Thermal Spreading and Contact Resistance E-Book

Yuri S. Muzychka

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley-ASME Press Series
• Sprache: Englisch

Seitenzahl: 690

Veröffentlichungsjahr: 2023

Beschreibung

Thermal Spreading and Contact Resistance: Fundamentals and Applications

Single source reference on how applying thermal spreading and contact resistance can solve problems across a variety of engineering fields

Thermal Spreading and Contact Resistance: Fundamentals and Applications offers comprehensive coverage of the key information that engineers need to know to understand thermal spreading and contact resistance, including numerous predictive models for determining thermal spreading resistance and contact conductance of mechanical joints and interfaces, plus detailed examples throughout the book.

Written by two of the leading experts in the field, Thermal Spreading and Contact Resistance: Fundamentals and Applications includes information on:

• Contact conductance, mass transfer, transport from super-hydrophobic surfaces, droplet/surface phase change problems, and tribology applications such as sliding surfaces and roller bearings
• Heat transfer in micro-devices and thermal spreaders, orthotropic systems, and multi-source applications for electronics thermal management applications
• Fundamental principles, thermal spreading in isotropic half-space regions, circular flux tubes and disc spreaders, and rectangular flux channels and compound spreaders
• Systems with non-uniform sink plane conductance, transient spreading resistance, and contact resistance between both non-conforming and conforming rough surfaces

Providing comprehensive coverage of the subject, Thermal Spreading and Contact Resistance: Fundamentals and Applications is an essential resource for mechanical, aerospace, and chemical engineers working on research in the fields of heat transfer, thermal management of electronics, and tribology, as well as thermal engineers and researchers in the field of thermal physics.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

About the Authors

Preface

Acknowledgments

Nomenclature

Greek Symbols

Subscripts

Superscripts

1 Fundamental Principles of Thermal Spreading Resistance

1.1 Applications

1.2 Semi‐Infinite Regions, Flux Tubes, Flux Channels, and Finite Spreaders

1.3 Governing Equations and Boundary Conditions

1.4 Thermal Spreading Resistance

1.5 Solution Methods

1.6 Summary

References

2 Thermal Spreading in Isotropic Half‐Space Regions

2.1 Circular Area on a Half‐Space

2.2 Elliptical Area on a Half‐Space

2.3 Method of Superposition of Point Sources

2.4 Rectangular Area on a Half‐Space

2.5 Spreading Resistance of Symmetric Singly Connected Areas: The Hyperellipse

2.6 Regular Polygonal Isoflux Sources

2.7 Additional Results for Other Source Shapes

2.8 Model for an Arbitrary Singly Connected Heat Source on a Half‐Space

2.9 Circular Annular Area on a Half‐Space

2.10 Other Doubly Connected Areas on a Half‐Space

2.11 Problems with Source Plane Conductance

2.12 Circular Area on Single Layer (Coating) on Half‐Space

2.13 Thermal Spreading Resistance Zone: Elliptical Heat Source

2.14 Temperature Rise of Multiple Isoflux Sources

2.15 Temperature Rise in an Arbitrary Area

2.16 Superposition of Isoflux Circular Heat Sources

2.17 Superposition of Micro‐ and Macro‐Spreading Resistances

References

3 Circular Flux Tubes and Disks

3.1 Semi‐Infinite Flux Tube

3.2 Finite Disk with Sink Plane Conductance

3.3 Compound Disk

3.4 Multilayered Disks

3.5 Flux Tube with Circular Annular Heat Source

3.6 Flux Tubes and Disks with Edge Conductance

3.7 Spreading Resistance for an Eccentric Source on a Flux Tube

3.8 Thermal Spreading with Variable Conductivity Near the Contact Surface

3.9 Effect of Surface Curvature on Thermal Spreading Resistance in a Flux Tube

References

4 Rectangular Flux Channels

4.1 Two‐Dimensional Semi‐Infinite Flux Channel

4.2 Three‐Dimensional Semi‐Infinite Flux Channel

4.3 Finite Two‐ and Three‐Dimensional Flux Channels

4.4 Compound Two‐ and Three‐Dimensional Flux Channels

4.5 Finite Two‐ and Three‐Dimensional Flux Channels with Eccentric Heat Sources

4.6 Rectangular Flux Channels with Edge Conductance

4.7 Multilayered Rectangular Flux Channels

4.8 Rectangular Flux Channel with an Elliptic Heat Source

4.9 Spreading in a Curved Flux Channel (Annular Sector)

4.10 Effect of Surface Curvature on Thermal Spreading Resistance in a Two‐Dimensional Flux Channel

References

5 Orthotropic Media

5.1 Heat Conduction in Orthotropic Media

5.2 Circular Source on a Half‐Space

5.3 Single‐Layer Flux Tubes

5.4 Single‐Layer Rectangular Flux Channel

5.5 Multilayered Orthotropic Spreaders

5.6 General Multilayered Rectangular Orthotropic Spreaders

5.7 Measurement of Orthotropic Thermal Conductivity

References

6 Multisource Analysis for Microelectronic Devices

6.1 Multiple Heat Sources on Finite Isotropic Spreaders

6.2 Influence Coefficient Method

6.3 Extension to Compound, Orthotropic, and Multilayer Spreaders

6.4 Non‐Fourier Conduction Effects in Microscale Devices

6.5 Application to Irregular‐Shaped Heat Sources

References

7 Transient Thermal Spreading Resistance

7.1 Transient Spreading Resistance of an Isoflux Source on an Isotropic Half‐Space

7.2 Transient Spreading Resistance of an Isothermal Source on a Half‐Space

7.3 Models for Transient Thermal Spreading in a Half‐Space

7.4 Transient Spreading Resistance Between Two Half‐Spaces in Contact Through a Circular Area

7.5 Transient Spreading in a Two‐Dimensional Flux Channel

7.6 Transient Spreading in a Circular Flux Tube from an Isoflux Source

7.7 Transient Spreading in a Circular Flux Tube from an Isothermal Source

7.8 Models for Transient Thermal Spreading in Circular Flux Tubes

References

8 Applications with Nonuniform Conductance in the Sink Plane

8.1 Applications with Nonuniform Conductance

8.2 Finite Flux Channels with Variable Conductance

8.3 Finite Flux Tube with Variable Conductance

References

9 Further Applications of Spreading Resistance

9.1 Moving Heat Sources

9.2 Problems Involving Mass Diffusion

9.3 Mass Diffusion with Chemical Reaction

9.4 Diffusion Limited Slip Behavior: Super‐Hydrophobic Surfaces

9.5 Problems with Phase Change in the Source Region (Solidification)

9.6 Thermal Spreading with Temperature‐Dependent Thermal Conductivity

9.7 Thermal Spreading in Spherical Domains

References

10 Introduction to Thermal Contact Resistance

10.1 Thermal Contact Resistance

10.2 Types of Joints or Interfaces

10.3 Parameters Influencing Contact Resistance or Conductance

10.4 Assumptions for Resistance and Conductance Model Development

10.5 Measurement of Joint Conductance and Thermal Interface Material Resistance

References

11 Conforming Rough Surface Models

11.1 Conforming Rough Surface Models

11.2 Plastic Contact Model for Asperities

11.3 Elastic Contact Model for Asperities

11.4 Conforming Rough Surface Model: Elastic–Plastic Asperity Deformation

11.5 Radiation Resistance and Conductance for Conforming Rough Surfaces

11.6 Gap Conductance for Large Parallel Isothermal Plates

11.7 Gap Conductance for Joint Between Conforming Rough Surfaces

11.8 Joint Conductance for Conforming Rough Surfaces

11.9 Joint Conductance for Conforming Rough Surfaces: Scale Analysis Approach

11.10 Joint Conductance Enhancement Methods

11.11 Thermal Resistance at Bolted Joints

References

12 Contact of Nonconforming Smooth Solids

12.1 Joint Resistances of Nonconforming Smooth Solids

12.2 Point Contact Model

12.3 Local Gap Thickness

12.4 Contact Resistance of Isothermal Elliptical Contact Area

12.5 Elastogap Resistance Model

12.6 Joint Radiative Resistance

12.7 Joint Resistance of Sphere‐Flat Contact

12.8 Joint Resistance for Contact of a Sphere and Layered Substrate

12.9 Joint Resistance for Elastic–Plastic Contact of Hemisphere and Flat in Vacuum

12.10 Ball Bearing Resistance

12.11 Line Contact Models

12.12 Joint Resistance of Nonconforming Rough Surfaces

12.13 System for Nonconforming Rough Surface Contact

12.14 Joint Resistance of Nonconforming Rough Surface and Smooth Flat Contact

References

Appendix A: Special Functions

A.1 Gamma and Beta Function

A.2 Error Function

A.3 Bessel Functions

A.4 Elliptic Integrals

A.5 Legendre Functions

A.6 Hypergeometric Function

References

Appendix B: Hardness

B.1 Micro‐ and Macro‐hardness Indenters

B.2 Micro‐ and Macro‐hardness Tests and Correlations

B.3 Correlation Equations for Vickers Coefficients

B.4 Temperature Effects on Vickers and Brinell Hardness

B.5 Nanoindentation Tests

References

Appendix C: Thermal Properties

C.1 Thermal Properties of Solids

C.2 Thermal Conductivity of Gases

C.3 Resistance of Thermal Interface Materials (TIMs)

References

Index

End User License Agreement

List of Tables

Chapter 2

Table 2.1 Dimensionless source temperature.

Table 2.2 Effect of boundary conditions on stationary circular heat source....

Table 2.3 Dimensionless spreading resistance of isothermal ellipse.

Table 2.4 Dimensionless spreading resistance of isothermal rectangular area....

Table 2.5 Dimensionless spreading resistance for isoflux rectangular area.

Table 2.6 Effect of on dimensionless spreading resistances for .

Table 2.7 Dimensionless resistance for isoflux hyperellipse contacts.

Table 2.8 Dimensionless spreading resistance for polygons.

Table 2.9 Correlation coefficients for doubly‐connected polygons.

Table 2.10 Coefficients for isoflux and isothermal heat sources.

Table 2.11 Coefficients for square and circle.

Table 2.12 Values of for several values of and .

Table 2.13 Results for isoflux sources.

Table 2.14 Comparison of Model 1 and Model 2.

Chapter 3

Table 3.1 Effect of on dimensionless spreading resistance .

Table 3.2 Coefficients for correlations of .

Table 3.3 Correlation coefficients for three flux distributions.

Table 3.4 Solutions obtainable from an isoflux source on a compound disk.

Chapter 4

Table 4.1 Dimensionless spreading resistance in flux channels.

Table 4.2 Coefficients for correlations of dimensionless spreading resistanc...

Table 4.3 Summary of solutions for isoflux source on a compound flux channel...

Chapter 6

Table 6.1 Thermal properties of select semiconductors.

Chapter 7

Table 7.1 Comparison of exact and approximate models for for an isoflux c...

Chapter 9

Table 9.1 Effect of boundary condition on a moving circular heat source.

Table 9.2 Effect of shape on isoflux moving heat sources.

Table 9.3 Dimensionless resistance of moving heat sources on a half‐space.

Table 9.4 Dimensionless parameters for an elliptic contact (theory) and rect...

Chapter 11

Table 11.1 Radiative conductances for black surfaces.

Table 11.2 Models and correlation equations for gap conductance for conformi...

Table 11.3 Summary of Hegazy (1985) experiments.

Table 11.4 Summary of Song (1988) experiments.

Table 11.5 Input parameters for a typical stainless steel nitrogen joint.

Table 11.6 Range of parameters for experimental data.

Table 11.7 Properties of interstitial gases.

Table 11.8 Assumed nominal property values of four coatings.

Table 11.9 Ranking the effectiveness of coatings . , ,  radians

Table 11.10 Thickness, surface roughness, and thermophysical properties of t...

Table 11.11 Surface roughness and grease thermal conductivity.

Chapter 12

Table 12.1 Hertz contact parameters and elastoconstriction parameter.

Table 12.2 Dimensionless load, constriction, radiative and joint resistances...

Table 12.3 Effect of gas pressure on gap and joint resistances for air.

Table 12.4 Parameter values for T1–T3 tests for non‐conforming rough model....

Table 12.5 Ranges of geometric, mechanical and thermal parameters of experim...

Table 12.6 Comparison between model and test results .

Table 12.7 Comparison between model and test results .

Table 12.8 Comparison between model and test results .

Table 12.9 Values of parameters in parametric studies.

Appendix A

Table A.1 Roots of the Bessel functions.

Appendix B

Table B.1 Ratio of Brinell to Meyer hardness versus contact strain.

Table B.2 Values of Knoop hardness number for selected materials.

Table B.3 Vickers correlation coefficients for four metals.

Table B.4 Modified Vickers correlation coefficients for four metals.

Table B.5 Comparisons of modified correlations with Vickers hardness values....

Table B.6 Temperature effects on yield strength and Vickers hardness.

Table B.7 Temperature effect on Brinell hardness of SS 304.

Table B.8 Temperature effect on Brinell hardness of Ni 200.

Table B.9 Temperature effect on Brinell hardness of Al 6061 T‐6.

Table B.10 Temperature coefficients for Brinell hardness of three metals.

Table B.11 Temperature effect on coefficients for Ni 200.

Table B.12 Temperature effect on coefficients for SS 304.

Table B.13 Temperature effect on coefficients for Al 6061 T‐6.

Table B.14 Parameter values for power–law fits of unloading curve.

Table B.15 Values of hardness and elastic modulus for aluminum.

Table B.16 Values of hardness and elastic modulus for tungsten.

Table B.17 Values of hardness and elastic modulus for quartz.

Appendix C

Table C.1 Thermal properties of metals at 300 K.

Table C.2 Thermal properties of nonmetals at 300 K.

Table C.3 Power–law and Sutherland formula conductivity parameters for gases...

Table C.4 Specific thermal resistance of greases and pastes.

Table C.5 Specific thermal resistance of interface materials.

List of Illustrations

Chapter 1

Figure 1.1 Thermal spreading (constriction resistance) from an isothermal – 

Figure 1.2 Applications of thermal spreading resistance. From top left to bo...

Figure 1.3 Thermal spreading problems with adiabatic boundaries. (a) Half‐sp...

Figure 1.4 Thermal spreading problems with edge conductance. (a) Half‐space,...

Figure 1.5 Thermal spreading problems in isotropic (a), orthotropic (b), and...

Chapter 2

Figure 2.1 Thermal spreading from a circular heat source on a half‐space. (a...

Figure 2.2 Thermal spreading resistance for elliptic heat sources.

Figure 2.3 Point heat source on a half space. (a) Point source within the he...

Figure 2.4 Triangular contacts. Temperature rise at point P for a triangular...

Figure 2.5 Dimensionless thermal spreading resistance for isothermal and iso...

Figure 2.6 Rectangular contact. Source: Yovanovich (1976c)/University of Wat...

Figure 2.7 Data for selected source shapes and aspect ratios of the hyperell...

Figure 2.8 Definitions of effective aspect ratio. is the nominal aspect ra...

Figure 2.9 Results for isoflux heat sources based on the maximum or centroid...

Figure 2.10 Thermal spreading from a circular source with conductance: (a) s...

Figure 2.11 Thermal spreading from a circular source with source plane condu...

Figure 2.12 Coated half‐space with equivalent isothermal flux distribution....

Figure 2.13 Superposition of seven isoflux circular sources in a strip.

Figure 2.14 Superposition of nine isoflux circular sources arranged in a squ...

Figure 2.15 Superposition of five isoflux circular sources arranged in a squ...

Figure 2.16 Superposition of four isoflux circular sources arranged in a tri...

Figure 2.17 Approximation of continuous arbitrary contact area by discrete c...

Figure 2.18 Discretization of square contact area. Source: After Yovanovich ...

Chapter 3

Figure 3.1 Semi‐infinite flux tube with a uniform heat flux source.

Figure 3.2 Finite isotropic disk spreader.

Figure 3.3 Compound thermal spreader.

Figure 3.4 Multilayered thermal spreader.

Figure 3.5 Isotropic spreader with edge conductance.

Figure 3.6 Variation of thermal conductivity for a diffused Nickel–Copper In...

Figure 3.7 Flux tube with a linearly varying thermal conductivity layer. Sou...

Figure 3.8 Thermal spreading from a curved boundary: (a) flat contact, (b) n...

Figure 3.9 Dimensionless thermal spreading resistance, theory, and numerical...

Figure 3.10 Dimensionless thermal spreading resistance, theory, and numerica...

Chapter 4

Figure 4.1 Two dimensional semi‐infinite flux channel.

Figure 4.2 Semi‐infinite flux channel with abrupt change in cross‐section.

Figure 4.3 Three dimensional semi‐infinite flux channel.

Figure 4.4 Superpositionof solutions.

Figure 4.5 Finite rectangular flux channels: (a) two dimensional finite flux...

Figure 4.6 Compound rectangular flux channels: (a) two dimensional finite fl...

Figure 4.7 Eccentric heat source on a rectangular flux channel.

Figure 4.8 Isotropic rectangular flux channel with edge conductance.

Figure 4.9 Multi‐layered rectangular flux channel with arbitrarily located h...

Figure 4.10 Thermal spreading in a compound annular sector.

Figure 4.11 Thermal spreading from a curved boundary: (a) one dimensional fl...

Figure 4.12 Dimensionless thermal spreading resistance versus contact ratio ...

Chapter 5

Figure 5.1 Effect of co‐ordinate transformations on effective thermal conduc...

Figure 5.2 General multi‐layered orthotropic rectangular flux channel. Sourc...

Figure 5.3 Schematic of test facility developed by Gaitonde et al. (2023). L...

Figure 5.4 Two dimensional measurement of an isotropic

polytetrafluoroethyle

...

Figure 5.5 Two dimensional measurement of a Temprion

Organic Heat Spreader

(

Chapter 6

Figure 6.1 Multiple heat sources on a rectangular flux channel. Source: Culh...

Figure 6.2 Single eccentric heat source (a) and multiple heat sources (b).

Figure 6.3 Device layout for a comparative thermal analysis of a GaN HEMT de...

Figure 6.4 Thermal profile for the device shown in Figure 6.3 along with the...

Figure 6.5 Comparison of computation time for analytical versus finite eleme...

Figure 6.6 Two dimensional system considered by Hua et al. (2019) and Shen e...

Figure 6.7 Dimensionless total thermal resistance as a function of spreader ...

Figure 6.8 Thermal conductivity of semiconductor materials. Source: Modified...

Figure 6.9 Total thermal resistance of a simple two dimensional thermal spre...

Figure 6.10 Dimensionless temperature in the source plane for a semiconducto...

Figure 6.11 Modeling an elliptic heat source on a rectangular spreader using...

Chapter 7

Figure 7.1 Transient thermal spreading resistance from a circular isoflux he...

Figure 7.2 Dimensionless heat flow from convex isothermal bodies in full spa...

Figure 7.3 Dimensionless transient spreading resistance of a circular and sq...

Figure 7.4 Dimensionless transient thermal spreading resistance for an isoth...

Figure 7.5 Dimensionless transient thermal spreading resistance for an isofl...

Figure 7.6 Dimensionless transient thermal spreading resistance for two half...

Figure 7.7 Dimensionless transient thermal spreading resistance for a circul...

Figure 7.8 Dimensionless transient thermal spreading resistance for a circul...

Chapter 8

Figure 8.1 Examples of nonuniform conductance in the sink plane of a thermal...

Figure 8.2 Variable height plate fin heat sink.

Figure 8.3 Symmetric distribution of conductance for Eq. (8.4). Source: Al‐K...

Figure 8.4 Plot of (a) and (b) with their approximate heat sink fin dist...

Figure 8.5 Flux channel with a symmetric distribution of conductance and its...

Figure 8.6 Three‐dimensional flux channel with central heat source and symme...

Figure 8.7 Flux tube with variable axisymmetric conductance distribution. So...

Figure 8.8 Flux and conductance distributions considered by Al‐Khamaiseh et ...

Chapter 9

Figure 9.1 A moving (frictional heat source) in contact with a stationary su...

Figure 9.2 Moving rectangular and elliptic sources on half‐space.

Figure 9.3 Dimensionless thermal resistance for a square and circular moving...

Figure 9.4 Typical pharmaceutical patch.

Figure 9.5 Two‐dimensional strip model.

Figure 9.6 Dimensionless concentration field for , , , , and for a str...

Figure 9.7 Mass diffusion from a circular source on a semi‐infinite flux tub...

Figure 9.8 Flux channel formed from a square pillar cell in a super‐hydropho...

Figure 9.9 Comparison of hydrodynamic slip length data of Enright et al. (20...

Figure 9.10 Elliptic and rectangular pillars.

Figure 9.11 Solidification of a drop on a layered half‐space. Source: Reprod...

Figure 9.12 Impact dynamics of a drop on a layered half‐space. (a) Pre‐impac...

Figure 9.13 Experimental and theoretical time to freeze as a function of spr...

Figure 9.14 System considered by Al‐Khamaiseh et al. (2019). Source: Reprodu...

Figure 9.15 Temperature profile along ‐axis in the source plane (at ) for ...

Figure 9.16 Temperature profile along ‐axis in the source plane (at ) for ...

Figure 9.17 Thermal spreading between polar contact regions in a hollow sphe...

Chapter 10

Figure 10.1 Timeline of thermal contact resistance research. Source: Yovanov...

Figure 10.2 IBM's Thermal Conduction Module (TCM). (a) TCM module, (b) TCM p...

Figure 10.3 Thermal contact resistance between a region of two contacting su...

Figure 10.4 Thermal contact resistance TRIAD. Source: Modified from Yovanovi...

Figure 10.5 Schematics of six types of joints. (a) Smooth‐smooth non‐conform...

Figure 10.6 Schematic of a contact resistance measurement facility. Source: ...

Figure 10.7 Thermal interface material test facility. Source: Culham et al. ...

Figure 10.8 Extrapolation of flux meter temperature gradient to obtain joint...

Chapter 11

Figure 11.1 Schematic of typical joint between conforming rough surfaces. So...

Figure 11.2 Vickers micro‐hardness versus indentation diagonal for four meta...

Figure 11.3 Vickers, Brinell, and Rockwell hardness versus indentation depth...

Figure 11.4 Comparison of plastic contact conductance model and vacuum data....

Figure 11.5 Gap conductance model and data for two large parallel isothermal...

Figure 11.6 Joint conductance model and data for conforming rough stainless ...

Figure 11.7 Gap conductance model and data for conforming rough Ni 200 surfa...

Figure 11.8 Dimensionless gap conductance model and data for conforming roug...

Figure 11.9 Comparison between scale analysis model and data for conforming ...

Figure 11.10 Nondimensional thermal joint resistance. Source: Bahrami et al....

Figure 11.11 Effect of load on thermal joint resistance. Source: Bahrami et ...

Figure 11.12 Effect of nitrogen pressure on thermal joint resistance. Source...

Figure 11.13 Comparison of model with Hegazy (1985) data. Source: Bahrami et...

Figure 11.14 Comparison of model with Song (1988) data. Source: Bahrami et a...

Figure 11.15 Comparison of model with Song (1988) and Hegazy (1985) data. So...

Figure 11.16 Vickers micro‐hardness of silver layer on nickel substrate. Sou...

Figure 11.17 Dimensionless contact conductance versus relative contact press...

Figure 11.18 Effect of layer thickness and contact pressure on joint conduct...

Figure 11.19 Dimensionless joint conductance for bare and silver layer on Ni...

Figure 11.20 Effect of layer thickness for four metallic layers. Source: Ada...

Figure 11.21 Dimensionless joint conductance versus dimensionless contact pr...

Figure 11.22 Specific joint resistance versus for grease. Source: Adapted ...

Chapter 12

Figure 12.1 Joint formed by elastic contact of sphere or cylinder with smoot...

Figure 12.2 Contact between hemisphere and layer on substrate. (a) Uncoated ...

Figure 12.3 Comparison of data and model for contact between rigid hemispher...

Figure 12.4 Comparison of data and model for elastic contact between hemisph...

Figure 12.5 Comparison of data and model for elastic–plastic contact between...

Figure 12.6 Schematic of the non‐conforming smooth – rough surface contact....

Figure 12.7 Comparison between model and test results. Source: Bahrami (20...

Figure 12.8 Comparison between model and test results. Source: Bahrami (20...

Figure 12.9 Comparison between model and test results. Source: Bahrami (20...

Figure 12.10 Comparison of – test data and model. Source: Bahrami (2004)/M...

Figure 12.11 Comparison of the scale analysis model and data for the conform...

Figure 12.12 Comparison of general model with all data. Source: Bahrami et a...

Figure 12.13 Comparison between general model and non‐conforming rough data....

Figure 12.14 Contact of non‐conforming rough surfaces with presence of inter...

Figure 12.15 Micro‐gap geometry. Source: Adapted from Bahrami et al. (2004b)...

Appendix A

Figure A.1 The Gamma function.

Figure A.2 The Gaussian error function.

Figure A.3 The Bessel function .

Figure A.4 The Bessel function .

Figure A.5 The Bessel function .

Figure A.6 The Bessel function .

Figure A.7 The Legendre function .

Figure A.8 The Legendre function .

Appendix B

Figure B.1 Brinell hardness (top) and Vickers hardness (bottom) indenters an...

Figure B.2 Vickers, Brinell, and Rockwell Hardness versus indentation depths...

Figure B.3 Vickers, Brinell, and Rockwell Hardness versus indentation depths...

Figure B.4 Vickers, Brinell, and Rockwell Hardness versus indentation depths...

Figure B.5 Vickers, Brinell, and Rockwell Hardness versus indentation diagon...

Figure B.6 Normalized Vickers micro‐hardness versus Vickers diagonals for fo...

Figure B.7 Plot of Vickers coefficients versus micro‐hardness. Source: Hegaz...

Figure B.8 Typical loading and unloading curves for nano‐indentation tests. ...

Figure B.9 Final unloading curves versus reduced displacements for six mater...

Figure B.10 Schematic of loading and unloading curves versus displacements s...

Figure B.11 Normalized final unloading curves versus displacements for six m...

Figure B.12 Calculated contact areas versus contact depths for six materials...

Appendix C

Figure C.1 Contact conductance of interface materials. Source: Adapted from ...

Guide

Cover Page

Series Page

Title Page

Copyright

About the Authors

Preface

Acknowledgments

Nomenclature

Table of Contents

Begin Reading

Index

WILEY END USER LICENSE AGREEMENT

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Wiley‐ASME Press Series

Thermal Spreading and Contact Resistance: Fundamentals and Applications

Yuri S. Muzychka and M. Michael Yovanovich

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This Work is a co-publication between John Wiley & Sons, Inc. and ASME Press.

Thermal Spreading and Contact Resistance

Fundamentals and Applications

Yuri S. MuzychkaDepartment of Mechanical and Mechatronics EngineeringFaculty of Engineering and Applied ScienceMemorial University of NewfoundlandSt. John's, NL, Canada

M. Michael YovanovichUniversity of WaterlooDepartment of Mechanical and Mechatronics EngineeringFaculty of EngineeringUniversity of WaterlooWaterloo, Ontario, Canada

Copyright © 2023 by John Wiley & Sons Inc. All rights reserved.

This Work is a co-publication between John Wiley & Sons, Inc. and ASME Press.

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About the Authors

Yuri S. Muzychka is a professor of mechanical engineering at Memorial University of Newfoundland (Canada). He joined the Faculty of Engineering and Applied Science at Memorial University of Newfoundland in 2000. He completed his PhD in 1999 at the University of Waterloo. Since joining Memorial University, he has focused his research efforts in several areas of heat transfer and fluid dynamics, namely Fundamentals of Convection and Conduction Heat Transfer, Thermal Management in Electronics, Transport Phenomena in Internal Flows, Multiphase Phase Flow, and most recently Marine Icing Phenomena. He has published over 250 papers in high‐quality journals and international conference proceedings, and five book chapters. He is a Fellow of the ASME, CSME, and the Engineering Institute of Canada (EIC).

M. Michael Yovanovich is a distinguished professor emeritus at the University of Waterloo (Canada). He joined the University of Waterloo in 1969 after a short period as an associate professor at the University of Poitiers (France). He completed his mechanical engineering studies at Queens University (Canada) and obtained his ScD from the Massachusetts Institute of Technology in 1967. He devoted his early research in the area of thermal contact resistance and constriction resistance analysis. Later in his career, he was active in thermal management in electronics, conduction heat ransfer, convective heat transfer, microfluidics, and thermal contact resistance. He has published seven book chapters, given over 150 Keynote Lectures, and published over 350 papers in international journals and conference proceedings. He is a fellow of ASME, CSME, AIAA, AAAS, and RSC.

Preface

Thermal spreading and contact resistance are fundamentally important topics in heat transfer. The topic appears in virtually all thermal management applications in engineering. Applications are found in mechanical, aerospace, chemical, nuclear, and countless other engineering fields. The fundamentals of this topic have grown out of research needs in electronics cooling, aerospace systems, and tribology applications over the past 60+ years. Much of the information in this field is found in research papers, compendiums, handbooks, unpublished reports, and other reference books in varying degrees of coverage. We have set out to provide the necessary fundamentals and applications in one comprehensive reference book which will enable current practitioners to apply the state‐of‐the‐art knowledge to their problems and also enable them to go further with new problems as technology demands lead to new areas of research.

Thermal engineering covers a broad area of applications. In nearly all applications, heat enters or leaves a system through a smaller area of contact. This area of contact may be perfect or imperfect. If it is perfect, thermal spreading (or constriction) resistance occurs. If it is imperfect, then thermal transport occurs over a much smaller region of real contact within the apparent area of contact. This is contact resistance. The topics covered in this book have been drawn from over 60 years of research and development applications in many engineering fields. Each decade has brought new applications and new solutions. Early applications focused on thermal interfaces of mechanical joints, followed by electronics cooling using heat sinks for discrete heat sources. Today, we see micro‐devices such as GaN high electron mobility transistors (HEMTs), LED lighting, and three‐dimensional chip stacks. Other areas such as conduction in roller bearings and mass transfer from pharmaceutical patches also contain spreading resistance. Presently, the emerging area of super‐hydrophobic (nonwetting) surfaces requires knowledge of thermal spreading at small scales. Even common problems such as heat loss from buildings and homes into the earth (a half‐space) are modeled using thermal spreading resistance. We expect there will always be a need for the useful knowledge collected in this unique manuscript.

The book is presented primarily as a research reference. It compiles historical solutions and summarizes key results from over 60 years of published works in the field of thermal constriction (spreading) and contact resistance studies. The book is divided into 12 chapters covering the fundamentals and applications in problems where heat transfer from discrete heat sources is prevalent. The book also provides numerous simple predictive models for determining thermal spreading resistance and contact conductance of mechanical joints and interfaces. The book also discusses advanced applications in contact resistance, mass transfer, transport from super‐hydrophobic surfaces, droplet/surface phase change problems, tribology applications such as sliding surfaces and roller bearings, heat transfer in micro‐devices, micro‐electronics, and thermal spreaders used in thermal management of electronics equipment. Solutions to fundamental problems are presented with extensive details which are often left out in handbook references. Fundamental concepts are explained clearly and concisely.

Much of the material that is presented in Chapters 10–12, along with key results in Chapters 2–4, was presented in a large number of short courses and book chapters dealing with Thermal Contact Resistance. The second author (M. Michael Yovanovich) presented several 3–5 day courses on Thermal Contact Resistance at various research labs in the 1970s, 1980s, 1990s, and early 2000s. He presented his first five‐day course on Theory and Applications of Thermal Contact Resistance to the engineers and scientists at IBM Research Laboratory in Paris‐Saclay, France, in 1968. Over the next 30 years, he presented numerous two to three‐day and five‐day courses at many companies and research establishments. Over 500 engineers and physicists attended these courses with most short courses attended by 35–40 engineers from industry and government research labs. Five‐day long courses were presented to IBM Research and Manufacturing at Burlington, Vermont, Kingston and Poughkeepsie, New York; Los Alamos National Laboratory, Los Alamos, New Mexico; Nokia Research, Helsinki, Finland; and Sandia National Laboratories, Albuquerque, New Mexico. Ten 3‐day courses were sponsored by Cooling Zone of Marlborough, Massachusetts, in the late 1990s and early 2000s. Finally, numerous two to three‐day short courses presented during conferences sponsored by the American Society of Mechanical Engineers (ASME) and the American Institute of Aeronautics and Aeronautics (AIAA). The content of the courses was based on research results of about 60 graduate students (masters and PhD), two‐to‐three postdoctoral fellows and faculty members.

The funded research topics arose from the Nuclear Industry (Atomic Energy of Canada Limited in Pinawa, Manitoba and Chalk River, Ontario); Aerospace Industry (Lockheed Missiles and Space Corporation, in Palo, Alto, California; Los Alamos Laboratory in Los Alamos, New Mexico; Sandia Laboratories in Albuquerque, New Mexico); Microelectronics industries (such as Nokia Research in Helsinki, Finland, and IBM Research in Binghamton, Kingston, Poughkeepsie in New York, and Burlington in Vermont). Much of the theoretical and practical materials found in this book are based on the aforementioned sponsored research.

Though it is written primarily as a research reference, a limited number of examples are included throughout each chapter. As a research reference its purpose is to collect, summarize, and disseminate the most useful results that engineers use in the many fields of application. The analysis is detailed, allowing readers to appreciate the limitations of the theory as well as adapt it to new problems as they arise in practice. We have specifically targeted researchers and engineers working in the fields of heat transfer, thermal management of electronics, and tribology. Practitioners in other fields such as mechanical, aerospace, or nuclear engineering will also benefit, as many applications occur in these broad areas of engineering. Readers can also use it to learn solution techniques to extend and adapt theory to new problems.

The topics are distributed over 12 chapters and 3 appendices. Chapter 1 discusses the fundamental concepts and definitions used throughout the text. Chapters 2 through 8 present a thorough discussion of thermal spreading resistance fundamentals and solutions for a wide array of applications including the semi‐infinite (or half‐space) domain, flux tubes and disks, rectangular flux channels, orthotropic systems, multisource systems, transient spreading solutions, and finally problems with variable sink conductance. Chapter 9 provides extensive discussion on a number of special applications including moving heat sources, mass diffusion, superhydrophobic surfaces, temperature dependent thermal conductivity, and problems in spherical domains. The remaining three chapters, Chapters 10–12, provide comprehensive coverage of thermal contact resistance theory and its applications. Finally, we include three appendices. The first contains a concise overview of the special mathematical functions that appear in many of the solutions. The second presents useful materials pertaining to hardness which is needed in the application of contact resistance models. Finally, the third provides useful data for thermal properties of common metals, nonmetals, gases, and thermal interface materials.

Throughout each chapter, there are a handful of illustrative examples that are collected from our research experience as well as the published literature. These are provided to assist the reader in the appropriate application of the concepts and to provide some calculation benchmarks. Over 400 works spanning six decades of research have been cited. While we have made every effort to review all of the relevant historical literature, we no doubt have missed some excellent works. We strived to present the most useful and recurring material, i.e. the results used most often in practice or applications that can be easily extended using the methods outlined in the book.

Yuri S. Muzychka, St. John's, NL, Canada, 2023

M. Michael Yovanovich, Waterloo, ON, Canada, 2023

Acknowledgments

The authors would like to acknowledge the contributions of many individuals whose work contributed immensely in the preparation of this text.

The first author is deeply indebted to the second author M. Michael Yovanovich for 30 years of mentorship and collaboration that ensued from the beginning of my research career as a graduate student at the University of Waterloo. I would also like to acknowledge the support and collaborations of R. Culham (University of Waterloo), M. Hodes (Tufts University), K. Bagnall (Device Lab at MIT), M. Razavi (Memorial University), B. Al‐Khamaiseh (Memorial University), L. Lam (Tufts University, Memorial University), and S. Goudarzi (Memorial University) on the many problems in thermal spreading resistance we tackled and solved together. Additional thanks to A. Etminan for the preparation of many excellent figures, plots, and drawings appearing throughout the text.

The second author would like to thank all the students who collaborated in thermal spreading and contact resistance research over the past five decades. These include G. Schneider, S. Burde, V. Antonetti, A. Hegazy, J. DeVaal, S. Song, K. Negus, R. Culham, K. Kno, M. Sridhar, M. Mantelli, Y. Muzychka, P. Teerstra, H. Attia, V. Cecco, C. Tien, K. Martin, W. Kitscha, G. McGee, J. Zwart, J. Saabas, N. Fisher, P. Turyk, T. Lemczyk, M. Stevanovic, I. Savija, and M. Bahrami. Their contributions to this text are numerous.

Additionally, the many faculty and industry collaborators through the last five decades: H. Cordier and J. Coutenceau (University of Poitiers, France); H. Fenech, W. Rohsenow, B. Mikic, M. Cooper, and J. Henry (Massachusetts Institute of Technology, USA); L. Fletcher, G. Peterson (Texas A&M University, USA); V. Antonetti (Manhattan College, USA); J. Beck (Michigan State University, USA); G. Schneider, L. Chow, G. Gladwell, A. Strong, J. Thompson, J. Tevaarwerk, R. Culham, and D. Roulston (University of Waterloo, Canada); J. Dryden (Western University, Canada); Y. Muzychka (Memorial University of Newfoundland, Canada); M. Bahrami (Simon Fraser University, Canada); M. Mantelli and H. Milanez (Federal University of Santa Catarina, Brazil); M. Shankula (AECL Canada); and D. Wesley (Babcock and Wilcox, USA).

Nomenclature

linear dimensions (m)

radial dimensions (m)

semi‐axes of an ellipse or rectangle (m)

contact spot radius (m)

elastic contact spot radius (m)

elastic–plastic contact spot radius (m)

Hertz contact spot radius (m)

plastic contact spot radius (m)

contact spot radius for layer (m)

contact spot radius for substrate (m)

area ()

apparent contact area ()

contact area ()

gap area ()

real contact area ()

flux tube cross‐sectional area ()

Fourier coefficients

hydrodynamic slip length

thermal slip length

effective CLA roughness ()

modified Fourier coefficients

Biot number ()

specific heat ()

Fourier coefficients

equation coefficients

concentration ()

dimensionless contact conductance ()

Center Line Average roughness (m)

plate separation (m)

diameter of contacting sphere (m)

mass diffusivity of in ()

influence coefficient (K/W)

exponential function ()

eccentricity (m)

Gaussian error function

complementary error function

inverse complementary error function

complete elliptic integral of the second kind

modulus of elasticity (GPa)

effective modulus of elasticity (GPa)

exponential integral

elastic–plastic contact parameter

applied load (force) (N)

incomplete elliptic integral of the first kind

effective radiative surface factor

Fourier number ()

superposition functions

convection film coefficient or conductance ()

sink plane conductance ()

contact conductance ()

edge conductance ()

gap conductance ()

joint conductance ()

maximum conductance ()

equivalent elastic micro‐hardness (GPa)

elasto‐plastic micro‐hardness (GPa)

Brinell hardness (GPa)

layer hardness (GPa)

micro‐hardness of softer substrate (GPa)

Vickers micro‐hardness (GPa)

equivalent plastic micro‐hardness (GPa)

effective micro‐hardness of coated surface (GPa)

integrated complementary error function

polar second moment of area

modified Bessel functions of the first kind of order 0 and 1

gap conductance integral

gap conductance integral for line contact

gap conductance integral for point contact

mass flux in ‐direction ()

Bessel functions of the first kind of order 0 and 1

thermal conductivity ratio

modified Bessel functions of the second kind of order 0 and 1

thermal conductivities ()

gas thermal conductivity ()

polymer thermal conductivity ()

effective contact spot thermal conductivity ()

reaction rate ()

complete elliptic integral of the first kind

Kirchoff transform

Knudsen number ()

arbitrary depth (m)

arbitrary length scale (m)

length (m)

latent heat of fusion (J/kg)

indices for summations

Hertz parameters

surface slope

rarefaction parameter (m)

dimensionless rarefaction parameter ()

normal direction (m)

hyper‐ellipse shape parameter

contact spot density ()

number of contact spots

number of heat sources

number of sides of a polygon

contact pressure (MPa)

gap gas pressure (Pa)

Peclet number ()

sink plane conductance distribution parameter

thermal conductivity coefficients

constant uniform heat flux ()

heat flux ()

dimensionless heat flux ()

heat flow rate (W)

heat flow rate per unit depth (W/m)

cylindrical or spherical radial coordinate (m)

inscribed radius of a polygon

thermal resistance (K/W)

gap thermal resistance (K/W)

radiation thermal resistance (K/W)

contact thermal resistance (K/W)

joint thermal resistance (K/W)

thermal spreading resistance (K/W)

total thermal resistance (K/W)

bulk resistance thermal resistance (K/W)

dimensionless thermal spreading resistance

specific thermal resistance ( [])

shape factor (m)

material flow stress (GPa)

Stefan number ()

time (s)

integration variable

total and layer thicknesses (m)

temperature (K)

bulk material temperature (K)

contact temperature (K)

joint temperature (K)

source temperature (K)

contact plane surface temperature (K)

sink temperature (K)

velocity (m/s)

elastic displacement (m)

velocity (m/s)

Kirchoff transform variable

velocity of heat sliding heat source (m/s)

Cartesian coordinate (m)

heat source centroid (m)

mean plane separation (m)

Cartesian coordinate (m)

Bessel function of the second kind of order zero

Cartesian coordinate (m)

thermal spreading zone for circle ()

thermal spreading zone for ellipse ()

Greek Symbols

thermal diffusivity ()

dimensionless conductivity ratio

accommodation coefficient

semi‐axes of an ellipse (m)

equation coefficients

eigenvalues

eigenvalues ()

angular measurement

Beta function

orthotropic conductivity variable ()

specific heat ratio

Gamma function

eigenvalues

perpendicular (m)

local gap thickness (m)

penetration depth (m)

aspect ratio ()

relative contact area ()

surface emissivity

dimensionless contact strain

dummy variable

dimensionless length ()

temperature excess ( [K])

mean temperature excess (K)

centroidal temperature excess (K)

constant uniform temperature excess (K)

source temperature excess (K)

complementary modulus ()

relative conductivity ()

integration variable

eigenvalues

relative mean plane separation ()

mean free path (m)

heat flux shape parameter

coefficient of dynamic friction

dynamic viscosity (Pa s)

Poisson's ratio

dimensionless length ()

denotes arbitrary eigenvalue in spreading function

orthotropic coordinate transformation variable

hydrodynamic spreading factor

density ()

segment length

relative position in polar coordinates ()

radius of curvature (m)

radii of curvature of contacting bodies (m)

RMS surface roughness (m)

Stefan–Boltzmann constant ()

wall shear stress (Pa)

dimensionless wall shear ()

relative thickness ()

area contact ratio ()

thermal spreading function

reciprocal of thermal spreading function

thermal constriction (spreading) parameter

angular measurement

thermal elastoconstriction parameter

angular measurement

thermal conductivity coefficients

omega function in point source method

Subscripts

0

denotes at centroid or reference value

denotes layer number

length scale used to define dimensionless resistance

apparent

base

bulk

centroid

contact

contact plane

critical

effective

edge

effective value

equivalent isothermal

gap

Hertz

denotes the and sources

inner

in plane

inscribed

joint

moving

macro

micro

macro

micro

outer

polymer

‐direction

radiation

real

denotes source

denotes sink

stationary

denotes source

denotes spreading

denotes total

thermal

denotes flux tube

through plane

viscous

associated with ‐direction

‐plane

associated with ‐direction

associated with ‐direction

Superscripts

denotes dimensionless variable

denotes derivative of function where specified

denotes effective variables where specified

mean value

centroid value

denotes the and sources

isoflux

isothermal

1Fundamental Principles of Thermal Spreading Resistance

Thermal spreading (or constriction) resistance results when heat is conducted away from a finite region into a larger region (spreading resistance) or from a larger region through a smaller region (constriction resistance). Spreading or constriction resistance can be determined by solving transient or steady‐state heat conduction in a solid region with appropriately applied boundary and initial conditions. When the contact areas result in imperfect contact at the microscopic level, we find an array of discrete points of contacts results. Heat transfer across the apparent contact region occurs through conduction at these discrete points of contact and across the gap if there is a gas or liquid present; otherwise, heat transfer through radiation may occur in a vacuum. In these instances, we have contact resistance. In this text, we strive to present the general theory and applications of thermal spreading and contact resistance.

The earliest known thermal spreading solution is the classic problem of an isopotential (or electrified) disk on a semi‐infinite region (see Figure 1.1). This problem was first examined by Weber (1873) (see also Gray and Mathews (1966)). Since that time, countless other solutions have been found for semi‐infinite (half‐space) and finite regions using several techniques. Kennedy (1960) obtained solutions for circular disk spreaders in semiconductor applications. These solutions represent the earliest known results used by engineers to predict heat flow from discrete heat sources. Additional results for a rectangular heat source also appeared in Carslaw and Jaeger (1959). Throughout the last six decades, many useful solutions have been obtained by researchers to aid in calculating heat flow or source temperature in a wide array of applications.

A significant number of these thermal spreading solutions are summarized in the two heat transfer handbook chapters by one of the authors: Yovanovich (1998) and Yovanovich and Marotta (2003). More recently, Razavi et al. (2016) provided a comprehensive review of the literature between 2003 and 2016. While these compendiums are useful, the limitations of space leave most of the details of solutions for the reader in the archival literature. For this reason, we have chosen to provide a comprehensive text on this important branch of heat transfer. We will develop the solutions to the most fundamental problems in great detail, while for others provide only the essential details to assist the reader in understanding the solution process. This approach is necessary, since many problems have rather direct easy‐to‐follow solutions, while others involve many additional steps and often require some numerical analysis.

In this chapter, we will introduce the reader to the most common geometries, governing equations, boundary conditions, solution methods, and applications of thermal spreading resistance. In Chapters 2–9, we go deeper into the details of many types of thermal spreading problems and their solutions. Finally, in Chapters 10–12, we deal specifically with thermal contact resistance principles and models.

Figure 1.1 Thermal spreading (constriction resistance) from an isothermal –  source or an isoflux –  source on a semi‐infinite region (half‐space).

1.1 Applications

Applications of thermal spreading resistance are numerous and include but are not limited to: thermal management in electronics devices and systems, thermal contact (interface) resistance, tribology applications such as roller bearings, phase change phenomena for droplets on surfaces, and thermal‐fluid transport on super‐hydrophobic surfaces and channels. Several examples are provided in Figure 1.2.

One of the earliest applications of thermal spreading/constriction resistance theory is in the area of thermal contact resistance [Yovanovich (1998)]. When two surfaces are brought into contact, one or more discrete points of contact result due to the geometry and/or inherent roughness of surfaces. The real contact area is much smaller than the nominal contact area and heat flow is disrupted while passing from one surface into the other surface, leading to an interface or contact resistance. This resistance is typically modeled using a contact conductance . This contact conductance depends on the mechanical loading of the surfaces (contact pressure), the surface properties (roughness), mechanical properties (hardness, Young's modulus), and thermal properties of the materials in contact (thermal conductivity).

The microelectronics cooling area is rich in applications involving discrete heat sources at package scale which requires thermal management using heat sinks. The heat sink baseplate acts as a thermal spreader for one or more heat sources, and usually has fins which are convectively cooled. These fins may be reduced to a single value of conductance and an appropriate thermal spreading model used to predict the average (or maximum) source temperature or the maximum heat transfer rate for a temperature‐constrained application.

At the microscale, device miniaturization produces thumbnail‐sized electronic chips with many distributed discrete heat sources. Some examples are high electron mobility transistor (HEMT) devices and LED lighting. Analysis of these devices using analytical methods has been quite fruitful, and many useful solutions are available for a wide range of applications.

Other heat transfer applications arise in condensation, evaporation, or solidification problems involving droplets on surfaces. In these applications, the substrate provides a conduction path for heat and controls the phase change process. These surfaces may be treated or coated with a thin resistive layer, i.e. paints or other protective or even thermally enhancing coatings. Other interesting applications involve superhydrophobic surfaces and channels which have precisely engineered microstructural surface features such as regularly dispersed pillars or transverse or longitudinal ridges. Thermal transport from these complex surfaces relies upon thermal spreading resistance solutions to predict the heat transfer characteristics across these complex boundaries into the fluid in contact.

In tribology applications where sliding or rolling contact occurs between surfaces, thermal spreading resistance plays an important role in predicting the microcontact temperatures. In these applications, one surface is often modeled as rough and flat and the other surface as smooth and flat. Heat generated through sliding or rolling friction is conducted into each surface, one of which is assumed to be in motion relative to the other surface. In this instance, a sliding thermal spreading resistance and stationary thermal spreading resistance are required to predict the average surface contact temperature. In applications involving roller bearings, thermal constriction zones are created between the ball and race.

Figure 1.2 Applications of thermal spreading resistance. From top left to bottom: (a) thermal contact resistance. Source: Hegazy (1985), (b) ball bearings. Source: Scanrail/Adobe Stock, (c) device cooling on heat sinks, Lee et al. (1995). Used with the permission of the American Society of Mechanical Engineers (ASME). (d) LED lighting, (e) phase change with droplets on surfaces. Source: Monsterkoi/Pixabay, and (f) super‐hydrophobic channels, Karamanis et al. (2018). Used with the permission of the American Society of Mechanical Engineers (ASME).

Additional applications can be found in other fields and will be dealt with as we progress through the text. One unique application is in mass diffusion from discrete regions containing a chemical species such as in pharmaceutical patches. With some modification, thermal spreading solutions can be easily used to model analogous mass diffusion applications.

1.2 Semi‐Infinite Regions, Flux Tubes, Flux Channels, and Finite Spreaders