Thermal Spreading and Contact Resistance

ISBN-13: 9781394187522 / Twarda / 2023

Muzychka

Spis treści

About the Authors xvPreface xviAcknowledgments xixNomenclature xx1 Fundamental Principles of Thermal Spreading Resistance 11.1 Applications 21.2 Semi-Infinite Regions, Flux Tubes, Flux Channels, and Finite Spreaders 41.3 Governing Equations and Boundary Conditions 61.3.1 Source Plane Conditions 61.3.2 Sink Plane Conditions 71.3.3 Interface Conditions 81.4 Thermal Spreading Resistance 81.4.1 Half-Space Regions 81.4.2 Semi-Infinite Flux Tubes and Channels 101.4.3 Finite Disks and Channels 111.5 Solution Methods 111.6 Summary 12References 122 Thermal Spreading in Isotropic Half-Space Regions 152.1 CircularAreaonaHalf-Space 152.1.1 Isothermal Circular Source 162.1.2 Isoflux Circular Source 172.1.3 Parabolic Flux Circular Source 192.1.4 Summary of Circular Source Thermal Spreading Resistance 202.2 Elliptical Area on a Half-Space 202.2.1 Isothermal Elliptical Source 202.2.2 Isoflux Elliptical Source 222.2.3 Parabolic Flux Elliptical Source 232.3 Method of Superposition of Point Sources 252.3.1 Application to a Circular Source 262.3.2 Application to Triangular Source Areas 282.4 Rectangular Area on a Half-Space 292.4.1 Isothermal Rectangular Area 292.4.2 Isoflux Rectangular Source 302.5 Spreading Resistance of Symmetric Singly Connected Areas: The Hyperellipse 332.6 Regular Polygonal Isoflux Sources 342.7 Additional Results for Other Source Shapes 362.7.1 Triangular Source 362.7.2 Rhombic Source 362.7.3 Rectangular Source with Rounded Ends 372.7.4 Rectangular Source with Semicircular Ends 372.8 Model for an Arbitrary Singly Connected Heat Source on a Half-Space 382.9 Circular Annular Area on a Half-Space 402.9.1 Isothermal Circular Annular Ring Source 402.9.2 Isoflux Circular Annular Ring Source 402.10 Other Doubly Connected Areas on a Half-Space 412.11 Problems with Source Plane Conductance 422.11.1 Isoflux Heat Source on a Convectively Cooled Half-Space 422.11.2 Effect of Source Contact Conductance on Spreading Resistance 442.12 Circular Area on Single Layer (Coating) on Half-Space 452.12.1 Equivalent Isothermal Circular Contact 452.12.2 Isoflux Circular Contact 472.12.3 Isoflux, Equivalent Isothermal, and Isothermal Solutions 472.12.3.1 Isoflux Contact Area 472.12.3.2 Equivalent Isothermal Contact Area 482.12.3.3 Isothermal Contact Area 482.13 Thermal Spreading Resistance Zone: Elliptical Heat Source 482.14 Temperature Rise of Multiple Isoflux Sources 522.14.1 Two Coplanar Isoflux Circular Sources 522.15 Temperature Rise in an Arbitrary Area 562.15.1 Temperature Rise at Arbitrary Point 562.15.2 Average Temperature Rise 572.16 Superposition of Isoflux Circular Heat Sources 582.16.1 Nine Coplanar Circles on Square Cluster 612.16.2 Five Coplanar Circles on Square Cluster 622.16.3 Four Coplanar Circles on Triangular Cluster 632.17 Superposition of Micro- and Macro-Spreading Resistances 64References 683 Circular Flux Tubes and Disks 713.1 Semi-Infinite Flux Tube 713.1.1 Isothermal Source on a Flux Tube 763.2 Finite Disk with Sink Plane Conductance 773.2.1 Distributed Heat Flux over Source Area 813.3 Compound Disk 823.3.1 Special Limits in the Compound Disk Solution 853.4 Multilayered Disks 853.5 Flux Tube with Circular Annular Heat Source 883.6 Flux Tubes and Disks with Edge Conductance 903.7 Spreading Resistance for an Eccentric Source on a Flux Tube 933.8 Thermal Spreading with Variable Conductivity Near the Contact Surface 943.9 Effect of Surface Curvature on Thermal Spreading Resistance in a Flux Tube 97References 994 Rectangular Flux Channels 1034.1 Two-Dimensional Semi-Infinite Flux Channel 1044.1.1 Variable Heat Flux Distributions 1064.2 Three-Dimensional Semi-Infinite Flux Channel 1084.2.1 Correlation Equations for Various Combinations of Source Areas and Boundary Conditions 1104.3 Finite Two- and Three-Dimensional Flux Channels 1114.4 Compound Two- and Three-Dimensional Flux Channels 1154.4.1 Special Limiting Cases for Rectangular Flux Channels 1184.5 Finite Two- and Three-Dimensional Flux Channels with Eccentric Heat Sources 1204.6 Rectangular Flux Channels with Edge Conductance 1244.7 Multilayered Rectangular Flux Channels 1264.8 Rectangular Flux Channel with an Elliptic Heat Source 1284.9 Spreading in a Curved Flux Channel (Annular Sector) 1304.10 Effect of Surface Curvature on Thermal Spreading Resistance in a Two-Dimensional Flux Channel 134References 1355 Orthotropic Media 1375.1 Heat Conduction in Orthotropic Media 1375.2 Circular Source on a Half-Space 1415.3 Single-Layer Flux Tubes 1435.3.1 Circular Flux Tubes with Edge Cooling 1445.4 Single-Layer Rectangular Flux Channel 1445.4.1 Rectangular Flux Channels with Edge Cooling 1465.5 Multilayered Orthotropic Spreaders 1475.5.1 Circular Flux Tubes 1485.5.2 Multilayered Orthotropic Flux Channels 1515.5.3 Multilayered Orthotropic Flux Channels with an Eccentric Source 1535.6 General Multilayered Rectangular Orthotropic Spreaders 1535.6.1 Coordinate Transformations for Fully Orthotropic Media 1555.6.2 General Solution for K X <> K Y <> K Z 1565.6.3 Total Thermal Resistance 1595.7 Measurement of Orthotropic Thermal Conductivity 160References 1636 Multisource Analysis for Microelectronic Devices 1676.1 Multiple Heat Sources on Finite Isotropic Spreaders 1686.1.1 Single Source Surface Temperature Distribution 1696.1.2 Multisource Surface Temperature Distribution 1706.2 Influence Coefficient Method 1726.2.1 Thermal Resistance 1746.2.2 Source Plane Convection 1746.3 Extension to Compound, Orthotropic, and Multilayer Spreaders 1756.3.1 Compound Media 1756.3.2 Orthotropic Spreaders 1776.3.3 Multilayer Isotropic/Orthotropic Spreaders 1786.4 Non-Fourier Conduction Effects in Microscale Devices 1816.5 Application to Irregular-Shaped Heat Sources 185References 1877 Transient Thermal Spreading Resistance 1897.1 Transient Spreading Resistance of an Isoflux Source on an Isotropic Half-Space 1897.1.1 Transient Spreading Resistance of an Isoflux Circular Area 1907.1.2 Transient Spreading Resistance of an Isoflux Strip on a Half-Space 1937.1.3 Transient Spreading Resistance of an Isoflux Hyperellipse 1947.1.4 Transient Spreading Resistance of Isoflux Regular Polygons 1947.1.5 Universal Time Function 1957.2 Transient Spreading Resistance of an Isothermal Source on a Half-Space 1957.3 Models for Transient Thermal Spreading in a Half-Space 1997.4 Transient Spreading Resistance Between Two Half-Spaces in Contact Through a Circular Area 2017.5 Transient Spreading in a Two-Dimensional Flux Channel 2027.6 Transient Spreading in a Circular Flux Tube from an Isoflux Source 2037.7 Transient Spreading in a Circular Flux Tube from an Isothermal Source 2057.8 Models for Transient Thermal Spreading in Circular Flux Tubes 207References 2118 Applications with Nonuniform Conductance in the Sink Plane 2138.1 Applications with Nonuniform Conductance 2138.1.1 Distributed Heat Transfer Coefficient Models 2148.1.2 Mixed-Boundary Conditions in the Source Plane 2168.1.3 Least Squares Approximation 2178.2 Finite Flux Channels with Variable Conductance 2188.2.1 Two-Dimensional Flux Channel 2188.2.2 Three-Dimensional Flux Channel 2218.3 Finite Flux Tube with Variable Conductance 225References 2289 Further Applications of Spreading Resistance 2319.1 Moving Heat Sources 2319.1.1 Governing Equations 2329.1.2 Asymptotic Limits 2339.1.3 Stationary and Moving Heat Source Limits 2349.1.3.1 Stationary Heat Sources (Pe --> 0) 2349.1.3.2 Moving Heat Sources (Pe --> infinity ) 2369.1.4 Analysis of Real Contacts 2389.1.4.1 Effect of Contact Shape 2389.1.4.2 Models for All Peclet Numbers 2409.1.5 Prediction of Flash Temperature 2419.2 Problems Involving Mass Diffusion 2439.2.1 Mass Transport from a Circular Source on a Half-Space 2449.2.2 Diffusion from Other Source Shapes 2459.2.2.1 Elliptic Source 2469.2.2.2 Rectangular Source 2469.3 Mass Diffusion with Chemical Reaction 2469.3.1 Diffusion from a 2D Strip Source with Chemical Reaction 2479.3.2 Circular Source on a Disk with Chemical Reaction 2499.3.3 Diffusion from a Rectangular Source with Chemical Reaction 2519.4 Diffusion Limited Slip Behavior: Super-Hydrophobic Surfaces 2549.4.1 Circular and Square Pillars 2569.4.1.1 Circular/Square 2569.4.1.2 Ridges 2579.4.2 Rectangular and Elliptical Pillars for phi s --> 0 2589.4.3 Effect of Meniscus Curvature 2619.5 Problems with Phase Change in the Source Region (Solidification) 2619.6 Thermal Spreading with Temperature-Dependent Thermal Conductivity 2639.6.1 Kirchoff Transform 2639.6.2 Thermal Conductivity Models 2659.6.3 Application for Thermal Spreading Resistance in a Rectangular Flux Channel 2669.7 Thermal Spreading in Spherical Domains 2689.7.1 Thermal Spreading in Hollow Spherical Shells 2689.7.2 Thermal Spreading in a Hollow Hemispherical Shell with Convection on the Interior Boundary 271References 27210 Introduction to Thermal Contact Resistance 27510.1 Thermal Contact Resistance 27510.2 Types of Joints or Interfaces 27810.2.1 Conforming Rough Solids 27910.2.2 Nonconforming Smooth Solids 28110.2.3 Nonconforming Rough Solids 28110.2.4 Single Layer Between Two Conforming Rough Solids 28110.3 Parameters Influencing Contact Resistance or Conductance 28210.4 Assumptions for Resistance and Conductance Model Development 28310.5 Measurement of Joint Conductance and Thermal Interface Material Resistance 283References 28511 Conforming Rough Surface Models 28711.1 Conforming Rough Surface Models 28811.2 Plastic Contact Model for Asperities 29011.2.1 Vickers Micro-hardness Correlation Coefficients 29311.2.2 Dimensionless Contact Conductance: Plastic Deformation 29311.3 Elastic Contact Model for Asperities 29411.3.1 Dimensionless Contact Conductance: Elastic Deformation 29511.4 Conforming Rough Surface Model: Elastic-Plastic Asperity Deformation 29611.4.1 Correlation Equations for Dimensionless Contact Conductance: Elastic-Plastic Model 29711.5 Radiation Resistance and Conductance for Conforming Rough Surfaces 30011.6 Gap Conductance for Large Parallel Isothermal Plates 30211.7 Gap Conductance for Joint Between Conforming Rough Surfaces 30311.8 Joint Conductance for Conforming Rough Surfaces 30611.9 Joint Conductance for Conforming Rough Surfaces: Scale Analysis Approach 31011.10 Joint Conductance Enhancement Methods 31711.10.1 Metallic Coatings and Foils 31711.10.2 Ranking Metallic Coating Performance 32511.10.3 Elastomeric Inserts 32611.10.4 Thermal Greases and Pastes 32811.10.5 Phase Change Materials (PCM) 33211.11 Thermal Resistance at Bolted Joints 332References 33212 Contact of Nonconforming Smooth Solids 33712.1 Joint Resistances of Nonconforming Smooth Solids 33812.2 Point Contact Model 33812.3 Local Gap Thickness 34112.4 Contact Resistance of Isothermal Elliptical Contact Area 34112.5 Elastogap Resistance Model 34212.6 Joint Radiative Resistance 34412.7 Joint Resistance of Sphere-Flat Contact 34512.7.1 Joint Resistance for Sphere-Flat in Vacuum 34512.7.2 Effect of Gas Pressure on Joint Resistance of a Sphere-Flat Contact 34612.8 Joint Resistance for Contact of a Sphere and Layered Substrate 34912.9 Joint Resistance for Elastic-Plastic Contact of Hemisphere and Flat in Vacuum 35212.9.1 Alternative Constriction Parameter for Hemisphere 35312.10 Ball Bearing Resistance 35612.11 Line Contact Models 35612.11.1 Contact Strip and Local Gap Thickness 35612.11.2 Contact Resistance at Line Contact 35712.11.3 Gap Resistance at Line Contact 35812.11.4 Joint Resistance at Line Contact 35812.12 Joint Resistance of Nonconforming Rough Surfaces 35912.13 System for Nonconforming Rough Surface Contact 36012.13.1 Vickers Micro-hardness Model 36012.13.2 Scale Analysis Results 36112.13.3 Contact of Smooth Hemisphere and Rough Flat 36312.13.4 General Micro-Macro Spreading Resistance Model 36412.13.5 Comparisons of Nonconforming Rough Surface Model with Vacuum Data 36512.13.6 General Model Obtained from Scaling Analysis and Data 36612.14 Joint Resistance of Nonconforming Rough Surface and Smooth Flat Contact 37012.14.1 Micro-gap Thermal Resistance 37112.14.2 Macro-gap Thermal Resistance 372References 374Appendix A Special Functions 379A. 1 Gamma and Beta Function 379A.. 1 Gamma Function 379A.1. 2 Beta Function 382A. 2 Error Function 382A. 3 Bessel Functions 384A.3. 1 Bessel Functions of the First and Second Kind 385A.3. 2 Zeroes of the Bessel Functions 387A.. 3 Modified Bessel Functions of the First and Second Kind 387A. 4 Elliptic Integrals 389A. 5 Legendre Functions 391A. 6 Hypergeometric Function 392A.6. 1 Relationship to Other Functions 393References 393Appendix B Hardness 395B. 1 Micro- and Macro-hardness Indenters 395B.. 1 Brinell and Meyer Macrohardness 395B.1. 2 Rockwell Macro-hardness 397B.1. 3 Knoop Micro-hardness Indenter and Test 398B.1. 4 Vickers Micro-hardness Indenter and Test 399B.1. 5 Berkovich Micro and Nano Hardness Indenter and Nano Hardness Tests 399B. 2 Micro- and Macro-hardness Tests and Correlations 400B.2. 1 Direct Approximate Method 402B.. 2 Vickers Micro-hardness Correlation Equations 403B. 3 Correlation Equations for Vickers Coefficients 406B. 4 Temperature Effects on Vickers and Brinell Hardness 407B.4. 1 Temperature Effects on Yield Strength and Vickers Micro Hardness of SS 304L 407B.4. 2 Temperature Effect on Brinell Hardness 407B.4. 3 Temperature Effect on Vickers Micro-hardness and Correlation Coefficients 409B. 5 Nanoindentation Tests 411References 416Appendix C Thermal Properties 419C.1 Thermal Properties of Solids 420C.2 Thermal Conductivity of Gases 420C.3 Resistance of Thermal Interface Materials (TIMs) 423References 423Index 425

O autorze

Yuri S. Muzychka is a Professor of Mechanical Engineering at Memorial University of Newfoundland, Canada. He is a Fellow of ASME, CSME, and the Engineering Institute of Canada (EIC) and has published over 250 journal and conference proceedings papers, in addition to three handbook chapters.M. Michael Yovanovich is a Distinguished Professor Emeritus at the University of Waterloo, Canada. He is a fellow of ASME, CSME, AIAA, AAAS, and RSC. He has published seven handbook chapters and over 350 journal and conference proceedings papers, and has given over 150 keynote lectures.