Macro- to Microscale Heat Transfer: The Lagging Behavior

Physical processes taking place in micro/nanoscale strongly depend on the material types and can be very complicated. Known approaches include kinetic theory and quantum mechanics, non-equilibrium and irreversible thermodynamics, molecular dynamics, and/or fractal theory and fraction model. Due to innately different physical bases employed, different approaches may involve different physical properties in describing micro/nanoscale heat transport. In addition, the parameters involved in different approaches, may not be mutually inclusive. Macro- to Microscale Heat Transfer: The Lagging Behavior, Second Edition continues the well-received concept of thermal lagging through the revolutionary approach that focuses on the finite times required to complete the various physical processes in micro/nanoscale. Different physical processes in heat/mass transport imply different delay times, which are common regardless of the material type. The delay times, termed phase lags, are characteristics of materials. Therefore the dual-phase-lag model developed is able to describe eleven heat transfer models from macro to nanoscale in the same framework of thermal lagging. Recent extensions included are the lagging behavior in mass transport, as well as the nonlocal behavior in space, bearing the same merit of thermal lagging in time, in shrinking the ultrafast response down to the nanoscale. Key features: Takes a unified approach describing heat and mass transport from macro, micro to nanoscale Compares experimental results for model validation Includes easy to follow mathematical formulation Accompanied by a website hosting supporting material  Macro- to Microscale Heat Transfer: The Lagging Behavior, Second Edition is a comprehensive reference for researchers and practitioners, and graduate students in mechanical, aerospace, biological and chemical engineering.

Table of Contents:
Preface xi Nomenclature xiii 1 Heat Transport by Phonons and Electrons 1 1.1 Challenges in Microscale Heat Conduction 2 1.2 Phonon–Electron Interaction Model 5 1.2.1 Single Energy Equation 10 1.3 Phonon-Scattering Model 11 1.3.1 Operator Method 13 1.3.2 Phonon Hydrodynamics 15 1.4 Phonon Radiative Transfer Model 18 1.5 Relaxation Behavior in Thermal Waves 24 1.5.1 Engineering Assessment of the Relaxation Time 26 1.5.2 Admissibility with Phonon Radiative Transport Phenomena 27 1.6 Micro/Nanoscale Thermal Properties 28 1.6.1 Heat Capacity 29 1.6.2 Thermal Conductivity 30 1.6.3 Normal and Umklapp Relaxation Times 34 1.7 Size Effect 37 1.8 Phase Lags 51 References 56 2 Lagging Behavior 61 2.1 Phase-Lag Concept 62 2.2 Internal Mechanisms 64 2.3 Temperature Formulation 66 2.4 Heat Flux Formulation 69 2.5 Methods of Solutions 70 2.5.1 Method of Laplace Transform 73 2.5.2 Separation of Variables 82 2.5.3 Method of Fourier Transform 87 2.6 Precedence Switching in Fast-Transient Processes 90 2.7 Rate Effect 91 2.8 Problems Involving Heat Fluxes and Finite Boundaries 92 2.9 Characteristic Times 99 2.10 Alternating Sequence 103 2.11 Determination of Phase Lags 104 2.12 Depth of Thermal Penetration 108 Appendix 2.1 FORTRAN Code for the Riemann-Sum Approximation of Laplace Inversion 117 Appendix 2.2 Mathematica Code for Calculating the Depth of Thermal Penetration 122 References 122 3 Thermodynamic and Kinetic Foundation 125 3.1 Classical Thermodynamics 126 3.2 Extended Irreversible Thermodynamics 131 3.3 Lagging Behavior 135 3.4 Thermomechanical Coupling 137 3.4.1 Rigid Conductors 141 3.4.2 Isothermal Deformation 142 3.5 Dynamic and Nonequilibrium Temperatures 143 3.6 Conductive and Thermodynamic Temperatures 146 3.7 Kinetic Theory 149 References 156 4 Temperature Pulses in Superfluid Liquid Helium 159 4.1 Second Sound in Liquid Helium 160 4.2 Experimental Observations 163 4.3 Lagging Behavior 164 4.4 Heating Pulse in Terms of Fluxes 167 4.5 Overshooting Phenomenon of Temperature 172 4.6 Longitudinal and Transverse Pulses 181 4.6.1 Lamé Potential 182 4.6.2 Helmholtz Potential 183 References 190 5 Ultrafast Pulse-Laser Heating on Metal Films 193 5.1 Experimental Observations 194 5.2 Laser Light Intensity 196 5.2.1 Gaussian Distribution 196 5.2.2 Alternate Form of Light Intensity 197 5.3 Microscopic Phonon–Electron Interaction Model 200 5.4 Characteristic Times – The Lagging Behavior 202 5.5 Phase Lags in Metal Films 204 5.6 Effect of Temperature-Dependent Thermal Properties 210 5.7 Cumulative Phase Lags 211 5.8 Conduction in the Metal Lattice 213 5.9 Multiple-Layered Films 219 5.9.1 Mixed Formulation 220 5.9.2 Initial Conditions for Heat Flux 221 5.9.3 Laplace Transform Solution 222 5.9.4 Surface Reflectivity 224 References 228 6 Nonhomogeneous Lagging Response in Porous Media 231 6.1 Experimental Observations 232 6.2 Mathematical Formulation 234 6.3 Short-Time Responses in the Near Field 236 6.4 Two-Step Process of Energy Exchange 240 6.5 Lagging Behavior 241 6.6 Nonhomogeneous Phase Lags 243 6.7 Precedence Switching in the Fast-Transient Process 249 References 253 7 Thermal Lagging in Amorphous Media 255 7.1 Experimental Observations 256 7.2 Fourier Diffusion: The t–1/2 Behavior 258 7.3 Fractal Behavior in Space 259 7.4 Lagging Behavior in Time 262 7.4.1 Classical Diffusion, Z = 1 264 7.4.2 Partial Expansions 265 7.4.3 Riemann-Sum Approximation 265 7.4.4 Real-Time Responses 269 7.5 Thermal Control 271 References 279 8 Material Defects in Thermal Processing 281 8.1 Localization of Heat Flux 282 8.1.1 Microcracks 284 8.2 Energy Transport around a Suddenly Formed Crack 288 8.3 Thermal Shock Formation – Fast-Transient Effect 290 8.3.1 Asymptotic Analysis 291 8.3.2 Subsonic Regime with M< 1 294 8.3.3 Supersonic Regime with M> 1 298 8.3.4 Transonic Stage with M= 1 301 8.4 Diminution of Damage – Microscale Interaction Effect 304 8.4.1 Eigenvalues 308 8.4.2 Eigenfunctions 308 8.5 High Heat Flux around a Microvoid 311 8.5.1 Mathematical Formulation 312 8.5.2 Linear Decomposition 314 8.5.3 Steady-State Solution 315 8.5.4 Fast-Transient Component 317 8.5.5 Flux Intensification 319 References 324 9 Lagging Behavior in other Transport Processes 327 9.1 Film Growth 328 9.1.1 Lagging Behavior 330 9.1.2 Thermal Oxidation of Silicon 336 9.1.3 Intermetallics 340 9.2 Thermoelectricity 343 9.2.1 Thermoelectric Coupling 344 9.2.2 Lagging Behavior 346 9.2.3 Dominating Parameters 348 9.3 Visco/Thermoelastic Response 351 9.4 Nanofluids 352 References 355 10 Lagging Behavior in Biological Systems 359 10.1 Bioheat Equations 360 10.1.1 Two-Equation Model 360 10.1.2 Three-Equation Model 363 10.2 Mass Interdiffusion 370 10.3 Lagging Behavior 376 10.3.1 Rapidly Stretched Springs 376 10.3.2 One-Dimensional Fins 378 References 379 11 Thermomechanical Coupling 381 11.1 Thermal Expansion 382 11.1.1 Mechanically Driven Cooling Phenomenon 385 11.1.2 Thermomechanical Coupling Factor 386 11.1.3 Apparent Thermal Conductivity 388 11.2 Thermoelastic Deformation 388 11.3 Mechanically Driven Cooling Waves 391 11.3.1 Heat Transport by Diffusion 396 11.3.2 Heat Transport by Thermal Waves 398 11.3.3 Lagging Behavior in Heat Transport 406 11.4 Thermal Stresses in Rapid Heating 408 11.4.1 Diffusion 413 11.4.2 CV Waves 414 11.4.3 Lagging Behavior 417 11.5 Hot-Electron Blast 419 References 422 12 High-Order Effect and Nonlocal Behavior 425 12.1 Intrinsic Structures of T Waves 426 12.1.1 Thermal Relaxation of Electrons 427 12.1.2 Relaxation of Internal Energy 431 12.1.3 Propagation of T Waves 436 12.1.4 Effect of τT 2 439 12.1.5 Effect of Microvoids on the Amplification of T Waves 443 12.2 Multiple Carriers 447 12.2.1 Two-Carrier System 448 12.2.2 Three-Carrier System 449 12.2.3 N-Carrier System 452 12.3 Thermal Resonance 453 12.4 Heat Transport in Deformable Conductors 458 12.4.1 Energy Equation 459 12.4.2 Momentum Equation 472 12.5 Nonlocal Behavior 473 12.5.1 Nonlocal Lengths 475 12.5.2 Thermomass Model 478 12.5.3 Deformable Conductors 486 12.5.4 Effect of Dual Conduction 488 References 490 13 Numerical Methods 491 13.1 Neumann Stability 492 13.1.1 Interfacial Resistance 495 13.2 Finite-Difference Differential Formulation 501 13.2.1 Mixed Formulation 503 13.3 Hot-Electron Blast 507 13.3.1 Full Coupling 520 13.4 Thermoelectric Coupling 531 13.4.1 The Case of Constant J 531 13.4.2 The Case of Constant E 533 Appendix 13.1 Mathematica Code for the Finite-Difference Differential Method: Equations (13.23)–(13.26) 535 Appendix 13.2 Mathematica Code for the Finite-Difference Differential Method: Equations (13.35), (13.37), and (13.38) 537 Appendix 13.3 Mathematica Code (V5.0) for the Finite-Difference Differential Method: Equations (13.51) and (13.52) 539 Appendix 13.4 Mathematica Code (V5.0) for the Finite-Difference Differential Method: Equations (13.62), (13.63) and (13.52) 541 Appendix 13.5 Mathematica Code (V5.0) for the Finite-Difference Differential Method: Equations (13.68) and (13.66) 543 Appendix 13.6 Mathematica Code (V5.0) for the Finite-Difference Differential Method: Equations (13.69) and (13.66) 544 References 545 Index 547

About the Author :
D. Y. “Robert” Tzou is a James C. Dowell Professor of Engineering in the Mechanical and Aerospace Engineering Department at the University of Missouri. He served as the Department Chairman in 1997-2012, and has become the Associate Dean for Academic Programs in the College of Engineering since 2012. Tzou is a Fellow of American Society of Mechanical Engineers. His research is in the general area of ultrafast thermomechanics, in which he delivered keynote speeches at a number of national and international professional society conferences and published more than 150 journal/conference articles. His research has been financed by the National Science Foundation, the Air Force Office of Scientific Research, the Army Research Office, Sandia National Laboratories, Los Alamos National Laboratory and the Air Force Research Laboratory. Tzou has pioneered on the lagging behavior during ultrafast heat/mass transport in micro/nanoscale since 1997. He is the founding chair for the ASME International Conference on Micro/Nanoscale Heat Transfer, http://www.asmeconferences.org/MNHMT2013/Organizers.cfm. Tzou worked at the University of New Mexico and Lehigh University before joining the Missouri faculty.