbitrary flux on a circular flux tube ±lux distributions of the form ( 1 u 2 ) µ Equivalent isothermal circular source Isoflux circular source Parabolic flux distribution Asymptotic values for dimensionless spreading resistances Correlation equations for spreading resistance Simple correlation equations 4.8.2
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