Thermal Management of Microelectronic Equipment Heat Transfer Theory, Analysis Methods, and Design Practices

THERMAL MANAGEMENT OF MICROELECTRONIC EQUIPMENT HEAT TRANSFER THEORY, ANALYSIS METHODS, AND DESIGN PRACTICES

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use ASME Press Book Series on Electronic Packaging Dereje Agonafer, Editor-in-Chief

Associate Technical Editors Administrative Editors

Cristina Amon, Carnegie Mellon University Damena Agonafer A. Haji-Sheikh, University of Texas, Arlington University of Texas, Arlington Yung-Cheng Lee, University of Colorado, Boulder Wataru Nakayama, ThermTech International Senayet Agonafer Shlomo Novotny, Sun Microsystems Princeton University Al Ortega, University of Arizona, Tucson Donald Price, Raytheon Electronic Systems Viswam Puligandla, Nokia Koneru Ramakrishna, Motorola, Inc. Gamal Refai-Ahmed, Ceyba Inc. Bahgat Sammakia, SUNY, Binghamton Roger Schmidt, IBM Masaki Shiratori, Yokohama National University Suresh Sitaraman, Georgia Institute of Technology Ephraim Suhir, Iolon, Inc.

About the Series

Electronic packaging is experiencing unprecedented growth, as it is the key enabling technology for applications ranging from computers and telecommunications, to automobiles and consumer products. Although technology improvements are still possible, these solutions are becoming quite expensive. In addition, technology enhancements seem to be reaching physics-based limits. Therefore, packaging can present an opportunity for performance improvements without the need for new CMOS technology. At InterPACK 1999, the flagship conference of the Electronic and Photonic Packaging Division, the electronic packaging business worldwide was estimated to be over $100 billion per year. The rapidly changing technology necessitates current and up-to-date technical knowledge on the subject for both practicing engineers and academic researchers. It was with this background that this book series was initiated.

The book series will cover broad topics in packaging ranging from electronic cooling, interconnects, thermo/mechanical challenges, including thermally induced stress and vibration, and various aspects of failures in electronic systems. The important field of design tools, including thermal, mechanical and electrical, and the corresponding important issues of integration of these various design tools will also be topics for future books. Other topics include the growing field of micro-electronics mechanical systems, optoelectronics, and nanotechnol- ogy packaging. Please contact the Editor-in-Chief or one of the outstanding associate technical editors listed above, or Mary Grace Stefanchik of ASME Press should you be interested in suggesting a particular book topic or have an interest in authoring a book.

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use THERMAL MANAGEMENT OF MICROELECTRONIC EQUIPMENT

HEAT TRANSFER THEORY, ANALYSIS METHODS, AND DESIGN PRACTICES

L. T. Yeh, Ph.D., P.E. R. C. Chu

ASME Press Book Series on Electronic Packaging Dereje Agonafer, Editor-in-Chief

ASME PRESS NEW YORK 2002

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All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.

Statement from By-Laws: The Society shall not be responsible for statements or opinions advanced in papers . . . or printed in its publications (B7.1.3)

INFORMATION CONTAINED IN THIS WORK HAS BEEN OBTAINED BY THE AMERI- CAN SOCIETY OF MECHANICAL ENGINEERS FROM SOURCES BELIEVED TO BE RELIABLE. HOWEVER, NEITHER ASME NOR ITS AUTHORS OR EDITORS GUARAN- TEE THE ACCURACY OR COMPLETENESS OF ANY INFORMATION PUBLISHED IN THIS WORK. NEITHER ASME NOR ITS AUTHORS AND EDITORS SHALL BE RESPON- SIBLE FOR ANY ERRORS, OMISSIONS, OR DAMAGES ARISING OUT OF THE USE OF THIS INFORMATION. THE WORK IS PUBLISHED WITH THE UNDERSTANDING THAT ASME AND ITS AUTHORS AND EDITORS ARE SUPPLYING INFORMATION BUT ARE NOT ATTEMPTING TO RENDER ENGINEERING OR OTHER PROFESSIONAL SER- VICES. IF SUCH ENGINEERING OR PROFESSIONAL SERVICES ARE REQUIRED, THE ASSISTANCE OF AN APPROPRIATE PROFESSIONAL SHOULD BE SOUGHT.

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Library of Congress Cataloging-in-Publication Data

Yeh, L.-T. (Lian-Tuu), 1944- Thermal management of microelectronic equipment: heat transfer theory, analysis methods, and design practices / L.-T. Yeh and R. C. Chu. p. cm. ISBN 0-7918-0168-3 1. Electronic apparatus and appliances – Cooling. 2. Electronic apparatus and appliances – Thermal properties. 3. Heat – Transmission. I. Chu, R. C. (Richard C.), 1933. II. Title.

TK7870.25.Y44 2002 621.381’04 – dc21 2001034086

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List of Figures xi List of Tables xvii Nomenclature xix Foreword xxiii Preface xxv

Chapter 1 Introduction 1 1.1 Need for Thermal Control ...... 1 1.2 Reliability and Temperature...... 3 1.3 Levels of Thermal Resistance...... 4 1.4 Thermal Design Considerations ...... 5 1.5 Optimization and Life-Cycle Cost ...... 6

Chapter 2 Conduction 9 2.1 Fundamental Law of Heat Conduction ...... 9 2.2 General Differential Equations for Conduction...... 10 2.3 One-Dimensional Heat Conduction...... 16 2.4 Thermal/Electrical Analogy ...... 17 2.5 Lumped-System Transient Analysis ...... 20 2.6 Heat Conduction with Phase Change...... 25

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Chapter 3 Convection 31 3.1 Flow and Temperature Fields...... 31 3.2 Heat Transfer Coefficient ...... 34 3.3 Parameter Effects on Heat Transfer...... 35 3.4 Pressure Drop and Friction Factor...... 43 3.5 Thermal Properties of Fluids ...... 46 3.6 Correlations for Heat Transfer and Friction ...... 47

CHAPTER 4 RADIATION 53 4.1 Stefan-Boltzmann Law ...... 53 4.2 Kirchhoff’s Law and Emissivity ...... 54 4.3 Radiation Between Black Isothermal Surfaces...... 55 4.4 Radiation Between Gray Isothermal Surfaces...... 58 4.5 Extreme Climatic Conditions ...... 61

Chapter 5 Pool Boiling 67 5.1 Boiling Curve...... 67 5.2 Nucleate Boiling...... 70 5.3 Incipient Boiling at Heating Surfaces ...... 72 5.4 Nucleate Boiling Correlations ...... 76 5.5 Critical Heat Flux Correlations ...... 77 5.6 Minimum Heat Flux Correlations (Leidenforst Point)...... 79 5.7 Parameters Affecting Pool Boiling...... 81 5.8 Effect of Gravity on Pool Boiling ...... 87

Chapter 6 Flow Boiling 95 6.1 Flow Patterns ...... 95 6.2 Heat Transfer Mechanisms ...... 95 6.3 Boiling Crisis ...... 98 6.4 Heat Transfer Equations ...... 99 6.5 Thermal Enhancement ...... 109 6.6 Pressure Drop ...... 109

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Chapter 7 Condensation 115 7.1 Modes of Condensation ...... 115 7.2 Heat Transfer in Filmwise Condensation...... 116 7.3 Improvements Over Nusselt Analysis...... 121 7.4 Condensation Inside a Horizontal Tube ...... 123 7.5 Noncondensable Gas in a Condenser ...... 127

Chapter 8 Extended Surfaces 131 8.1 Uniform–Cross Section Fins ...... 131 8.2 Fin Efficiency ...... 134 8.3 Selection and Design of Fins ...... 137

Chapter 9 Thermal Interface Resistance 141 9.1 Factors Affecting Thermal Contact Resistance...... 141 9.2 Joint Thermal Contact Resistance ...... 145 9.3 Methods of Reducing Thermal Contact Resistance ...... 147 9.4 Solder and Epoxy Joints...... 159 9.5 Practical Design Data...... 160

Chapter 10 Components and Printed Circuit Boards 169 10.1 Chip Packaging Technology ...... 169 10.2 Chip Package Thermal Resistance...... 172 10.3 Chip Package Attachment...... 173 10.4 Board-Cooling Methods ...... 176 10.5 Board Thermal Analysis ...... 177 10.6 Equivalent Thermal Conductivity...... 178

Chapter 11 Direct Air Cooling and Fans 185 11.1 Previous Work ...... 185 11.2 Heat Transfer Correlations ...... 187 11.3 Pressure Drop Correlations...... 190 11.4 Heat Transfer Enhancement...... 194 11.5 Fans and Air-Handling Systems...... 197

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Chapter 12 Natural and Mixed Convection 213 12.1 Parallel Plates ...... 214 12.2 Straight-Fin Arrays ...... 220 12.3 Pin-Fin Arrays ...... 229 12.4 Enclosures...... 234 12.5 Mixed Convection in Vertical Plates ...... 237

Chapter 13 Heat Exchangers and Cold Plates 243 13.1 Compact Heat Exchangers...... 243 13.2 Flow Arrangement of Heat Exchangers ...... 244 13.3 Overall Heat Transfer Coefficient ...... 244 13.4 Heat Exchanger Effectiveness ...... 245 13.5 Heat Exchanger Analysis ...... 246 13.6 Heat Transfer and Pressure Drop ...... 248 13.7 Geometric Factors ...... 250 13.8 Cold-Plate Analysis...... 251 13.9 Correlations...... 255

Chapter 14 Advanced Cooling Technologies I: Single-Phase Liquid Cooling 261 14.1 Coolant Selection...... 261 14.2 Natural Convection...... 265 14.3 Forced Convection ...... 267

Chapter 15 Advanced Cooling Technologies II: Two-Phase Flow Cooling 283 15.1 Figure of Merit...... 283 15.2 Direct-Immersion Cooling ...... 285 15.3 Enhancement of Pool Boiling ...... 287 15.4 Flow Boiling ...... 300

Chapter 16 Heat Pipes 309 16.1 Operation Principles ...... 309 16.2 Useful Characteristics...... 309

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16.3 Construction...... 311 16.4 Operation Limits ...... 312 16.5 Materials Compatibility ...... 318 16.6 Operating Temperatures ...... 320 16.7 Operation Methods ...... 321 16.8 Thermal Resistances...... 323 16.9 Applications ...... 325 16.10 Micro Heat Pipes ...... 330

Chapter 17 Thermoelectric Coolers 335 17.1 Basic Theories of Thermoelectricity ...... 335 17.2 Net Thermoelectric Effect...... 337 17.3 Figure of Merit...... 338 17.4 Operation Principles ...... 339 17.5 System Configurations...... 339 17.6 Performance Analysis ...... 340 17.7 Practical Design Procedure...... 344

Appendices 349 A. Material Thermal Properties ...... 349 B. Thermal Conductivity of Silicon and Gallium Arsenide...... 351 C. Properties of Air, Water, and Dielectric Fluids ...... 353 D. Typical Emissivities of Materials ...... 371 E. Solar Absorptivities and Emissivities of Common Surfaces ...... 373 F. Properties of Phase-Change Materials...... 375 G. Friction Factor Correlations...... 377 H. Heat Transfer Correlations ...... 381 I. Units Conversion Table ...... 403

Index 405

About the Authors 413

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1.1 Increase in circuit complexity...... 2 1.2 Major causes of electronics failures...... 3 1.3 Effect of part placement on component junction temperature...... 6 1.4 Example of current thermal design practice...... 7 2.1 Cylindrical and spherical coordinates...... 12 2.2 Thermal resistance factor Φ at heat source center for rectangular solids with W = L...... 13 2.3 Thermal resistance factor Φ at heat source center for cylindrical solids...... 14 2.4 Thermal network for a multilayer composite wall...... 19 2.5 Temperature profile in a thermal-energy storage system...... 25 2.6 Moving solid-liquid interface for melting and solidification...... 26 2.7 Temperature profile with natural convection in liquid region...... 27 2.8 Cross section of microencapsulated phase-change material particle. . . . . 28 3.1 Boundary-layer development for fluid in a tube...... 32 3.2 Flow regimes for flow over a flat plate...... 33 3.3 Boundary-layer formation and separation on a circular cylinder in crossflow...... 33 3.4 Velocity profile and flow separation for a circular cylinder in crossflow...... 34 3.5 Boundary layer over a vertical plate in natural convection...... 39 3.6 Friction factor versus Reynolds number for round tubes...... 45 4.1 Geometrical relation between two radiative surfaces...... 55 4.2 View-factor algebra for pairs of surfaces...... 56 4.3 Hottel crossed-string method...... 58 4.4 Equivalent network for radiative heat exchange between two gray surfaces...... 59 4.5 Solar heat flux and ambient air temperature...... 62 5.1 Constant-pressure heating process...... 68 5.2 Regimes in pool-boiling heat transfer...... 68 5.3 Boiling mechanism for heat flux–controlled and temperature-controlled conditions...... 70 5.4 Nucleation from a cavity...... 71 5.5 Bubble shapes with various values of contact angle (surface tension).. . . 72 5.6 Vapor trapped in cavities by advancing liquid...... 73

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5.7 Conditions required for nucleation in a temperature gradient...... 74 5.8 Schematic representation of wall superheat excursion (overshoot). . . . . 75 5.9 Predictions of critical heat flux for various heater configurations...... 78 5.10 Effect of pressure on pool-boiling curve...... 82 5.11 Influence of subcooling on pool boiling...... 83 5.12 Effect of surface roughness on boiling curve...... 84 5.13 Effect of dissolved gas on boiling curve...... 85 5.14 Effect of dissolved gas and subcooling on CHF under atmospheric pressure...... 86 5.15 Effect of agitation in methanol boiling at 1 atm...... 87 5.16 Thermal hysteresis in pool boiling of R-113...... 88 5.17 Liquid nitrogen pool-boiling curve at earth and near zero gravity...... 89 5.18 Reduced-gravity CHF data for horizontal surfaces and wires...... 90 5.19 Reduced-gravity CHF data for vertical wires and spheres...... 90 6.1 Typical flow patterns for a horizontal tube...... 96 6.2 Heat transfer mechanisms and flow regions in upward vertical flow...... 97 6.3 Heat transfer regions under various heat fluxes...... 98 6.4 Effects of velocity and subcooling on flow boiling...... 101 6.5 F factor for Chen correlation...... 103 6.6 S factor for Chen correlation...... 103 ∆ ∆ 6.7 Relationship between p and Tsat...... 104 6.8 Parametric effects on critical heat flux...... 105 6.9 Boiling in smooth and multilead ribbed tubes...... 110 7.1 Velocity and temperature profiles in liquid film...... 117 7.2 Film condensation in horizontal tubes...... 120 7.3 Heat transfer for condensation on a vertical plate...... 122 7.4 Baker flow regime map (horizontal flow)...... 124 7.5 F factor in a horizontal tube...... 125 7.6 Circumferentially local heat transfer coefficient...... 126 7.7 Effect of noncondensable gas in condensation...... 127 8.1 Model for fin analysis...... 132 8.2 Various types of fin configuration...... 135 8.3 Fin efficiency of commonly used fins...... 137 9.1 Two nominally flat surfaces in contact...... 142 9.2 Various types of surfaces in contact...... 143 9.3 Effect of pressure on interface resistance...... 144 9.4 Increasing contact surface with same projected area...... 148 9.5 Contact resistance versus various foil thicknesses...... 150 9.6 Insertion of a filler at interface...... 151 9.7 Contact of bare and coated joints...... 152 9.8 Thermal contact conductance of various coating materials...... 155 9.9 Effect of metallic coating thickness on contact conductance...... 156 9.10 Insertion of phase-change materials at interface...... 157 9.11 Microcapillary thermal interface concept...... 158 9.12 Measured and calculated thermal resistances of an epoxy joint...... 160 9.13 Contact resistance for transistor mounting...... 162 9.14 Thermal contact conductance for various materials...... 163

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10.1 Scale comparison of MCM and SCP. (a) Dual in-line package (DIP) and pin-grid array. (b) MCM containing various types of individual chips...... 170 10.2 Drawings of common packages: (a) DIP; (b) PGA...... 171 10.3 Interconnection between pin and board: (a) DIP; (b) PGA...... 172 10.4 Thermal conductivity and CTE of various materials...... 174 10.5 Thermal conductivity and CTE of copper-invar...... 175 10.6 Thermal path for a common board in an electronic box...... 176 10.7 Typical PCB construction...... 178 10.8 Dimensions of board and card guides for Example 10.2...... 181 11.1 Array of rectangular modules in a channel...... 186 11.2 Airflow over the PCB of Example 11.1...... 192 11.3 Local heat transfer coefficient over the DIPs of Example 11.1...... 193 11.4 Diagrams of heat sinks applied to PGAs: (a) pin-fin, (b) disk-fin, and (c) straight-fin heat sinks...... 195 11.5 Flow patterns in a heat sink: (a) free plates, s >> δ (L); (b) developing flow, s ≈ δ (L); (c) interfering, s << δ (L)...... 196 11.6 Typical fan curves and system operation point...... 199 11.7 Range of specific speed for various types of fan...... 201 11.8 Effect of altitude or air density on system operation point...... 201 11.9 Performance of fans in series or parallel...... 202 11.10 Turbulent flow pattern over a sharp bend...... 205 11.11 Flow in sharp bends with various types of vanes installed: (a) concentric splitters; (b) circular vanes; (c) airfoil vanes...... 205 11.12 Vanes in a diffuser...... 206 11.13 Improvement on vanes in a diffuser...... 208 11.14 Four types of manifolds: (a) inlet manifold (divided manifold), (b) outlet manifold (combined manifold), (c) S-type manifold (parallel flow), and (d) U-type manifold (reverse flow)...... 208 11.15 Pressure distribution in manifolds...... 209 12.1 Flow between parallel plates...... 214 12.2 Natural convection between vertical parallel plates: (a) nondimensional form; (b) dimensional form...... 217 12.3 Effects of edge blockage on heat transfer in parallel plates...... 218 12.4 Continuous and discrete vertical plates: (a) continuous-plate array; (b) in-line discrete-plate array; (c) staggered discrete-plate array. . . . . 220 12.5 Comparisons between staggered discrete and continuous parallel plates: (a) heat transfer rates vs. (Dh/H) Ra; (b) inlet velocity vs. (Dh/H) Ra...... 221 12.6 Heat transfer rate ratios vs. (Dh/H) Ra at fixed ∆T and area: (a) in-line discrete plates vs. continuous parallel-plate channel; (b) in-line discrete plates vs. staggered discrete plates...... 222 1/2 12.7 Temperature ratios vs. (Dh/H)(Ram) at fixed Q and area: (a) in-line discrete plates vs. continuous parallel-plate channel; (b) in-line discrete plates vs. staggered discrete plates...... 223 12.8 Array height ratios vs.Qˆ at fixed Q and ∆T: (a) in-line discrete plates vs. staggered discrete plates; (b) in-line discrete plates vs. continuous parallel-plate channel...... 224

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12.9 A vertical straight-fin array...... 225 12.10 U-channel apparent emittance εa for different values of ε...... 227 12.11 Horizontal fin array configuration...... 228 12.12 Heat transfer coefficients for pin-fin arrays: (a) arrays with vertical base plate; (b) arrays facing up; (c) arrays facing down...... 230 12.13 Pin-fin configurations...... 233 12.14 Dimensionless shape factors for conduction...... 236 12.15 Flow reversal of buoyancy-assisted flow in vertical plates...... 238 12.16 Reversed-flow zone in buoyancy-assisted flow...... 238 12.17 Nusselt numbers and flow reversal for isoflux-isoflux...... 240 13.1 Effectiveness of parallel- and counterflow heat exchangers...... 248 13.2 Effectiveness of various flow arrangements at Cmin/Cmax = 1...... 249 13.3 Sketch of a heat exchanger...... 251 13.4 Electronics on a finned cold plate...... 252 13.5 Cross-sectional view of the heat exchanger of Example 13.1...... 256 14.1 Sketch of a counterflow system for electronic cooling...... 267 14.2 Cold-wall temperature distribution for unidirectional and counterflow...... 268 14.3 Schematic of heat source array with finned pins...... 269 14.4 Average Nusselt number for unaugmented heat source array...... 270 14.5 Average Nusselt number for each row of pin-fin heat source...... 271 14.6 Heat transfer data for smooth and microstud surfaces...... 272 14.7 Fin dimensions and cutaway view of thick offset fins...... 273 14.8 Rectangular (thin) offset fins...... 273 14.9 Friction factor and Nusselt number for offset fins in a narrow passage...... 274 14.10 Friction factor vs. Reynolds number in microchannels...... 276 14.11 Variations of heat transfer coefficient with wall temperature...... 277 15.1 Pool-boiling data for a plain tube...... 286 15.2 Typical boiling for small heaters immersed in R113...... 288 15.3 Pool-boiling curves for plain and ABM surfaces...... 289 15.4 Bubble generation heater and bubble paths for various spacings. . . . 290 15.5 Falling liquid film...... 291 15.6 Liquid-vapor exchange at a nucleation site. (a) Pool boiling on a vertical surface. (b) Boiling in a falling liquid film...... 291 15.7 Boiling curve of a falling liquid film...... 292 15.8 Surface enhancement for falling liquid film...... 293 15.9 Boiling data for microfin and microstud surfaces...... 293 15.10 Boiling data for a microfin surface with and without a deflector.. . . . 294 15.11 Jet-impingement boiling curve for R-113...... 296 15.12 Boiling curves for jet-impingement and spray cooling...... 297 15.13 Variation of peak heat flux with liquid mass flux in spray cooling.. . . 299 15.14 Curved-surface semiconductor cooling system...... 301 15.15 Flow boiling over a small heater...... 302 15.16 Boiling curves of smooth and microstud surfaces...... 303 15.17 Boiling curves for mini- and microchannel heat sinks...... 304 15.18 Fluid exit temperature for mini- and microchannels...... 304

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15.19 Comparison of performance between mini- and microchannel heat sinks...... 306 16.1 Heat pipe structure...... 310 16.2 Heat pipe nomenclature...... 313 16.3 Heat pipe operating limits...... 319 16.4 Heat pipe operating temperatures...... 320 16.5 Control techniques for heat pipe operations...... 322 16.6 Electrothermal analog for a heat pipe...... 323 16.7 Effect of orientation on heat pipe performance...... 325 16.8 Performance of double-condenser heat pipe...... 326 16.9 Performance of circuit-card heat pipes...... 327 16.10 Cutaway view of heat pipe–cooled SEM...... 327 16.11 Edge-cooled heat pipe printed wiring board...... 328 16.12 Direct heat pipe cooling of a semiconductor...... 328 16.13 Temperatures of a PCB rack with and without heat pipes...... 329 16.14 Heat pipe heat exchanger for cooling enclosed electronics...... 330 16.15 Micro heat pipes...... 332 16.16 Data for rectangular and triangular heat pipe arrays...... 333 17.1 Base configuration of a Peltier couple...... 340 17.2 Interconnected thermoelectric couples...... 340 17.3 Thermoelectric device for gas flow systems...... 341 17.4 Two-stage cascaded thermoelectric module...... 341 17.5 Cooling capacity and COP of a typical thermoelectric device...... 343 17.6 Thermoelectric module for cooling electronics inside an enclosure. . . 345 17.7 Temperature difference vs. net heat transfer rate...... 346

1.1 Effect of Thermal Conditions on Part Failure...... 4 2.1 Equations of Heat Conduction in Various Geometrical Configurations...... 16 2.2 Analogous Quantities between Thermal and Electrical Systems. . . . . 18 2.3 Conduction Thermal Resistance ...... 21 3.1 Nusselt Number and Friction Factor in Fully Developed Laminar Flow ...... 41 3.2 Nusselt Number NuT for Rectangular Tubes in Fully Developed Laminar Flow at T Condition...... 42 3.3 Nusselt Number NuH1 for Rectangular Tubes in Fully Developed Laminar Flow at H1Condition ...... 43 3.4 Absolute Surface Roughness ...... 45 4.1 View Factor for Two Finite Surfaces ...... 57 4.2 Hot Dry Daily Cycle of Air Temperature and Solar Radiation ...... 63 5.1 Surface-Fluid Parameters...... 76 9.1 Typical Ranges of Parameters in Equations 9.2 and 9.3...... 147 9.2 Thermal Ratings for Solid Metal Interstitial Materials ...... 149 9.3 Thermal Resistances with Alloy at Interface ...... 157 9.4 Typical Values of Thermal Contact Resistance ...... 161 9.5 Properties of Thermal Compounds ...... 161 9.6 Maximum Torque Values for Alloy Steel Screws...... 162 10.1 Thermal Conductivity of Basic PCB Materials ...... 177 11.1 Loss Coefficients for Various Physical Configurations...... 204 12.1 Heat Transfer Correlations for Parallel Plates...... 219 12.2 Nusselt Numbers for Isoflux-Isoflux ...... 239 13.1 Fouling Factors for Some Fluids ...... 245 14.1 Coolant Figures of Merit ...... 265 14.2 Empirical Constants for Heat Transfer ...... 278 14.3 Empirical Constants for Friction Factor ...... 279 15.1 Differences between Jet-Impingement and Spray Cooling ...... 299 15.2 Summary Results of Experimental Data on Mixing of Dielectric Fluids ...... 300 15.3 Heat Transfer from a Small Heated Patch: Test Conditions and Key Boiling Characteristics...... 302 16.1 Expressions for Effective Capillary Radius ...... 314

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16.2 Expressions for Wick Permeability ...... 315 16.3 Effective Thermal Conductivity of Liquid and Wick ...... 318 16.4 Heat Pipe Material Compatibility ...... 319 16.5 Comparative Values of Heat Pipe Resistances ...... 324 C.1 Properties of Air ...... 355 C.2 Properties of Water...... 356 C.3 Properties of FC-43...... 357 C.4 Properties of FC-75...... 358 C.5 Properties of FC-77...... 359 C.6 Properties of FC-78...... 360 C.7 Properties of Coolanol-20...... 361 C.8 Properties of Coolanol-25...... 362 C.9 Properties of Coolanol-35...... 363 C.10 Properties of Coolanol-45 ...... 364 C.11 Properties of Freon E2 ...... 365 C.12 Properties of Glycol-Water Mixture...... 366 C.13 Properties of PAO (2 CST) ...... 367 C.14 Properties of PAO (4 CST) ...... 368 C.15 Properties of PAO (6 CST) ...... 369 C.16 Properties of PAO (8 CST) ...... 370 H.1 Common Correlations for External Forced Convection ...... 382 H.2 Common Correlations for Internal Forced Convection...... 386 H.3 Common Correlations for External Free Convection ...... 392 H.4 Common Correlations for Free Convection in Enclosures or from Parallel Vertical Plates ...... 394 H.5 Common Correlations for Forced Convection in Tube Banks...... 398

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Letter Symbols

A cross-sectional area or surface area, ft2 2 Af fin surface area, ft 2 At total heat transfer surface area, ft cp specific heat, Btu/(lbm·°F) D diameter, ft Dh hydraulic diameter, ft E emissive power, Btu/(hr·ft2) f Fanning friction factor, F black body view factor for radiation g acceleration of gravity = 32.1739 ft/s2 gc gravitational constant or conversion factor 2 8 2 = 32.1739 lbm·ft/(lbf·s ) = 4.17 × 10 lbm·ft/(lbf·hr ) 2 G mass velocity, lbm/(hr·ft ) h heat transfer coefficient, Btu/(hr·ft2·°F) 2 hc contact conductance, Btu/(hr·ft ·°F) hfg latent heat of vaporization, Btu/lbm 2 hg gap conductance, Btu/(hr·ft ·°F) H hardness of solid, psi I electrical current, A ℑ gray body view factor for radiation J radiosity, Btu/(hr·ft2·°F) k thermal conductivity, Btu/(hr·ft·°F) ks harmonic mean thermal conductivity, Btu/(hr·ft·°F) kxy thermal conductivity in xy plane, Btu/(hr·ft·°F) kz thermal conductivity in z direction, Btu/(hr·ft·°F) K fluid loss coefficient L length or characteristic length, ft L latent heat of fusion or solidification, Btu/lbm m mass flow, lbm/hr m RMS absolute surface asperity slope Ns specific speed of fan P fluid pressure or apparent contact pressure, psi ∆Pa pressure drop due to flow acceleration (or momentum change), psi

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∆Pf pressure drop due to flow friction, psi ∆Pg pressure drop or gain due to gravity in vertical flow, psi ∆Ps pressure drop due to inlet, exit, expansion, contraction, bends, fittings, etc., psi ∆Pt total pressure drop, psi q heat transfer rate, Btu/hr q″ heat flux, Btu/(hr·ft2) R thermal resistance, °F·hr/Btu R electrical resistance, Ω s fin spacing, in T temperature, °F t fin thickness, in U overall heat transfer coefficient, Btu/(hr·ft·°F) V velocity of fluid, ft/hr V voltage, V w plate width or heat sink width, in

Greek Symbols

α absorptivity of radiation α Seebeck coefficient α thermal diffusivity, ft2/hr β coefficient of thermal expansion, °R–1 δ boundary-layer thickness, ft ε emissivity ε heat exchanger effectiveness ε a apparent emissivity ηf fin efficiency η0 overall surface efficiency θ angle, degrees θ time, hr µ viscosity of fluid, lbm·ft/s ν dynamic viscosity, ft2/s ξ pump efficiency π Peltier coefficient 3 ρ density of fluid, lbm/ft σ RMS surface roughness, ft σ Stefan-Boltzmann constant = 0.1714 × 10–8 Btu/(h·ft2·°R4) σ surface tension, lbf·ft τ Thomson coefficient τ transmissivity of radiation

Dimensionless Groups

B Biot number = hL/k Gr Grashof number = βρ2gL3 ∆T/µ2

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Gz Graetz number = mcp/kL J Colburn factor = Nu/(Pr Re) Pr2/3 Nu Nusselt number = hL/k Pe Peclet number = Pr Re Pr Prandtl number = µcp/k Ra Rayleigh number = Gr Pr Re Reynolds number = ρVL/µ St Stanton number = Nu/(Pr Re) or h/ρVcp

Subscripts

amb ambient condition c cold side of heat exchanger CHF critical heat flux f liquid condition fg difference between liquid and vapor states g gas or vapor condition l liquid condition sat saturation condition sub subcooling condition w wall condition w value for wick material in heat pipe

Thermal Management of Microelectronic Equipment: Heat Transfer Theory, Analy- sis Methods, and Design Practices is the first book in this series and is on the very important subject of electronic cooling. It has been well documented that over 50 percent of electronic failures are thermally related. According to National Elec- tronics Manufacturing Report (NEMI 2000), electronics cooling is a key enabling technology in the development of advanced packaging systems and has helped sustain the technology trends as predicted by “Moore’s Law”. The report predicts that maximum chip power for high performance chips will be approaching 300 watts in the next decade. This increasing power trend extends to all key products including computers, telecommunication, handhelds, automotive, and military electronics. This book is written to serve engineers in industry, as well for use in academia for a senior/graduate level course. Dr. Yeh and Dr. Chu have done an outstanding job of including both the fundamental and practical aspects of thermal management of electronic systems.

Dereje Agonafer University of Texas, Arlington

Developing reliable microelectronic devices capable of operating at high speeds with complex functionality requires a better understanding of the factors that govern the thermal performance of electronics. With an increased demand on system reliability and performance combined with miniaturization of the devices, thermal consideration has become a crucial factor in the design of electronic packages, from chips to systems. The authors understand that the challenge in the field of thermal management of electronic systems resides not only with very-high-performance and high-heat- dissipation devices, but also with the intermediate and lower-power devices, where improved reliability objectives require cooler operation of chips. The authors further realize that no one design method is best suited for all applications. It is common to employ several different heat transfer modes simultaneously in a system; therefore, this book includes a wide range of subjects related to various heat transfer technologies that can be utilized for thermal design of electronic equipment. This book places a great deal of emphasis on understanding the physics of each subject, and thus a high level of mathematics is kept to a minimum. There are 17 chapters in the book. Among them, Chapters 2 through 8 deal with the fundamentals of various heat transfer modes, while the rest focus on specific subjects of practical importance to the design of electronic systems. For examples, Chapter 9 gives a detailed discussion of the thermal interface resistances. (Daily cycles of the ambient air and solar flux on a hot, dry day are also given in Chapter 4 to be used in the design of outdoor equipment.) Appendix 8 presents a comprehensive catalog of topics in convective heat transfer that includes heat transfer correlations for various physical configurations and thermal boundary conditions. Property tables for solids and 16 types of fluids such as air, water, and dielectric fluids are also included, in appendices. This book has been developed to serve many types of readers. It will guide upperclass undergraduates or graduate students through practical approaches to solving real-world problems of vast complexity. Professional engineers will find this book a valuable reference resource for both refreshing their knowledge of the fundamentals in heat transfer and reading up on the latest thermal control technology. Furthermore, this book also contains a great deal of practical design data and correlations to be used in the analysis and design of electronic equipment.

L. T. Yeh R. C. Chu

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INTRODUCTION

Electronics are the heart of any modern equipment. Thermal control of electronic equipment has long been one of the major areas of application of heat transfer technologies. Many improvements in reliability, power density, and physical miniaturization of electronic equipment over the years can be attributed in part to improved thermal analysis and design of systems. Improved thermal design has been made possible through advances in heat transfer technologies as well as computational methods and tools. The primary function of cooling systems for electronic equipment is to provide an acceptable thermal environment in which the equipment can operate. To achieve this goal, it is necessary to maintain a thermal path with a minimum resistance from equipment heat sources to the ultimate heat sink.

1.1 NEED FOR THERMAL CONTROL

Because of advances in circuit and component technologies, electric circuits have become more efficient, and thus heat dissipation from individual devices such as transistors has also become less. Miniaturization of circuits greatly decreases the size of individual devices, however, and increases the number of such devices that can be integrated on a single chip. The net result is that the chip heat flux (heat flow per unit surface area) has significantly increased in recent decades as chip development has moved from small-scale integration (SSI) to very large-scale integration (VLSI), and further to ultra-large-scale integration (ULSI). The chronology of advances in the integrated circuit (IC) is outlined in the following steps, with the trends shown in Figure 1.1.

1960—Small-scale integration (SSI), fewer than 100 devices per chip 1966—Medium-scale integration (MSI), fewer than 1000 devices per chip 1969—Large-scale integration (LSI), fewer than 10,000 devices per chip 1975—Very-large-scale integration (VLSI), fewer than 107 devices per chip 1990—Ultra-large-scale integration (ULSI), more than 107 devices per chip

Decreasing the temperature of a component increases its performance as well as its reliability. In addition to lowering the junction temperatures within a

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FIGURE 1.1 Increase in circuit complexity.

component, it is sometimes also important to reduce the temperature variation between components that are electronically connected in order to obtain optimum performance. Thermal considerations become an important part of electronic equipment because of increased heat flux. An increased demand on system performance and reliability also intensifies the need for good thermal management of electronic equipment. Further evidence of the importance of thermal management to electronic systems is shown in Figure 1.2, based on a survey by the U.S. Air Force and indicating that more than 50% of all electronics failures are caused by shortcomings in temperature control [1]. Although a great deal of attention has recently been paid to high-heat-flux systems, the opportunity also exists to improve thermal performance and reliability in low-heat-flux equipment. Therefore, the challenge in the field of thermal management of electronic equipment resides not only with very-high- performance (high-heat-dissipation) devices, but also with intermediate- and

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FIGURE 1.2 Major causes of electronics failures [1].

lower-power components where improved reliability objectives require cooler operation of chips.

1.2 RELIABILITY AND TEMPERATURE

Reliability—that is, the reciprocal of the mean time between failures (MTBF)—is described as the statistical probability that a device or system will operate without failure for a specific time period. Each component or device has its failure rate curve. The reliability of a system is determined by combining many individual part failure rates in series and/or parallel. System reliability is also affected by the relative importance of the individual part to the system, and it can be improved by redundancy at the component, packaging, or system level. Table 1 lists the possible effect of temperature on the part failure under various thermal conditions such as thermal shock, temperature under continuous operation (steady state), and temperature cycling. In the temperature range specific to electronic equipment, it is an established fact that the reliability of electronics is a strongly inverse function (a near exponential dependency) of a component’s temperature. The reliability of a silicon chip is decreased by about 10% for every 2°C of temperature rise [2]. A typical temperature limit for a silicon chip is 125°C; however, a much lower design limit is commonly sought to maintain the desired reliability, especially in military products. The component failure rate is also related to temperature cycling, which results from either device power cycling or cycling of environmental conditions. In a U.S. Navy–sponsored study [3], an eightfold increase in failure rate was

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Table 1.1 Effect of Thermal Conditions on Part Failure

Condition Cause Effect Possible failure

Steady-state Ambient exposure, Aging—discoloration Alteration of (hot) equipment- Insulation properties induced deterioration Shorting Oxidation Rust Expansion Physical damage, Viscosity decrease increased wear Softening/hardening Loss of lubrication Evaporation/drying Physical breakdown Chemical changes Dielectric loss Steady-state Ambient exposure, Contraction Wear, structural (cold) mission- Viscosity increase failure, binding induced Embrittlement Loss of lubricity Ice formation Structural failure, cracked parts Alteration of electric properties Loss of resilience— seal leaks Thermal cycling Ambient-induced, Temperature Mechanical failure of mission gradients parts, solder joints, operation Expansion/ and connections contraction Delimitation of bonding line Thermal shock Mission profile High temperature Mechanical failure, gradients cracks, rupture

encountered in equipment subjected to a deliberate temperature cycling of more than 20°C.

1.3 LEVELS OF THERMAL RESISTANCE

The component-level resistance, which is often referred to as a component’s internal resistance, is defined as the thermal resistance from the device junction (heat source) to some predetermined reference point on the outside surface of the component or package. In most component assemblies, heat must be transferred through a number of different materials and interfaces. The total thermal resistance is the sum of several individual resistances in series and/or parallel. The primary heat transfer mode inside the component or package is conduction. In well-designed component packaging, the number of interfaces should be minimized, and bonding and sealing operations should also be selected to create the lowest interface resistance. The coefficient of thermal expansion (CTE) is a major concern at the interfaces of all materials laminated together. Any mismatch of the coefficient of thermal expansion would increase the thermal resistance, and even

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cause part failure. Note that the thermal designer may have no control over the internal resistance if commercial components are used. The packaging-level resistance is generally defined as the thermal resistance from the aforementioned surface reference point (the terminal point for measuring component-level resistance) to some designated point in a convective stream. For example, this designated point may be the local air temperature over a given component or the local fluid or wall temperature of a cold plate. The system-level resistance is defined as the thermal resistance from the designated terminal point for measuring the packaging-level resistance to the ultimate heat sink of the equipment or system. The ultimate heat sink is the environment in which the system operates. Sometimes, the combined resistance of packaging and system levels is referred to as the external thermal resistance. The foregoing definitions are somewhat arbitrary; however, they give an insight into the contribution of each level of resistance to the total resistance of a system, and thus provide thermal engineers an opportunity to optimize the overall thermal resistance by working at the individual levels.

1.4 THERMAL DESIGN CONSIDERATIONS

Thermal control of electronic equipment may employ several different heat transfer modes simultaneously and may consist of up to three tasks: (1) removing heat from the sources (components), (2) transporting heat to system internal heat sinks, and (3) rejecting heat from the internal heat sinks to the ultimate heat sink, the environment. Sometimes, tasks 2 and 3 are lumped into a single task. The purpose of a thermal design is to provide equipment that will induce thermal energy to flow properly from heat sources to the ultimate heat sinks. The goal of thermal control is to prevent part failures and also to achieve desired system performance and reliability. Another goal of thermal control is to obtain an optimum design. The optimum system will result from a series of trade-off studies considering several factors:

1. Performance. The primary and foremost consideration is that the system must be able to perform its required functions and the specific tasks for which it is designed. 2. Producibility. The system under consideration must be producible without involving a very complicated manufacturing process. 3. Serviceability (or maintainability). The equipment under design must be readily and easily accessible for testing, repair, or replacement. The design approach may also be affected by the system maintenance methods—e.g., whether the equipment will be maintained in the field or in the shop. 4. Compatibility. The system, including the coolant used, must be compatible with the environment in which it is being used. 5. Cost. The final product must be cost-effective. Cost may be the most important factor in product design. In determining the cost of a product, one should consider not only manufacturing but also maintenance costs.

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FIGURE 1.3 Effect of part placement on component junction temperature.

The relative importance of these five individual factors is mission-related; for example, the cost may become a secondary issue for some military equipment, where performance and reliability are the major concerns. A concurrent engineering approach must be adopted in the development of a new product. The design philosophy or basic rule is that the best design is the simplest and lowest- cost one that will meet the specified requirements.

1.5 OPTIMIZATION AND LIFE-CYCLE COST

Because of the strong dependence of microelectronic device reliability on temperature, it is important and desirable in the thermal management of electronic systems to pursue temperature distributions that are not merely acceptable but are optimal. Mayer [4] discusses the opportunities for optimization at all packaging levels. These opportunities include (1) optimal thermal placement of electronic components on circuit boards and optimal arrangement of boards within boxes or assemblies, and (2) optimal distribution of thermally conductive layers and optimal distribution of coolant flow rates. As an example of the first type, Figure 1.3 compares junction temperatures with and without optimization at different component positions on a printed circuit board cooled by forced-air convection. The left diagram in the figure is for the purpose of illustration and does not include all components. By switching component locations, a significant improvement in overall temperature (near 40°F reduction for half of the components) is found with optimization, even though the maximum temperature is about the same for both cases. This in turn improves the overall system reliability. It is important to involve thermal design in the early phase of the equipment design process. The design approach should place emphasis at the system level for optimizing the thermal control unit. However, since for most electronics there

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FIGURE 1.4 Example of current thermal design practice.

is a direct relationship between a component’s junction temperature and its failure rate, the current design practice is often to establish an upper temperature limit that is below the maximum allowable temperature set by manufacturers, ensuring that no component junction temperatures exceed this temperature limit. Currently, an upper limit of 105°C for semiconductor devices is being used by the U.S. Air Force and Navy [4, 5]. In theory, the actual temperature limit should be determined by the required reliability of each individual piece of equipment. As pointed out by Berger [6], the problem with this temperature limit design approach stems from the attempt to analyze the reliability of equipment by look- ing at the individual components in terms of their temperature rather than their actual part failure rate. The goal should not be to minimize component temperatures but to minimize component failure rates that directly impact the reliability of the equipment. A simplified example given by Berger as shown in Figure 1.4 illustrates the preceding statement. Component B, in the center of the layout diagrammed in the figure, is at 110°C and thus is the hottest component. A redesign is in order, to get the temperature of all parts below 105°C, if that particular standard is to be applied. One of the simplest ways to redesign is just to switch component B with component A at left, thereby placing the hot component close to the heat sink. Acceptable design criteria would be met under the standard if components A and B both turned out to be at 100°C. The graph in the figure reveals, however, that the increase in the failure rate of component A for a 10°C rise outweighs the benefits obtained for a 10°C reduction in temperature for component B. The man- dated redesign actually results in a reduction of the overall reliability of the equipment. Any design change, therefore, should not be initiated unless it will be an overall improvement in equipment reliability and the change can also be justi- fied on the basis of a system life-cycle cost (LCC) reduction. In short, the equip-

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ment design should be thermally optimized to minimize the LCC and maximize the reliability and operation life of the system.

REFERENCES

1. U.S. Air Force Avionics Integrity Program (AVIP) notes, 1989. 2. A. Bar-Cohen, A. D. Kraus, and S. F. Davidson, “Thermal Frontiers in the Design and Packaging of Microelectronic Equipment,” Mech. Eng., June 1983. 3. W. F. Hilbert and F. H. Kube, “Effects on Electronic Equipment Reliability of Temperature Cycling in Equipment,” Report No. EC-69-400 (Final Report), Grumman Aircraft Corp., Bethpage, NY, February 1969. 4. A. H. Mayer, “Opportunities for Thermal Optimization in Electronics Packag- ing,” Proc. 1st Int. Electronics Packaging Soc. Conf., 1981. 5. “Thermal Design, Analysis, and Test Procedures for Airborne Electronic Equipment,” MIL-STD-2218, 1987. 6. R, L. Berger, “A System Approach—Minimizing Avionics Life-Cycle Cost,” SAE Technical Paper Series 831107, 13th Intersociety Conf. on Environmental Systems, San Francisco, 1983. 7. L. T. Yeh, “Future Thermal Design and Management of Electronic Equip- ment,” in Wei-Jei Yang and Yasuo Mori, eds., Heat Transfer in High Technol- ogy and Power Engineering, Hemisphere, New York, 1987. 8. A. D. Kraus, Cooling Electronic Equipment, Prentice-Hall, Englewood Cliffs, NJ, 1965. 9. J. H. Seely and R. C. Chu, Heat Transfer in Microelectronic Equipment: A Prac- tical Guide, Marcel Dekker, New York, 1972. 10. A. W. Scott, Cooling of Electronic Equipment, Wiley, New York, 1974. 11. D. S. Steinberg, Cooling Techniques for Electronic Equipment, Wiley, New York, 1980. 12. A. D. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment, Hemisphere, New York, 1983. 13. G. N. Ellison, Thermal Computation for Electronic Equipment, Van Nostrand Reinhold, New York, 1984. 14. L. T. Yeh, “Review of Heat Transfer Technologies in Electronic Equipment,” J. Electronic Packaging 117, 1995. 15. M. Pecht, “Why the Traditional Reliability Prediction Models Do Not Work— Is There an Alternative?” Electronics Cooling 2(1), 1996.

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CONDUCTION

Heat conduction is a process by which heat flows from a region of higher temperature to a region of lower temperature through a solid, liquid, or gaseous medium, or between different media in intimate contact. Generally, heat conduction is due to movement and interaction of molecules. Therefore, a solid is superior to a liquid in transferring heat by conduction because of its shorter distance between molecules. Similarly, a liquid has better thermal conduction characteristics than a gas. Generally, a good conductor of electricity is also a good conductor of heat.

2.1 FUNDAMENTAL LAW OF HEAT CONDUCTION

The fundamental law of heat conduction is attributed to Fourier [1], who stated that the rate of heat conduction in a medium is proportional to the product of the area normal to the path and the temperature gradient along the path. Mathemat- ically, the Fourier law for the one-dimensional case can be expressed as follows:

dT qA∝− (2.1) dx

where q is the heat transfer rate, A is the area normal to the heat flow path, and dT is the temperature gradient over the distance dx. The negative sign is assigned to allow a positive heat flow when the temperature decreases along the path. With an insertion of a proportionality constant, Equation 2.1 becomes

dT qkA=− (2.2) dx

where k is the proportionality constant, also referred to as the thermal conductivity of the medium. Thermal conductivity is a material property. The thermal conductivity of gases is an order of magnitude smaller than that of liquids, and likewise that of liquids is an order of magnitude smaller than that of solids. Thermal conductivity of

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materials within a limited temperature range is generally a function of temperature and can be approximately expressed in the form

k(T) = k0(1+βT) (2.3)

where k0 is the thermal conductivity of a medium at T = 0 °F and β is the temperature coefficient of thermal conductivity. Generally, the value of β is negative for most pure metals and is positive for most alloys, gases, and liquids with an exception in high temperature ranges for some alloys and liquids.

2.2 GENERAL DIFFERENTIAL EQUATIONS FOR CONDUCTION

With the aid of Fourier’s law along with the assumption of energy balance on a differential element of a solid, the general differential equations of heat conduction can be developed. Without including the detailed derivations, which are available in most textbooks, we can show the general differential equation for heat conduction as

∂  ∂T  ∂  ∂T  ∂  ∂T  ∂T  k  +  k  +  k  + qc′′′ = ρ ∂x  xyz∂xy ∂  ∂yz ∂  ∂z  ∂t (2.4)

where the internal heat generation and thermal conductivity can be functions of time, space, and temperature. With proper initial and boundary conditions, the differential equation can be solved analytically or numerically. For the case with constant thermal properties, Equation 2.4 can be simplified to the form

∂2T ∂2T ∂2T q′′′ 1 ∂T + + + = (2.5) ∂x2 ∂y2 ∂z2 k α ∂t

where α (= k/ρc) is thermal diffusivity, which represents an approximate measure of the ratio of heat conducted to heat stored. Equation 2.5 can further be reduced under the following assumptions:

Fourier equation (no internal heat generation):

∂2 ∂2 ∂2 ∂ T + T + T = 1 T (2.6a) ∂x2 ∂y2 ∂z2 α ∂t

Poisson equation (steady state):

∂2 ∂2 ∂2 ′′′ T + T + T + q = 0 (2.6b) ∂x2 ∂y2 ∂z2 k

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Laplace equation (no internal heat generation and steady state):

∂2 ∂2 ∂2 T + T + T = 0 (2.6c) ∂x2 ∂y2 ∂z2

The above discussions are limited to the Cartesian coordinate system. Again, without going through the detailed derivations, the final form of the general differential equation for heat conduction in cylindrical or spherical solids can be expressed as follows:

For cylindrical systems:

∂  ∂  ∂  ∂  ∂  ∂  ∂ 11T + T + T + ′′′ = ρ T rkrz  kθ   k  qc (2.7) rr∂  ∂r  r2 ∂θθ ∂  ∂z  ∂z  ∂t

For spherical systems:

∂  ∂  ∂  ∂  112 T + T rkr   kθ  r2 ∂r  ∂r  r22sin θ ∂θ  ∂θ  1 ∂  ∂T  ∂T (2.8) +  kφ sinφ  + qc′′′ = ρ r22sin φ ∂φ  ∂φ  ∂t

It should be noted that Equations 2.7 and 2.8 can also be obtained from Equa- tion 2.4 through the following transformations with the coordinates given in Figure 2.1:

Cylindrical systems Spherical systems x = r cos θ x = r sin φ cos θ y = r sin θ y = r sin φ sin θ z = zz= r cos φ

The solution to any three-dimensional heat conduction problem is always complicated, even in the steady-state condition. The problem becomes even more complex where multiple heat sources are applied to the top surface of the multilayer structure to simulate heat conduction in microelectronics. The closed-form solution, which is typically expressed by tediously long infinite series either analytically or approximately, is not practical for engineering applications. The steady-state solution for the simple case where a single heat source is applied to the top surface of a medium was presented in papers by Lindsted and Surty [2] and Kennedy [3] for rectangular and cylindrical solids, respectively. The maximum temperature occurs at the center of the heat source, and its corresponding temperature rise over the isothermal surface can be determined by

∆ Tm = Rmq (2.9)

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FIGURE 2.1 Cylindrical and spherical coordinates.

where q is heat dissipation and Rm is the maximum thermal resistance. Rm is defined for the two cases as follows:

 Φ  rectangular solid (2.10a)  ( ) = kABH Rm   Φ  cylindrical solid (2.10b)  kAπ

where Φ is the thermal resistance factor, k is the thermal conductivity of the material, and linear dimensions A, B, and H are as listed in Figures 2.2 and 2.3.

Example 2.1 A chip with thermal conductivity of 2 W/(in ·°C) dissipates 0.2 W at the junction. The bottom of the chip is assumed to be isothermal. Find the maximum temperature rise in the chip over the isothermal surface for the following cases:

Case I Chip size: 0.1 by 0.1 by 0.05 in; thus, chip cross-sectional area = 0.01 in2 Heat source size: 0.005 by 0.005 in; thus, heat input area = 2.5 × 10–5 in2 Heat source perimeter: 0.02 in, with • W = L = 0.1/2 = 0.05 in • A = B = 0.005/2 = 0.0025 in • B/W = 0.0025/0.05 = 0.05 • W/H = 0.05/0.05 = 1 • A/B = 1

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FIGURE 2.2 Thermal resistance factor at heat source center for rectangular solids with W = L.

From Figure 2.2 with A/B = 1, we have

Φ = 0.014

From Equation 2.10a, one obtains

= 0.014 =° Rm 56 C/W 2( 0.0025× 0.0025 0.005)

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FIGURE 2.3 Thermal resistance factor at heat source center for cylindrical solids.

The maximum temperature rise for the chip can readily be computed according to Equation 2.9:

∆Tm = 56 × 0.2 = 11.2°C

Case II Chip size: 0.1 by 0.1 by 0.05 in (chip cross-sectional area = 0.01 in2, same as previous case)

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Heat source size: 1.18 × 10–2 by 2.236 × 10–3 in (heat input area = 2.5 × 10–5 in2, same as previous case) Heat source perimeter: 0.0268 in, with •W= L = 0.1/2 = 0.05 in •A= 1.118 × 10–2/2 = 5.59 × 10–3 in •B= 2.236 × 10–3/2 = 1.118 × 10–3 in •B/W = 1.118 × 10–3/0.05 = 0.0236 •W/H = 0.05/0.05 = 1 •A/B = 5

From Figure 2.2 with A/B = 5, we have

0.012 Φ=0.012 ⇒R = =°48 C/W m −− 2( 5.59×× 1033 1.118 × 10 0.05)

and

∆ Tm = 48 × 0.2 = 9.6°C

Case III Cylindrical chip with same cross-sectional areas for chip and heat source as the previous two cases:

Chip diameter (2B): 0.11283 in (chip cross-sectional area = 0.01 in2, same as previous cases) Heat source diameter (2A): 0.0056419 in (heat input area = 2.5 × 10–5 in2, same as previous cases) Heat source perimeter: 0.0177 in, with •A= 0.0056419/2 = 0.002821 in •B= 0.112838/2 = 0.056419 in •H/B = 0.05/0.0056419 = 8.86 •A/B =0.05

From Figure 2.3, we have

1.5 Φ=1.5 ⇒R = =84.6 °C/W m 2π 0.002821

and

∆Tm = 84.6 × 0.2 = 16.9°C

Comment: The above results clearly indicate that for the same heat source area, the thermal resistance decreases as the perimeter of the heat source increases.

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Table 2.1 Equations of Heat Conduction in Various Geometrical Configurations L = thickness of plane wall or length of cylinder.

Geometry Temperature profile Heat flow

TT− x TT− 1 =− q = 12 TT− L LkA/ Plane wall 12 TT− ln(r/r ) TT− i = i q = io TT− ln(r /r ) ln(r /r )/2π kL Hollow cylinder io io oi TT− 11/r− /r TT− i = i q = io TT− 11/r− /r (r− r )/4π krr Hollow sphere io io oi io

2.3 ONE-DIMENSIONAL HEAT CONDUCTION

Under certain limiting conditions, heat conduction in a three-dimensional system can be approximated by a one-dimensional problem. For example, a one- dimensional heat conduction problem may be assumed if the lateral dimensions of a wall are very large as compared with its thickness. Another case is if thermal resistances in lateral directions are large as compared with the conductive resistance through the thickness of the wall.

2.3.1 Steady-State Condition

For a steady-state condition, the temperature of a medium is independent of the time, and thus the energy entering must be equal to the energy leaving the system. The analysis of Equations 2.4, 2.7, and 2.8 is greatly simplified when heat conduction becomes a one-dimensional steady-state condition. For cases without internal heat generation and with constant thermal properties, the temperature profile and the heat flow for a plane wall, hollow cylinder, and hollow sphere with prescribed wall temperatures can be summarized as in Table 2.1. The temperature profile of a given geometry can easily obtained by a double integration of Equation 2.4, 2.7, or 2.8 with the aid of the boundary conditions, i.e., prescribed temperature at the two surfaces of the wall. Once the temperature profile is known, the heat flux can be determined with Fourier’s law.

2.3.2 Transient Condition

For the case with constant properties, the differential equation for one-dimensional heat conduction is

∂  ∂  ′′′ ∂ 11n−1 T + q = T −  x  (2.11) xn 1 ∂x  ∂x k ∂α t

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where

1 for slabs  n = 2 for cylinders  3 for spheres

Again, with proper initial and boundary conditions, Equation 2.11 can be solved either analytically or numerically. Solutions for the above problems are generally available and are often expressed in graphical form for practical applications. For the case without internal heat generation, the charts of Heisler [4] can be applied for a convection boundary condition. In the case with thermal radiation at the boundary, the problem becomes nonlinear and is typically solved numerically. The labor involved in repeatedly solving these nonlinear problems further warrants the construction of solution charts, which will be of interest to thermal engineers in various fields. The graphical charts of the temperature history are available for a one-dimensional slab [5], cylinder[6], and sphere [7]. Chung and Yeh [8, 9] extended the above solutions by including the internal heat generations for a slab, cylinder, and sphere under heating or cooling conditions. The heating case represents the condition where the environment temperature is higher than the initial temperature of the solid. On the other hand, the cooling case implies that the solid is initially hotter than its environment. Numerical computations reveal that for the same parameters, the sphere cools most rapidly, followed by the cylinder, and finally the slab. The same sequence is found for radiant heating.

2.4 THERMAL/ELECTRICAL ANALOGY

Heat flow within a medium due to a temperature gradient is similar to the case of an electrical system, where the current flows in a circuit according to Ohm’s law, which can be stated as

∆V = IR (2.12)

where ∆V is the voltage (electrical potential), I is the current, and R is the electrical resistance. The following relation may express the temperature in a heat transfer process:

∆T = qR (2.13)

where q is the heat transfer rate and R is the thermal resistance, equivalent to the denominators of the heat flow equations in Table 2.1. Analogous quantities between the two kinds of systems are listed in Table 2.2. The purpose of employing the analogy between electrical and thermal systems is to simplify the calculation procedure for the latter. This approach is often used to develop a general thermal analyzer for analysis and design of a

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Table 2.2 Analogous Quantities between Thermal and Electrical Systems

Thermal system Electrical system Potential ∆T, °K or °R ∆V, volts Flow q, W or Btu/hr I, amperes Resistance R, °K/W or °R/(Btu/hr) R, ohms Capacitance C, kJ/°K or Btu/°R C, farads

thermal system. For physical systems encountered in practice, one rarely finds a problem that can be solved analytically. A complicated three-dimensional system may be approximated by subdividing it into a large number of small elements or nodes which have no internal temperature gradient. These nodes are then interconnected by thermal paths (thermal resistances) to form a network. Nodes represent discrete quantities of mass that may gain or lose thermal energy from or to neighboring nodes, or may gain energy from internal heat sources. The thermal resistance for different heat transfer modes may be defined as follows:

L Conduction: R = (2.14a) kA

where

L = conduction length A = cross-sectional area normal to heat conduction path

1 Convection: R = (2.14b) hA

where

h = heat transfer coefficient A = heat transfer surface area (fluid-wall contacting surface)

TT− Radiation: R = 12 (2.14c) σℑ−4 4 AT( 1 T2 )

where

σ = Stefan-Boltzmann constant ℑ = gray body view factor

As can be seen from Equation 2.14c, the radiation thermal resistance is explicitly expressed as a nonlinear function of the surface temperature that is somewhat

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FIGURE 2.4 Thermal network for a multilayer composite wall.

different from the cases of the conduction and convection thermal resistances. The heat transfer coefficient under natural convection is also a nonlinear function of temperature. Convection and radiation heat transfer will be discussed in Chapters 3 and 4, respectively. As does an electrical network, a thermal network consists of series and parallel paths. For example, the thermal network of a multiple-layer composite wall is shown in Figure 2.4. It should be noted that a two-dimensional heat flow might result if the thermal conductivities of materials C, D, and E are significantly different. Also, the thermal contact resistances at the interfaces are not shown in Figure 2.4. Generally, a thermal resistance exists at the interface when two bodies are in contact. A detailed discussion of the interface thermal contact resistance will be presented in Chapter 9.

Example 2.2

Referring to Figure 2.4, let L1, L2, and L3 be 0.5, 0.8, and 1 ft, respectively. It is assumed the total heat transfer rate, q, is 100 Btu/hr. Other assumptions are listed in the accompanying table. Determine the temperature difference between T1 and T4.

ABC DE Thermal conductivity, Btu/(h · ft · °F) 50 80 100 100 100 Cross-sectional area, ft2 1 1 0.3 0.4 0.3

Solution:

L 05. 08. R == ==001. R = 001. ABkA 50× 1 80× 1 1 1 RR== ==0. 033 R = 0. 025 CE100× 0. 3 D100× 0. 4

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For parallel paths between T3 and T4, the net thermal resistance is

1 R′ = =°001. F/(Btu/hr) ++ 111RRRCDE

The total thermal resistance becomes

Rt = RA + RB + R′ = 0.01 + 0.01 + 0.01 = 0.03°F/(Btu/hr)

The temperature difference between T1 and T4 is

T1 – T4 = qRt = 100 × 0.03 = 3°F

To facilitate computations, the thermal resistance due to heat conduction for several common geometrical configurations is listed in Table 2.3. By applying proper values of θ and φ in Equations 3 and 5 in Table 2.3, one obtains the following equations:

 ln(rr)  oi (2.15a)  hollow cylinder = 2πkL Ri   rr− oi hollow sphere (2.15b)  π 4 krio r

2.5 LUMPED-SYSTEM TRANSIENT ANALYSIS There is no material with an infinite thermal conductivity. However, many transient problems may be treated approximately as a lumped system by assuming uniform temperature within the system at any instant. This simplification is jus- tified only for the case where the external thermal resistance between the system and its surroundings is so large compared with the internal thermal resistance that the latter can be neglected. The Biot number, hLc/k, which is the ratio of the internal to the external resistance, is often used to measure the relative importance of the external and internal thermal resistances; h is the average heat transfer coefficient, k is the thermal conductivity of the solid, and Lc is the characteristic length defined as the ratio of the volume of the body to its surface area. For regular-shaped bodies such as plates, cylinders, and spheres, the error induced by the assumption of a uniform internal temperature in the solid is less than 5% if the Biot number is less than 0.1. The following are characteristic lengths for several common geometric shapes:

 3 (43)π r r  = sphere (2.16a)  4πr2 3  π 2 = rL= r Lc  cylinder (2.16b) 22πrL  L3 L  = cube (2.16c)  2 6 6L

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Table 2.3 Conduction Thermal Resistance

Geometry Thermal resistance

A A L R = (1) kA L

Lw/wln()21 W1 R = (2) kA()− A A 21 A1 2

ln()r/r R = oi r θ (3) r0 kL θ Rθ = (4) θ r i kLln() roi /r

θ − θ rroi 1 2 R = r θθφφ− − (5) krio r ()cos1221 cos () θθ ln[] tan()()21//22 tan / R = θ − φφ− (6) kr()oi r ()21 φφ− r = 21 i Rφ (7) kr()− rln[] tan ()()θθ /22 / tan / ro oi 21 φ 1 φ 2

where θ1, θ2, φ1, and φ2 are angles in radians

To understand the solution to the lumped-system transient, let us consider a solid body with an internal heat generation, q. The body is initially at a uniform temperature of Ti and is suddenly exposed to an airstream at a temperature of Ta. The heat transfer coefficient is assumed to be relatively small compared with the thermal conductivity of the body, so that the Biot number is less than 0.1. From the energy balance equation, we have

dT cVρ=−−qhATT( ) (2.17) dt a

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with the following initial boundary condition:

T = Ti at t = 0 (2.18)

where c is the specific heat, ρ is the density, and V is the volume of the body. Equation 2.17 can be rewritten as follows:

dT dT = (2.19) TT− ∞ τ

where q T∞ =+T (2.20) hA a

and

cVρ τ = (2.21) hA

Let us assume that T∞ and τ are constant. The solution of Equations 2.19 to 2.21 becomes

–t/τ T–T∞ = (Ti–T∞)e (2.22)

where τ is usually referred to as the time constant. Equation 2.22 indicates that the transient response decreases when τ increases. In other words, the rate of the temperature change is slow when the body heat capacitance is large or the product of the heat transfer coefficient and the surface area is small. A similar solution can be obtained for a more complicated case which includes multiple heat transfer paths from the body to various heat sinks at different temperatures. The basic energy equation takes the form

dT cVρ=qhATTi− ∑ ( − ) = 123,,,… , n (2.23) dt ii ai,

and the solution is identical to that of Equation 2.22, with T∞ and τ modified as follows:

+ qhAT∑ iiai, T∞ = (2.24) ∑ hAii

and

cVρ τ = (2.25) ∑ hAii

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Lumped-system analysis is very handy in making a quick estimate of the transient response of a body or an electronic component. The following example will illustrate the usefulness of the approach.

Example 2.3 A relay whose weight is 0.5 oz is mounted on a printed circuit board at 185°F. The relay is initially at 170°F and is suddenly exposed to an airstream at 140°F. It is assumed that the heat transfer coefficient h1 between the relay and the airstream 2 is 5 Btu/(hr · ft ·°F) and the conductance h2 from the relay to the printed circuit board, including conduction from leads to the board, is 50 Btu/(hr · ft2 ·°F). The specific heat and heat dissipation of the relay are 0.2 Btu/(lbm ·°F) and 20 Btu/hr, respectively. It is further assumed that the heat transfer surface area for h1 and h2 is 0.003 ft2 (neglecting the convection from the sides of the relay because components are close-packed). Determine the relay temperature at the end of 1 minute.

Solution: From Equations 2.23 and 2.24, we have

+ qhAT∑ iiai, T∞ = ∑ hAii +× × + × × = 20 5 140 0.. 003 50 185 0 003 5×+× 0.. 003 50 0 003 =°302. 1 F ρ τ = cV ∑ hAii (0516..) × 02 = 5×+× 0.. 003 50 0 003 = 0. 0379 hr

At the end of 1 minute, t = 1/60 = 0.0167 hr. The corresponding temperature is obtained from Equation 2.22 as follows:

–t/τ T–T∞ = (Ti–Ta)e T–302.1 = (170-302.1)e–(0.0167/0.0379) T = 217˚F

It should be noted that the value of 217°F represents the case temperature of the relay (not a junction temperature). A more general equation for a system in the process of heating or cooling down can be expressed as follows:

dT cVρα=+qqAhATT′′ −−∑ ( ) dt aiia,i −−σε 44 = (2.26) ∑ FAiei( T T, ) i 123,,,… ,n

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where

F = radiation view factor q″ = external heat flux, e.g., solar flux αa = solar absorptivity of surfaces

Equation 2.26 can be rewritten in the form

dT = ftT( , ) (2.27) dt

where

+ ασ′′ −−−−ε44 qqAhATTaiia∑ ( ,,i) ∑ FATT ie( i) ftT( , ) = (2.28) cVρ

Since Equation 2.27 is a nonlinear problem, it can be solved numerically, e.g., by the fourth-order Runge-Kutta method described as follows:

= kHftT1 ( , old ) (2.29a)  H k  kHft=+ ,T +1  (2.29b) 2  22old   H k  kHft=+ ,T +2  (2.29c) 3  22old   H k  kHft=+ ,T +3  (2.29d) 4  22old 

and

kkkk+++22 TT=+1234 (2.30) new old 6

where H is a small time increment. The calculation begins by setting Told equal to the initial temperature of the system. A new temperature is then obtained at the end of the time step, and this process is repeated until the specified time is reached. The thermal conductivities of silicon (Si) and gallium arsenide (GaAs) are a strong function of temperature. Generally, the thermal conductivity of a pure single-crystal silicon material is zero at 0 K and rises approximately exponentially to a maximum value near 20 K, then falls somewhat faster than 1/T to about 500 K, and then varies approximately as 1/T to the melting point. Virtually all semiconductor materials—materials of the same family as Si—exhibit this general trend of thermal conductivity as a function of temperature.

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FIGURE 2.5 Temperature profile in a thermal-energy storage system.

2.6 HEAT CONDUCTION WITH PHASE CHANGE

Heat transfer involving melting and solidification has many engineering applications. One of the most noticeable applications is in thermal-energy storage (TES) systems. Compared with sensible-energy storage systems, using a phase-change material (PCM) for energy storage has some advantages:

1. Larger heat capacity in the storage system (the latent heat is much larger than the specific heat) 2. Lower temperature gradient in the storage system, as shown in Figure 2.5 (theoretically, phase change takes place at a constant temperature)

Heat transfer involving melting and solidification is a complex problem because of its moving boundary condition. Figure 2.6 shows the moving boundary condition at the interface between the solid and liquid. Mathematically, the boundary condition at the interface can be described in the following equation by assuming no energy storage at the interface:

ds( t) qqmL′′ − ′′ = ρ s 1 dt (2.31)

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FIGURE 2.6 Moving solid-liquid interface for melting and solidification.

where

L = latent heat ρ = density (assuming the same value for the solid and the liquid) −1 for melting m =   1 for solidification

In addition, qs″ and q1″ are heat fluxes, respectively, at the solid and liquid sides of the interface pointing in the direction of the cooled surface as shown in Figure 2.6. The location of the solid-liquid interface measured from the origin is given by s and is a function of time. Basically, the left-hand side of Equation 2.31 represents the net energy removed from (or supplied to) the interface per unit area and per unit time in the direction of the cooled surface. The right-hand side of the equation corresponds to the energy released (or gained) at the interface per unit area and per unit time as a result of release of the latent heat during solidification (or melting). Equation 2.31 can be rewritten as follows:

 ∂T ∂T  ds( t) k s − k 1  = mLρ  s ∂ 1 ∂  (2.32) x s xs= (t) dt

Another boundary condition at the interface is the continuity of temperature where the interface temperature is equal to the melting or freezing temperature, i.e., Ts = Tl at x = s. To simplify the problem, solutions are often obtained by assuming no natural convection in the liquid region. Even with this assumption, the problem of transient heat conduction involving melting and freezing, which was first studied by Stefan [10], is still complex. Therefore, exact solutions are limited to only a few very simple cases [10, 11]. Approximate and numerical methods become very

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FIGURE 2.7 Temperature profile with natural convection in liquid region.

important and practical. Due to extensive studies on this subject, there are too many published papers to be listed here; however, some basic information can be found in Stefan’s paper [10] and in works of Carslaw and Jaeger [11] and Ozisik [12]. The temperature profile for the case including natural convection in the liquid region is illustrated schematically in Figure 2.7. Representative papers in this area have been written by Sparrow and others [13, 14]. Because of their ability to absorb a relatively large amount of heat with a small temperature gradient, phase-change materials have been used in thermal control of electronics. Examples of this application are given in Estes [15] and Pal and Yoshi [16]. Various types of paraffin wax are commonly used because of their low cost and chemical stability during thermal cycling. Some thermophysical properties of a representative type (P116 paraffin wax) are as shown here [17]:

Specific heat of solid 2.95 kJ/(kg · °C) Specific heat of liquid 2.51 kJ/(kg · °C) Density of solid 818 kg/m3 Density of liquid 760 kg/m3 Thermal conductivity of solid 0.24 W/(m · °C) Thermal conductivity of liquid 0.24 W/(m · °C) Heat of fusion (latent heat) 262 kJ/kg Melting point 44°C

A common characteristic of most phase-change materials is low thermal conductivity in both liquid and solid phases. To enhance the heat transfer rate, an extended surface such as plate or honeycomb-type fins which extend from the heat exchange surface through the phase-change material is often applied. Recently, a microencapsulation of PCMs has been developed to enhance heat transport in liquid [18]. Figure 2.8 shows a single micro-PCM particle. Particle

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FIGURE 2.8 Cross section of microencapsulated phase-change material particle.

diameters can range from one to several hundred microns. The core PCM consti- tutes approximately 80 to 85% of the total weight, and the thickness of shell is typically less than one micron. When it is heated, the PCM inside the shell is changed from solid to liquid. The particle flexible wall can accommodate the volumetric expansion (less than 15%). Another type of phase-change is ablation, or the change of a material from solid to gas. The process is similar to the melting process except that the liquid phase as shown in Figure 2.6 no longer exists and the heat supply, q(t), is always applied to the solid-gas interface. To reflect this change, Equation 2.32 is rewritten as follows:

 ∂T  ds( t) qt( ) + k s  = ρL s  ∂  (2.33) x xs= (t) dt

The assumption that the boundary surface is completely free from the ablated material is not always correct. This is because a porous charred layer may be formed on the heating surface during the ablation of many materials consisting of organic substances such as plastic or resins. Heat transfer with ablation is commonly employed in thermal protection systems. For example, a thin insulation layer is frequently applied to the external surface of a high-speed object such as a missile to protect the internal structures from aerodynamic heating on the missile skin.

REFERENCES

1. J. B. J. Fourier, The Analytical Theory of Heat, Dover, New York, 1955. 2. R. D. Lindsted and R. J. Surty, “Steady State Junction Temperature of Semi- conductor Chips,” IEEE Trans. Electron Devices 19(1), 1972. 3. D. P. Kennedy, “Spreading Resistance in Cylindrical Semiconductor Devices,” J. Appl. Phys. 31(8), 1960 4. M. P. Heisler, “Temperature Charts for Induction and Constant-Temperature Heating,” Trans. ASME 26, 1947.

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5. R. D. Zerkle and J. E. Sunderland, “The Transient Temperature Distribution in a Slab Subject to Thermal Radiation,” J. Heat Transfer 87, 1965. 6. D. L. Ayers, “The Transient Temperature Distribution in a Radiating Cylin- der,” ASME Paper No. 67-HT-71, 1967. 7. D. L. Ayers, “Transient Cooling of a Sphere in Space,” J. Heat Transfer 92, 1970. 8. B. T. F. Chung and L. T. Yeh, “Heat Conduction in Slabs with Uniform Heat Sources and Radiative Boundary Conditions,” J. Spacecraft and Rockets 10, 1973. 9. B. T. F. Chung and L. T. Yeh, “Temperature Distribution in Radiating Cylin- ders and Spheres with Uniform Heat Generation,” ASME Paper No. 73- WA/HT-2, 1973. 10. J. Stefan, “Über die Theorie der Eisbildung, insbesondere über die Eisbil- dung im Polarmeere,” Ann. Phys. Chem. N.S. 42, 1891. 11. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford Univer- sity Press, London, 1947. 12. M. N. Ozisik, Boundary Value Problems of Heat Conduction, International Textbook, Scranton, PA, 1968. 13. E. M. Sparrow, S. V. Patankar, and S. Ramadhyani, “Analysis of Melting in the Presence of Natural Convection in the Melt Region,” J. Heat Transfer 99, 1977. 14. E. M. Sparrow and P. Souza Mendes, “Natural Convection Heat Transfer Coefficients Measured in Experiments on Freezing,” Int. J. Heat Mass Trans- fer 25, 1982. 15. R. C. Estes, “The Effect of Thermal Capacitance and Phase Change on Out- side Plant Electronic Enclosures,” IEEE Trans. Components, Hybrid and Manufacturing Technol. 5, 1992. 16. D. Pal and Y. K. Joshi, “Transient Thermal Management of an Avionics Mod- ule Using Solid-Liquid Phase Change Materials (PCMs),” 7, ASME Proc. 31st Nat. Heat Transfer Conf., ASME HTD-329, 1996. 17. A. Haji-Sheikh, J. Eftekhar, and D. Y. S. Lou, “Some Thermophysical Proper- ties of Paraffin Wax as a Thermal Storage Medium,” in P. E. Bauer and H. E. Collicott, eds., Spacecraft Thermal Control, Design, and Operations, Vol. 86 of Progress in Astronautics and Aeronautics, AIAA, New York, NY, 1983. 18. C. P. Colvin, J. C. Mulligan, and Y. G. Bryant, “Enhanced Heat Transport in Environmental Systems Using Microencapsulated Phase Change Materials,” 22nd Int. Conf. on Environmental Systems, Seattle, July 13–16, 1992.

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CONVECTION

Convection is a heat transport process in a fluid involving the combined action of heat conduction and energy storage when the fluid is undergoing a mixing motion. Since convection heat transfer is closely related to fluid motion, prior knowledge of the flow field is prerequisite before the mechanism of heat transfer in the fluid can be investigated. Generally, the motion of a fluid, either a gas or a liquid, can be divided into two basic flow conditions, namely, laminar and turbulent. In laminar flow, the fluid particles follow smooth and continuous paths in an orderly fashion. The fluid particles in any one layer do not move across layers, and particles remain in an orderly sequence within each layer without passing one another. Therefore, heat is transferred purely by molecular conduction among the particles through direct contact. Heat is also exchanged through the interface between the fluid and the surface of the walls. On the other hand, turbulent flow is irregular, and the fluid velocity components are random, varying with time and space. Turbulent eddies of the fluid move across the streamlines. Energy is transferred by mixing the fluid particles in motion. Thus, increasing the mixing rate will increase the heat transfer rate. Convection heat transfer can be classified into forced and free (or natural) convection. In the former process, the fluid motion is induced by some external means such as a pump, blower, fan, or wind. In the latter process, the fluid motion is generated from the density difference caused by a temperature variation in the fluid.

3.1 FLOW AND TEMPERATURE FIELDS

Due to the viscous effect, the velocity boundary layer of an internal flow begins forming at the inlet when a fluid with a uniform velocity enters a duct. The hydrodynamic boundary-layer thickness increases along the flow path as shown in Figure 3.1. For a symmetrical tube flow, the edge of the boundary layer coin- cides with the axis at some distance from the entrance, and this distance is often referred to as the hydrodynamic entry length. The flow continues to develop throughout the entire hydrodynamic entry length. After the boundary layer merges, the flow becomes fully developed hydrodynamically, which implies that the velocity profile is invariant with the axial distance.

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FIGURE 3.1 Boundary-layer development for fluid in a tube.

Similar trends are found for the temperature field in a heat transfer process. The boundary layer is now referred to as the thermal boundary layer, and the distance for the thermal boundary layer to develop is called the thermal entrance length. In a thermally developed flow, the temperature profile of the fluid is invariant with the axial distance. The ratio of the tube length x to the diameter Dh is often adopted as an indicator to express the flow condition, i.e., fully developed or developing flow. This ratio is sometimes misleading. A better way to indicate the flow regime is to use * + the respective dimensionless parameters x [= x/(Re Pr Dh)] thermally and x [= x/(Re Dh)] hydrodynamically. The hydraulic diameter Dh is equal to 4 times the hydraulic radius, which is defined as the ratio of the cross-section area to the perimeter of a tube. Re and Pr are the Reynolds and Prandtl numbers, respectively. Both these dimensionless parameters take into account the tube geometry (x and Dh), whereas Pr takes into account the fluid property and Re the flow condition. A small value of x* or x+ represents a developing flow, while a large value of x* or x+, on the other hand, corresponds to a fully developed flow. The problem of fluids flowing over or across solids in the case of an external flow has also frequently occurred in many engineering applications. Figure 3.2 shows one of the simple cases where fluid flows over a flat plate. There is a significant difference between the internal and the external flow cases in the development of the boundary layer. As shown in Figure 3.2, the hydrodynamic boundary layer continues to grow over the plate. Additionally, the flow may separate from the wall of a nonslender body. The following are the major differences between flow over a bluff body and flow over a flat plate or a streamlined body:

1. As noted in Figure 3.3, the velocity outside the boundary layer, u∞, which is different from the upstream or approaching velocity V, depends on the distance x from the stagnation point. The velocities of u∞ and V are identical in the case of flow over a flat plate.

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FIGURE 3.2 Flow regimes for flow over a flat plate.

2. The most important difference between these two cases lies in the behavior of the boundary layer. For a streamlined body, the flow separation from the solid wall, if it happens at all, occurs nears the rear. For a bluff body, on the other hand, the point of flow separation often takes place not far from the leading edge. The separation as indicated in Figure 3.4 is a condition in which the boundary detaches from the surface of a solid and a wake is then formed in the downstream. At the separation point, the velocity gradient at the wall surface becomes zero. 3. In the case of flow over a streamlined body, the pressure drop is mainly caused by the boundary-layer surface shear stress, i.e., skin friction drag. For a bluff body, however, the skin friction drag is small

FIGURE 3.3 Boundary-layer formation and separation on a circular cylinder in crossflow.

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FIGURE 3.4 Velocity profile and flow separation for a circular cylinder in crossflow.

compared with the form drag. The form or pressure drag arises from the separation of the flow, which induces a low-pressure region in the rear of a solid body. A pressure difference is generated in the flow direction when the pressure over the rear of the body is lower than that over the front.

3.2 HEAT TRANSFER COEFFICIENT

The practical concern in heat transfer by convection is to determine the heat transfer coefficient. The governing equation for convection can be expressed as follows:

q = hA(Tw – Tref) (3.1)

where

q = heat transfer rate h = heat transfer coefficient A = heat transfer surface area Tw = wall temperature Tref = reference temperature of fluid

The heat transfer coefficient is a proportional constant and does not have any physical meaning by itself. However, it is a function of the fluid flow, fluid properties, and the geometry of the system. Note that in defining the heat transfer

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coefficient, the reference temperature can be the bulk fluid temperature or the inlet fluid temperature for an internal flow, and the free stream temperature for an external flow. Therefore, one should apply the heat transfer coefficient according to its definition. The heat transfer coefficient is frequently expressed in a dimensionless form called the Nusselt number, Nu = hLc/k, which is, in turn, a function of various dimensionless groups as follows:

For forced convection:

Nu = F1(Re)F2(Pr) (3.2)

For free or natural convection:

Nu = F3(Gr)F4(Pr) (3.3)

where

Re = Reynolds number Pr = Prandtl number Gr = Grashof number

The details of each dimensionless group will be discussed later. The characteristic length, Lc, used to define the Nusselt number can have different meanings under various physical conditions. For example, it can be the hydraulic diameter for a duct flow or it can be the height of a vertical plate in natural convection. The heat transfer coefficient can be obtained either mathematically or experimentally; however, neither approach is generally enough to cover all possible physical problems. Mathematically, the heat transfer coefficient can be obtained by knowing the fluid temperature distribution that is determined by solving a set of differential equations involving conservation of mass, momentum, and energy. Analytical solutions are confined to some very simple cases. For a more complex problem, numerical solutions are needed. The accuracy and validity of numerical solutions depend on both the mathematical model used to describe the physical problem and numerical methods used in the analysis. The computer simulations of computational fluid dynamics (CFD), therefore, cannot replace experiments totally because physical phenomena still cannot be modeled accurately and completely, such as the turbulent flow. It should be noted that all empirical correlations should be applied only to the conditions under which the tests were conducted.

3.3 PARAMETER EFFECTS ON HEAT TRANSFER

Since there is no simple way to predict heat transfer coefficients under various thermal and physical conditions, the following discussions place emphasis on understanding the physical phenomena in convection heat transfer and the effects of various parameters on the heat transfer coefficient.

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3.3.1 Effects of the Reynolds Number In the case of forced convection, the Nusselt number depends on the flow conditions, which can be characterized by the Reynolds number Re, defined as follows:

ρVL Re = c (3.4) µ

where ρ = density of fluid V = velocity of fluid Lc = characteristic length µ = viscosity of fluid

The Reynolds number also represents the ratio of the inertia force to the friction (or viscous) force. In laminar flow, which corresponds to the case with small Reynolds numbers, there is no mixing in fluid particles, and heat transfer takes place only by molecular conduction due to contact between fluid particles. Since all fluids (with the exception of liquid metals) have a small thermal conductivity, the heat transfer coefficients in laminar flow are relatively small. As the Reynolds number (or velocity) increases, the flow regime is changed gradually from laminar to transition flow, in which a certain amount of mixing takes place between the fluid particles. The mixing motion, even on a small scale, accelerates the heat transfer rate considerably. As the Reynolds number is further increased, the flow becomes fully turbulent except for a very thin layer of fluid immediately adjacent to the wall. In this thin layer, which is called the laminar sublayer, the viscous force predominates near the wall surface, and heat transfer takes place mainly by conduction. In the turbulent core, the mixing due to eddies is so efficient that heat is transferred very rapidly between the edge of the laminar sublayer and the turbulent core of the fluid. Although the thermal resistance of the laminar sublayer controls the rate of heat transfer, the overall heat transfer in turbulent flow is still much higher than that in laminar flow. This phenomenon results from the small thickness of the laminar sublayer as compared with the thickness of the entire thermal boundary layer in laminar flow. It is found that increasing the Reynolds number decreases the thickness of the laminar sublayer and at the same time increases the eddy mixing in the turbulent core. The net result is that the heat transfer rate goes up with the Reynolds number.

3.3.2 Effect of the Prandtl Number The Prandtl number is defined as the ratio of the kinematic viscosity of the fluid to the thermal diffusivity of the fluid; that is, ν Pr = α (3.5) µc = p k

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where cp = specific heat of fluid k = thermal conductivity of fluid

The kinematic viscosity, ν =µ/ρ, is often referred to as the molecular diffusivity of momentum because it is a measure of the rate of momentum transfer between the molecules. On the other hand, the thermal diffusivity α = k/ρcp of a fluid usually is called the molecular diffusivity of heat, which is a measure of the ratio of the heat transmission and energy storage capacities of the molecules. The heat and momentum are diffused through the fluid at the same rate if the Prandtl number is unity (1). If the Prandtl number is greater than 1, the velocity profile will develop more rapidly than the temperature profile, whereas the opposite trends are found in a fluid with Prandtl number less than 1. It should be noted that the temperature profile starts to develop at the point where the heat transfer between the fluid and wall actually takes place. If the Prandtl number is greater than 5, the velocity profile generally will develop much faster than the temperature profile, so that a solution based on a fully developed velocity profile will apply quite accurately, even without accounting for the hydrodynamic starting length. Such a condition frequently prevails in the dielectric fluids that are widely used for electronic equipment, because the Prandtl number for most dielectric fluids is, in general, greater than 20 at room temperatures. On the other hand, for the case of Pr → 0, the velocity profile never actually develops, while the temperature profile does develop. A uniform velocity profile (slug flow), therefore, can be assumed over the entire thermal boundary layer. At very low Prandtl numbers, such as for liquid metals, the thermal resistance is distributed over the entire fluid. The temperature profile for a fluid with a low Prandtl number is similar to that in laminar flow. In addition, the thermal entry length and the thermal response of the fluid to axial changes in surface temperature are slow and are similar to those characteristics of laminar behavior. For fluids with high Prandtl numbers, very short thermal entry length and quick thermal response to axial surface temperature changes are found. Both are attributed to the concentration of the thermal resistance in the thin laminar sublayer near the wall and the rapid transfer of heat from the edge of the sublayer over the entire fluid because of effective mixing of fluid particles in the turbulent core.

3.3.3 Effect of the Grashof Number

The Grashof number represents the ratio of buoyant to viscous forces and is defined as follows:

gTLρβ ∆ 3 Gr = c µ2 (3.6) = ∆ 3 BTLc

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where

g = gravitational constant β = fluid thermal expansion coefficient ∆T = temperature difference between wall and ambient B = property parameter ρβ2 = g µ2

As the Reynolds number characterizes the flow condition in forced convection, the flow condition—i.e., laminar or turbulent flow—in free or natural convection can be characterized by the Grashof number, or sometimes by the Rayleigh number, defined as

Ra = Gr Pr (3.7)

In natural convection, flow can be laminar or turbulent, depending on the value of the Grashof or Rayleigh number. For example, the flow over a vertical plate becomes turbulent as Ra ≥ 109. Figure 3.5 shows the velocity profile over a vertical plate in natural convection. The velocity profile is different from that in forced convection. At the wall, the velocity is zero because of the nonslip condition. The velocity increases to a maximum value and then decreases to zero again at the edge of the boundary layer because of the stationary air that exists outside the velocity boundary layer in the natural convection. Natural convection is not a very effective heat transfer process; it tends to give a low heat transfer coefficient. The effect of radiation is important in many cases of natural convection and should not be neglected. This situation is especially true for natural convection occurring in high altitudes, where the air density is low.

3.3.4 Effect of Boundary Conditions

In laminar flow or for fluids with a Prandtl number less than 1, the heat transfer coefficient depends on the thermal boundary conditions. The reason for this dependency is that the heat transfer mechanism is primarily due to molecular diffusion. Thus, the thermal resistance is distributed over the entire thermal boundary-layer thickness, and different boundary conditions or different shapes of the tube cross section simply generate different temperature profiles. For high–Prandtl number fluids or turbulent flow, the thermal resistance is mainly in the laminar sublayer near the wall, which yields a square temperature profile in a tube or a very flat temperature profile over the wall in an external flow case. The above discussion suggests that the Nusselt number for a high Prandtl number or for any fluids in turbulent flow is relatively insensitive to variations of the wall surface temperature or heat flux in the flow direction. On the other

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FIGURE 3.5 Boundary layer over a vertical plate in natural convection.

hand, the Nusselt number for a low–Prandtl number fluid or any fluid in laminar flow is strongly dependent on the axial variations of the wall surface temperature and/or heat flux. As pointed out in a 1984 paper by Yeh [1], based on the subject discussed here, it is concluded that the heat transfer correlations for a given shaped tube cannot be applied to other shaped tubes in laminar flow or to low–Prandtl number fluids. The two basic thermal boundary conditions are constant axial wall temperature (T condition)and constant axial wall heat flux (H condition) Both are real- ized in many practical applications—the former in condensers and evaporators with high liquid flow rates, and the latter in electrical heating as well as in a counterflow heat exchanger with equal thermal capacity rates. It should be noted that, in both cases, the wall temperature around the periphery of a duct is also uniform. In laminar flow, the heat transfer coefficients for constant axial wall heat flux and constant axial wall temperature conditions represent the upper and the lower limits, respectively. In other words, bounded between the heat transfer coefficient in laminar flow for other boundary conditions are these two limits.

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For symmetrically heated noncircular tubes (with corners or variable peripheral curvature), two common cases are:

1. H1 condition: constant axial wall heat flux with uniform peripheral wall temperature (variable peripheral heat flux). This is identical to the above-mentioned H condition. 2. H2 condition: constant axial and peripheral wall heat fluxes (variable peripheral wall temperature).

The H1 condition is a good approximation for a tube with very highly conductive materials (e.g., copper or aluminum) or with a very thick wall. On the other hand, the H2 condition is an approximation for a tube with very poorly conductive materials or a very thin wall. It should be noted that the H1 and H2 conditions are identical for circular tubes or parallel plates, but the heat transfer coefficient for the H2 condition is generally less than that of the H1 condition for noncircular tubes. Like the heat transfer process, the friction factor in laminar flow or for fluids with a low Prandtl number also depends on the shape of the tube. Excellent information about heat transfer and flow friction for fluids in laminar flow with various shaped tubes is presented in Shah and London [2]. It is a common mistake to use the round-tube heat transfer (or friction) correlation for rectangular ducts in laminar flow. Table 3.1 illustrates the dependence of the Nusselt number and friction factor on the shape of duct and on thermal boundary conditions. Information contained in Table 3.1 is primarily from Shah and London [2]. The aspect ratio α is defined as the ratio of the small to large sides of a rectangular tube. The aspect ratio is bounded by 1 (representing a square duct) and 0 (representing parallel plates). The Reynolds and Nusselt numbers are based on the hydraulic diameter. The values listed in the table are valid only for the case with all sides heated uniformly.

Example 3.1 Compare the heat transfer coefficients h of circular and square tubes carrying the same fluid under the fully developed laminar flow condition. Assume that the diameter of the round tube is D and that this is also the length of each side of the square tube.

Solution The hydraulic diameter for the square tube is determined as follows:

4D 2 D = = D s 2(DD+ )

Since both tubes have the same hydraulic diameter, the ratio of the heat transfer coefficients is equal to the ratio of the Nusselt numbers (assuming thermal conductivity k is constant). From Table 3.1, we have:

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Table 3.1 Nusselt Number and Friction Factor in Fully Developed Laminar Flow

Tube shape f · Re NuT NuH1 NuH2 Rectangular tube 2b 2a

Aspect ratio α = 2a/2b: 1 14.227 2.976 3.608 3.091 0.9 14.261 — 3.620 — 0.8 14.378 — 3.664 — 0.714 14.565 3.077 3.734 — 0.667 14.712 3.117 3.790 — 0.6 14.980 — 3.895 — 0.5 15.548 3.391 4.123 3.020 0.4 16.368 — 4.472 — 0.333 17.090 3.956 4.795 2.970 0.3 17.512 — 4.990 — 0.25 18.233 4.439 5.331 2.940 0.2 19.071 — 5.738 2.930 0.167 19.702 5.137 6.049 2.930 0.125 20.585 5.597 6.490 2.940 0.1 21.169 — 6.785 2.950 0.05 22.477 — 7.451 — 0 24.000 7.541 8.235 8.235 Circular tube 16.000 3.657 4.364 4.364 Triangle tube (equilateral) 13.333 2.470 3.110 1.892

T condition:

h (round tube) Nu 3. 657 r ==r =123. hs(square tube) Nus 2. 976

H1 condition:

h (round tube) Nu 4. 364 r ==r =121. hs(square tube) Nus 3. 608

The above results clearly indicate that the heat transfer coefficient for a round tube is different from that of a square one, and an improper use of the correlation will result in a significant error. Rectangular tubes have been used extensively in cooling of electronics, especially in cold-plate cooling. Tables 3.2 and 3.3 present the Nusselt number with one or more walls transferring heat under constant–wall temperature and constant–wall heat flux conditions, respectively [2].

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Table 3.2 Nusselt Number NuT for Rectangular Tubes in Fully Developed Laminar Flow at T Condition

2b Dimensions: 2a

Adiabatic wall: Heat transfer wall:

NuT for wall configuration

2b/2a 0.0 7.541 7.541 7.541 0.0 4.861 0.1 5.858 6.091 6.399 0.457 3.823 0.2 4.803 5.195 5.703 0.833 3.330 0.3 4.114 4.579 5.224 1.148 2.996 0.4 3.670 4.153 4.884 1.416 2.768 0.5 3.383 3.842 4.619 1.647 2.613 0.7 3.083 3.408 4.192 2.023 2.443 1.0 2.970 3.018 3.703 2.437 2.375 2.0 3.383 2.602 2.657 3.185 2.613 2.5 3.670 2.603 2.333 3.390 2.768 3.33 4.114 2.703 1.946 3.626 2.996 5.0 4.803 2.982 1.467 3.909 3.330 10.0 5.858 3.590 0.843 4.270 3.823 ∞ 7.541 4.861 0.0 4.861 4.861

3.3.5 Other Effects

Another factor that can influence heat transfer and friction considerably is the variation of physical properties with temperature. In general, the properties of most fluids are temperature-dependent and will vary through the boundary layer or over the flow cross section of a tube. Further, if the viscosity of fluids varies significantly, the velocity profile can be altered, which, in turn, changes the heat transfer characteristics. As stated by Kays [3], the specific heat of a gas varies only slightly with temperature, but the viscosity and thermal conductivity are a strong function of temperature to about the 0.8 power of the absolute temperature while the density varies inversely with the first power of the absolute temperature. The Prandtl number, however, is nearly independent of temperature. On the other hand, the specific heat and thermal conductivity of most liquids are relatively insensitive to temperature variation, but the viscosity decreases very significantly as the temperature increases. Thus, the Prandtl number varies with temperature in the same manner as the viscosity. The density of liquids varies little with temperature. In most cases, a constant-property assumption is reasonably acceptable for general engineering calculations. The effects of physical property variations

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Table 3.3 Nusselt Number NuH1 for Rectangular Tubes in Fully Developed Laminar Flow at H1 Condition

2b Dimensions: 2a

Adiabatic wall: Heat transfer wall:

NuT for wall configuration

2b/2a 0.0 8.235 8.235 8.235 0.0 5.385 0.1 6.700 6.939 7.248 0.538 4.410 0.2 5.704 6.072 6.561 0.964 3.914 0.3 4.969 5.393 5.997 1.712 3.538 0.4 4.457 4.885 5.555 1.004 3.279 0.5 4.111 4.505 5.203 1.854 3.104 0.7 3.740 3.991 4.662 2.263 2.911 1.0 3.599 3.556 4.094 2.712 2.836 2.0 4.111 3.146 2.947 3.539 3.104 2.5 4.457 3.159 2.598 3.777 3.279 3.33 4.969 3.306 2.182 4.060 3.538 5.0 4.704 4.636 1.664 4.441 2.914 10.0 6.700 4.252 0.975 4.851 4.410 ∞ 8.235 5.385 0.0 5.385 5.385

become important when the temperature difference between the fluid and the solid surface is large. Two possible ways are commonly applied to correct the constant-property results. One is called the reference temperature approach, and another is referred to as the property ratio method. The former chooses a reference temperature to evaluate all properties. This reference temperature can be the wall temperature, the bulk temperature, or the film temperature. The film temperature, which is an average of the wall and bulk temperatures, is most commonly used in empirical correlation of heat transfer and friction data. In the property ratio approach, all properties are computed at the free-stream temperature or the mixed mean temperature. Then all of the variable-property effects are lumped into a function of the ratio of some pertinent property evaluated at the surface to that property determined at the free-stream or mixed mean temperature.

3.4 PRESSURE DROP AND FRICTION FACTOR

In addition to the heat transfer coefficient, the friction factor is also needed in computing the pressure drop in a system. The general relation for pressure drop in a duct can be expressed as follows:

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∆Pt = ∆Pf + ∆Ps + ∆Pa ±∆Pg (3.8)

where

∆ = Pt total pressure drop ∆ = Pf pressure drop due to friction L G2 = 4f ρ Dhmc2 g ∆ = Ps pressure drop due to additional losses such as entrance, exit, bends, fittings, valves, etc. G2 = ∑ K ρ 2 mcg ∆ = Pa pressure drop due to flow acceleration (or momentum changes)  11 G2 =−  ρρ  21 2gc ∆ =+ − Pg pressure drop ( , for upward flow) or gain ( , for downward flow) due to gravity in a vertical flow and where

= Dh hydraulic diameter f = Fanning friction factor G = mass velocity = ρVV ( is velocity) g = gravitational acceleration = gc conversion factor =⋅⋅=×⋅⋅28 2 32.1739 lbmf ft/(lb s ) 4.17 10 lbmf ft/(lb hr ) H = tube vertical height K = loss coefficient

For a single-phase fluid and incompressible flow, ∆Pa is very small and can be neglected. As is the heat transfer coefficient in the thermal entrance region, the friction factor in a hydrodynamically developing flow is much higher than that of the fully developed flow. In addition, rough surface in a tube will increase the pressure drop in turbulent flow. On the other hand, the roughness will also increase the heat transfer coefficient. In addition to the Reynolds number, the friction factor in turbulent flow also depends on the surface roughness of a tube. Typically, the effect of roughness is expressed by the relative roughness ε/D, where ε is the absolute (average) surface roughness and D is the tube inner diameter. A simplified Moody chart for friction factor [4] is shown in Figure 3.6. The absolute surface roughness for pipes of different materials included in the same reference is presented in Table 3.4

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FIGURE 3.6 Friction factor versus Reynolds number for round tubes.

The turbulent flow in the Moody chart is further divided into transition and complete turbulent flow zones. The friction factor for the former is a complicated function of the Reynolds number and the relative roughness, while the friction factor for the latter is nearly independent of the Reynolds number, depending only on the relative surface roughness of the tube. In summary, the heat transfer coefficient (or Nusselt number) for laminar flow or for low–Prandtl number fluids in a tube depends not only on the fluid property

Table 3.4 Absolute Surface Roughness

Material ε, ft Riveted steel 0.003–0.03 Concrete 0.001–0.01 Wood stave 0.0006–0.003 Cast iron 0.00085 Galvanized iron 0.0005 Asphalted cast iron 0.0004 Commercial steel or wrought iron 0.00015 Drawn tubing 0.000005

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and the flow condition but also on the shape of the cross section and the thermal boundary conditions. The adoption of the heat transfer correlations for a round tube via the concept of the hydraulic diameter to other shaped tubes may, therefore, result in a significant error. There is only a weak influence of the duct shape on the heat transfer and friction in turbulent flow or for high–Prandtl number fluids. Consequently, for turbulent flow in a noncircular tube, the round-tube correlations for heat transfer coefficient and friction factor, in general, will yield accurate results if the hydraulic diameter is used as a characteristic dimension and if the Prandtl number of the fluids is not very small. The same conclusion can also be applied to the use of the friction factor correlations. From Table 3.1, one can clearly see that the Nusselt number is constant. This result implies that the heat transfer coefficient h depends only on thermal conductivity k and hydraulic diameter Dh and is independent of the other fluid properties and velocity. This unique feature results from the fact that in fully developed laminar flow heat transfer occurs as purely molecular conduction. Similarly, the friction factor is a linear function of the Reynolds number. It should be noted that any methods used to enhance the heat transfer would generally also increase the pressure drop of the system.

3.5 THERMAL PROPERTIES OF FLUIDS

Before all heat transfer and flow friction calculations, the properties of the fluids involved must be known. Air and water are the most popular working fluids used in industry. The former is a primary coolant in much electronic equipment while the latter is probably the most effective and cheap liquid coolant. Some dielectric fluids—such as silicate ester fluids marketed as Coolanols by Exxon, Fluorinert chemicals (FC-xx) made by 3M, and synthetic fluids with a polyalphaolefin (PAO) base made by Chevron, Mobile, Dow Corning, and others—are often used to cool high-heat-dissipation electronics. Silicate ester–based fluids such as Coolanol-25R are most widely used in the cooling of electronics by the U.S. Navy and Air Force. A decade ago, these fluids were found to cause significant and sometimes catastrophic problems under consistent high-temperature use and in high-voltage electronics fields. These problems are due to the fluids’ hygroscopic nature and subsequent formation of flammable alcohol and silica gel. Efforts have been made by the U.S. military to replace Coolanol-25R with PAO since 1989. Electronic equipment is often required to operate at high altitudes without conditioned cooling air. At high altitudes, the air density is low. The mass flow rate of cooling air for a given volumetric flow rate under forced convection may be reduced sharply; hence the equipment may be damaged as a result of over- heating. A reduction of air density in high altitudes also has a great effect on free convection; therefore, accurate prediction of the air density is required. For engineering application purposes, air can be treated approximately as an ideal gas, and its density can be obtained from the ideal gas law in the form

ρ = P RT (3.9)

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where

ρ = 3 air density, lbm /ft = 2 P air pressure, lbf /ft R = gas constant =⋅⋅° 53.34 ft lbfm /lb R T =air ( absolute ) temperature , ° R

As the altitude changes, the atmospheric pressure will vary accordingly. Based on the tabular values presented in Reference 5, the correlations for atmospheric pressure as a function of altitude are obtained as follows:

2116..HH 2−≤ 0 091 0 (3.10a) P =  2116..H 2−≤≤ 0 071 0 H 5000 (3.10b)

and

=−43 + log10( P.A.A) 019036 2 20331 (3.10c) −++8.A.A. 565332 1114878 3 32556 H ≥ 5000

where

= 2 P atmosphere pressure, lbf /ft (or psf) H = altitude, ft = AHlog10( )

According to the ideal gas assumption, the properties of gases (except the density) are a function of temperature only. In other words, all air properties except the density are independent of the altitude. For liquids, the properties given in Appendix C are limited to saturated conditions. If a fluid is not at the saturated state, the effect of pressure on its properties is very significant; therefore, it cannot be neglected. For example, the properties of water at a superheated or subcooled condition are functions of temperature and pressure. In addition, all correlations developed are valid within certain temperature ranges (generally between the freezing and boiling points of liquids) as listed in the same appendix. To simplify calculations for the Grashof number, the B parameter defined in Equation 3.6 is also included in the property tables of Appendix C.

3.6 CORRELATIONS FOR HEAT TRANSFER AND FRICTION

Some of the friction factors in fully developed laminar flow have been included in Table 3.1 The friction factor correlations for various configurations under fully developed flow conditions are summarized in Appendix G.

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On the basis of an extensive review and evaluation of the available literature in the field of convection heat transfer, the authors have prepared a comprehensive convection heat transfer catalog for practical applications. This catalog presents the correlations for both forced and free convention under various physical and thermal constraints. The correlation for flow parallel to or across a tube bank is also included. This convective heat transfer catalog, presented in Appendix H, includes the following tables:

Table H.1: Common Correlations for External Forced Convection Table H.2: Common Correlations for Internal Forced Convection Table H.3: Common Correlations for External Free Convection Table H.4: Common Correlations for Free Convection in Enclosures or Vertical Parallel Plates Table H.5: Common Correlations for Forced Convection in Tube Banks

Example 3.2

An airstream with a mass flow rate of m = 400 lbm/hr flows through an uninsu- lated metal tube with diameter D = 0.4 ft. The inlet and outlet air temperatures are 210 and 170°F, respectively. The tube, with a length L = 20 ft, is located in a still-air environment at 40°F. Determine the heat flux and the tube surface temperature at the end of the tube (x = L). Assumptions made in the analysis are as follows:

1. Steady-state conditions 2. Uniform and constant free convection coefficient at the outer surface of the tube 3. Tube-wall conduction having no bearing and therefore ignored 4. Fully developed flow inside the tube

Solution

As a first step, determine the inside convection heat transfer coefficient hi for forced convection in a tube, on the basis of the following:

πD2204. A ==314159. ×=01257. ft2 4 4 m˙ 400 G == =31831. lb /(ft2 ⋅ hr) A.01257 m 210+ 170 T = =°190 F ave 2 µ =⋅=⋅⋅°= 0.0517 lbm /(hr ft); k.0 0174 Btu/(hr ft F); Pr 0. 714 GD 3183.. 1× 0 4 Re == =>24, 628 2300 (turbulent flow) µ 0. 0517

From index 6 of Table H.2 in Appendix H (table of correlations for internal forced convection),

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=+ 0.83 + NuL 5 0.. 012Re (Pr0 29 ) =+5 0.012(24,628)0.83 (0.714+ 0.29) = 58.19 k 5819..× 0 0174 h ==Nu =⋅⋅253. Btu/(hr ft2 ° F) i D 04.

The internal thermal resistance per unit area becomes

==11 = RI 0. 395 h.i 253

The next step is to determine the outside convection heat transfer coefficient he for free convection over a tube. In order to determine the free convection coefficient, we need a value for the tube-wall temperature Tw. Let Tw = 120°F. Then:

TT+ 120+ 40 T = wa,e= =°80 F ave 2 2 K.=⋅0 015 Btu/(hr ft⋅°= F) Pr 0 . 718 ρµ==⋅3 0.. 743 lbm /ft 0 045 lbm /(hr ft) − β =°=×0.B 001866 (F)170. 2125 10

From Equation 3.6,

gTLρβ23∆ Gr = c µ2 = ∆ 3 BTLc =××−0.. 2125 1073( 120 40) 0 4 7 =× 1088. 10

and

Ra==×× GrPr 1.088 107 0.718 =×<7. 8118 1069 10 (laminar flow)

At the constant–surface temperature condition, from index 7 of Table H.3 in Appendix H, we have

1 4 =+ 0. 518Ra NuT 036. [1+ (0.559 Pr)916 ] 4 9 0.. 518() 7 8118× 10614/ =+036. [(1+ 0.. 559 0 718 )916 ] 4 9 = 211.

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k 0. 015 h ==×=Nu 211. 0. 79 Btu/(hr ⋅⋅° ft2 F) TTD 04.

On the other hand, at the constant–heat flux condition, from index 8 of Table H.3 in Appendix H, we have

1 4 =+ 0. 579Ra NuH 036. [1+ (Pr)] 0.442 916 4 9 0.. 579() 7 8118× 10614/ =+036. [(1+ 0.. 442 0 718 )916 ] 4 9 = 2416.

k 0. 015 h ==Nu 2416. ×=0. 906 Btu/(hr ⋅⋅° ft2 F) HHD 04.

Since the problem is characterized by neither the constant surface nor the constant heat flux condition, an average value is adopted, i.e.,

0.. 79+ 0 906 h = =⋅⋅°0. 848 Btu/(hr ft2 F) e 2

The external thermal resistance per unit area is

==11 = Re 1. 179 he 0. 848

Based on the assumption of no thermal resistance across the tube wall, the heat flux q″ at x = L can be expressed in the following form:

∆T q′′ = + RRie

where

∆ =−=−=° TTa,i T a,e 170 40 130 F ∑ =+= + = RRie R0 . 395 1179 . 1575 .

From the above equation, the heat flux q″ is calculated to be 82.5 Btu/hr·ft2.

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Since it is at the steady state, the heat flux at x = L can also expressed in terms of the tube-wall temperature as follows:

TT− TT− q′′ = ai,, w = wae 11hi he

Tw can be computed from the above equation by knowing either the internal or the external air temperature as follows:

Tw = 170 – 82.5 × 0.395 = 170 – 32.6 = 137.4°F

Tw = 40 + 82.5 × 1.179 = 40 + 97.3 = 137.3°F

Since the calculated Tw is different from the assumed value (Tw was assumed to be 120°F), additional iterations are required to obtain the final solution.

REFERENCES

1. L. T. Yeh, “Validity of One Heat Transfer Correlation to Various Shaped Ducts via Hydraulic Diameter Concept,” Proc. 4th Int. Electronics Packaging Conf., Baltimore, 1984. 2. R. K. Shah and A. L. London, “Laminar Flow Forced Convection in Ducts,” Advances in Heat Transfer, Supplement 1, Academic Press, New York, 1978. 3. W. M. Kays, Convective Heat and Mass Transfer, McGraw-Hill, New York, 1966. 4. L. F. Moody, “Friction Factors for Pipe Flow,” Trans. ASME 66, 1944. 5. F. Kreith, Principles of Heat Transfer, International Textbook, Scranton, PA, 1965.

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RADIATION

Another mechanism of heat transfer in addition to conduction and convection is radiative heat transfer, which involves the transport of thermal energy by electromagnetic waves. Since the application of thermal radiation to electronic equipment generally occurs in a relatively simple form, this chapter will be limited to basic knowledge of thermal radiation. Gases that do not absorb thermal radiation are known as nonparticipating gases. Generally no significant absorption takes place when thermal radiation passes through air. Participating media, or fluids that will absorb some thermal radiation, include carbon monoxide, water vapor, and liquids.

4.1 STEFAN-BOLTZMANN LAW

For practical engineering purposes, thermal radiation is limited to wavelengths ranging from 0.1 to 100 micrometers (µm), where 1 µm (formerly known as the micron, or µ) is equal to 10–6 m or 3.94 ×10–5 in. The total amount of thermal radiation emitted by a body per unit surface and time is called the total emissive power E, which depends on the temperature and surface characteristics of the body. Furthermore, the radiation energy emitted per unit surface at a given temperature is different at different wavelengths. The total emissive power of an ideal radiator at a given temperature can be obtained by integrating Planck’s law over all wavelengths, which results in

4 Eb = σT (4.1)

Equation 4.1 is also known as the Stefan-Boltzmann law. Eb has units of energy per unit surface area, and σ is the Stefan-Boltzmann constant, with numerical value 0.1714 × 10–8 Btu/(hr·ft2·°R4) or 5.67 × 10–8 W/(m2·K4). The total thermal radiation energy from a surface is

4 q = EbA = σT A (4.2)

It should be noted that the absolute temperature must be used in all radiation heat transfer computations.

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4.2 KIRCHHOFF’S LAW AND EMISSIVITY

Radiative energy is transmitted when internal energy within a body is converted into electromagnetic waves that are emitted from the surface and travel through space and finally strike another body or bodies. Some of the thermal radiation is then absorbed while the remaining energy is reflected from or transmitted through the body. The relation between the absorbed, reflected, and transmitted energy is as follows:

α + ρ + τ = 1 (4.3)

where

α = absorptivity, fraction of incident radiation energy absorbed by body ρ = reflectivity, fraction of incident radistion energy reflected from body τ = transmissivity, fraction of incident radiation energy transmitted through body

The relative magnitudes of absorptivity, reflectivity, and transmissivity not only depend on a body’s material, thickness, and surface finish, but also vary with the wavelength of the radiation. Bodies that do not transmit radiation energy are described as opaque; they exhibit a value of τ equal to zero. Many liquids and gases, by contrast, are transparent. Reflection of radiation can either be specular or diffuse. If a surface is highly polished and smooth, the reflection is similar to the reflection of a light beam from a mirror. In this case, the angle of incidence is equal to the angle of reflection. On the other hand, the radiation from a rough surface goes practically in all directions and is said to be diffuse. The ideal radiator or black body is theoretically treated as a body that absorbs all radiation incident on its surface or a body that emits the maximum possible amount of thermal radiation at a given temperature. The black body, therefore, is used as a standard to measure the characteristics of thermal radiation of other bodies. Kirchhoff’s law states that at thermal equilibrium, the ratio of the emissive power of a surface to its absorptivity is the same for all bodies or materials; i.e.,

EE E 1 ===2 … n αα α (4.4) 1 2 n

The maximum emissive power occurs when α is equal to 1. This implies the black body condition, where the emissive power is designated as Eb, with the subscript b. For all practical materials, the emissive power is less than that of the black body, and the ratio of the emissive power of a given material to that of the black body is defined as the emissivity ε = E/Eb. Furthermore, from Equation 4.4 with αb = 1, the ratio E/Eb is also equal to the absorptivity. Hence, one can conclude that at thermal equilibrium, the absorptivity of a material is equal to its emissivity, i.e., α = ε.

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FIGURE 4.1 Geometrical relation between two radiative surfaces.

4.3 RADIATION BETWEEN BLACK ISOTHERMAL SURFACES

In order to calculate heat exchange between black isothermal surfaces, one must first determine the black body view factor (or shape or configuration factor). The net heat exchange due to radiation between two surfaces as shown in Figure 4.1 can be expressed as follows:

ϕϕ =−cos1212 cos dA dA dq12→ () Ebb 1 E 2 (4.5) πr2

ϕϕ =− cos1212 cos dA dA qEE12→ ()bb 1 2 ∫∫ AA 21 πr2 (4.6) =− =− ()()EEFAEEFAbb1212112212−→ bb

where Fi - j is the black body view factor based on Ai. Physically, F1-2 represents the fraction of the total radiant energy leaving surface area A1 which is directly intercepted by A2. Similarly, F2-1corresponds to the fraction of radiant energy directly striking A1 from A2, and the following relationship holds:

F1–2A1 = F2–1A2 (4.7)

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FIGURE 4.2 View-factor algebra for pairs of surfaces.

In addition to the reciprocal property given in Equation 4.7, the additive relation among the view factors as shown in Figure 4.2 is as follows:

F1–2 = F1–(3 + 4) = F1–3 + F1–4 (4.8a)

F1–3 = F1-2 – F1–4 (4.8b)

and

    = A1 = A1 − F3–1   F1–3  ()FF1–21–4 (4.8c)  A3   A3 

Another useful relation for the view factor in a complete enclosure is

== … ∑ Fjij– 1 1, 2, 3, , n for any given i (4.9)

Generally, the determination of a view factor except for a few simple geometries is rather complex. A catalog of radiation view factors presented by Howell [1] is probably the best reference for collected information available in this field. Table 4.1 shows a few examples taken from Reference 1.

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Configuration View Factor

Coaxial parallel 112 []AB222()++ F =  ln squares 12– 2 22 πA  ()()YX++22   A2 −−Y X ++()tanYY2124 /  11− X tan  b  ()Y 212+ 44// ()Y 212+  b    A −−X Y  1 ++()XX2124 /  tan 11− Y tan   a  ()X 212+ 44/ (X 2+ ))12/   a ()AB 2 F =

Identical parallel   ++2212/ = 2 ()()11XY rectangles F12–  ln   πXY   1++X22 Y  A2 −−X Y b ++XY()1 212//tan 1 ++YX(1 212 ) tan 1 a ()1+ Y 212/ ()1+ X 212/ a  A1 −− −−XXYYtan11 tan  b  where X = a/c Y = b/c c = distance between two plates

Perpendicular  rectangles = 1  −−1111+−+2212/ −11 F12–  W tanH tan ()HW tan πW  W H ()HW2212+ / A2 h cc  W2 H 2   1 ()()(11++WH22 WWH2221++ )  HWH222()1++   + ln        A 22 222 222  w 1 4  1++WH ()()1++WW H (1+ HHWH)(+ )    where H = h/c W = w/c c = window of rectangles

Unequal coaxial 12/   disks   2       1  R 2  A r =−− 2    1 1 F12– XX4   2    R1          A2 r2 == where rr12radius of small disk radius of large disk a = distance between two disks 1+ R 2 R===+ r /a R r /a X 1 2 11 2 2 2 R1

Right isosceles   =−1 1 = A A F.12– 1  0 2113 2 1 2  312/ 

Source: From Howell [1].

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FIGURE 4.3 Hottel crossed-string method.

For two-dimensional systems that are characterized by areas of infinite extent in one direction, the view factor can be determined by the Hottel crossed-string method [2]. Consider two arbitrary surfaces as shown in Figure 4.3. The view factor from surface 1 (AH) to surface 2 (DE) can be obtained from the following equation, representing the crossed-string method:

()()HD+− AE ABCD + HGFE F = (4.10) 12– 2

The sum of the lengths of crossed strings stretched between the ends of the lines representing the two surfaces minus the sum of the lengths of uncrossed strings similarly stretched between the surfaces is divided by 2.

4.4 RADIATION BETWEEN GRAY ISOTHERMAL SURFACES

Radiation from a gray surface, which is defined as a surface whose absorptivity and emissivity are less than 1, to another gray surface is expressed by the radiosity J. The radiosity, which is defined as the radiative heat flux leaving a surface, is the sum of radiation emitted, reflected, and transmitted:

J = ρG + εEb + τG′ (4.11)

where G is irradiation, or incident radiative heat flux, and G′ is incident flux from the opposite side of the body. The last term of the equation will disappear for an opaque body. The net radiative heat flux for a gray surface becomes

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FIGURE 4.4 Equivalent network for radiative heat exchange between two gray surfaces.

dq =−JG (4.12) dA

For an opaque gray body (ρ + α = ρ + ε = 1), we have

dq  ε  ()1− ρ J =   E − dA  ρ  b ρ (4.13a)  ε  =  ()EJ−  ρ  b

 ε  qEJA=  ()−  ρ  b (4.13b)

With the aid of Figure 4.4, the net radiative heat exchange between two gray surfaces can then be expressed by the following equation:

EE− q = bb12 12– + + (4.14) RR123 R

where ρ R = 1 1 ε 11A ==11 R2 AF112–– A 2 F 21 ρ R = 2 3 ε 22A

By substituting the temperature difference into the emissive power equation, one obtains:

=ℑσ 4 −4 qA12– 1 12– () TT 1 2 (4.15)

where

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ℑ= 12– gray body view factor = 1 (4.16) ρ ερε++ 11//1 FAA 122122– /

For an opaque body, i.e., ρ + ε =1, Equation 4.16 becomes:

ℑ= 1 12– − εε++−ε ε (4.17) ()//11111FAA 12– ( 2122 )/

For concentric cylinders of infinite length, or concentric spheres (F1-2 = 1), Equa- tion 4.17 is reduced to the following expression:

ℑ= 1 12– εεε+− (4.18) 1/(12121 )/AA2

For two parallel and infinite plates (A1 = A2), Equation 4.18 is further simplified as follows:

ℑ= 1 12– εε+− (4.19) 11//12 1

Equation 4.19 can also be approximated to any two parallel plates when end effects are relatively small. The temperature gradient ∆T in a system is the main reason for heat transfer. For convection, ∆T is the primary factor governing the heat transfer rate if the heat transfer coefficient and area remain unchanged. The heat transfer rate in radiation, however, depends on the absolute temperature of two surfaces, not on the temperature gradient, because of the nonlinear nature of radiation heat transfer. The following example will illustrate this statement.

Example 4.1 Assuming the surface area, view factor, and emissivity of two surfaces are constant, compare the heat transfer rates for the following two cases:

Case 1 (Tw)1 = 200°F(Tw)2 = 180°F ∆T = 20°F Case 2 (Tw)1 = 60°F(Tw)2 = 40°F ∆T = 20°F

The two conditions give

4 4 10 q1 ∝ [(460 + 200) – (460 + 180) ] = 2.197524 × 10

and

4 4 10 q2 ∝ [(460 + 60) – (460 + 40) ] = 1.061616 × 10

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With the same temperature gradient, the heat transfer rate of case 1 is about twice that of case 2. This result explains the fact that the radiation heat transfer is much more effective in higher-temperature environments. Another important factor in radiation heat transfer is that the absolute temperature must be used in calculations.

4.5 EXTREME CLIMATIC CONDITIONS Most outdoor equipment is subjected to climatic conditions such as solar radiation and ambient temperature. The wind speed is also an important factor to be considered for outdoor electronics. Outdoor electronics are designed to be used under extremely hot environmental conditions. According to U.S. Army Regula- tion for environmental conditions [3], the maximum solar flux on a horizontal surface occurring at noon is assumed to be 355 Btu/(hr·ft2), and the maximum ambient air temperature at 4:00 P.M. is 120°F. In many cases, solar heating is a major heat load upon a system. Therefore, one should always include hot climatic conditions in the design for any outdoor electronic equipment. Only the solar flux upon a horizontal surface is available in Reference 3; however, the solar flux upon a vertical surface is also often needed to complete the analysis because the equipment typically consists of horizontal and vertical surfaces. In order to derive the incident solar flux on a vertical surface, one must know how the data for a horizontal surface were obtained. Unfortunately, this is never mentioned in Reference 3. The following derivations are obtained according to Kreith [4]. The solar flux normally impinging on a surface placed at the outer fringes of the earth’s atmosphere is known as the solar constant (Go) with the numerical value of 442 Btu/(hr·ft2). Some losses occur at the earth’s surface, however, due to absorption and scattering when solar radiation travels through the approximately 90-mile-thick layer of air, water vapor, carbon dioxide, and dust particles enveloping the earth. The actual solar flux that reaches the surface of the earth is less than the solar constant and can be expressed as follows:

= τ m GGnoa (4.20)

where

m = relative air mass defined as ratio of actural path length to shortest possible path τa = transmission coefficient for unit air mass

The transmission coefficient τa depends not only on the season of the year, but also on the condition of the sky; it ranges from 0.62 on a cloudy day to 0.81 on a clear day. A mean value of 0.7 is considered to be acceptable for most practical applications. The value of the relative air mass m is dependent on the position of the sun given by the zenith distance z, which is the angle between the zenith and the direction of the sun. Furthermore, the relative air mass can be approximated as the secant of z if the thickness of the atmosphere is negligible as compared with the radius of the earth.

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FIGURE 4.5 Solar heat flux and ambient air temperature.

Under worst-case conditions, when the sky is clear, Equation 4.20 can be rewritten as:

sec z Gn = Go(0.81) (4.21)

If the receiving surface on the earth is not normal to the sun, the maximum incident radiation per unit area upon horizontal and vertical surfaces is approximately as follows:

sec z Gn = Go(0.81) cos z horizontal surface (4.22a) sec z Gn = Go(0.81) sin z vertical surface (4.22b)

The values of z at various local times were taken from the solar angle chart for a latitude of 25º north in Reference 4. Results obtained from equations 4.22a and 4.22b in conjunction with these z values are plotted in Figure 4.5. As can be seen from the figure, an excellent agreement for the horizontal flux is found between the calculated values and those from Reference 3. The ambient temperature for a hot, dry day per Reference 3 is also plotted in the figure. Based on findings, solar fluxes upon a horizontal and upon a vertical surface are calculated and presented in Table 4.2. The solar fluxes in the morning are assumed to be the mirror image of those in the afternoon. Also shown in the table are the z values, as well as the ambient temperatures and the solar fluxes on a horizontal surface from Reference 3 for a hot dry day.

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Table 4.2 Hot Dry Daily Cycle of Air Temperature and Solar Radiation

Solar flux, Btu/(hr·ft2) Ambient Solar flux, Btu/(hr·ft2) (horizontal surface) Local time, air, ∞F (vertical surface) hour AR70-38 AR70-38 Eq. 4.22a Eq. 4.22b 195000 294000 393000 492000 591000 6 (z = 82∞)90 18 14 96 7 (z = 68∞) 91 85 94 234 8 (z = 55∞) 95 160 176 251 9 (z = 42∞) 101 231 247 223 10 (z = 28∞) 106 291 307 164 11 (z = 15∞) 110 330 343 92 12 (z = 0∞) 112 355 358 0 13 116 355 358 0 14 118 330 343 92 15 119 291 307 164 16 120 231 247 223 17 119 160 176 251 18 118 85 94 234 19 114 18 14 96 20 108 0 0 0 21 105 0 0 0 22 102 0 0 0 23 100 0 0 0 24 98 0 0 0

Finally, the daily-cycle ambient air temperature listed in Table 4.2 can be correlated into the following equation:

T = 105 + 15 sin θ (4.23)

where

T = ambient air temperature, ˚F θ = 15(t–10), degrees t = local time, hr

For hot, dry day conditions, nominal accompanying wind speeds at the time of high temperature are up to 13 ft/s (or 4 m/s). The maximum ground surface temperature is assumed to be 145°F (or 63°C). At ground elevations above 3000 ft (or 915 m), maximum air temperatures will be lower by approximately 5°F per 1000 ft (or 9.1°C per 1000 m), and solar radiation fluxes may be higher by approximately 4 Btu/(hr·ft2) per 1000 ft (or 43 W/m2 per 1000 m) to 15,000 ft.

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There is a common misconception about radiation exchange between a body and the ambient air. The air (atmosphere) is essentially transparent, i.e., a nonparticipating gas. The radiation exchange actually takes place between the body and its surrounding structures as well as the sky. The sky temperature is related to the local air temperature by the following simple equation due to Swinbank [5]:

1.5 Tsky = 0.0552(Tair) (4.24)

where Tsky and Tair are both in Kelvins. An alternative expression is due to Bliss [6]:

 T − 273025. =+dp TT.sky air 08  (4.25)  250 

where Tdp is the dew-point temperature and all temperatures are again in kelvins. Both equations give about the same results when the relative humidity is approximately 25%. Equation 4.24 gives a small difference between the sky temperature and the local air temperature. The range of the differences as determined by Equation 4.25 is from 10°C in a hot, moist climate to 30°C in a cold, dry climate. The net radiation to the sky from a surface (or body) with emissivity ε and area A at temperature T is

=−εσ 4 qATT()4 sky (4.26)

For a cloudy day, the ambient air temperature Tair can replace Tsky. The infrared (IR) energy emitted by the sun is at much shorter wavelengths than those emitted by a body near room temperature. This distinct characteristic allows for the use of some thermal-control coatings that are very reflective in the solar spectrum but highly emissive to room-temperature (long-wavelength) IR. In other words, the values of solar absorptivity and longer-wavelength emissivity of the coated surface are quite different. This method provides the primary thermal control of spacecraft by using various coatings with high longer-wavelength emissivity and low solar absorptivity; however, the same concept may also be applied to ground systems.

REFERENCES

1. J. R. Howell, A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1982. 2. H. C. Hottel and A. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. 3. Research, Development, Test and Evaluation of Materials for Extreme Cli- matic Conditions, U.S. Army AR 70-38, 1979. 4. F. Kreith, Chapter 5, Principles of Heat Transfer, International Textbook, Scran- ton, PA, 1958.

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5. W. C. Swinbank, “Long Wave Radiation from Clear Skies,” Quar. J. Royal Mete- orol. Soc. 89, 1963. 6. R. W. Bliss, “Atmospheric Radiation Near the Surface of the Ground,” Solar Energy 5, 1961. 7. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, McGraw-Hill, New York, 1972. 8. D. G. Gilmore, “Satellite Thermal Environments,” Chapter II, Satellite Thermal Control, D. G. Gilmore (ed.), Aerospace Corporation Press, El Segundo, CA, 1994.

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POOL BOILING

From the viewpoint of thermodynamics, when a body of fluid in the purely liquid phase is heated at a constant pressure, its temperature and volume will both increase, as reflected in the line from A to B in Figure 5.1. Point B represents the maximum temperature and volume of saturated liquid at the given pressure, and between points B and C, any additional heat added to the system results in a change from the liquid phase to the vapor phase. In this region, liquid and vapor phases coexist in equilibrium. The amount of heat required to convert a unit of mass from liquid to vapor is called the latent heat of vaporization hfg (= hg – hf), where hg and hf are the enthalpy of saturated vapor and liquid, respectively. In this region, the addition of heat increases the vapor fraction and the volume of the mixture; however, the temperature of the two-phase mixture still remains at the saturation temperature. At point C, all the liquid has vaporized, and the vapor is now referred to as saturated vapor. Further heating will result in a temperature rise of the vapor (superheated vapor) along the constant-pressure line. Boiling is the process of evaporation associated with vapor bubbles in liquid. The change of phase from liquid to vapor is caused by heat transfer or by pressure changes. The two basic types of boiling are pool boiling and flow boiling. The former is boiling on heated surfaces submerged in a pool of stationary liquid, while the latter is boiling on the surface of a flowing stream of liquid. Boil- ing can also be classified according to the liquid temperature, into subcooling boiling, in which the bulk fluid temperature is below the saturation temperature; and saturated boiling, in which the bulk fluid temperature is uniformly at the saturation temperature.

5.1 BOILING CURVE

When a pool of liquid at saturation temperature heated by an electric heater, representing a heat flux–controlled condition, the typical boiling curve for heat flux q″ versus ∆Tsat (= Tw – Tsat) is plotted in Figure 5.2. The solid line in the figure represents pool boiling in Figure 5.2. Physically, the basic regions and points on the boiling curve given in Figure 5.2 are described as follows:

1. Single-phase natural convection (AB). At low superheat of the liquid ∆ ( Tsat), heat transfer is governed by single-phase natural convection. When ∆Tsat is large enough at point B, bubble nucleation takes place.

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FIGURE 5.1 Constant-pressure heating process.

FIGURE 5.2 Regimes in pool-boiling heat transfer.

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2. Nucleate boiling (BC). As the heat flux is increased continuously, the number of nucleation sites on the heating surface increases until the maximum heat flux is reached at point C. The maximum heat flux is often called the critical heat flux (CHF). At point C, the vapor generation rate is so high that the surface is blanketed with a vapor film which prevents liquid from contacting the surface. 3. Partial film boiling (CD). In this region, a portion of the heated surface will undergo film boiling while other portions are under nucleate boiling; therefore, the vapor film is unstable. A smooth transition from nucleate to film boiling can occur only under temperature-controlled conditions, which can be accomplished by condensing vapors or by convection from high-temperature fluids with high mass flow rates. 4. Film boiling (DE). With electric heating, it is impossible to operate in region CD. When the power is increased, the operating point shifts rapidly from point C to C′, which is at a much higher temperature. A burnout could occur if point C′ is above the melting point of the solid surface. If the temperature of point C′ is below the melting point, operations can be maintained along the curve DE. The region DE is called film boiling because a layer of vapor film continuously covers the surface and liquid does not contact the surface. Because of high surface temperature in this region, radiation becomes an important factor. The operating condition will change directly from point D to nucleate boiling at point D′ if the heat flux is reduced below point D. Then, power again must be increased to point C before the process can shift to film boiling.

The dashed line shown for pool boiling of subcooled liquid lies near the curve for saturated liquids. The degree of subcooling is defined as the difference between the saturation temperature and the bulk fluid temperature (i.e., ∆ Tsub = Tsat – Tbulk). As can be seen from Figure 5.2, the slope of pure natural convection is much smaller than that of nucleate boiling. This result implies that nucleate boiling is a more effective heat transfer mode. The major difference between saturated and subcooled boiling is that no net vapor is generated and no bubble is detached from the heated surface in subcooled boiling. In this form of boiling, bubbles grow and collapse while attached to the heated surface. Figure 5.3 depicts the mechanism by which the transition from nucleate to film boiling occurs for heat flux–controlled and temperature-controlled conditions. At a heat flux–controlled surface with a constant heat flux q″, when excessive vapor forms at the surface, the heat transfer coefficient hb drops, and the wall temperature Tw becomes extremely high in order to maintain the heat flux: q″ = hb[Tw – Tvapor (or Tsat)]. In this case, the surface temperature is so high that physical burnout may occur. On the other hand, at a temperature-controlled surface, when the vapor film forms, the surface temperature Tw goes up; therefore, the heat flux q″ [= hf (Tf – Tw)] drops because hf and Tf are constant. Consequently, the process in this case is self-regulating.

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FIGURE 5.3 Boiling mechanism for heat flux–controlled and temperature-controlled conditions.

5.2 NUCLEATE BOILING

Nucleate boiling involves two separate processes: the formation of bubbles (nucleation), and the subsequent growth and motion of these bubbles. For the case of boiling at solid heating surfaces (heterogeneous nucleation), most of the proposed theories assume that nucleation is initiated in cavities in which a gas or vapor preexists. A simple model, which is represented by a conical cavity in the surface with a spherically shaped bubble emerging from it, is shown in Figure 5.4. The contact angle β is the angle measured from the solid surface through the liquid to a tangent from the bubble. The solid angle increases with the surface tension σ and is an indicator of how well the liquid wets the solid. From a force balance of a static spherical bubble, one obtains

2σ PP−= (5.1) vlr

where

Pv = pressure in vapor Pl = pressure in liquid σ = surface tension of liquid r = radius of hemispherical bubble on a surface

To obtain the ∆T corresponding to ∆P (= Pv – Pl) given in Equation 5.1, one can integrate the Clausius-Clapeyron relation along the P-T saturation curve. The relation is

dT Tvfg = (5.2) dP hfg

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FIGURE 5.4 Nucleation from a cavity.

where vfg (= vv – vf) is the difference in specific volume between the saturated vapor and liquid. Assuming Tvfg/hfg is constant and T ≈ Tsat, the integration of Equation 5.2 becomes

 Tv 2σ TT−=sat fg   =−TT (5.3) v sat    l sat  hrfg 

The subscripts v and l are for vapor and liquid phases, respectively. In addition to the above assumption, many other approximations have also been used [1]. As pointed out by Rohsenow [1], however, no significant difference in the results among all approximations was found. The vapor in a cavity on a heated wall grows by evaporation at the vapor-liquid interface in the cavity when heat is added. At this point, the important parameters are the wall superheat and the surface tension. With sufficiently high superheat, the bubble emerges from the cavity and starts to grow on the flat surface outside the cavity. The bubble will grow and detach from the surface as the buoyant force exceeds the surface tension force if a liquid is at or above the saturation temperature. On the other hand, if the liquid is highly subcooled, the bubble will collapse without leaving the heated surface, or it will detach and immediately collapse into the subcooled liquid. The surface tension relations on the surface generate three different characteristic shapes for the bubble [2], as shown in Figure 5.5. For a large surface tension, labeled case A, the liquid does not wet the surface, and the edge of the bubble is drawn between the liquid and the heated surface in the form of a fine wedge. In this case, the bubble is difficult to detach from the surface. For a small surface tension as in case C, the surface is substantially wetted, and the bubbles are in general small and in a near spherical shape. For case B, the bubble represents an intermediate shape between cases A and C.

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FIGURE 5.5 Bubble shapes with various values of contact angle (surface tension).

Once the bubbles are detached, depending on the contact angle, which is related to the surface tension of the liquid, the inflow of liquid may trap gas and vapor in the cavities for the next nucleation sites. As shown in Figure 5.6 [3], it can be seen from the geometry relation that liquid will fill up the cavity completely (no gas or vapor trapped) if β ≤ φ. On the other hand, if β > φ, some vapor or gas will be trapped under the liquid in the cavity and the cavity will remain active for the next nucleation. Inert gas is frequently present in boiling systems, either in solution or trapped in surface cavities. The previous analysis can be modified to account for the presence of an inert gas, and Equations 5.1 and 5.3 become:

2σ PP−= −P (5.4) vlr g

and

 Tv 2σ  TT−=sat fg  − PTT =− v sat   gl sat (5.5)  hrfg 

which indicates that the superheat required for a bubble of a given size to grow is decreased by the presence of the gas partial pressure Pg.

5.3 INCIPIENT BOILING AT HEATING SURFACES

The previous topic, e.g., Equation 5.5, is referred to as nucleation in isothermal surroundings, i.e., in the uniform-temperature region of liquid and solid surfaces. For a bubble growing at a heated surface, a temperature gradient can exist in the liquid. In the region near the wall, a linear profile of the liquid temperature can be assumed, i.e.,

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FIGURE 5.6 Vapor trapped in cavities by advancing liquid.

 TT−  qk′′ =−  wl (5.6a) l  y 

=−′′ y TTlw q (5.6b) kl

where kl is thermal conductivity of the liquid and y is the distance from the wall. The postulated criterion proposed by Bergles and Rohsenow [4] as illustrated in Figure 5.7 is that nucleation takes place when the temperature curve, Equation 5.6b, in the liquid is tangent to the temperature of the vapor represented by Equation 5.5. At the point of tangency, the radius of the first cavity to nucleate obtained by solving Equations 5.5 and 5.6b is:

 σ  05. 2 Tvksat fg l r =   ′′ (5.7)  hqfg 

and

2  q  hkT()− T q′′ =   = fg l w sat   σ (5.8) A i 8 Tvsat fg

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FIGURE 5.7 Conditions required for nucleation in a temperature gradient.

The subscript i indicates the incipient condition. Equation 5.8 is for the case of heterogeneous nucleation and can be rewritten in the following form in terms of the wall superheat for boiling incipience:

 8qTv′′σ  05. −= sat fg ()TTwisat   (5.9)  hkfg l 

As pointed out in Bar-Cohen and Simon [3], because of the small wetting angle of most candidates for direct-immersion cooling fluids (dielectric fluids), as well as the smoothness and relatively small dimensions of microelectronics packages, heterogeneous nucleation may not take place. As a consequence, under the imposed heat flux, a much higher wall superheat may be required for nucleation. The liquid superheat required for initiating boiling in this fashion is equal to the maximum superheat that the liquid can sustain. The maximum superheat for this type of nucleation (homogeneous nucleation) is suggested by Lienhard [5] as follows:

    8  T T  ()TT−≤ T0.. 905 −+sat 0 095 sat  (5.10) wicsat      TccT 

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FIGURE 5.8 Schematic representation of wall superheat excursion (overshoot).

where the temperature is expressed in kelvins and Tc is the critical temperature. Assuming surface, liquid, and vapor properties to be invariant with temperature, the correlation for heat flux versus wall superheat in nucleate pool boiling can be written in the form:

∆ =− = ′′ 033. TTsw Tsat Cq() (5.11)

where the expression for q″ will be given in Equation 5.14 of the next section. With the aid of Equation 5.11, the wall superheat excursions [3] for homogeneous, such as highly wetting liquids, and heterogeneous nucleation are as follows, respectively:

    8  T T  033. ()TT−≤ T0.. 905 −+sat 0 095 sat  − Cq()′′ (5.12) wicsat      TccT 

and

 Tv 2σ  ()TT−=sat fg  − PCq − ()′′ 033. (5.13) wisat   g   hrfg 

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Table 5.1 Surface-Fluid Parameters

Surface-fluid combination Csf Water-nickel 0.006 Water-platinum 0.013 Water-copper 0.013 Water-brass 0.006 Carbon tetrachloride–copper 0.013 Benzene-chromium 0.010 n-Pentane–chromium 0.015 Ethyl alcohol–chromium 0.0027 Isopropyl alcohol–copper 0.0025 35% potassium carbonate–copper 0.0054 50% potassium carbonate–copper 0.0027 n-Butyl alcohol–copper 0.003

The schematic representation of the wall superheat excursion as shown in Figure 5.8 is given in Bar-Cohen and Simon [3].

5.4 NUCLEATE BOILING CORRELATIONS

As stated before, boiling is the process of evaporation associated with vapor bubbles in liquid. Based on the postulate that the observed nucleate boiling heat flux is associated with bubble pumping of hot liquid away from the heated surface, Rohsenow [6] obtained a correlation of nucleate pool boiling in the form

  − n cTpl() w Tsat  q′′  g σ 05.  = C   c   (Pr)m (5.14) h sf µ h g()ρρ− fg  lfg lv 

where

= cpl specific heat of saturated liquid = gc conversion factor g = gravitational acceleration ρ = l density of saturated liquid µ = l viscosity of saturated liquid ρ = v density of saturated vapor

Csf is a constant reflecting the condition of a particular fluid and surface combination as tabulated in Table 5.1. For a general application, n is commonly set at 0.33, m = 1 for water, and m = 1.7 for other fluids.

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5.5 CRITICAL HEAT FLUX CORRELATIONS

The critical or burnout or peak heat flux is generally the most important piece of information required for design of any heat transfer system involving boiling. Any heat flux above this value will result in an extremely high wall temperature. In spite of intensive research efforts, there is still some disagreement regarding the actual mechanism of the critical heat flux (CHF). Among the alternatives, two models from Rohsenow and Griffith [8] and Zuber [9] may be briefly described as follows:

1. Bubble packing [8]: Because of high heat flux, the number of nucleation sites becomes so numerous that the neighboring bubble or vapor column coalesces, causing a vapor to blanket the heating surface. 2. Hydrodynamic instability [9]: At high heat flux, the number of nucleation sites is so large and the vapor generation rate is so high that the area between the bubble columns for liquid flow to the heating surface is reduced. The relative velocity is then so large that the liquid-vapor interface becomes unstable, essentially starving the surface of the liquid and causing the formation of a vapor blanket.

Numerous semiempirical correlations for the critical heat flux have been proposed by using the above models. Based on the first model [8], a simple relation is:

 ρρ−  06.  g  025. qh′′ = 143 ρ lv crit fg v ρ    (5.15)  vc  g 

where all quantities are in British units. The acceleration of gravity and the gravitational constant are g and gc, respectively. The following combined equation from References [10-13] has been proposed according to the assumption of hydrodynamic instability [9]:

σρ()− ρ gg 025. qKh′′ = ρ  lvc (5.16) crit vfg ρ 2   v 

The constant K varies from 0.12 to 0.16, but 0.18 agrees well with most data for a horizontal heating body. For a vertical body, Chang [14] proposed a value of K = 0.098. Note that while the above correlation, Equation (5.16), predicts well for the selected sets of data, it is not accurate for all fluids in all systems. For example, the predicted critical heat fluxes are widely divergent for a number of cryogenic fluids such as liquid hydrogen, liquid nitrogen, and liquid oxygen [15]. Significant progress has also been made in accounting for the effect of heater geometry. Lienhard and coworkers [16–18] performed a series of investigations and found that heater geometry and size affected the CHF. The general correlation is given as follows:

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FIGURE 5.9 Predictions of critical heat flux for various heater configurations.

q′′ crit = fL()′ ′′ (5.17) ()qcrit F

where (qcrit″ )F is given in Equation 5.16 and L′ is a characteristic length defined by

 g()ρρ− 05. LL′ =  lv (5.18) σ  gc 

The data, as shown in Figure 5.9, indicate that the curves for a horizontal cylinder, a vertical plate with both sides heated, a vertical plate with one side insulated, and a sphere vary according to geometry. The results approach the Zuber relation asymptotically as L′ increases toward 10. It is interesting to note that at very small values of L′ (e.g., L′<0.01), which are frequently encountered in the case of boiling from surfaces of microelectronics packages, nucleate boiling and the associated hydrodynamic instability vanish. As heat flux is increased, natural convection will proceed directly into film boiling when the first bubble is generated. This effect is due to the fact that the first

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bubble may blanket the heater (microelectronics circuit) because of the extremely small size of the circuit. Under these circumstances, it is difficult to define the CHF [19], and extrapolation of those curves presented in Figure 5.9 to values of L′ less than 0.1 must be made with caution.

5.6 MINIMUM HEAT FLUX CORRELATIONS (LEIDENFORST POINT)

As mentioned previously, a smooth transition from nucleate boiling to film boiling can occur only under temperature-controlled conditions. In the transition region between C and D in Figure 5.2, the vapor film is unstable. As the surface temperature is increased, the average wetted area of the heater surface decreases. The corresponding heat flux is also decreased and finally reaches a minimum value, which is often referred to as the Leidenforst point. Physically, a continuous film covers the heater’s surface at this condition, which is the minimum film boiling condition. Numerous analyses have been made to predict the minimum heat flux. The result for a horizontal flat plate is correlated into the following general equation:

σρρgg ()− 025. qCh′′ = ρ  cl v (5.19) min fg v  ρρ+ 2   ()lv 

where C is a constant with various values such as 0.177 [10], 0.13 [9], or 0.09 [20]. The minimum wall superheat required to sustain film boiling is also given as follows [20]:

()ρ h  gg()(ρρ− 23/  / g)σ 12/ ∆T = 0. 127 vf fg lv  c  min ρρ+ ρρ  ()kvf  lv  lv  (5.20)  ()µ 13/ ×  l f  σρ− ρ  gclv()

where the subscript f means that the properties are evaluated at the mean film temperature, i.e., Tf = 0.5(Tw + Tsat). For the case of small-diameter tubes, Lienhard and Wong [21] included the effects of the curvature and surface tension in the transverse direction on the Tay- lor instability of the interface and obtained a semiempirical equation:

 ρ h  2g()ρρ− σ 12/ q′′ = 0. 057 vfg  lv+  min    ρρ+ ρρ+ 2  R  lv ()lv (5.21)  g()ρρ+ 05. −34/ ×  lv+   σ R2 

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where R < 0.03. For large tubes, Equation 5.19 may still be applied; however, the accuracy is questionable for water on a clean surface at pressures above atmospheric [22]. Also note that the above equations do not take into consideration the effects of impurities in liquids and surface contamination such as oxidation and scaling. These effects will increase (q″)min significantly. A stable film boiling will be developed if the surface temperature is increased continuously beyond the Leidenforst point. The analytical solution becomes possible in this region because the flow pattern is relatively simple. The result for vertical surfaces can be determined as an inversion problem of the classical Nus- selt equation for laminar film condensation [23]. The average heat transfer coefficient over a plate width L is given by:

 kgh3 ′()ρρρ− 14/ h = 0. 943 vlvv (5.22a)  µ −   LTTvw()sat 

where

 −  ′ =+05. cTpv() w Tsat hhfg 1  (5.22b)  hfg 

where all vapor properties are determined at the mean film temperature Tf. Sometimes, under certain conditions, Equations 5.22a and 5.22b can be approximately applied to the case of horizontal tubes by replacing L with πD/2; however, a more accurate expression that includes the effects caused by interfacial shear and curvature is presented by Breen and Westwater [24]:

 λ   kgh3 ′()ρρρ− 34/ h =+0.. 59 0 069   vlvv    λµ −  (5.23) D  vw()TTsat 

where

σ 12/ λπ= ( ggc / ) 2  ρρ−   lv

The heat transfer coefficient for film boiling from a horizontal flat plate is as follows [20]:

14/  kgh3 ′()ρρρ−  h = 067.  vlvv (5.24)  λµ −   vw()TTsat 

Because of the high temperature in film boiling, thermal radiation cannot be neglected. The total heat transfer coefficient is expressed as:

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 13/ = h +≈+ hht   hhrr075. h (5.25)  ht 

where

σFT()4 − T4 h = w sat (5.26) r − TTw sat

The view factor F is approximately equal to 1, and σ is the Stefan-Boltzmann constant.

5.7 PARAMETERS AFFECTING POOL BOILING

A number of variables that can affect the heat transfer capability of a boiling system will be discussed in this section.

5.7.1 Pressure

Less wall superheat is required for a given heat flux when the pressure is increased in a boiling system. That means the nucleate boiling regime on the boiling curve moves to the left as shown in Figure 5.10 [25].

5.7.2 Subcooling

The effect of subcooling on the position of the boiling curve seems to be geometry-dependent. This effect may be due to variable natural convection depending on the geometry prior to boiling. Figure 5.11 shows that for water boiling on a 0.0646-in-diameter stainless steel tube [26], the boiling is shifted to the right with increasing subcooling. Conversely, the opposite trend was found with a flat, horizontal heating surface as presented in Figure 5.11 [27]. This may be due to differences in natural convection based on the geometry prior to boiling. The effect of subcooling is negligible over most of the nucleate boiling regime; however, the CHF is increased approximately as a linear function of the subcooling. This fact is probably attributable to an increase in the number of bubbles formed and their frequency of departure or collapse as the degree of subcooling is increased. This result, in turn, creates greater turbulence in the boundary layer near the heated surface. Another reason is that the heat capacity increases as the subcooling increases. The enhanced heat transfer, combined with the higher heat capacity of the colder liquid, allows much higher heat flux to be handled in a system before the vapor blankets the heated surface. The effect of subcooling on the CHF has been studied by a number of investi- gators. Among them, Ivey [28] proposed the following simple correlation:

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FIGURE 5.10 Effect of pressure on pool-boiling curve.

ρ 3/. 4 0 273 − ()q′′    g  cTpl()sat T b crit sub =+101.  l    ′′ ρ (5.27) ()qcrit sat  vc  g  hfg

For water at subatmospheric pressures, the following equations proposed by Ponter and Haigh [29] should be used:

()q′′  TT−  crit sub =+1.. 06 0 015 sat b  ′′ 0474. (5.28) ()qcrit sat  p 

° ″ where Tsat – Tb is in C and p is in torr. (qcrit)sat in the preceding equations is determined by Equation 5.16.

5.7.3 Surface Conditions

Surface Finish

The surface finish (roughness) can shift the position of the boiling curve because different surface characteristics are associated with different sizes and numbers

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FIGURE 5.11 Influence of subcooling on pool boiling.

of cavities. The effect due to the surface roughness is quantitatively represented in Figure 5.12. Rougher surfaces are more effective in the nucleate boiling region. As previous discussions related to Figure 5.6, small cavities (smooth surface) more easily trap vapor than large cavities; thus, they will favor the possibility of nucleation. As a consequence, a rough surface enhances the nucleate boiling process; meanwhile it also increases the possibility of thermal hysteresis (wall superheat excursion). Therefore, it appears that there exists an optimum surface for any application, and this can only be found by experiment.

Aging

Experiments indicate that aged metal surfaces have higher required superheat ∆T for a given heat flux. The reason is that a scale or deposit may form on the surfaces from liquid boiling, or a film may also form on the surface from oxidation or other chemical reactions. In either case, any impurities may reduce the size of the cavities and thus increase the required superheat for boiling. In any coated

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FIGURE 5.12 Effect of surface roughness on boiling curve.

surface, an additional temperature difference caused by heat conduction causes the boiling curve (q″ versus ∆T) to be shifted further to the right and at a lower slope than that of clean surfaces.

5.7.4 Dissolved Gases

Dissolved gas bubbles will generally form on the heated surface, and they tend to shift the boiling curve to the left as illustrated in Figure 5.13 [30]. The dissolved gas bubbles, however, will detach from the surface due to the action of the buoyant forces or fluid dynamic force. Therefore, any effects of the dissolved gases are temporary because each detached bubble carries away some of the inert gas and eventually the gas is depleted completely. The effect of noncondensable gases on the CHF in pool boiling tends to reduce the magnitude of the CHF for a given condition but to increase CHF as subcooling increases, as indicated in Figure 5.14 [31].

5.7.5 Wettability

The degree of wetting between the heated surface and a liquid is an important parameter in boiling heat transfer. To ensure that all cavities are active, it is desirable that the liquid not wet the cavity surface too well. Otherwise, the

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FIGURE 5.13 Effect of dissolved gas on boiling curve.

cavities may not trap the vapor and thus may become deactivated, or “snuffed out.” On the other hand, once a bubble forms and begins to grow on the surface, it is better for the liquid to wet the surface to prevent the vapor from blanketing the surface. To balance these two opposite effects, Hummel [32] applied a nonwetting epoxy resin to coat stainless steel strips and then sanded the surface. By doing so, he removed the nonwetting coating from the surface but not from the cavities. Bergles et al. [33] used stainless steel coated with Teflon in the boiling of water and of two dielectric fluids, R-113 and FC -78. Spots of Teflon in pits were found to be very effective in augmenting heat transfer for pool boiling or low-velocity flow boiling of water, but were not effective for boiling of R-113 and FC-78. The contact angle of water to Teflon was much higher than that of water to uncoated stainless steel; while the contact angles formed by both R-113 and FC-78 were just a few degrees with both Teflon and stainless steel.

5.7.6 Heater Geometry The geometry effect is significant for the region of single-phase natural convection along the boiling curve; however, size and orientation appear to have very

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FIGURE 5.14 Effect of dissolved gas and subcooling on CHF under atmospheric pressure.

little effect on the q″ versus ∆T curve in the nucleate boiling region. On the other hand, the geometry effect is pronounced for the CHF. According to Bernath’s studies [34, 35], the variation in CHF is large for flat-plate data but is small for vertical-plate results. CHF increases as the heater diameter increases up to about 0.1 in (from 0.025 to 0.25 cm) and then levels off beyond that size. For sizes below 0.025 cm down to 0.0076 cm, the magnitude of CHF levels off again [36]. It was also found that in the range where CHF varies with diameter of the wire, the bubble sizes are on the order of the wire diameter.

5.7.7 Agitation

The effect caused by agitation, such as with a propeller, is very similar to the effect whereby q″ increases with velocity. This effect, indicated in Figure 5.15, was investigated by Pramuk and Westwater [37]. They found that CHF also increases as the rotation speed (rpm) increases.

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FIGURE 5.15 Effect of agitation in methanol boiling at 1 atm.

5.7.8 Hysteresis

Thermal hysteresis is a phenomenon characterized by a deviation from the normal boiling curve. This phenomenon often occurs in boiling of dielectric liquids or liquid metals because of their extremely high wettability. An example of this wall temperature overshoot is presented in Figure 5.16 [33] for pool boiling of R-113. There are many factors such as the following that contribute to this phenomenon:

1. Liquid property. High wettability caused by a low liquid surface tension. 2. Heated wall surface. The smoother the surfaces, the less wall superheat overshoot. 3. Hydrodynamic conditions. Thermal hysteresis occurring in general only in very low-velocity flow boiling or pool boiling.

5.8 EFFECT OF GRAVITY ON POOL BOILING

The effect of reduced gravity on heat transfer for both single- and two- phase fluids was documented in detail by Siegel [38]. The effect of gravity on pool boiling is summarized as follows.

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FIGURE 5.16 Thermal hysteresis in pool boiling of R-113.

5.8.1 Nucleate Pool Boiling

Siegel and Vsiskin [39, 40] indicated that only a very small g field is needed to sustain nucleate boiling, and thus, the heat flux q″ versus the wall superheat (Tw – Tsat) in the nucleate pool-boiling region is insensitive to gravity. A typical pool-boiling curve for liquid nitrogen is shown in Figure 5.17 [41].

5.8.2 Critical Heat Flux for Pool Boiling

The theories and correlations as given in Equations 5.15 and 5.16 indicate that the CHF depends on g1/4. If the fluid properties are independent of gravity, the CHF ratio can be written as:

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FIGURE 5.17 Liquid nitrogen pool-boiling curve at earth and near zero gravity.

()qg′′  g 14/ CHF =   ′′ (5.29) ()qgCHF EE g 

where gE is the gravity of the earth. Equation 5.29 suggests that as gravity approaches zero, it is no longer possible to maintain the nucleate pool boiling because the peak heat flux or CHF will approach zero. Figures 5.18 and 5.19 [39] indicate that for the range of 0.01gE < g < 1gE, the one-quarter power dependency appears to be a reasonable approximation for engineering applications.

5.8.3 Transition Region for Pool Boiling

On the basis of the limited information presented in Figure 5.17, it appears that the curve q″ versus (Tw – Tsat) in the transition region is insensitive to gravity reductions, as it is generally in nucleate boiling.

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FIGURE 5.18 Reduced-gravity CHF data for horizontal surfaces and wires.

5.8.4 Minimum Heat Flux

For a horizontal plate, a relation for the minimum heat flux was given by Equa- tion 5.19. The correlation predicts that heat flux will vary as gravity to the one- quarter power; however, the same correlation may not be valid for other geometries.

FIGURE 5.19 Reduced-gravity CHF data for vertical wires and spheres.

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5.8.5 Film Pool Boiling

As shown in Equations 5.22a and 5.24, the heat transfer coefficient decreases as gravity to the one-quarter power for a laminar vapor film; however, the exponent of the gravity ratio will increase when the vapor film becomes turbulent, as discussed by Hsu and Westwater [46].

REFERENCES

1. W. M. Rohsenow, “Nucleation with Boiling Heat Transfer,” ASME paper No. 70-HT-18, 1970. 2. J. H. Seely and R. C. Chu, Heat Transfer in Microelectronic Equipment: A Prac- tical Guide, Marcel Dekker, New York, 1972. 3. A. Bar-Cohen and T. W. Simon, “Wall Superheat Excursions in the Boiling Incipience of Dielectric Fluids,” AIAA/ASME Conf. 1, 1986. 4. A. E. Bergles and W. M. Rohsenow, “The Determination of Forced Convection Surface Boiling Heat Transfer,” J. Heat Transfer 86, 1964. 5. J. H. Lienhard, “Correlation of Limiting Liquid Superheat,” Chem. Eng. Sci- ences 31, 1976. 6. W. M. Rohsenow, “A Method of Correlating Heat Transfer Data for Surface Boiling of Liquids,” Trans. ASME 74, 1952. 7. W. M. Rohsenow, “Pool Boiling,” in G. Hetsroni, ed., Handbook of Multiphase Systems, Hemisphere/McGraw-Hill, New York, 1982. 8. W. M. Rohsenow and P. Griffith, “Correlation of Maximum Heat Transfer Data for Boiling Saturated Liquids,” Chem. Eng. Progress Symp. Ser. 52(18), 1956. 9. N. Zuber, “On the Stability of Boiling Heat Transfer,” J. Heat Transfer, 80(3), 1958. 10. N. Zuber, “Hydrodynamic Aspects of Boiling Heat Transfer,” U.S. AEC Report AEC-4439, 1959 (doctoral dissertation, University of California at Los Ange- les, 1959). 11. S. S. Kutateladze, “Heat Transfer in Condensation and Boiling,” U.S. AEC Report AEC-TR-3770, 1952. 12. Y. P. Chang and N. W. Synder, “Heat Transfer Saturated Boiling,” Chem. Eng. Progress Symp. Ser. 56(30), 1960. 13. N. Zuber, M. Tribus, and J. W. Westwater, “The Hydrodynamic Crisis in Pool Boiling of Saturated and Subcooled Liquids,” Int. Dev. Heat Transfer, Part II, ASME, New York, 1961. 14. Y. P. Chang, “Some Possible Critical Conditions in Nucleate Boiling,” J. Heat Transfer 86, 1964. 15. J. D. Seader et al., “Boiling Heat Transfer for Cryogenics,” NASA CR-243, 1965. 16. K. H. Sun and J. H. Lienhard, “The Peak Pool Boiling Heat Flux on Horizon- tal Cylinders,” Int. J. Heat and Mass Transfer 10, 1970. 17. J. H. Lienhard and V. K. Dhir, “Hydrodynamic Prediction of Peak Pool Boil- ing Heat Fluxes from Finite Bodies,” J. Heat Transfer 95, 1973. 18. J. H. Lienhard, V. K. Dhir, and D. M. Ricerd, “Peak Pool Boiling Heat Flux Measurements on Finite Horizontal Flat Plate,” J. Heat Transfer 95, 1973.

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19. A. E. Bergles, “Burnout in Boiling Heat Transfer, Part 1: Pool Boiling Sys- tems,” Nuclear Safety 16, 1975. 20. P. J. Berenson, “Transition Boiling Heat Transfer from a Horizontal Surface,” J. Heat Transfer 83, 1961. 21. J. H. Lienhard and P. T. Y. Wong, “The Dominant Instable Wavelength and Minimum Heat Flux during Film Boiling on a horizontal Cylinder,” J. Heat Transfer 86, 1964. 22. S. A. Kovalev, “An Investigation of Minimum Heat Fluxes in Boiling of Water,” Int. J. Heat and Mass Transfer 9, 1966. 23. C. A. Bromley, “Heat Transfer in Stable Film Boiling,” Chem. Eng. Progress Symp. Ser. 46(5), 1950. 24. B. P. Breen and J. W. Westwater, “Effect of Diameter of Horizontal Tubes on Film Boiling Heat Transfer,” Chem. Eng. Progress Symp. Ser. 58(7), 1962. 25. M. T. Cichelli and C. F. Bonilla, Trans. AIChE 41, 1945. 26. A. E. Bergles and W. M. Rohsenow, ‘The Determination of Forced-Convection Surface-Boiling Heat Transfer,” J. Heat Transfer 86, 1964. 27. E. E. Duke and V. E. Shrock, “Void Volume, Site Density, and Bubble Size for Subcooled Nucleate Pool Boiling,” Proc. Heat Transfer and Fluid Mech. Inst., Stanford University Press, Palo Alto, CA, 1961. 28. H. J. Ivey, “Acceleration and the Critical Heat Flux in Pool Boiling Heat Transfer,” Chartered Mechanical Engineer 9, 1962. 29. A. B. Ponter and C. P. Haigh, “The Boiling Crisis in Saturated and Subcooled Pool Boiling at Reduced Pressures,” Int. J. Heat and Mass Transfer 12, 1969. 30. W. H. McAdams, W. E. Kennel, C. S. Minden, C. Rudolf, and J. E. Dow, “Heat Transfer at High Rates to Water with Surface Boiling,” Ind. Eng. Chem. 41, 1959. 31. M. Jakob and W. Fritz, “Versuche über den Verdampfungsvorgang,” Forsch. Gebiete Ing. 2, 1931. 32. R. L. Hummel, “Means for Increasing the Heat Transfer Coefficient between a Wall and Boiling Liquid,” U.S. Patent No. 3,207,209, 1962. 33. A. E. Bergles, N. Bakhru, and J. W. Shires, Jr., “Cooling of High-Power-Density Computer Components,” Report No. DSR 70712-60, Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, 1968. 34. L. Bernath, “Prediction of Heat Transfer Burnout,” Heat Transfer Symp., AIChE National Meeting, Louisville, KY, 1955. 35. L. Bernath, “Theory of Local Boiling Burnout and Its Application to Existing Data,” Chem. Eng. Progress Symp. Ser. 56(30), 1960. 36. C. C. Pitts and G. Lappert, Int. J. Heat and Mass Transfer 9, 1966. 37. F. S. Pramuk and J. W. Westwater, “Effect of Agitation on the Critical Tem- perature Difference for a Boiling Liquid,” Chem. Eng. Progress Symp. Ser. 53(18), 1956. 38. R. Siegel, “Effects of Reduced Gravity on Heat Transfer,” in J. P. Hartnett and T. F. Irvine, Jr., eds., Advances in Heat Transfer, Vol. 4, Academic Press, New York, 1967. 39. R. Siegel and C. Vsiskin, “A Photographic Study of Boiling in the Absence of Gravity,” J. Heat Transfer 81, 1959. 40. C. Vsiskin and R. Siegel, “An Experimental Study of Boiling in Reduced and Zero Gravity Fields,” J. Heat Transfer 83, 1961.

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41. H. Merte, Jr., and J. A. Clark, “Boiling Heat Transfer with Cryogenic Fluids at Standard, Fractional, and Near-Zero Gravity,” J. Heat Transfer 86, 1964. 42. C. M. Usiskin and R. Siegel, “An Experimental Study of Boiling in Reduced and Zero Gravity Fields,” J. Heat Transfer 83, 1961. 43. J. E. Sherley, “Nucleate Boiling Heat Transfer Data for Liquid Hydrogen at Standard and Zero Gravity,” Advances in Cryogenic Engineering, K. D. Tim- merhaus, (ed.), Vol. 8, Plenum, New York, 1963. 44. R. G. Clodfelter, “Low Gravity Pool-Boiling Heat Transfer,” APL-TDR64-19 (DDC No. AD-437803), 1964. 45. R. Siegel and J. R. Howell, “Critical Heat Flux for Saturated Pool Boiling from Horizontal and Vertical Wires in Reduced Gravity,” NASA Tech. Note TND-3123, 1965. 46. Y. Y. Hsu and J. W. Westwater, “Approximate Theory for Film Boiling on Ver- tical Surfaces,” Chem. Eng. Progress Symp. Ser. 56, 1960. 47. A. E. Bergles, J. G. Collier, J. M. Delhaye, G. F. Hewitt, and F. Mayinger, Two- Phase Flow and Heat Transfer in the Power and Process Industries, Hemi- sphere, New York, 1981. 48. W. Frost, Heat Transfer at Low Temperature, Plenum, New York, 1975. 49. W. M. Rohsenow, J. P. Hartnett, and E. N. Ganic, Chapter 12, Fundamental Handbook of Heat Transfer, 2nd ed., McGraw-Hill, New York, 1985. 50. A. D. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment, Hemisphere, New York, 1983.

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FLOW BOILING

The previous chapter dealt with boiling in a pool of stationary liquid. Flow boiling is different from pool boiling in that the fluid is moving with a velocity caused by natural circulation in a loop or by an external force such as a pump. The basic heat transfer mechanisms in flow boiling are similar to those discussed in pool boiling; however, the fluid velocity enhances heat transfer in flow boiling. The boiling curve, therefore, is shifted to the upper left corner as compared with the case of pool boiling diagrammed in Figure 5.2. Another distinguishing feature of flow boiling is varying flow regions along the tube in the flow direction. In addition to heat transfer in flow boiling, the pressure drop for two-phase flow will also be discussed in this chapter.

6.1 FLOW PATTERNS

Because of the effect of gravity, flow regimes in horizontal and vertical tubes are different. The typical flow regimes in a horizontal tube are shown in Figure 6.1 [1]. In a horizontal tube, gravity tends to separate the liquid and vapor (or gas) phases. Because of gravity, the bubbles in a horizontal tube move to the upper section of the tube in bubbly flow. Plug flow follows when the gas flow rate is increased. For low liquid and gas flow rates, a stratified flow takes place with a smooth interface. On the other hand, a wavy interface (wavy flow) occurs at higher gas flow rates. The wave propagates to the top wall of the tube and forms a slug flow. Finally, the combination of high gas flow with low liquid flow gives rise to an annular flow with a thicker liquid film at the lower section of the tube. The flow patterns in upward vertical flow are shown in Figure 6.2. A different flow pattern will exist in downward flow. In upward flow, bubbles spread over the entire tube cross section, whereas bubbles gather near the center of the tube in downward flow.

6.2 HEAT TRANSFER MECHANISMS

Attention will first focus on a uniformly heated vertical tube with upward flow. The representative regions under this condition are illustrated in Figure 6.2. Ini- tially, single-phase (liquid) forced convection takes place. As the heat flux is

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FIGURE 6.1 Typical flow patterns for a horizontal tube.

increased over a certain value, the elevated wall temperature superheats the liquid on the wall surface and activates the nucleation sites. Bubbles are generated, and the boiling process is initiated. The boiling process in this region is called partial nucleate boiling; single-phase forced convection still exists between nucleation sites, and the bulk fluid temperature is below the saturation temperature. As the heating process continues, the number of nucleation sites increases until fully developed nucleate boiling is reached and the entire heating surface is in nucleate boiling. Nucleate boiling can also be divided into two regions, namely, subcooled and saturated nucleate boiling. The thermodynamic (mass) quality is less than 1 in the region of subcooled nucleate boiling. The quality also represents the fraction of vapor by weight and is given as follows:

hh− x = l (6.1) hfg

where

h = local enthalpy of fluid = hl enthalpy of saturated liquid = hfg latent heat of vaporization

As heating is continued, the flow undergoes a transition to the two-phase forced convective boiling region, where evaporation from the thin liquid layer is

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FIGURE 6.2 Heat transfer mechanisms and flow regions in upward vertical flow.

the dominant mode of heat transfer. Further heating causes the thin liquid film to dry out at some point along the tube. Single-phase forced convection of vapor flow then follows after the dryout. Generally, a liquid-deficient region, where a small quantity of liquid droplets is entrained in the vapor flow, exists between the regions of liquid film dryout and pure single-phase vapor flow.

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FIGURE 6.3 Heat transfer regions under various heat fluxes.

The temperature profiles of the wall and the fluid are also shown in Figure 6.2. The decrease in the saturation temperature, which will affect the tube wall temperature, is caused by the reduction in the system pressure along the flow direction. The above discussion is limited to low–heat flux conditions. For a high–heat flux case, a transition from the nucleate boiling region to film boiling will take place. This phenomenon is called departure from nucleate boiling (DNB), which is similar to the critical heat flux condition in saturated pool boiling. The result of DNB is a rapid increase in the wall temperature. The effect of varying heat flux on heat transfer coefficient is illustrated in Figure 6.3.

6.3 BOILING CRISIS

Boiling crisis is characterized by a sudden decrease in the heat transfer coefficient due to a change in boiling heat transfer mechanism. Boiling crisis often results in a very rapid increase in the wall temperature. The heat flux just before the boiling crisis is referred to as the critical heat flux (CHF) or peak heat flux. Several names such as departure from nucleate boiling (DNB), dryout, burnout, and boiling crisis have been used to describe this boiling phenomenon. The first two are the most commonly used in the literature. The mechanism of heat transfer when the critical heat flux due to dryout or DNB has been exceeded depends on the initial condition, i.e., in the process of evaporation from the thin liquid film or in the process of nucleate boiling. The

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former will initiate the liquid-deficient region, whereas film boiling will follow in the latter case. Although the terms dryout and DNB have been used interchangeably, the heat transfer mechanisms are somewhat different. As can be seen from Figure 6.3, DNB occurs only at relatively high heat fluxes. The high heat flux causes intensive boiling, and the bubbles are generated so fast that they are crowded near the heated surface. Such local voidage impairs surface cooling by reducing the amount of incoming liquid. This type of boiling crisis usually takes place at a high flow rate, and the fluid near the heated surface may not be in thermodynamic equilibrium with the bulk stream such as in subcooled boiling. The magnitude of the critical heat flux in the low-quality region depends on the “surface proximity” much more than it depends on the subcooling of the bulk fluid. The surface proximity parameters include the surface condition, local voidage, and boundary-layer superheat. In this type of boiling crisis, the wall surface temperature rises rapidly to a very high value at which physical burnout may occur. This type of boiling crisis is sometimes called fast burnout. On the other hand, dryout often takes places at a lower heat flux than does DNB. The total mass flow rate may be small, but the vapor velocity is still high owing to the high void fraction. The flow pattern is generally annular as opposed to the bubbly flow in DNB. The thin liquid film will break down and boiling crisis (dryout) will occur if there is excessive evaporation on the liquid layer. The magnitude of the critical heat flux in this high-quality region depends strongly on the flow parameters, such as void fraction, slip ratio (ratio of vapor to liquid velocity), vapor velocity, and liquid film thickness. Because of the relatively high heat transfer coefficient of a fast-moving vapor, the wall temperature rise after boiling crisis in the high-quality region is usually slower than that in a subcooled boiling crisis of DNB. This type of boiling crisis is sometimes referred to as slow burnout because physical burnout does not necessarily occur.

6.4 HEAT TRANSFER EQUATIONS

6.4.1 Single-Phase Forced Convection (Liquid)

Initially, heat transfer in a liquid takes place by forced convection. Therefore any suitable heat transfer correlations, such as those listed in Appendix H, can be used in this region. For example, the Dittus-Boelter equation is a popular one for turbulent flow in a smooth tube:

Nu = 0.023 Re0.8 Pr0.4 (6.2)

6.4.2 Nucleate Boiling

Onset of Subcooled Nucleate Boiling

As in pool boiling, nucleate boiling will occur if there is enough wall superheating. Davis and Anderson [2] proposed the following equation as valid for all fluids:

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 8σqT′′  05. ()TT−=sat wsat onb  ρ  (6.3)  khlfgg

This equation is practically identical to Equation 5.9 if ρg is replaced by 1/vfg. Equation 6.3 will predict the onset of nucleate boiling accurately if there is a full range of cavity sizes and the liquid temperature is linear. However, this equation will not work for very well wetting (very small surface tension) liquids.

Heat Transfer Equation

In the subcooled region, single-phase (liquid) forced convection is still valid between nucleation sites, and the heat transfer can be expressed by the following equation:

′′ =− qhTT()wl =− +− (6.4) hT[(w Tsat ) onb ( T sat Tl )]

Solving Equations 6.3 and 6.4 simultaneously gives the heat flux q″ and (Tw – Tsat)onb required for the onset of boiling. As the surface temperature increases, a transition from the partial boiling to the fully developed boiling region takes place. In this region, the single-phase convection heat transfer reduces to zero and nucleate boiling is relatively insensitive to the degree of subcooling or the fluid velocity, as shown in Figure 6.4; however, the critical heat flux increases with subcooling and velocity. The most commonly used heat transfer correlations for water are due to Jens and Lottes [3] and Thom [4]. The heat transfer equation by Jens and Lottes [3] is as follows:

∆ 0.25 (–p/62) Tsat = 25(q″) e (6.5)

2 where ∆Tsat is in °C, q″ is in MW/m , and p is bars. Equation 6.5 was based on the test data for water flowing upward in uniformly heated stainless steel or nickel tubes with inside diameters between 3.63 and 5.74 mm. The system pressures ranged from 7 to 172 bars, water temperatures from 115°C to 340°C, mass velocities from 11 to 1.05 × 104 kg/(s·m2), and heat fluxes up to 12.5 MW/m2. Equation 6.5 was modified by Thom [4] into the following expression:

0.5 (–p/87) ∆Tsat = 22.65(q″) e (6.6)

The values of ∆Tsat, and thus the heat transfer coefficient h (= q″/∆Tsat), calculated from the above two equations are not much different from those based on pool- boiling correlations. It is felt that, in the absence of available information, one may use saturated nucleate pool-boiling data at the same pressure to evaluate fully developed subcooled flow boiling for fluids other than water.

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FIGURE 6.4 Effects of velocity and subcooling on flow boiling.

It has been found from various test data, however, that the heat transfer in nucleate boiling can generally be correlated into the following equation:

′′ ∝ ∆ n qT()sat (6.7)

where n = 3 to 3.33. The heat transfer mechanism in saturated nucleate boiling is essentially identical to subcooled nucleate boiling except the quality is greater than 1. The heat transfer equations for subcooled nucleate boiling are also valid for saturated nucleate boiling.

6.4.3 Two-Phase Forced Convection

The two-phase forced convection region is generally associated with annular flow. Heat transfer takes place between the wall and liquid layer by convection or conduction, and between the vapor core and liquid layer by evaporation.

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Extremely high heat transfer occurs in this region. The heat transfer equation proposed by Chen [5] is considered to be one of the best available. He suggested that the heat transfer should include both single-phase forced convection and nucleate boiling; Chen’s correlation is expressed as follows:

htp = hnb + hfc (6.8)

where htp is the total (or two-phase) heat transfer coefficient, hnb is the nucleate boiling heat transfer coefficient, and hfc is the forced convection heat transfer coefficient. hfc is defined as follows:

hfc = hlF (6.9)

where hl is the single-phase convection heat transfer coefficient and can be determined from Equation 6.2. The Reynolds number Re should be based on the liquid flow rate alone, not the total flow rate. F is the two-phase heat transfer coefficient multiplier and is a function of the Martinelli parameter Xtt as indicated in Figure 6.5. The Martinelli parameter is commonly used in calculation of two-phase pressure drop and is defined as follows:

2 = ()dp/dz l ()Xtt (6.10) ()dp/dz g

where the right-hand side of the equation represents the ratio of the pressure gradient in the liquid phase to the pressure gradient of the gas phase in turbulent flow. If the friction factor for each phase is assumed to be proportional to Re–0.2, then the Martinelli parameter can be computed from the following expression:

 ρ  05.   1 − x 09. g µ 01. X =    l  tt    ρ  µ (6.11) x  l   g 

On the other hand, the nucleate boiling heat transfer coefficient is defined as follows:

= ′ hShnb (6.12)

where S is the suppression factor and is given in Figure 6.6. In Equation 6.11, h′ is the nucleate boiling heat transfer coefficient based on the Forster-Zuber equation as follows:

′ ∆ 0.24 0.75 h = 0.00122( Tsat) (∆psat) f(p) (6.13a)

where ∆psat is the difference in saturation pressure due to the change of ∆Tsat (= Tw – Tsat) and is illustrated in Figure 6.7. The net effect f(p) due to all thermodynamic properties which are functions of the system or local pressure is as follows:

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FIGURE 6.5 F factor for Chen correlation.

FIGURE 6.6 S factor for Chen correlation.

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FIGURE 6.7 Relationship between p and Tsat.

045. ρ 049. 079. ()ckpl ()()l l fp()= (6.13b) σµρ05. 024. 029. 024. ()()()hfg l g

The suppression factor S is a function of the heat transfer multiplier and two- phase Reynolds number, which can be calculated from the following expression:

1.25 Reth = Rel F (6.14)

where Rel is liquid Reynolds number in two-phase flow [= G(1 – x)d/µl]. According to Bergles et al. [1], the curve fittings for the F and S factors are as follows:

 1 1 for ≤ 01. (6.15a)  Xtt F =    1  0. 736 1 235.  + 0. 213 for ≤ 01. (6.15b)   Xtt  Xtt

and

1 S = +×−6117. (6.16) 1 2. 53 10 Reth

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FIGURE 6.8 Parametric effects on critical heat flux.

6.4.4 Critical Heat Flux

Under some operating conditions in flow boiling the heat transfer coefficient may fall rapidly while at the same time the wall temperature rises rapidly. This phenomenon is known by many names (e.g., boiling crisis), as indicated previously. The parametric effect of the critical heat flux for a uniformly heated round tube is summarized in this section and illustrated graphically in Figure 6.8 [1]. It should also be noted that the critical heat flux represents the maximum heat flux limit for the system design. The relations between the critical heat flux and other quantities may be summed up as follows: ∆ The critical heat flux varies linearly with the inlet subcooling hi (= hf – h) if other variables remain constant. The critical heat flux decreases and is asymptotic to zero when the tube length increases; however, the power input [PCHF = (q″)CHFldπ] increases with the tube length, eventually approaching a constant.

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The critical heat flux increases with either mass velocity or the tube diameter if other variables remain constant. A vast amount of data has been gathered for a steam-water mixture in upward flow. One popular correlation is due to Bowring [6]:

A+ 025 . dG∆ h ()q′′ = i (6.17) CHF BL+

where

d = tube diameter, m G =⋅total mass velocity or flux, kg/(m2 s) L = tube length, m ∆ = hi enthalpy of inlet subcooling, J/kg ′′ = 2 (q )CHF critical heat flux, W/m

Also, A and B are given by:

0. 5792 hfg dGC1 A = (6.18a) + 05. 1 0.CdG 0143 2

and

0.CdG 077 B = 3 + n (6.18b) 1 0.CG/ 3474() 1356

C1, C2, C3, C4, and n are functions of a dimensionless pressure p′: p p′ = (6.19) 69

and

n = 2 – 0.5p′ (6.20)

where p is the system pressure in bars. By defining p″ =1 – p′, C1, C2, C3, and C4 can be expressed as follows:

For p′<1: ′′ ()pe′ 18942..p 20 8 + 0 . 917 C = (6.21a) 1 1917.

= 1309.C1 C2 ′′ (6.21b) ()pe′ 1316..p 2 444 + 0 . 309

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′′ ()pe′ 17..p 023 16 658 + 0 . 667 C = (6.21c) 3 1667.

1.649 C4 = C3(p′) (6.21d)

For p′>1:

–0.368 0.648p″ C1 = (p′) e (6.22a) –0.448 0.245p″ C2 = (p′) e (6.22b) 0.219 C3 = (p′) (6.22c) 1.649 C4 = C3(p′) (6.22d)

The root-mean-square (RMS) error for this correlation is about 7%; however, much larger errors are expected for conditions outside the test data ranges. Ranges of the test data from Bowring for vertically upward steam-water flow are as follows:

Pressure = 2–190 bars Tube diameter = 2–45 mm Tube length = 0.15–3.7 m Total mass velocity = 126–18,600 kg/(m2·s)

For fluids other than steam-water mixtures, Ahmad [7] gives scaling rules based on the dimensional analysis that may be applied. He used five dimensionless groups, as follows, to characterize the critical heat flux:

L/d = ratio of tube length to diameter ρρ = lg/ ratio of liquid to gas (vapor) density ∆ = h/hifg ratio of enthalpy of inlet subcooling to latent heat of vaporization ′′ = (q )CHF /Ghfg boiling number

and

ψ = Ahmad parameter

 2  23/  µ 18/  Gd µ g (6.23) =  l   µ  σρ  µ   l  d l   l 

To determine the critical heat flux for fluid x from the data of the steam-water mixture, the procedures are as follows:

1. Setting L/d to the same value for both systems. Typically, the two systems have the same physical dimensions. 2. Setting ρl/ρg to the same value for both systems. One can then determine the operating pressure for the water system from this ratio.

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3. Setting ∆hi/hfg to the same value for both systems. This ratio is used to determine the water inlet subcooling enthalpy. 4. Setting ψ to the same value for both systems. The mass velocity of the water system can be found on the basis of the value of ψ. 5. Computing the critical heat flux for the water system from the existing correlations such as Equation 6.17. 6. Calculating the critical heat flux for fluid x from the following equation:

()q′′   ()q′′   CHF  =  CHF  (6.24)  Ghfg  x  Ghfg water

6.4.5 Post-Burnout Heat Transfer

Since there are two types of boiling crisis, namely, DNB and dryout, the general post-dryout heat transfer can also be divided into two different types:

1. Film boiling. This region generally occurs after the subcooled DNB critical heat flux has occurred. 2. Post-dryout. This region occurs after the high-quality dryout critical heat flux has occurred.

In case 1, the heat transfer mechanism is very similar to the condition of pool boiling. Therefore, the heat transfer equations, e.g., Equations 5.22a and 5.22b for film boiling in pool boiling, can still be valid and be applied if no other correlation is available. In case 2, the post-dryout is often called the liquid-deficient region. One of the popular equations for the heat transfer is due to Groeneveld [8]. The Groeneveld correlation is for tubes and annuli and is given as follows:

  ρ  b Nu=+ax Re g ()1 − x Pr cd Y gg ρ  g (6.25)   l  

where

 ρ  04. Yx=−101.  l − 11()−  ρ (6.26)  g  

and

Tubes/annuli Tubes Annuli a = 0.00327 0.00109 0.052 b = 0.901 0.989 0.688 c = 1.32 1.41 1.26 d = –1.5 –1.15 –1.06

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The above correlation is based on the following range of data:

Flow direction Vertical/horizontal tube Vertical annuli d, cm 0.25 to 2.5 0.15 to 0.63 p, bars 68 to 215 34 to 100 G, kg/(m2·s) 700 to 5300 800 to 4100 x 0.1 to 0.9 0.1 to 0.9 q″, kW/m2 120 to 2100 450 to 2250 Nug 95 to 1770 160 to 640  ρ  g Re xx+ ()1 − × 4 × 6 × 5 × 5 g  ρ  6.6 10 to 1.3 10 1 10 to 3.9 10  l 

Prg 0.88 to 2.21 0.91 to 1.22 Y 0.706 to 0.976 0.61 to 0.963

6.4.6 Single-Phase Forced Convection (Vapor or Gas) Finally, a saturated or superheated vapor region is reached if the heating process is continued. In this region, the heat transfer mechanism is just single-phase (vapor) forced convection. Any suitable heat transfer correlations listed in Appendix H can be used. For turbulent flow in a smooth round tube, Equation 6.2 may still be applied; however, the constant in the equation is changed from 0.023 to 0.021 for gas flow.

6.5 THERMAL ENHANCEMENT Many augmentation techniques have been used to enhance heat transfer in single-phase forced convection. Heat transfer is so effective in boiling that the improvement from enhanced heat transfer (combination of heat transfer coefficient and surface area) may be limited. The focal point of thermal enhancement may be delaying (or eliminating) the occurrence of the boiling crisis (DNB or dryout) and/or increasing the value of the critical heat flux. One possible way to delay or avoid the boiling crisis is to utilize centrifugal force. The multilead ribbed tube used by Babcock & Wilcox in steam generating systems and illustrated in Figure 6.9 is a good example. The helical ribs on the inside surface of the tube create a swirling flow. The resulting centrifugal force causes liquid droplets to move toward the heated inner tube surface and prevents the formation of a vapor film. The multilead ribbed tube, therefore, can maintain nucleate boiling to much higher (steam) qualities and delay the boiling crisis. A comparison of results between multilead ribbed and smooth tubes is also given in Figure 6.9 [9]. As can be seen from the figure, DNB has been delayed from point D to D1. The friction pressure drop of the ribbed tubes is only slightly higher than that of cold-finished smooth-bore tubes.

6.6 PRESSURE DROP Unlike the case of pool boiling, the pressure drop in a flow system is also an important consideration in flow boiling. The pressure will affect the saturation

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FIGURE 6.9 Boiling in smooth and multilead ribbed tubes.

temperature, which will in turn have a great impact on the temperature of the heated wall. In addition, the fluid properties are a also function of the pressure. The basic components in Equation 3.8 used to calculate the pressure drop for single-phase flow are still valid; however, consideration must be extended to the mixture of two phases. The differential form of the total pressure drop can be expressed as follows:

dP dP dP dP dP t =++±f asg (6.27) dz dz dz dz dz

where the four pressure drop components are identical to the case of a single- phase flow given in Equation 3.8 and dz is the differential length of the tube. The pressure drop due to friction is a major contributor to the total pressure drop either in single-phase or two-phase flow. Generally, the frictional pressure drop in two-phase flow is expressed by the product of frictional pressure drop of single- phase flow and a two-phase multiplier. Calculating the pressure drop in two-phase flow is complicated. Different models have been proposed. For a homogeneous flow assumption, two-phase flow is treated as a single-phase fluid with a single velocity for both phases. Another approach, called the separated flow assumption, considers different velocities for each phase in a two-phase flow. The latter approach is a better representation of the physical flow condition and thus is adopted in this chapter. Several terms are often used in conjunction with two-phase flow, as follows:

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s (slip ratio)= velocity ratio of vapor (or gas) to liquid u (6.28a) = g ul x (quality)= mass (or mass velocity) ratio of vapor to total flow   m g Gg (6.28b) = or  mt  Gt 

α (void fraction)= cross - sectional area ratio (or volume ratio) of vapor to total flow = 1 (6.28c)  1 − x ρ  1 + s  g   ρ  x  l 

where s will become 1 for the homogeneous model and α can be simplified accordingly. The individual components in Equation 6.27 for the separated flow are as follows:

dP g = pressure drop or gain due to gravity dz =+−αρ α ρ θ [(gl1 )]g sin dP a = pressure drop due to flow acceleration dz  xx22()1 −  =+G2   αρ− α ρ  gl()1  dP s = pressure drop due to entrance, exit, bends, fittings, dz and changes of cross - sectional area dP f = pressure drop due to friction dz

where θ is an inclined angle measured from the horizontal plane. Since the frictional pressure drop is the major factor, many different models and correlations have been proposed in the past. Among them, only the Friedel correlation [10] is presented here. As in many frictional pressure drop correlations, it is written in terms of a two-phase multiplier, which is defined as follows:

  dPf −   dz  φ 2 = th (6.29) lo   dPf −   dz  lo

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3 where (dPf/dz)th is the frictional pressure gradient (N/m ) in two-phase flow and 3 (dPf/dz)lo is the frictional pressure gradient (N/m ) in single-phase liquid flow with the same mass flow rate as the total two-phase flow rate. The minus sign in Equation 6.29 indicates that the system pressure decreases in the flow direction. The Friedel correlation is described as follows:

φ 2 =+ 324. BC lo A (6.30) Fr0.. 045+ We 0 035

where

 ρ  f  Axx=−()1 22 + l g  ρ   (6.31a)  g  fl  Bx=−0.. 78()1 x 0 224 (6.31b)

  091.  µ  019.  µ  07. ρ g g C = l 1 − (6.31c)  ρ   µ   µ   g   l   l  G2 Fr = (6.31d) ρ 2 gd h Gd2 We = σρ (6.31e) h ρ = 1 h ρ +− ρ (6.31f) xx/ g (1 ) / l

The units for the above variables are as follows:

d = tube diameter, m = fg friction factor for total mass flow with vapor or gas phase alone, dimensionless = fl friction factor for total mass flow with liquid phase alone, dimensionless G = mass velocity, kg/m2 g = acceleration due to gravity, 9.8 m/s2 x = quality, dimensionless ρ = 3 g vapor or gas phase density, kg/m ρ = 3 l liquid phase density, kg/m µ = 2 g vapor or gas viscosity, N· s/m µ = 2 l liquid viscosity, N· s/m σ = surface tension, N/m

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This correlation is generally applicable to vertical upward flow and to horizontal flow; however, this method does not work well when the viscosity ratio of liquid to vapor (or gas) exceeds 1000.

REFERENCES

1. A. E. Bergles, J. G. Collier, J. M. Delhaye, G. F. Hewitt, and F. Mayinger, Two- Phase Flow and Heat Transfer in the Power and Process Industries, Hemi- sphere, Washington, 1981. 2. E. J. Davis and G. H. Anderson, “The Incipience of Nucleate Boiling in Forced Convection Flow,” AIChE J. 12(4), 1966. 3. W. H. Jens and P. A. Lottes, “Analysis of Heat Transfer Burnout, Pressure Drop, and Density Data for High Pressure Water,” USAFC Report ANL-4627, U. S. Atomic Energy Commission, 1951. 4. J. R. S. Thom, W. M. Walker, T. A. Fallon, and G. F. S. Reising, “Boiling in Subcooled Water during Flow up Heated Tubes or Annuli,” presented at the Symposium on Boiling Heat Transfer in Steam Generating Units and Heat Exchangers, Manchester, Institute of Mechanical Engineers (London), Paper No. 6, September 15–16, 1965. 5. J. C. Chen, “Correlation for Boiling Heat Transfer to Saturated Liquids in Convective Flow,” ASME Paper No. 63-HT-34, 1963. 6. R. W. Bowring, “A Simple but Accurate Round Tube, Uniform Heat Flux, Dryout Correlation for Pressure in the Range 0.7–17 MN/m2, ” AEEW-R789, United Kingdom Atomic Energy Authority, 1972. 7. S. Y. Ahmad, “Flow to Fluid Modeling of Critical Heat Flux: A Compensated Distortion Model,” Int. J. Heat Mass Transfer, 1973. 8. D. C. Groeneveld, “Post Dryout Heat Transfer at Reactor Operating Condi- tions,” AECL-4513, ANS Meeting on Water Reactor Safety, Salt Lake City, UT, 1973. 9. W. Wiener, “Latest Developments in Natural Circulation Boiler Design,” presented to American Power Conference, Chicago, April 18–20, 1977. 10. L. Friedel, “Improved Friction Pressure Drop Correlations for Horizontal and Vertical Two Phase Flow,” presented to European Two Phase Group Meeting, Ispra, Italy, 1979.

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CONDENSATION

The opposite of boiling is the process of condensation, which changes vapor to liquid by removing heat from the vapor to the surroundings or to bounding surfaces. Like boiling, condensation may occur homogeneously within vapor or heterogeneously on entrained solid particles or on the surfaces of structures in the system.

7.1 MODES OF CONDENSATION

There are several possible modes of condensation by which vapor may be converted into a liquid phase. Four common modes are dropwise condensation, filmwise condensation, direct-contact condensation, and homogeneous condensation.

7.1.1 Dropwise Condensation

Condensation occurs when a saturated or superheated vapor comes in contact with a surface below the saturation temperature of the vapor at the pressure existing in the vapor phase. Initially, tiny drops of condensate form at numerous nucleation sites on the surface. The droplets then grow by further condensation from the surrounding vapor and become larger through contact with their neigh- bors. The droplets are held by the surface tension forces until they grow large enough to run off the surface if the surface is not wettable. The area swept clean by the departure of a large droplet immediately becomes the next nucleation site. Because the vapor is in direct contact with the cold solid surface, the heat transfer rates in dropwise condensation are extremely high, an order of magnitude more effective than filmwise condensation, but dropwise condensation is very difficult to maintain for practical applications. Special coatings and addi- tives have been used to promote dropwise condensation, but the effectiveness of such surface treatments generally decreases with time.

7.1.2 Filmwise Condensation

In filmwise condensation, as in dropwise condensation, small drops are initially created at nucleation sites. If the liquid wets the surface, the condensed liquid

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drops will rapidly coalesce into a continuous liquid film on the surface. The latent heat released at the liquid-vapor interface is conducted through the thickness of the liquid film and is removed from the wall. Filmwise condensation should be applied in normal design practice and is the only type to be discussed at length in this chapter.

7.1.3 Direct-Contact Condensation

Direct-contact condensation takes place when a subcooled liquid is sprayed directly into the vapor space so that vapor is condensed on the liquid interface. The heat released from vapor during the process is conducted into the liquid. Direct-contact condensation is useful, but its applications are limited.

7.1.4 Homogeneous Condensation

Homogeneous condensation may occur in principle when a vapor is changed to liquid by means of strong subcooling in the absence of any solid surfaces. For example, a large mass of vapor-laden air is cooled below its saturation temperature (by radiation to the night sky) to form atmospheric fog. In direct contrast to the large superheating required in homogeneous boiling, extremely high subcooling is required in the atmosphere for water vapor condensation.

7.2 HEAT TRANSFER IN FILMWISE CONDENSATION

As explained in Section 7.1, the mode of condensation that is of greatest practical importance is filmwise condensation. Heat transfer correlations for several different geometries of practical importance are presented in this section.

7.2.1 Condensation on a Vertical or Inclined Plate

Heat transfer in filmwise condensation over a vertical or inclined plate was first analyzed by Nusselt [1] based on a simple force balance on the liquid film. The other major assumptions in his analysis are as follows:

1. Laminar flow in the liquid film driven by gravity 2. Constant fluid properties 3. No subcooling in the condensate 4. No shear force exerted by vapor flow on the liquid film 5. Heat transfer by conduction only 6. Linear temperature profile in the liquid film

The velocity and temperature profiles in the Nusselt model are given in Figure 7.1. A detailed analysis can be found in reference 1 or any of the heat transfer books listed in the references. The local heat transfer coefficient obtained by Nusselt is

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FIGURE 7.1 Velocity and temperature profiles in liquid film.

defined as hz = kl/δ, where δ is the film thickness at a distance z from the starting point (δ = 0 at z = 0), as given in the following equation:

 4µ kTz∆ 14/ δ =  ll sat  (7.1) ρρ− ρ θ  ll() gghsin fg

Once δ is known, the local heat transfer coefficient becomes:

 ρρ− ρ θ3 14/ ll() ggkhsin lfg h =   z  µ ∆  (7.2)  4 l Tzsat 

where z is the distance from the incipience of condensation, θ is the angle of inclination, and ∆Tsat = Tsat – Tw. The mean value of the heat transfer coefficient over the entire plate can easily be determined by integrating Equation 7.2 over the length of the plate and is:

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 ρρ− ρ θ3 14/ ll() ggkhsin lfg h = 0. 943   µ∆  (7.3)  l TLsat 

where L is the plate length. The fluid properties should be evaluated at the arithmetic average value of the vapor and wall temperature. Equation 7.3 can also be rewritten as a function of the mass condensation flow rate per unit width of the film, Wc, and becomes

 ρρ− ρ θ3 13/ ll() ggksin l h = 0. 925  (7.4)  µ   lcW 

where

ρρ()− ρg sin θδ3 W = ll g (7.5) c µ 3 l

The above equation derived for a vertical or inclined plate generally is also applicable to condensation inside or outside a vertical tube if the previous assumptions are met. This is because the film thickness is so small as compared with the diameters of typical tubes that the effect of curvature is negligible.

7.2.2 Condensation on Outside Surface of a Horizontal Tube

The Nusselt analysis can be extended to other geometries, such as condensation on the outside of a horizontal tube. The mean heat transfer coefficient for this case is as follows:

 ρρ− ρ 3 14/ ll() ggk lfg h h = 0. 725   µ∆  (7.6)  l TDsat 

where D is the tube outer diameter. Examining Equations 7.3 and 7.6, the following relation between a vertical plate and a horizontal tube exists in order to have the same heat transfer coefficient under the same flow conditions and subcooling:

0.. 943 0 725 ==or LD287. (7.7) LD14// 14

In other words, a higher heat transfer coefficient is found for the horizontal tube than that of the vertical plate if the length of the vertical plate or tube is larger

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than 2.87 times the outer diameter of the tube. This is because the path of the condensate is shorter and the film is thinner for a horizontal tube. Since this (L > 2.87D) often occurs in many practical applications, there is a considerable advantage to placing the tubes in a horizontal position. In a bank of horizontal tubes, the condensate may run off the bottom of the upper tube directly onto the next tube below. Jakob [2] studied this problem and found that the mean heat transfer coefficient for a system consisting of n tubes in a vertical column can be estimated by replacing D by nD. This finding leads to the following equations:

()h n −14/ = n (7.8a) hl

and

h 34// 34 n =−−nn()1 (7.8b) hl

where

= ()hnn mean heat transfer coefficient for system with tubes = hnn heat transfer coefficient for th tube in column = hl heat transfer coefficient for first tube on top

Experimental results indicated that the heat transfer coefficient is higher than that predicted by Equations 7.8a and 7.8b. Possible reasons are that a certain amount of turbulence seems unavoidable in this type of arrangement and also the condensate does not fall from one tube to another as a continuous sheet as given in Figure 7.2. Kern [3] has suggested that Equations 7.8a and 7.8b be modified as follows:

()h − n = n 16/ (7.9a) hl

and

h n =−−nn56//()1 56 (7.9b) hl

Equation 7.6 assumes no shearing force exists. If the interfacial shear force occurs only as a result of the momentum lost by condensing vapor, the heat transfer coefficient for vapor flow downward across a horizontal tube can be expressed as follows [4]:

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FIGURE 7.2 Film condensation in horizontal tubes.

 k  hC=  l  Re05. (7.10)  D

6 where Re = ρlUvD/µl and Uv is the approaching vapor velocity. C is 0.9 for Re < 10 and is 0.59 for Re > 106. This is because separation of the boundary layer takes place at high Reynolds numbers. For the low-velocity case, where both gravitational and shearing forces are important, the heat transfer coefficient becomes:

=+2441 +//212 hh[05. shear ( 025. hhshear gravity ) ] (7.11)

where hshear and hgravity are obtained from Equations 7.10 and 7.6, respectively. Another application, for condensation on horizontal low-fin tubes, was experimentally studied by Beatty and Katz [5]. The correlation developed by them is as follows:

 ρρ− ρ 3 14/  14/ ll() ggk lfg h 1 h = 0. 689     µ∆  (7.12)  l TDsat   eq 

where

 14/   14/   14/ 1 0. 943 Af  11 Ap     =  η   +          Deq  0. 725  ALtm   ADtr 

and where

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= af area of one side of a fin = Af actual fin area = At total outside tube area, including fins =+η AAfp = Ap area of plain tube of diameter equal to root diameter of finned tube = Do outside fin diameter = Dr root diameter of finned tube = Lm mean effective height of fins = a/Dfo η = fin efficiency

7.3 IMPROVEMENTS OVER NUSSELT ANALYSIS

Many improvements have been made over Nusselt analysis, taking into account the effects of many factors, including those of film roughness, fluid property variations, subcooling, turbulence, and vapor velocity. These effects will be described in this section.

7.3.1 Effect of Film Roughness

Experimental data [6] indicate that the mean heat transfer coefficient is generally higher than those obtained from the Nusselt analysis. This is probably because film surfaces are not smooth. Ripples on the surface of a film greatly enhance the heat transfer rate; therefore it is recommended that the value computed from Equation 7.3 be multiplied by a factor of 1.2.

7.3.2 Effect of Variation of Properties

One method for considering variations in fluid properties is to evaluate all fluid properties at the effective film temperature, defined as follows:

Tf = Tw + C ∆Tsat (7.13)

where the value of the constant C is 0.31 [7].

7.3.3 Effect of Subcooling

To include the effect of subcooling in the liquid, Rohsenow [8] proposed that the latent heat (or heat of vaporization) hfg in Equations 7.3 and 7.4 be replaced by (h′)fg, which is defined as follows:

′ =+ ∆ hhfg fg068. cT pl sat (7.14)

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FIGURE 7.3 Heat transfer for condensation on a vertical plate.

7.3.4 Effect of Turbulence

The condensed film may go into turbulent flow at the lower section of a long vertical plate. This effect generally occurs at values of the film Reynolds number, Rec = 4Wc/µl, greater than 1800. Turbulence increases the heat transfer rate, and the heat transfer coefficient becomes larger than that of Equation 7.3. Colburn [9] applied the boundary-layer theory to this problem. The local heat transfer coefficient is:

 ρρ()− ρ gk3 13/   02.  ll g l 4Wc 13/ h = 0056.   Pr (7.15) z  µ2   µ  l  l  l

To obtain the mean heat transfer coefficient, one must integrate the local heat transfer coefficients over the entire range of the condensation Reynolds number, i.e., integrate Equation 7.2 up to a Reynolds number of 2000, and integrate Equa- tion 7.15 above 2000. The results of the integration for various Prandtl numbers are given in Figure 7.3. The correlation agreed well with the test data at high Prandtl numbers, but the heat transfer coefficient falls below the Nusselt results at very low Prandtl numbers. This result may be due to the thicker film in turbulent flow.

7.3.5 Effect of Vapor Velocity

The frictional drag between the condensate and the vapor was not included in the Nusselt analysis. For a vertical plate, the film thickness of the condensate increases with upward vapor flow and decreases with downward vapor flow. In addition, the transition from laminar to turbulent flow takes place in the low Reynolds numbers when the vapor velocity is high.

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Carpenter and Colburn [10] also studied the effect of vapor velocity on film condensation inside tubes. The authors proposed a simplified equation for the average heat transfer coefficient:

    12/ cGpl m ρ h = 0. 046  f  l  (7.16)  12/   ρ   Prl    g 

where

f = Fanning friction factor in a pipe evaluated at mean vapor velocity = Gm mean vapor velocity  GGGG2 ++2 12/ =  1 12 2    3  = G1 vapor mass velocity at inlet = G2 vapor mass velocity at exit

The fluid properties of the liquid are evaluated at a reference temperature equal to 0.25Tsat + 0.75Tw.

7.4 CONDENSATION INSIDE A HORIZONTAL TUBE

Condensation inside a horizontal tube is common in industrial applications. Various flow regimes may exist throughout the length of the tube, and thus different heat transfer correlations are required. Generally, the Baker flow map [11] is used to determine the condensation path in a condenser. The Baker map is presented in Figure 7.4, and the variables on the map are defined as follows:

 ρ ρ 12/ λ = g l  ρ ρ  (7.17a)  air water 

 213/ σ µ  ρ  ψ = water  l  water   σ  µ  ρ   l  water l  gas mass flow rate (7.17b) G = g tube cross - sectional area liquid mass flow rate G = l tube cross - sectional area

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FIGURE 7.4 Baker flow regime map (horizontal flow).

7.4.1 Stratified Flow

For low-quality and interfacial shear stress, a stratified flow often occurs. In this region, it is generally assumed that no condensation takes place in the stratified layer and condensation occurs only in the top of the tube. The Nusselt analysis discussed previously can be applied. Therefore, the heat transfer coefficient in stratified flow can be expressed as follows:

 ρρ− ρ 3 14/ ll() ggk lfg h hF=    µ ∆  (7.18a)  l TDsat 

The factor F is equal to 0.725 (Equation 7.6) for condensation outside a horizontal tube. For condensation inside a horizontal tube, the factor F is a function of the angle as shown in Figure 7.5.

7.4.2 Wavy, Slug, and Plug Flow

Some condensation occurs near the bottom of the tube for wavy, slug, or plug flow. In addition, some effect of vapor shear on the film also exists. Experimental

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FIGURE 7.5 F factor in a horizontal tube.

data show that the heat transfer coefficient varies around the tube as shown in Figure 7.6, and the value at the top of the tube can be determined from the following equation in conjunction with Equation 7.18a:

= 012. F 031.Reg (7.18b)

where

GD Re = g g µ g

7.4.3 Annular Flow

In annular flow, one simple method described by Whalley [12] is to determine the condensation heat transfer coefficient from the single-phase liquid flow as follows:

h =≈d 1 δα− (7.19) hl 4 1

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FIGURE 7.6 Circumferentially local heat transfer coefficient.

where h is the condensation heat transfer coefficient, hl is the single-phase liquid heat transfer coefficient, δ is the mean liquid film thickness, d is the tube inside diameter, and α is the void fraction. Another method, proposed by Ananiev et al. [13], is in the following correlations for the local and average condensation heat transfer coefficients, respectively:

 ρ 12/ hh= l zl ρ  (7.20)  m 

and

  12/   ρ   hh=+05. 1  l  (7.21) l   ρ    g 

where hl is the single-phase liquid heat transfer coefficient at the same total flow rate as the condensate in the tube and ρm is the homogeneous mean density of the vapor-liquid mixture.

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FIGURE 7.7 Effect of noncondensable gas in condensation.

7.5 NONCONDENSABLE GAS IN A CONDENSER

The presence of even a small amount of noncondensable gas in a condenser has a tremendous effect on its performance. Generally, noncondensable gas is carried by the vapor toward the interface where the gas is accumulated. The pressure and temperature distributions for the vapor and gas are presented in Figure 7.7. Since the system total pressure is constant, the vapor partial pressure is then reduced due to the existing noncondensable gas. Consequently, the interface temperature Ti will be lower than that of the case without noncondensable gas, (T′)i. Therefore, the driving force for condensation, Ti – Tw [< (T′)i – Tw], is also reduced because of the presence of noncondensable gas. Heat transfer from the bulk mixture to the condensation interface consists of two components, the latent heat released due to condensation of the vapor and the sensible heat transfer to the interface. When the vapor-gas mixture is stationary, the total heat flux is:

(q″)i = hi(Tv – Ti) (7.22)

and

jh+− jc() T T h = fg pv v i i − (7.23) TTvi

where hi is the heat transfer coefficient from the vapor to the condensate interface. The mass condensation flux (per unit area of the wall) j can be determined by the following equation according to Reference 14:

 pp−  Dp j =  i  v (7.24)  mA/  2 G RRTvGv

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where

D = gas diffusivity CT 32/ = v p 0./M/M 0069() 1+ 1 05. C = vg 13//+ 132 ()VVvg where MV is molecular weight and is atomic volume = mG mass flow rate of noncondensable gas A = condenser area = Rv gas constant for vapor = RG gas constant for noncondensable gas P = total pressure or condenser pressure = pTiipartial vapor pressure corresponding to = pTvvpartial vapor pressure corresponding to

If the vapor-gas mixture flows parallel to the interface, the total heat flux is then expressed as follows:

(q″)i = hi(Tv – Ti) (7.25)

as in Equation 7.22, and

jh+− h() T T h = fg v v i (7.26) i − TTvi

where hv is the heat transfer coefficient due to the flow of the vapor-gas mixture and can be calculated from the correlation appropriate to the flow condition (laminar or turbulent) and geometry (in a tube or over a plate or tube). The mass flux j can be found from the following expression:

Kppρ ()− j = vv v i (7.27) pam

where

= pam log mean partial pressure of noncondensable gas − = ppai av =− =− where ppppppai i; av v ln()p/pai av = K v mass transfer coefficient h  Pr  23/ = v   ρ   vpvc Sc Sc = Schmidt number

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The heat flux at the wall is

(q″)w = h(Ti – Tw) (7.28)

where h is the condensation heat transfer coefficient, which can be computed from previous equations, such as Equation 7.3. The calculation must be repeated for other locations in the condenser. The overall heat transfer coefficient due to two thermal resistances in series is defined as follows:

111 =+ (7.29) hhhti

where h and hi are given in in Equation 7.33 or 7.26, respectively. To solve the problem, an iteration process, which starts with a guess value of the interface temperature Ti (Tw < Ti < Tv) is generally required until the calculated heat flux from Equation (7.28) is equal to that from Equation (7.25). The iterative steps are briefly described as follows:

1. Calculate q” from Equation (7.28) with a predetermined value of h 2. Compute pi with the assumed Ti based on the saturation condition at the interface 3. Compute Kv with a predetermined value of hv 4. Calculate j and then hi from Equations (7.27) and (7.26), respectively 5. Compute q” from Equation (7.25) 6. Repeat the above steps until q” from Step 5 equal to that from Step 1

The effect due to noncondensable gas on heat transfer for the forced convection flow condition is less significant, but it will have a profound influence when the vapor-gas mixture is stagnant. Because of an extremely high solubility of dielectric fluids for air, special attention must be paid to the design of the condenser for use in electronics cooling systems. All previous discussions are theoretically limited to the case when a saturated vapor is being condensed. The preceding correlations may also be used with reasonable accuracy for condensation of superheated vapors; however, the following adjustment must be made in determining the heat transfer rate:

″ q = h(Tsv – Tw) (7.30)

where Tw is the cold-wall temperature and Tsv is the saturation temperature corresponding to the prevailing system pressure at the superheated condition.

REFERENCES

1. W. Nusselt, “Die Oberflachenkondensation des Wasserdampfes,” Zeitschr. Ver. Deutsch. Ing. 60, 1916.

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2. M. Jakob, Heat Transfer, Vol. 1, Wiley, New York, 1949. 3. D. Q. Kern, “Mathematical Development of the Tube Loading in Horizontal Condensers,” AIChE J. 4, 1958. 4. I. G. Shekriladz and V. I. Gomelauri, “Theoretical Study of Laminar Film Condensation of Flowing Vapor,” Int. J. Heat Mass Transfer 9, 1966. 5. K. O. Beatty and D. L. Katz, “Condensation of Vapors on Outside of Finned Tubes,” Chem. Eng. Prog. 44, 1948. 6. W. H. McAdams, Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954. 7. W. J. Minkowycz and E. M. Sparrow, “Condensation Heat Transfer in the Presence of Non-Condensable, Interfacial Resistance, Superheating, Variable Properties and Diffusion,” Int. J. Heat Mass Transfer 9, 1966. 8. W. M. Rohsenow, “Heat Transfer and Temperature Distribution in Laminar Film Condensation,” Trans. ASME 78, 1956. 9. A. P. Colburn, “The Calculation of Condensation Where a Portion of the Con- densate Layer Is in Turbulent Motion,” Trans. AIChE 30, 1933. 10. E. F. Carpenter and A. P. Colburn, “The Effects of Vapor Velocity on Conden- sation inside Tubes,” Inst. Mech. Eng., ASME Proc. General Discussion on Heat Transfer, 1951. 11. O. Baker, “Simultaneous Flow of Oil and Gas,” Oil Gas J. 53, 1954. 12. P. B. Whalley, Boiling, Condensation, and Gas-Liquid Flow, Clarendon Press, Oxford, 1987. 13. E. P. Ananiev, L. D. Boyko, and G. N. Kruzhilin, “Heat Transfer in the Pres- ence of Steam Condensation in Horizontal Tubes—Part II,” Paper 34, Int. Heat Transfer Conf., Boulder, CO, 1961. 14. W. M. Rohsenow and J. P. Hartnett, Chapter 11, Fundamental Handbook of Heat Transfer, McGraw-Hill, New York, 1973.

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EXTENDED SURFACES

Extending the surface of a body in contact with a fluid can enhance heat transfer in convection. Extended surfaces in practice often consist of small–cross section solids that protrude from the main body, such as fins attached to the walls of heat transfer equipment. The main reasons for using fins to enhance the heat transfer rate are (1) to increase the heat transfer surface area, (2) to break up the boundary layer of the flow, and (3) to increase the turbulence as flow passes over or across the fins.

8.1 UNIFORM–CROSS SECTION FINS The transverse dimensions of a fin are typically much smaller than its length. To simplify the analysis, it is generally assumed that heat transfer is one-dimensional in the fin along its axial length, as shown in Figure 8.1. Other assumptions are as follows: 1. Constant cross-sectional area A 2. Constant perimeter P 3. Constant physical properties 4. Uniform convection heat transfer coefficient h (except perhaps at the end of the fin) 5. Constant fin base temperature Tb By taking an energy balance on a given differential element, we have

dT dT dkAdTdx(− / ) −=−+kA kA dx+− hP dx() T T (8.1) dx dx dx f

2 dT=−2 mT() Tf (8.2) dx2

where Tf is the fluid temperature and

hP m2 = (8.3) kA

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FIGURE 8.1 Model for fin analysis.

The general solution for Equation 8.2 is

mx –mx T – Tf = C1e + C2e (8.4)

where C1 and C2 depend on the type of thermal boundary condition. The assumption of T being equal to Tb at the root of the fin (x = 0) leads to the following expression:

Tb – Tf = C1 + C2 (8.5)

One more boundary condition is required to complete the solution. The additional boundary condition and its corresponding solution for three different cases are given below:

Case 1—Infinite-length fin, i.e., fin tip temperature equal to fluid temperature:

Boundary Condition: Tx–∞ = Tf

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Solution

–mx Temperature: T – Tf = (Tb – Tf)e (8.6a) 0.5 Heat flow: q = (hPka) (Tb – Tf) (8.6b) 1 Effectiveness: η = (8.6c) f mL

Case 2—Finite-length fin with insulated tip, i.e., no heat loss at the fin tip:

 dT  Boundary Condition:   = 0   dx xL=

Solution −− ()TTbfcosh [( mLx )] Temperature: TT−= (8.7a) f cosh ()mL 0.5 Heat flow: q = (hPkA) (Tb – Tf) tanh (mL) (8.7b) tanh()mL Effectiveness: η = (8.7c) f mL h = heat transfer coefficient for lateral surfaces

Case 3—Finite-length fin without insulation, i.e., heat loss at the fin tip:  dT  Boundary Condition: −K  =−hTL[]() T   Lf dx xL=

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Solution

Temperature: T – Tf = (Tb – Tf)F1(x) (8.8a) where

h cosh [mL( −+ x)] L sinh [(mL− x )] Fx()= mk 1 h cosh ()mL + L sinh ()mL mk

0.5 Heat flow: q = (hPkA) (Tb – Tf)F2(x) (8.8b) where

h sinh [(mL−+ x )] L cosh [(mL− x )] Fx()= mk 2 h cosh ()mL + L sinh ()mL mk

F ()0 Effectiveness: η = 2 (8.8c) f mL

hL = heat transfer coefficient for tip

8.2 FIN EFFICIENCY

One way to evaluate the thermal performance of a fin is to determine the fin efficiency ηf. The fin efficiency as given in the three cases at the end of the last section is defined as the ratio of the actual heat transfer rate of the fin to the heat transfer rate that would occur if the entire fin surface were at the base temperature. This temperature condition can take place only if the fin thermal conductivity is infinite; for this case, the heat transfer rate would be expressed as follows:

q∞ = hPL(Tb – Tf) (8.9)

According to the definition, the fin efficiency would be:

η = q f (8.10) q∞

where q is as expressed in Equations 8.6b, 8.7b, and 8.8b for the respective conditions specified previously. Figure 8.2 shows various types of fins most applicable to electronic equipment. Among them, the straight rectangular fin is probably most commonly used because of its thermal effectiveness and also because of the ease of manufacture. The fin perimeter P can be expressed approximately in the following form if the width of the fin is much larger than its thickness:

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FIGURE 8.2 Various types of fin configuration.

P = 2(b + t) ≈ 2b (8.11a)

and

A = bt (8.11b)

With the aid of Equations 8.11a and 8.11b, we have

 2h 05. m ≈   (8.12)  kt 

For practical applications, the following approximate equation can be employed to determine the fin efficiency for all rectangular fins regardless of the type of boundary condition:

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tanh()mL η = (8.13) f mL

The triangular fin as shown in Figure 8.2 is also of practical interest because it yields the maximum heat flow per unit weight. The temperature profile for a triangular fin is:

− 05. TTf IBx()2 = 0 (8.14) TT− 05. bfIBL0()2

2 where B = 2Lh/(kt). I0 is the modified zero-order Bessel function of the first kind. The total heat transfer rate of the fin and the fin efficiency are as follows:

 dT  IBL()2 05. qkA=   =−()(2hkt05. b T T )1 (8.15)  dx  bf 05. xL= IBL0()2

and

η = 1 ImL1()2 f (8.16) mL ImL0()2

where

= I1 modified first - order Bessel function  2h 05. m =    kt 

For a circular pin fin with diameter D and length L, the fin efficiency in Equation 8.13 can still be applied with m = [4h/kD]0.5 [1]. For convenience, the approximate efficiencies of various types of fins are plotted in Figure 8.3. To determine the overall efficiency of a finned surface, one may consider the unfinned portion of the surface to be at 100% efficiency while the finned area is at efficiency ηf, or:

Atηo = (At – Af) + Afηf (8.17)

where At and Af are the total area and finned area, respectively. Equation 8.17 can be rewritten for the surface efficiency as

A η =−f −η o 11()f (8.18) At

The total heat transfer rate is

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FIGURE 8.3 Fin efficiency of commonly used fins.

q = hηoAt(Tw – Tf) (8.19)

where Tw and Tf are the wall (or surface) and fluid temperatures, respectively.

8.3 SELECTION AND DESIGN OF FINS

A fin can affect the heat transfer rate of a surface in two ways. One is by reducing the convective thermal resistance 1/hA because it increases the heat transfer area as well as the heat transfer coefficient. On the other hand, the fin can increase the conductive thermal resistance L/kA because it increases the conduction length. Theoretically, under certain conditions, the addition of fins may actually decrease the heat transfer rate. The maximum length of a fin which will increase the heat transfer rate can be obtained by setting dq/dL equal to zero. For illustration, let’s take the fin of case 3 in Section 8.1 as an example. From Equation 8.8c, the total heat transfer rate is

h sinh()mL + L cosh()mL 05. mk q=−()() hPkA T T (8.20a) bf h cosh()mL + L sinh()mL mk

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h tanh()mL + L =−05. mk q()() hPkA Tbf T (8.20b) h 1 + L tanh()mL mk

Assuming that all variables with the exception of q and L are constant and setting dq/dL = 0, we have

2 hL m −=0 (8.21) mk2

By substituting the value of m from Equation 8.3 along with the assumption of h = hL, Equation 8.21 is reduced to

hA = 1 (8.22) Pk

Equation 8.22 implies that hA/Pk must be less than 1 in order to increase heat transfer by the addition of fins. For practical application, hA/Pk is frequently set as less than 0.25. According to this statement, rectangular fins should be added only if the following condition is satisfied:

ht < 0.5 (8.22) k

where t is the thickness of the fin. The dimensionless group ht/k is often referred to as the Biot number, which represents the ratio of the internal to the external thermal resistance of a solid. Based on the previous discussion, the following conclusion may be made:

1. The use of fins in forced convection is generally advantageous for cases with small heat transfer coefficients. Therefore, fins effectively increase the heat transfer to or from a gas, and are less effective in a liquid, but offer little advantage in heat transfer involving boiling and condensation. 2. Fins should be placed on the side of a heat exchanger surface where the heat transfer coefficient is the lowest. 3. A large number of thin, slender, and closely spaced fins are superior to fewer and thicker fins for heat transfer. 4. It is also desirable to use materials with high thermal conductivity for fins.

Example 8.1 An aluminum rectangular fin with base at 200°F is cooled by an airstream at 120°F. The estimated heat transfer coefficient is 1.5 Btu/(hr·ft2·°F). The thermal conductivity of the fin is 100 Btu/(hr·ft·°F). The dimensions of the fin as

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shown in Figure 8.2 are b = 5 in, L = 1 in, and t = 0.05 in. Determine the fin efficiency and the total heat transfer rate. Repeat the problem with a stainless steel fin where k = 8 Btu/(hr·ft·°F).

Solution

 2h 05.. 15.  05 m =   =× 2  = 2. 68 1/ft  kt   100× 0./ 05 12 =× 1 = 0.2236 mL 2.68 12

From Figure 8.3, the fin efficiency is

ηf = 0.98

To calculate the heat transfer rate, one can use Equation 8.20b; however, a simpler approach is to utilize the fin efficiency obtained previously:

q=−=×××××−η hPL() T T 0.. 98 1 5() 2 5 1 (200 120 ) fbf 12 12 = 817. Btu/hr

Identical steps can be taken for the stainless steel fin, and the results are as follows:

m = 9.49 1/ft

and

mL = 0.79 ⇒ ηf = 0.84

The heat transfer rate is

q = 6.59 Btu/hr

Example 8.2 An upward forced air flow over an aluminum finned plate has dimensions of 8 in (height) by 10 in (width). The center-to-center spacing between fins is 0.5 in. The length and thickness of the fin are L =1.0 in and t = 0.1 in, respectively. The heat transfer coefficient is estimated to be 5 Btu/(hr·ft2·°F). Determine the total heat transfer rate if the base plate and air temperatures are set at 200°F and 100°F, respectively.

Solution For the fin efficiency:

 5  05. mL =×2  ×=1 0.. 289 1/ft ⇒ η = 0 97  100× 0./ 1 12 12 f

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To determine the number of fins, we assume that the first and the last fins are located at both edges of the plate:

(n – 1)(0.5 – 0.1) + 0.1n = 10 ⇒ n = 20.8

Let n = 21. The total heat transfer surface area of fins is

2 Af = 21(1 × 2 + 0.1) × 8 = 352.8 in

Total heat transfer surface area is

2 At = (10 + 21 × 2 × 1)8 = 416 in

For the surface efficiency:

A η =−f −η =−352. 8 − = o 11()f 1 (1 0. 97) 0. 975 At 416

The total heat transfer rate is

416 qhATT=−=×××−η ()0. 975 5 ()200 100 otb f 144 = 1408. 3 Btu/hr

The heat transfer rate will be increased when the number of fins per unit length (or area) increases. The pressure drop will also increase with the number of fins, and it leads to a large fan or pump. For forced convection flow, the increase in the heat transfer rate with denser fins can generally be expected as long as there is enough pumping power to overcome the system pressure drop. In the natural convection environment, on the contrary, increasing the number of fins may actually reduce the heat transfer rate, because the driving force of the flow in natural convection is normally limited (or fixed) for given thermal conditions and physical dimensions as well as the orientation of the solid. Therefore, the available head for circulating flow may not be large enough to overcome the system pressure drop due to increasing the number of fins. This result, in turn, reduces the mass flow rate of the circulating flow, resulting in a lower heat transfer coefficient. Detailed information will be presented in Chapter 12.

REFERENCES

1. F. Kreith, Chapter 2, Principles of Heat Transfer, International Textbook, Scran- ton, PA, 1963. 2. J. H. Seely and R. C. Chu, Chapter III, Heat Transfer in Microelectronic Equip- ment: A Practical Guide, Marcel Dekker, New York, 1972.

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THERMAL INTERFACE RESISTANCE

It is common practice to use brackets, heat sinks, and circuit boards for mounting components that are to be cooled by conduction. The primary heat transfer mode inside a component is also heat conduction. Cooling by conduction often requires the transfer of heat across various materials and interfaces that may be laminated, bolted, riveted, clamped, or bonded together. A high temperature gradient may occur if a large amount of heat is transferred across these interfaces. Since there is no perfect surface in the real world, an actual surface always contains peaks and valleys. The surface irregularities are an inherent result of manufacturing processes. Surface irregularities with large wavelength are usually referred to as waviness. The length of the waves can vary from 0.1 to 1 cm, while the height can vary from 20 to 40 µm. On the other hand, the depth due to the peaks and valleys is often called surface roughness and can vary from about 0.05 µm for a very smooth surface to 25 µm for a very rough surface. As shown in Figure 9.1, only a fraction of points on two nominally flat contacting surfaces are actually contacting each other. There- fore, additional thermal resistances, including constriction and spreading resistances, are created because of reduction of heat transfer area at the interface of two bodies. It is not necessary to examine in detail the local thermal transport process between two contacting surfaces; the heat transfer mechanism at a joint can be described by the thermal contact conductance hc, defined by the following equation:

qA/ h = c − (9.1) TT12

where q/A is the heat flow per unit area and T1 and T2 are the average temperatures at the interface of the two contacting bodies.

9.1 FACTORS AFFECTING THERMAL CONTACT RESISTANCE

The interface thermal contact resistance is a complex function of many parameters such as geometric, physical, and thermal properties of the contacting solids

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FIGURE 9.1 Two nominally flat surfaces in contact.

as well as interstitial substances. The important factors affecting thermal contact resistance as stated by Yovanovich [1] are summarized in this section.

9.1.1 Surface Roughness and Waviness

In the absence of surface waviness and curvature, the contact resistance at the interface is directly proportional to the roughness and inversely proportional to the asperity slope. The roughness is generally expressed in terms of the root- mean-square (RMS) roughness of two contacting surfaces. On the other hand, the contact resistance due to surface waviness in the absence of surface roughness is proportional to the one-third power of the local surface curvature. Examination of Figure 9.2 reveals that the number of direct contact points for two solids in contact decreases as the surfaces roughness increases. It should also be noted that the overall flatness of the surface is even more critical than surface roughness or finish. This result is due to the fact that no improvement in finish will help if the surfaces do not touch each other at all, as illustrated in case C of

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FIGURE 9.2 Various types of surfaces in contact.

Figure 9.2. Consequently, air gaps may be present between two wavy surfaces. In short, the thermal interface resistance increases with increasing surface roughness and waviness.

9.1.2 Mechanical Properties of Solids In mechanical contacts, the important physical properties are the material hardness, the modulus of elasticity, and Poisson’s ratio. During the first loading cycle, the contact asperities undergo plastic deformation that creates the actual contact area. The contact area is inversely proportional to the material hardness; therefore, when two nominally flat, rough surfaces are placed in direct contact by means of mechanical load, the most important physical property is the material hardness of the softer material of the two solids. On the other hand, the pertinent mechanical properties will be the modulus of elasticity and Poisson’s ratio if the contacting surfaces are smooth but curved. The maximum contact pressure according to the theory of elasticity is proportional to the modulus of elasticity to the one-third power. In the absence of the surface roughness, the contact resistance is also directly proportional to the effective modulus of elasticity to the same power. Both hardness and elasticity effects must be taken into consideration in determining the contact resistance when solids with waviness and roughness are placed in contact. It may be concluded that the thermal interface resistance increases with material hardness and modulus of elasticity.

9.1.3 Thermal Properties of Solids A very important thermal parameter is the thermal conductivity of the contacting solids and the interstitial materials. In a vacuum environment, contact resistance varies inversely as the harmonic mean thermal conductivity of two solids. As should be expected, high conductivity of solids and interstitial materials will result in a low contact resistance. The effect of mean free path, accommodation coefficients, and the Prandtl number must be taken into consideration when the interstitial material is a gas.

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FIGURE 9.3 Effect of pressure on interface resistance.

9.1.4 Apparent Contact Pressure

One of the most important parameters to have a significant effect upon the contact resistance is the mechanical load or apparent contact pressure. It has been found that the contact resistance decreases as the contact pressure increases. This is because an increase in contact pressure will increase the actual contact area and also will bring two surfaces closer together (reducing the gap size). It should also be noted that increasing contact pressure has a much greater effect in reducing the contact resistance for rough surfaces than for two smooth surfaces. Curves ab and bc in Figure 9.3 indicate the effect of pressure on the contact resistance during the first loading and unloading cycles, respectively, when two nominally flat surfaces are pressed in contact. During the first loading cycle, a plastic or permanent deformation of solids occurs at the contact points, which increases the actual contact area for heat transfer. During the first unloading cycle, the deformation is essentially elastic and the process is reversible along the curve bc. Figure 9.3 also shows that the reduction in the thermal interface resistance is large initially and becomes smaller as the apparent contact pressure continuously increases.

9.1.5 Void Spaces and Filler Materials

The average thickness of the void space and filler materials will have a direct impact on the contact resistance. The filler materials can be solid, liquid, or gas.

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The requirements for a good filler are high thermal conductivity and filling ability. A good indicator of filling ability is softness of the material, in the case of a solid filler; or wetting ability, for a liquid filler.

9.2 JOINT THERMAL CONTACT RESISTANCE

For a vacuum environment, if radiation across voids between two surfaces is neglected, heat transfer between two contacting surfaces occurs only by means of conduction through numerous contact points at the interface of the two solids. The contact resistance can be improved by adding filler materials in the void space, because part of the heat flows through the interface by means of the interstitial materials. Yovanovich [2] developed a general correlation to determine the joint contact resistance for nominally flat, rough surfaces in mechanical contact with gas, liquid, or thermal grease as the interstitial substance. It is assumed that the mechanical load is relatively light, which is applicable to microelectronics, and that the contacting solids are experiencing the first loading cycle. It is usual to consider the constriction resistance across the microcontact spots and the resistance across gas or grease to be parallel thermal resistances. It is further assumed that the interstitial material can be considered to be a continuum; therefore, the joint thermal contact conductance is the sum of two components, namely, the solid contact conductance and the gap conductance. According to Yovanovich [2], the solid contact conductance, which is inverse of the contact resistance, can be expressed by the following equation:

095.  km P h = 125.  s   (9.2) c  σ  H 

where

=⋅° ks harmonic mean conductivity, W/(m C) = 2kk12 + kk12 m = RMS absolute surface asperity slope =+2 205. ()mm1 2 σ = RMS surface roughness, m =+σσ2 205. ()1 2 P = apparent contact pressure, N/m2 H = hardness of the softer solid, N/m2

The gap conductance can also be obtained as follows:

k h = g (9.3) g YM+ σ

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where

=⋅° kg gas or filler conductivity, W/(m C) M = effective gas parameter αβΛ = σ

and

Y = separation distance between mean planes of contacting surfaces, m (9.4)  3132.P0. 547 =−1184. σ  ln   H 

and where

α = effective accommodation parameter − α − α = 221 + 2 α α 1 2 β = gas parameter γ = 2 Pr(γ +1 ) γ = specific heat ratio of constant pressure to constant volume Λ = molecular mean free path, m Pr = Prandtl number

Equation 9.4 can also be approximated in a much simpler form as

 P  −0. 097 Y.= 153σ  (9.4a)  H 

The subscripts 1 and 2 appended to many of the quantities represent the properties of solids 1 and 2, respectively. The value of m in Equation 9.2 can vary from 0.01 for a very smooth surface to 0.13 for a very rough surface. The molecular mean free path Λ is dependent on the gap mean temperature T and the gap pressure Pg, and it can be expressed in the following form:

  P  ΛΛ= T g0 0   (9.5)  T0  Pg 

where the subscript 0 refers to some reference conditions at 15°C and 1 atmosphere.

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Table 9.1 Typical Ranges of Parameters in Equations 9.2 and 9.3

10 –5 < P/H < 10 –2 1 < β < 2 4.26 > Y/σ > 2.34 0.04 µm < Λ0 < 0.19 µm 0.2 µm < σ < 4 µm 0.01 < Λ0/σ < 1.0 –4 –2 0.03 < m < 0.3 10 < kg0/k < 2 × 10 2 < α < 10

Typical values for parameters in Equation 9.3 were presented in Reference 2 and are reproduced in Table 9.1. For practical applications, the values of M in Equation 9.3 for air at 1 atmosphere and temperatures of 15°C and 100°C are 0.277 and 0.081, respectively. Furthermore, M is equal to zero for liquids, oils, and greases. Finally, the total or joint thermal contact conductance becomes

==+1 hj hhcg (9.6) Rj

where the unit of conductance is W/(m 2·°C). Several different models [3, 4] were proposed to refine the values of hardness in the above equations. Other, similar correlations for the solid contact conductance are those given by Cooper et al. [5]:

 km P0. 985 h = 145.  s   (9.7) c  σ  H 

and by Tien [6]:

 km P085. h = 055.  s   (9.8) c  σ  H 

The amount of heat that can be transferred between two clamped surfaces is sharply reduced at high altitudes above 80,000 ft and will reach a minimum conductance represented by Equation 9.2 in a vacuum. If heat transfer due to radiation is neglected, no heat will be transferred between two adjacent surfaces in a vacuum environment if they are not in intimate contact. It can be seen from Equations 9.3 and 9.5 that gap conductance becomes negligibly small as gap gas pressure Pg approaches zero. Therefore, special care must be taken at the thermal interfaces of electronic assemblies that will be cooled by conduction in outer space applications.

9.3 METHODS OF REDUCING THERMAL CONTACT RESISTANCE

The best way to reduce the thermal contact resistance at the interface of two solids is to solder or braze the joint in question. When a mechanical joint cannot

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FIGURE 9.4 Increasing contact surface with same projected area.

be made permanent by soldering or brazing, the thermal contact resistance can often be minimized by using various techniques. Generally, the interface contact resistance can be improved by reducing roughness and waviness of contacting surfaces, by increasing contact pressure, or by increasing contacting surface area. If space is limited, the contacting surface area can be increased without changing the footprint or projected area as illustrated in Figure 9.4. The contact resistance can also be reduced by preloading the interface pressure to some value much higher than the expected operation pressure, then releasing the load without disturbing the contacting solids, and, finally, setting the load to the required operation pressure. Other possible methods of reducing the thermal contact resistance are presented in the balance of this section.

9.3.1 Thermal Greases

Thermal greases that are available in various formulations are in general made of silicone loaded with thermally conductive metal oxides. The reduction in contact resistance occurs because the grease will fill the voids between two contacting surfaces. Only a small amount of grease should be used, to just fill the void spaces and still allow the high points of the metal surfaces to touch each other. Too much grease may create voids within the heat compound and increase the thermal resistance across the thickness of the grease. Care must be taken that no foreign objects get into the grease. Thermal greases have a tendency to migrate in high temperatures and to evaporate in low-pressure or vacuum environments. Thermal greases are also subject to contamination. In addition, it is very difficult to apply thermal grease uniformly over a large area.

9.3.2 Interstitial Filler

Interstitial materials in the form of metal foils may be employed to either enhance or reduce the rates of heat transfer across two contacting surfaces. The material hardness H and thermal conductivity k will in general dictate which process occurs. According to Snaith et al. [7], to enhance the thermal conductance across a bare joint, the following performance ratio must exceed unity:

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Table 9.2 Thermal Ratings for Solid Metal Interstitial Materials

S ratio Base materials Insertion Conductivity, Hardness, material W/(m·°C) MN/m2 Steel Aluminum Indium 81.7 10 978 41 Silver 429 245 209 8.8 Aluminum 237 157 181 7.5 Copper 400 294–392 163–122 6.8–5.1 Tin 67 59 135 5.7 Lead 35 39 107 4.5 Gold 317 589–687 64–55 2.7–2.3 Magnesium 156 392 48 2.0 Zinc 120 343 42 1.8 Al 6061-T6 200 1000 23.9 — Cartridge brass 110 784–981 16.8–13.4 0.7–0.56 Nickel 90.6 686–784 15.8–13.8 0.66–0.58 Tungsten 174 9810 2.1 0.09 Invar steel 15 1570 1.1 0.05 Titanium 17 1962 1.04 0.04 Inconel 600 12 1472 0.98 0.04 SS 304 15 1795 — 0.04 Nichrome 12 1766–2158 0.81–0.67 0.033–0.028

kHfm S = (9.9) kHmf

where the subscripts f and m refer to the filler and base material, respectively. Table 9.2, which is reproduced from Reference 7, gives the ratings for a range of interstitial materials between joints of stainless steel 304 with stainless steel 304 and aluminum 6061-T6 with aluminum 6061-T6. Thermal isolation at the interface can also be achieved when S is less than unity. Yovanovich [8] presented the results of a series of experiments to determine the effect of metallic foils upon the joint resistance of an interface formed by the mechanical contact of a lathe-turned surface and an optically flat surface. The foils tested were lead, tin, aluminum, and copper, and their thickness ranged from 10 to 500 µm. The test results indicate that there is an optimum foil thickness defined as that thickness which yields the minimum joint resistance. The ratio of optimum thickness to RMS surface roughness is about 2 for lead and tin, about 0.48–0.58 for aluminum, and about 0.68 for copper. The results as given in Figure 9.5 reveal that to reduce the contact resistance, tin is superior to lead, lead is better than aluminum, and aluminum is superior to copper. The ratios of thermal conductivity, in W/(cm·°C), to hardness, in kg/mm2, for tin, lead, aluminum, and copper are 0.1132, 0.0875, 0.0756, and 0.048, respectively. Therefore, the ratio of thermal conductivity to hardness of materials is proposed to rank the effectiveness of the foils. The larger this ratio, the greater will be the reduction in the bare-joint resistance. The softer foils yielded a lower joint

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FIGURE 9.5 Contact resistance versus various foil thicknesses.

resistance than did the harder foils. In other words, the hardness of foil material is more important than thermal conductivity of the foil in reducing the joint resistance. Finally, an empirical correlation was developed by Yovanovich [8] for predicting the dimensionless minimum joint thermal resistance for the optimum foil thickness:

R R* = m R b (9.10) = 1 exp[0. 0072 P+ 15 . 5()] k/H 092.

where

2 Rm = minimum thermal resistance with foils, ˚C·cm /W 2 Rb = bare-joint contact resistance, ˚C·cm /W P = contacting pressure, kg/cm2

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FIGURE 9.6 Insertion of a filler at interface.

H = foil hardness, kg/mm2 K = foil thermal conductivity, W/(cm·˚C)

Similar tests were later performed by Peterson and Fletcher [9]. They con- firmed the method used to rank the effectiveness of foils according to the ratio of foil thermal conductivity to hardness. In addition, they also found an optimum roughness when an optimum foil thickness is applied to a joint consisting of one rough and one smooth surface. O’Callaghan et al. [10] also found that an optimum filler thickness is expected to occur when the filler thickness δ is of the order of the surface RMS roughness σ. The thermal resistance decreases with increasing foil thickness for δ << σ. This reduction can be attributed to two factors. As shown in Figure 9.6, first, additional contact areas are created across the filler around the actual contact points. Second, the void spaces or gaps between two contacting surfaces are significantly reduced by a filler that has a higher thermal conductivity than gas (air) or vacuum. On the other hand, the thermal resistance increases with increasing δ for δ >> σ. It must also be stressed that the insertion of a filler that is harder than the base materials (contacting solids) will generally result in an increase in overall contact resistance, regardless of the thickness or thermal conductivity of the filler materials.

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FIGURE 9.7 Contact of bare and coated joints.

The insertion of a metal foil will greatly enhance the thermal contact conductance; however, the foil must be very thin and soft to be effective. Foils are extremely difficult to handle. The thermal contact resistance due to wrinkles or folds may actually increase if foils are not properly applied. In addition, foils in general cannot be reused.

9.3.3 Soft Metallic Coatings

As indicated earlier, application of thermal greases or soft metal foils can improve the thermal contact resistance within the joint. Thermal greases are sometimes objectionable because of the possibility that grease might evaporate and cause contamination. Foils may not be attractive for practical applications because soft foils tend to wrinkle and are difficult to handle. Soft metallic coatings are basically free from the problems associated with thermal greases or soft metal foils. Application of metallic coatings to minimize thermal contact resistance has been reported [11–13]. The authors proposed the following equation to predict the thermal contact conductance of the coated joint as indicated in Figure 9.7:

 H 093. hh′ =   k′ (9.11a) c H′

where

kk+ k′ = 12 + (9.11b) ck21 ck 12

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and where hc is the uncoated-contact conductance as presented in Equation 9.2, H is the hardness of the softer of two base materials or substrates, H′ is the effective hardness of the layer-substrate combination, k1 and k2 are thermal conductivities of the base material, and c1 and c2 are the constriction parameter correction factors which account for heat spreading in the coated material, defined as follows:

ψεφ( `, ) c = ψε() (9.12)

where

2 λ εφγρ 16 J1 (``) ψ (`εφ,= ) ∑ (9.13) πε λλ3 2 ` (`)J0 (`)

and

++− −ω φ = KK[(11 ) ( Ke ) ] −ω ()()11+−−KKe γ = 1 for isothermal contact point λε ρ = 05. sin ( ` `) λε J1(``)  P  −0. 027 ψ (ε)=076.    H   P 05. ε =    H  λλ= root of J1()

and where

 t  τ ' ==relative layer thickness    a  τρ 0. 097 a = 077.    = contact spot radius  mH  ωλετ= 2 ``` K = thermal conductivity ratio of substrate or base material to coating layer = Jnn th - order Bessel function of the first kind

The symbol ` represents the conditions for the coating layer. If the coating is applied to only one of the base materials, then either c1 or c2 will be equal to 1. H′ must be obtained empirically for the particular layer-substrate combination under consideration. Expressions for different layer-substrate combinations are given as follows:

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Silver-coated nickel [11-13]

 t   t  t HH′ =−1181 + .H  < 10. (9.14a) SL d  d d  t  t H.H.H′ =−181 0 208 − 1 10. ≤≤ 4 9. (9.14b) LL d  d

Tin-coated aluminum [14]

 t   t  t HH′ =−125 + . H   < 10. (9.14c) SL d  d d  t  t HH.H′ =−25.. 03 − 1 10. ≤≤ 60 (9.14d) LL d  d

Indium-coated aluminum [14]

 t   t  t HH′ =−140 + . H   < 10. (9.14e) SL d  d d  t  t HH.H′ =−40.. 077 − 1 10. ≤≤ 49 (9.14f) LL d  d

Lead-coated aluminum [14]

 t   t  t HH′ =−1325 + . H   < 10. (9.14g) SL d  d d  t  t HH.H′ =−3.. 25 0 833 − 1 10. ≤≤ 4 9 (9.14h) LL d  d

where

= HS microhardness of the substrate or base material = HL microhardness of the coating layer t = thickness of the coating layer d = equivalent Vickers indentation depth

It should be noted when t/d > 4.9, the effective microhardness approaches the microhardness of the coating layer, i.e., H′ = HL. Combining Equations 9.2 and 9.11a, one obtains

 km′  P095/  H′002. h′ = 125.      (9.15)  σ  H′  H 

Equation 9.15 can be further simplified to the following equation by letting (H′/H)0.02 be equal to unity because the error introduced will be minimal:

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FIGURE 9.8 Thermal contact conductance of various coating materials.

 km′  P095/ h′ = 125.    (9.16)  σ  H′

Equation 9.16 for coated contacts is identical to the correlation for a bare joint presented in Equation 9.2, except that the effective thermal conductivity and microhardness of the coated surfaces have replaced the usual values for a bare joint. The representative results of Antonetti and Yovanovich are presented in Figure 9.8. The authors stated that for a given coating layer thickness, the smoother the bare contacting surface, the greater the enhancement will be.

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FIGURE 9.9 Effect of metallic coating thickness on contact conductance

Kang et al. [14] performed an experimental study on other metallic coatings at the interface of contacting aluminum 6061-T6 surfaces. Their results as shown in Figure 9.9 verified the existence of an optimum coating thickness. The optimum coating thickness is in the range of 2.0 to 3.0 µm for indium, 1.5 to 2.5 µm for lead, and 0.2 to 0.5 µm for tin, and the enhancements over the bare joint are 700, 400, and 50%, respectively. Based on the experimental data, the hardness of the coating material appears to be the most important parameter affecting the thermal contact resistance.

9.3.4 Phase-Change Materials Another concept, using phase-change materials (PCMs) for reducing thermal contact resistance at the interface, was reported by Cook et al. [15]. Three

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FIGURE 9.10 Insertion of phase-change materials at interface.

different configurations given in Figure 9.10 were studied. One used a porous metal to contain low–melting temperature alloy and another applied the alloy to both sides of the solid metal. The third one directly inserted the alloy into the interface of two contacting surfaces. The results of the tests on 6061-T6 aluminum disks are given in Table 9.3. The key feature of this method is the incorporation of a PCM such as a low–melting temperature metallic alloy as an interstitial material. The selected alloy has a melting point below the operating temperature of the joint. During operation, heat transfer to the joint raises the interface temperature and causes the alloy to melt. The liquid metal then flows freely to fill the interstitial voids,

Table 9.3 Thermal Resistances with Alloy at Interface

Contact resistance at 13 psi, Test specimen °F·ft2/(Btu/min) Porous carrier 0.012 Solid carrier 0.0054 Alloy (no carrier) 0.0038 Bare joint 0.11

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FIGURE 9.11 Microcapillary thermal interface concept.

providing a continuous metallic heat transfer path. Consequently, the thermal contact resistance can be significantly reduced. The liquid metal may also flow to other, undesired locations during the operation. The molten alloy then solidifies when the system is not in operation. The parts will not easily be separated because the joint is basically soldered together.

9.3.5 Capillary Action A limited amount of liquid will partially fill a series of microscopic reentrant cavities at an interface, as shown in Figure 9.11 [16]. Through capillary action, the liquid will be pulled inward from the cavities and will fill up the voids at the interface, allowing heat to be transferred across a thin liquid film with little resistance.

9.3.6 Other Filler Materials

It is popular to apply a silicone or urethane elastomer binder with thermal conductive filler to the interface of electronic equipment. These materials have high dielectric strength, high thermal conductivity, and sufficient pliancy to conform to both microscopic and macroscopic surface irregularities. They are used to fill up the air gap, and they are much thicker than soft metallic foils. Typical thickness is about 0.005 to 0.02 in, and thermal conductivity is about 1 to 2 Btu/(hr·ft·°F). A relatively high and uniform pressure must be applied at the

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interface to achieve a reduction in the overall thermal contact resistance. The thickness should not be too great; otherwise, an increase in the thermal resistance can occur. This type of material is also frequently used for the purpose of electrical isolation.

9.4 SOLDER AND EPOXY JOINTS

The total resistance across the interface of two solids bonded by a solder or epoxy is the sum of the resistances at two interfaces and the resistance across the thickness of the solder or epoxy. The effects of soldering and brazing on the contact resistance were studied by Yovanovich and Tuarze [17, 18]. These authors found that the actual value of the thermal contact resistance was considerably larger than the theoretical value solely based on the thickness of the solder. This is because additional resistances occurred at two interfaces. For example, thermal resistances ranged from 0.025°C·cm2/W for the best joint (brass-brass) to 0.14°C·cm2/W for a bad solder joint. These values considerably exceed the theoretical values of 0.00246°C·cm2/W calculated for an average solder thickness of 15 µm. Similar trends for epoxy bonding materials were also found in References 19 and 20. In general, the total thermal resistance across a joint filled with an epoxy or other fillers can be expressed as follows:

Rt = Rth + Ri1 + Ri2 (9.17)

where

= Rt total thermal resistance = Rth resistance due to thickness of epoxy = Ri1 interface resistance between epoxy and solid 1 = Ri2 interface resistance between epoxy and solid 2

Rth is a function of void fraction and its distribution, thermal conductivity of the epoxy, and the thickness of the epoxy. Ri1 and Ri2, which can be determined only by experiment, depend on the material of epoxy and the bonding process involved. It is difficult to determine separate values for Ri1 and Ri2; for a given process and epoxy, Ri1 and Ri2 can be combined into a single value, and Equation 9.17 can be rewritten as:

  =+C RRt th 1  (9.18)  Rth 

where C represents the combined resistance at two interfaces. The value of C becomes significant when Rth is small; this case implies that the thermal conductivity of the epoxy is very large or the thickness of the epoxy is relatively small. On the other hand, the combined interface resistance C is negligibly small when Rth

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FIGURE 9.12 Measured and calculated thermal resistances of an epoxy joint.

is large. This condition indicates that the thermal conductivity of the epoxy is very small or the thickness of the epoxy is very large. For example, Peterson and Fletcher [19] indicated that the combined interface resistance C could vary from about 10% to 80% of the theoretically calculated values based on the epoxy thickness as the epoxy conductivity increased from 0.27 to 1.93 W/(m·°C). Their experimental data are presented in Figure 9.12. The following correlation was developed by the authors to predict the total thermal resistance over an epoxy with thickness ranging from 0.018 to 0.022 mm:

2 Rt = 0.26(Rth) + 0.62Rth + 0.15 (9.19)

where the unit for the resistance is cm2·°C/W. Schwinkendorf and Moss [20] reported that no interface resistance was observed for unfilled resin. This is because unfilled resin has a better wetting ability and also the thermal conductivity of an unfilled resin is very low, i.e., Rth is high. The ratio C/Rth becomes very small; therefore, the measured resistance is not far from the calculated value as presented in Equation 9.18.

9.5 PRACTICAL DESIGN DATA

Although the equations presented in the previous sections can be used to predict the thermal contact resistance at an interface, information about surface

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Table 9.4 Typical Values of Thermal Contact Resistance

Interface Resistance, in2·°C-W Ceramic-ceramic 0.56–2.50 Ceramic-metal 0.18–1.11 Graphite-metal 0.27–0.56 Steel-steel 0.18–0.91 Aluminum-aluminum 0.06–0.56 Metal-metal (filled with 0.018–0.055 metal foils or greases)

characteristics is often not readily available in the design and analysis phase. It is common to apply some empirical data for practical design in industry. This section is devoted to some practical design data. The information was obtained cumulatively over the years in various industries, but unfortunately some of the sources are unknown. The ranges of typical thermal resistance for different contacting surfaces are listed in Table 9.4. The properties of common thermal compounds are given in Table 9.5. For aluminum structures with 125-µin surface finish, the following empirical correlation can be used to determine the thermal contact resistance:

R = 6.5P–0.72 for P < 300 psi (9.20)

where R is the resistance, in2·F/(Btu/hr), and P is the contact pressure, psi. The following equation [21] can be applied to compute the working torque required for the force when the size of the individual fastener is known:

M = 0.2Fd (9.21)

This equation can further be rewritten in the following form:

5nM P = (9.22) Ad

Table 9.5 Properties of Thermal Compounds

Wakefield Emerson & Dow Corning General Electric 120 Cuming TC4 DC340 G641 Conductivity k, 0.43 0.68 0.11 0.36 Btu/(hr·ft·°F) 3 Density ρ, lbm/in 0.083 0.076 0.083 0.094

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Table 9.6 Maximum Torque Values for Alloy Steel Screws

Screw size* Nominal diameter, in Maximum torque range, in·lbf 2-56 0.086 2.5–2.6 4-40 0.112 4.4–5.2 6-32 0.138 8.8–9.6 8-32 0.164 15.4–19.8 10-32 0.190 25.3–31.7 1/4-28 0.250 62.3–94.0 5/16-24 0.313 126.5–150.0

*The second number in each listing is the number of threads per inch

where

= F loading force, lbf P = mounting interface pressure, psi N = number of screws A = mounting interface area, in2 d = nominal screw diameter, in =⋅ M working torque per screw, in lbf

The values of the torque M are presented in Table 9.6. The thermal contact resistance for a transistor mounting to a metal heat sink or structure is given in Figure 9.13. In addition, the contact resistances for selective joints with various filler materials are presented in Figure 9.14.

FIGURE 9.13 Contact resistance for transistor mounting.

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FIGURE 9.14 Thermal contact conductance for various materials.

EXAMPLE 9.1 1. Two identical aluminum (Al 6061-T6) plates are mechanically joined together. The contact pressure is 100 psi, and hardness of the plate is 1.97 × 105 psi. The surface finish is 25 µin and the surface asperity slope is 0.05. The thermal conductivity of the plate is 110 Btu/(hr·ft·°F). Determine the joint contact conductance.

Solution The first step is to compute the solid contact conductance from Equation 9.2. The values to insert into the equation are

= ks harmonic mean conductivity 2×× 110 110 = =⋅⋅110 Btu/(hr ft° F) 110+ 110

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σ = RMS surface roughness −− − =×[(25 1062 ) +× ( 25 106205 ) ]. = 35 . 355 × 10 6 in m = RMS absolute surface asperity slope =+(.0052205 005 . ). == 0 07071.

The solid contact conductance is

 ×  095. = 110 0. 07071 100 hc 125.  −    35./. 355× 1065 12197× 10  =⋅2447. 8 Btu/(hr ft2 ⋅° F)

The second step is to determine the gap conductance from Equation 9.3. Assume air at 1 atmosphere and a temperature of 80°F, a value of M of about 0.277, and thermal conductivity kg of 0.01497 Btu/(hr·ft·°F). Also,

Y  100  −0. 097 ≈ 153.   σ 197. × 105  = 3. 193

The gap conductance is

k = g = 0. 01497 hg − σσ(+)YM/ (.35 355××+ 106 /)(. 12 3 193 0 . 277 ) =⋅1464. 2 Btu/(hr ft2 ⋅° F)

The joint thermal conductance from Equation 9.6 is

2 hj = hc + hg = 2447.8 + 1464.2 = 3912.0 Btu/(hr·ft ·°F)

2. Repeat the same problem if the surface finish is 125 µin.

Solution Applying the procedure shown in the first part of the example with the new value of σ = 176.777 × 10–6 in, we have

2 hc = 489.6 Btu/(hr·ft ·°F)

and

2 hg = 292.8 Btu/(hr·ft ·°F)

The joint contact conductance is

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2 hj = 782.4 Btu/(hr·ft ·°F)

or the joint thermal resistance is

==1 2 ⋅° Rj 0 184. in F/(Btu/hr) hj

It is of interest to compare this analytical result with the value obtained from the empirical correlation from Equation 9.20. The difference between the analytical prediction and the result from the empirical correlation is about 28%:

–0.72 –0.72 2 Rj = 6.5P = 6.5 × 100 = 0.236 in ·°F/(Btu/hr)

3. Repeat the second part of the example with the contact pressure increased to 200 psi.

Solution Again, following the same procedure, we have

2 hc = 670.1 Btu/(hr·ft ·°F)

and

2 hg = 311.4 Btu/(hr·ft ·°F)

The joint contact conductance is

2 2 hj = 981.5 Btu/(hr·ft ·°F) ⇒ Rj = 0.1487 in ·°F/(Btu/hr)

From Equation 9.20, one obtains

2 Rj = 0.143 in ·°F/(Btu/hr)

The difference in the results from the analytical prediction and the empirical correlation is less than 3% when the pressure is increased from 100 to 200 psi.

Comments: 1. The thermal contact conductance increases with pressure. 2. The thermal contact conductance decreases with increasing surface roughness.

REFERENCES

1. M. M. Yovanovich, “Thermal Contact Resistance in Microelectronics,” National Electronic Packaging and Production Conference (NEPCON), Ana- heim, CA., 1978.

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2. M. M. Yovanovich, “Thermal Contact Correlations,” Spacecraft Radiative Transfer and Temperature Control 83, Progress in Astronautics and Aeronau- tics, 1982. 3. M. M. Yovanovich, A. H. Hegazy, and J. DeVaal, “Surface Hardness Distribu- tion Effects upon Contact, Gap and Joint Conductances,” AIAA-82-0887, 1982. 4. M. M. Yovanovich and A. H. Hegazy, “Experimental Verification of Contact Conductance Models Based upon Distributed Surface Micro-Hardness,” AIAA-83-0532, 1983. 5. M. G. Cooper, B. B. Mikic, and M. M. Yovanovich, “Thermal Contact Resis- tance, Int. J. Heat Mass Transfer 12, 1969. 6. C. L. Tien, “A Correlation for Thermal Contact Conductance of Nominally Flat Surfaces in a Vacuum,” Proc. 7th Conf. Thermal Conductivity, National Bureau of Standards, Gaithersburg, MD, November 1967. 7. B. Snaith, P. W. O’Callaghan, and S. D. Probert, “Use of Interstitial Materials for Thermal Contact Conductance Control,” AIAA-84-0145, 1984. 8. M. M. Yovanovich, “Effect of Foils upon Joint Resistance: Evidence of Opti- mum Thickness,” AIAA Progress in Astronautics and Aeronautics, Thermal Control and Radiation 3, 1973. 9. G. P. Peterson and L. S. Fletcher, “Thermal Contact Conductance in the Pres- ence of Thin Metal Foils,” AIAA-88-0466, 1988. 10. P. W. O’Callaghan, B. Snaith, and S. D. Probert, “Prediction of Interfacial Filler Thickness for Minimum Thermal Contact Resistance,” AIAA J. 21(9), 1983. 11. V. W. Antonetti and M. M. Yovanovich, “Using Metallic Coating to Enhance Thermal Contact Conductance of Electronic Packages,” Heat Transfer in Elec- tronic Equipment, ASME-HTD, 28,1983. 12. V. W. Antonetti and M. M. Yovanovich, “Enhancement of Thermal Contact Conductance by Metallic Coatings: Theory and Experiment,” J. Heat Transfer 107, 1985. 13. V. W. Antonetti, “On the Use of Metallic Coatings to Enhance Thermal Con- tact Conductance,” Ph.D. Thesis, University of Waterloo, 1983. 14. T. K. Kang, G. P. Peterson, and L. S. Fletcher, “Effect of Metallic Coatings on the Thermal Contact Conductance of Turned Surfaces,” ASME Paper No. 89- HT-18, 1989. 15. R. S. Cook, K. H. Token, and R. L. Calkins, “A Novel Concept for Reducing Thermal Contact Resistance,” presented at the 3rd AIAA/ASME Joint Ther- mophysics, Fluid, Plasma and Heat Transfer Conference, St. Louis, June 7–11, 1982. 16. D. H. Tuckerman, “Heat Transfer Microstructures for Integrated Circuits,” Ph.D. Thesis, Stanford University, 1984. 17. M. M. Yovanovich and M. Tuarze, “Experimental Evidence of Thermal Resis- tance at Solder Joints,” J. Spacecraft and Rockets 6(7), 1969. 18. M. M. Yovanovich, “A Correlation of Minimum Thermal Resistance at Solder Joints,” J. Spacecraft and Rockets 7(8), 1970. 19. G. P. Peterson and L. S. Fletcher, “Thermal Contact Resistance of Silicon Chip Bonding Materials,” Proc. of the 1st International Symposium on Cooling Technology for Electronic Equipment, Hawaii, March 17-21, 1987.

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20. W. E. Schwinkendorf and E. Moss, “Thermal Conductivity and Interface of Particle-Filled Silicon Resin,” ASME Paper No. 84-TH-88, 1984. 21. J. E. Shigley, Mechanical Engineering Design, 3rd Edition, McGraw Hill, Inc., New York, 1977. 22. S. R. Mirmira and L. S. Fletcher, “Comparison of Effective Thermal Conduc- tivity and Contact Conductance of Fibrous Composites,” J. Thermoptics 13(2), 1998. 23. A. Bejan and A. M. Morega, “Thermal Contact Resistance between Two Flat Surfaces That Squeeze a Film of Lubricant,” J. Heat Transfer 115, 1993. 24. S. K. Parihar and N. T, Wright, “Thermal Contact Resistance of Silicone Rub- ber to AISI 304 Contacts,” J. Heat Transfer 121, 1999.

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COMPONENTS AND PRINTED CIRCUIT BOARDS

Since the invention of the integrated circuit (IC) in the latter part of the 1950s, the development of new integrated circuits of greater complexity as shown in Figure 1.1 (Chapter 1) has progressed very rapidly. Along with greater numbers of components per integrated circuit have come greater challenges in cooling of electronic equipment. Generally, heat is generated from metallized areas on a small, thin, and fragile silicon die or chip inside a package. The heat transfer engineer must address the issue of cooling of electronics at all levels of packaging. The chip package that is the housing for the silicon die is the most fundamental level of packaging for solid-state electronic equipment; it serves to protect the chip from the environment and to facilitate handling during manufacturing as well as chip interconnection. The next higher level of packaging is the circuit board. The printed circuit board (PCB), which is also often referred to as the printed wiring board (PWB), is a major element of any electronic equipment. It serves as a mounting surface for most of the basic components (chip packages) and provides wiring channels that act as conduits for chip-to-chip connections. The PCB is also frequently considered the primary field-replaceable unit and provides a test bed of accessible points for making circuit checks.

10.1 CHIP PACKAGING TECHNOLOGY

There are two major categories of chip package: first, the single-chip package (SCP), and second, the multichip module (MCM), or multichip package (MCP). In the SCP, each chip is in a self-contained unit. In MCMs, in contrast, all chips are packaged inside a single housing. Individual chips in an MCM are not encap- sulated or enclosed for mechanical protection. In addition, it is typical to use a common substrate or thermal spreader for the entire multichip package. Figure 10.1 [1] gives a quantitative comparison in scale of a typical SCP—which may be a dual in-line package (DIP) or a pin-grid array (PGA)—and a typical MCM. Various MCMs and their benefits and drawbacks can be classified as follows, according to type of process and substrate material:

1. MCM-C: Ceramic substrate (alumina) with thick film • Moderate cost • Mature technology

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FIGURE 10.1 Scale comparison of MCM and SCP. (a) Two basic single- chip packages, the dual in-line package (DIP) and the pin-grid array. (b) MCM containing various types of individual chips.

2. MCM-D: Organic thin film deposited on a silicon, ceramic, or metal substrate • Most expensive • High interconnection density 3. MCM-L: Organic substrate (e.g., FR-4 and polyimide) with laminated copper • Poor thermal and electrical characteristics • Very inexpensive

From a construction point of view, the package can also be classified as plastic (nonhermetic) or ceramic (hermetic). The latter costs more and is often used in demanding environments where reliability is a major concern. To the contrary, the plastic package offers lower cost but is less reliable, having a higher package thermal resistance. Another difference between plastic and ceramic packages is in the heat transfer path. In the ceramic package, the silicon die is typically placed in a cavity and is bonded to a substrate. Heat flows from the chip or die through the bonding material to the substrate and finally to the housing or case. Mean- while, some of the heat is transferred from the leads to the printed circuit board. High thermal conductivities of substrate and die bond are a major thermal consideration. A good match in coefficient of thermal expansion between substrate and die is equally important. In the plastic package, on the other hand, most of the heat is transferred from the chip through the plastic molding compound—which has a relatively low thermal conductivity—to the case. The desired thermal properties are high thermal conductivities of the plastic molding compound and lead frames.

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FIGURE 10.2 Drawings of common packages: (a) DIP; (b) PGA.

Figure 10.2 shows two of the most popular packages, the dual in-line package (DIP) and the pin-grid array (PGA). The pins that provide input and output (I/O) functions for a DIP are arranged along the long sides of the rectangular package. The typical center-to-center distance between pins for a standard DIP is 100 mils (0.254 cm), which severely limits the I/O count. The maximum I/O count for a standard DIP is therefore 64. A PGA is much larger than a DIP, and its silicon die is usually located in a cavity. Although a PGA package has a large number of pins, none of them is directly under the silicon chip. Because of the much greater number of pins it allows, the PGA is the package design used for the ASIC (application-specific integrated circuit). The main drawbacks of PGAs are high cost of pin materials such as Alloy 42 and Copper, and mounting technology of having through-holes on the PCB which is not compatible to the surface mount technology. To overcome the problems, a new package, Ball Grid Array (BGA), was recently developed by replacing the pins of PGAs with small solder balls. Solder balls are generally cheaper than Alloy 42 (or Copper) pins and are also easier to attach to the bottom of the package. Through solder balls, the BGA is then attached to the copper pads on the PCB. The mounting approach is compatible to the surface mount technology. Unlike a PGA where there is no pin under the die area, the solder balls are spreading over the entire bottom of the BGA package. The interconnection between pins of a DIP or PGA and the PCB is illustrated in Figure 10.3. Because of manufacturing tolerances, the space between the bottom of the chip package (or component) and the top surface of the printed circuit board can vary from 0.015 to 0.060 in for a DIP and 0.033 to 0.068 in for a PGA. This space will sometimes create a thermal problem for a conduction-cooled board. The varying gap sizes under components will also cause an additional complication in the manufacturing process when a heat sink or thermal spreader is required to fill up the gaps between the board and components.

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FIGURE 10.3 Interconnection between pin and board: (a) DIP; (b) PGA.

10.2 CHIP PACKAGE THERMAL RESISTANCE

The chip package or component thermal resistance θj-c, which is often also referred to as internal resistance, is defined as the thermal resistance between the junction (metallized area) on the silicon die and some reference point on the outside of the chip package or component case. The value of θj-c can vary if different reference points are used. Therefore, caution must be taken to use this value in a correct way. In most component assemblies, heat must be transferred through a number of different materials and interfaces, and the total component thermal resistance is the net value of several individual thermal resistances in series or parallel. Once θj-c is known, the temperature rise from the case to the device junction can be determined as follows:

∆Tj-c = qθj-c (10.1)

where q is component heat dissipation. A typical unit for θj-c is °C/W. In the thermal analysis, the component case temperature is usually computed and Equation 10.1 is then used to predict the component junction temperature. A direct modeling of a chip package is possible if the materials and internal structures of the package are known. The important thermal design considerations in a chip package are as follows:

1. Spreading in die or chip: a. Die material: silicon or gallium arsenide (GaAs). b. Heat source: • Uniform versus nonuniform heat source: high thermal resistance for nonuniform heat sources. • Shape of heat source: low thermal resistance for a heat source with a larger perimeter.

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2. Die bond: The thermal resistance of the die bond is especially important for hermetic packages, where heat from the die is first transferred through the die bond to the substrate and then to the housing or case. a. Cracks: Caused by excessive mechanical stresses between the chip and substrate, or metallurgical fatigue during temperature cycling. b. Voids: Created either during the chip attachment process, or during cooling down, or in storage. The thermal resistance is affected by the size, shape, and distribution of the voids. For a fixed void fraction in the die bond, a few large, contiguous voids give a higher thermal resistance. Both cracks and voids increase the thermal resistance across the die bond materials. 3. Spreading in substrate and plastic molding: a. High–thermal conductivity materials. b. Good match in coefficient of thermal expansion between materials.

10.3 CHIP PACKAGE ATTACHMENT

Packages can also be divided according to the method of chip attachment, into through-hole and surface-mount packages (or components). DIPs and PGAs are typically mounted to the printed circuit board using the through-hole technology. The disadvantages of this approach are poor area efficiency and limited I/O counts. The surface-mount technology can significantly increase the pin count and board area efficiency, and improve the package density as well as electrical performance, because of the component size (surface-mount components are several times smaller in size than DIP equivalents). This approach also allows the components to be mounted on both sides of a printed circuit board. There are some challenges, as follows, involved in the use of surface-mount components:

1. Thermal expansion matching: The coefficient of thermal expansion (CTE) of the chip, substrate, and board may be quite different. 2. Thermal management: As component density increases, the heat load on the printed circuit board increases. The problem becomes even worse when components are mounted on both sides of the board. 3. Mechanical rigidity: The board must be very flat, without any warpage or twist, because the solder joints are the only mechanical attachment medium for surface-mounted components. The board must also be very rigid to resist flexing during handling or under vibration or shock conditions in order to prevent stress cracking in the solder joints. 4. Dimensional stability: The board must be able to withstand high assembly temperatures without any deformation, delamination, or via plating fracture.

The thermal conductivity and the coefficient of thermal expansion are probably the most important properties for surface-mount technology. Figure 10.4 shows both properties for various materials. For a PCB, multilayer dielectric materials may be adhesively bonded to a metal core using copper-invar (Cu-Inv). The Cu-Inv can provide the mechanical rigidity required for a typical PCB, and it

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FIGURE 10.4 Thermal conductivity and CTE of various materials.

also enables achieving the proper value of the coefficient of thermal expansion by adjusting the percent of invar involved. Cu-Inv-Cu is a metallurgically bonded tri- layer consisting of a core of invar (36% nickel/64% iron alloy) bonded on both sides with equal thicknesses of oxygen-free copper. This material has been made in thicknesses from 5 to 52 mils and in width up to 24 in. Figure 10.5 [2, 3] shows the lateral thermal conductivity and the coefficient of thermal expansion for various percents of copper involved.

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FIGURE 10.5 Thermal conductivity and CTE of copper-invar.

The following equations can be used to determine the thermal conductivity in the xy plane and z direction for Co-Inv combinations. For the xy plane (lateral):

Kxy = akCu + (1 – a)kinv (10.2a)

For the z direction:

 −  −1 =+a 1 a Kz   (10.2b)  kCuk inv 

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FIGURE 10.6 Thermal path for a common board in an electronic box.

where a is the fraction of copper involved and kCu and kinv are thermal conductivities of copper and invar, respectively.

10.4 BOARD-COOLING METHODS

Board-level thermal analysis includes conduction cooling and direct air cooling of components. The board and components are cooled by air through convection for the latter case, which will be discussed in the next chapter. On the other hand, heat generated from components, for the former case, is conducted through the PCB and is then transferred to the chassis at the edges of the board as illustrated in Figure 10.6. Typically, the board is attached to the chassis by a card retainer. The card retainer provides a better thermal path by

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forcing the board into contact with the chassis. A parametric study of a conduction-cooled board is provided in Laderman et al. [4]. The parameters in the study include PCB construction, chip carrier material, lead count, chip size, and thermal via density. To improve board thermal performance, a thermal spreader or plane (often also called a heat sink) is added to the board. For surface-mount components, the thermal spreader is generally added to the back side (opposite side from the components) of the board; however, the thermal spreader is usually added to the component side of the board for through-hole components such as DIPs and PGAs. In this case, many cutouts on the thermal spreader must be made for component leads.

10.5 BOARD THERMAL ANALYSIS

A typical printed circuit board consists of a multilayer of dielectric material and copper film. The copper layers are used to provide the chip-to-chip connections for transmitting signals, for power supply, and for grounding. The basic concern in thermal analysis is how to model the board accurately. Although the majority of the material of a PCB (probably over 95% by volume) is insulation materials, which have a very low thermal conductivity, one can never omit the contribution due to the copper layers in the thermal analysis. Any omission of the copper layer in the analysis will result in excessively high component temperatures, especially for the case of conduction-cooled PCBs. The high temperature is due to lack of heat conduction spreading because of the low thermal conductivity of the board. The thermal conductivities of the basic materials used to construct a PCB are listed in Table 10.1. The terminology typically used to describe the thickness of a copper layer is expressed as “x-oz,” which represents the weight in ounces of a copper layer with 1 ft2 surface area. For example, the thickness of a 1-oz copper layer is about 0.00135 in. In theory, one can include every layer of a given PCB in the thermal model, but this approach is time-consuming, and above all, it is not practical. This is because the construction of a PCB is very complex. As shown in Figure 10.7, a typical PCB frequently has more than 10 layers of each type of material (dielectric material and copper) laminated together, with the total thickness of the board less than 0.1 in. In addition, the coverage of the copper layers is not always 100%. There are often many cutouts over a copper layer, and sometimes only traces of the copper exist on some given layers.

Table 10.1 Thermal Conductivity of Basic PCB Materials

Material W/(m·°C) Btu/(hr·ft·°F) Copper layer 386.00 223.180 Dielectric layer: G10 0.30 0.173 FR-4 0.35 0.202 Polyimide 0.52 0.301

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FIGURE 10.7 Typical PCB construction.

The practical approach in the thermal analysis is to use the so-called equivalent thermal conductivities for the xy plane and the z direction (across the thickness) of the board. The basic treatment is to assume that the PCB consists of a single layer of material with equivalent thermal conductivities of Kxy and Kz.

10.6 EQUIVALENT THERMAL CONDUCTIVITY

As noted in the previous section, the two equivalent thermal conductivities of a printed circuit board (PCB) are the planar (lateral) thermal conductivity Kxy and the normal thermal conductivity (in the z direction). These can be determined through the respective methods described in the rest of this section.

10.6.1 Planar Thermal Conductivity Kxy

The laminated layers are considered as parallel paths in calculation of the equivalent thermal conductivity. For a parallel network, the total resistance is

1111 =+++… (10.3) RRRRxy 123

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where

==th Ri thermal resistance for the i layer, i,,,123… = Li kAciii = th Lii length of layer = th Aii cross - sectional area of layer (normal to heat flow) = th kii thermal conductivity of layer = th cii fraction of total coverage for layer (e.g., 08. if the cutout for a given layer is 20%)

Rearranging Equation 10.3, we have

KAxy  ckA  ckA = ∑  iii + ∑  iii     (10.4) L Li in Li Cu

where the subscripts “in” and “Cu” represent the dielectric insulation material and the copper, respectively. A is the total cross-sectional area over the thickness of a PCB; i.e., A = t (thickness) times w (width). Since L = L1 = L2 = L3 = … , Equa- tion 10.4 can be rewritten as follows:

 ckA  ckA K = ∑  iii + ∑  iii xy     (10.5) A in A Cu

Assuming there is 100% coverage on each layer (ci = 1), Equation 10.5 can be reduced to the following form:

    = tin + tCu Kkxy in   kCu  (10.6)  tPCB   tPCB 

where

= tin total thickness of dielectric material = tCu total thickness of copper film = tPCB total thickness of printed circuit board =+ ttin Cu

10.6.2 Normal Thermal Conductivity Kz

In considering the heat flow normal to each layer, it is assumed that the resistances are in series, as follows:

=+++ RRRRz 123… (10.7)

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t  t   t  PCB = ∑  i  + ∑  i  (10.8)     KAzz ckAiii in ckAiii Cu

Since Az = Ai, the equivalent thermal conductivity in the z direction is

t K = PCB z + (10.9) ∑(/tckiii )in ∑ (/ tck iii )Cu

Again, if ci = 1 (100% coverage on each layer), we have

t K = PCB z + (10.10) tkin// in t Cu k Cu

For ci = 1, Equations 10.5 and 10.9 are also reduced to those presented in Guenin [5].

Example 10.1

1 A PCB with thickness 0.1 in consists of 12 layers of /2-oz copper and polyimide. Determine the equivalent thermal conductivities Kxy and Kz.

Solution Assuming each layer has 100% coverage,

0. 00135 t =×12 =0.t... 0081 in and =−01 0 0081 = 0 0919 in Cu 2 in

From Table 10.1, we have

kCu = 386 W/(m·°C) = 9.66 W/(in·°C)

and

kin = 0.52 W/(m·°C) = 0.0132 W/(in·°C)

From Equation 10.6,

 0. 0132  966.  K = 0. 0919  + 0. 0081  =⋅0. 794 W/(in° C) xy  01.   01. 

and from Equation 10.10,

01. K = =⋅°0. 0144 W/(in C) z 0./../. 0919 0 0132+ 0 00819 66

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FIGURE 10.8 Dimensions of board and card guides for Example 10.2.

Comments: 1. The present example is relatively simple because each layer has 100% coverage. 2. Units such as W/(in·°C) for thermal conductivity and W/(in2·°C) for heat transfer coefficient are often used in the electronics industry for convenience because the heat dissipation is typically expressed in watts while drawing dimensions are frequently given in inches. 3. For printed circuit board analysis, one can consider the present PCB a single, homogeneous layer with equivalent thermal conductivities Kxy and Kz.

Example 10.2 The board of Example 10.1 is clamped into a chassis similar to that shown in Fig- ure 10.6. The board dimensions and the length of the card guide (or retainer) are given in Figure 10.8. Determine the temperature rise from the center of the board to the chassis if 12 W of heat is uniformly distributed over the board.

Solution Because of the symmetry, the analysis can be performed on half the configuration:

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q ==12 6 W 2 82/ L ==2 in 2 A..=×60106 = in2 L 2 R == =°42. C/W pcb × KAxy 0 . 794 0 . 6

The temperature rise from the center to the edge of the board is

∆T1 = qRPCB = 6 × 4.2 = 25.2°C

Assume the thermal resistance of the card guide is 2.5°C·in/W. (This is a very typical value.) Then

25. R ==°05. C/W guide 5

The temperature rise from the board to the chassis is

∆T2 = qRguide = 6 × 0.5 = 3°C

Therefore, the total temperature rise is

∆T = ∆T1 + ∆T2 = 25.2 + 3 = 28.2°C

Comments: 1. The actual thermal resistance of the card guide depends on the construction of the card guide, and the value can be obtained from the manufacturer’s data sheet. 2. The suggested card guide thermal resistance from the manufacturer should be applied only to the region directly covered by the card guide because the contact pressure will decrease in the region outside the range (length) of the card guide.

REFERENCES

1. C. Maxfield and D. Kuk, “A Multichip Module Primer,” Printed Circuit Des., June 1992. 2. F. J. Dance, “Can PWB’s Meet the Surface Mount Challenge?” Part 1, Electron- ics, May 1983. 3. F. J. Dance, “Can PWB’s Meet the Surface Mount Challenge?” Part 2, Electron- ics, June 1983. 4. A. J. Laderman, D. B. Osborn, R. M. Grabow, and M. C. Bury, “Parametric Study of Conduction Cooling for High Power Density Electronics,” Proc. Int. Symp. Cooling Technology for Electronic Equipment, Honolulu, March 17–21, 1987.

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5. B. M. Guenin, “Conduction Heat Transfer in a Printed Circuit Board,” Elec- tronic Cooling Mag. 4(2), 1998. 6. R. J. Hannemann, “Physical Technology and the Air Cooling and Interconnec- tion Limits for Mini- and Microcomputers,” Chapter 1 in A. Bar-Cohen and A. D. Kraus, eds., Advances in Thermal Modeling of Electronic Components and Systems, Vol. 2, ASME Press, New York, 1990. 7. J. W. Dally, Packaging of Electronic Systems: A Mechanical Engineering Approach, McGraw-Hill, New York, 1990.

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DIRECT AIR COOLING AND FANS

Simplicity and easy maintenance make direct air cooling a most attractive approach in cooling of electronics. Direct air cooling can take an active form, represented by forced convection; or a passive form, as free or natural convection. Passive cooling has always been the preferred choice. However, it is generally limited to systems with very low heat dissipation because of the poor heat transfer coefficients being characteristic of free convection. Forced-air cooling over printed circuit boards (PCBs) is often encountered in electronic equipment. Variations in component size and spacing, cause flow separation over components and recirculation between components. Analysis of the flow field is a fully three-dimensional (3D) problem and extremely complex. Heat transfer involves convection from components and the PCB to the airstream, conduction from components to the PCB, and heat spreading along the PCB, and also radiation exchange among components and from components to surroundings. Theoretically, the only possible way to analyze components over a PCB is by means of a 3D computational fluid dynamics (CFD) program. The thermal model with this approach may become very large, because of the large number of components involved in a typical board. The number of components on a PCB can be over one hundred. A simple analysis approach, utilizing a general-purpose thermal analyzer, is sometimes used to analyze the entire board. The most difficult task in using such a program is estimating the heat transfer coefficients over the components and the board, which are required as input, because of the complicated flow field.

11.1 PREVIOUS WORK

Heat transfer involving flow over solid objects has a great engineering application. Because of its importance, many efforts have been made to examine this type of problem. Most experimental studies idealize the IC components as rectangular blocks as shown in Figure 11.1. In the simplest case [1], the board is fully populated with components in an array with all of equal height t and equal spacing s, whereas in the more general case, the components are of varying heights and there are some “missing” modules [1–4]. The basic findings from References 1 through 4 can be summarized as follows:

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FIGURE 11.1 Array of rectangular modules in a channel.

1. Within an array of equal-height components, fully developed flow occurs at about the fifth row. 2. The heat transfer coefficient at neighboring modules is increased when there is a missing module in the array. The greatest enhancement of about 40% takes place on the module located immediately downstream of the missing module. The effect gradually decreases for the further downstream modules. 3. The enhancement of heat transfer coefficient due to the missing module is greater for flow at low Reynolds numbers, because the flow at a low Reynolds number penetrates the cavity created by the missing module while a high–Reynolds number flow tends to skim over the cavity.

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4. A missing module located at the side and downstream of the monitored module generates only a modest enhancement. 5. The application of a barrier in an array increases the heat transfer coefficient. The greatest improvement, which is about a factor of 2, occurs at the second row downstream of the barrier. 6. The heat transfer coefficient at a monitored module is increased when the monitored module is either taller or shorter than other modules in the array. Enhancement at a taller module is caused by direct impingement of the airflow on the portions of the module protruding above other modules. On the other hand, the improvement at the shorter module results from the recirculation zone in the cavity created by the gap in height in the array. The enhancement (up to 80%) for a taller module increases as the module height increases. The enhancement for a shorter module is less pronounced (up to 26%) and is increased as the Reynolds number increases. 7. The largest enhancement (up to 50%) of the heat transfer coefficient due to a taller module over the array occurs on the modules by the side of the taller module, and the second best location is just downstream of the taller module.8. 8. The entry region for a regular array with equal-height modules is generally characterized by higher heat transfer coefficients than is the developed-flow region. In an array containing a tall module, on the other hand, the heat transfer coefficient becomes relatively higher downstream as the tall module is moved along streamwise locations. 9. Any enhancement of the heat transfer coefficient also increases the pressure drop.

The preceding discussions provide an in-depth understanding of the physical phenomena associated with direct air–cooled components on a PCB. Further- more, this information can be used as a guideline for placing components on a board so as to improve thermal performance.

11.2 HEAT TRANSFER CORRELATIONS

For an array of equal-height modules, the Nusselt number for fully developed flow can be represented as follows [1]:

= 072. NuLH0 . 0935 Re (11.1)

where

hL Nu = L k ρVH Re = H µ H = height of flow passage (above module)

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h = heat transfer coefficient L = side dimension of square module V = mean velocity based on flow cross section above module k = thermal conductivity ρ = density µ = viscosity

Another correlation can be expressed as follows [4]:

 H  −0543.. L 148 Nu = 0. 075 Re0.653    (11.2) tt t   t 

where

ht Nu = t k ρVt Re = t µ t = module height

Equation 11.1 was obtained from experiments with the following parameters: 3 1 t/L = /8, s/L = /4, and (H + t)/L = 1. On the other hand, the module and array configurations for Equation 11.2 are (H + t)/L = 10/20 and 5/30, s/L = 8/20 and 12/30, and H/t = 5/10 and 12.8/5. The heat transfer coefficient h computed from Equa- tion 11.1 increases when the module length L decreases, while the opposite trends are found from Equation 11.2 if other parameters remain constant. ReH (= ρVH/µ) defined in Equation 11.1 can be rewritten as mw/µ, where m is the mass flow rate and w is the spanwise width. The Reynolds number becomes independent of the gap height H. Therefore, the Nusselt number or the heat transfer coefficient in Equation 11.1 is also independent of the gap height. For a given flow rate and channel width w, the velocity of the airflow increases when the gap height decreases. The heat transfer coefficient should also increase as the velocity increases. In other words, the heat transfer coefficient depends indirectly on the gap height for a given flow rate; however, the effect due to the gap height cannot be found in the present form of Equation 11.1. A large number of thermal tests were conducted by the Cranfield Institute of Research in England on ceramic dual in-line packages (DIPs) soldered onto PCBs in a variety of array configurations. Based on the test data, the following correlations [5] were obtained for air velocities ranging from 0 to 8 m/s:

• For PCBs fitted into card guides,

 b 62.()[()]ρ V08.. d/b −1013 h.= 53  + (11.3)  d ()nd 036.

where

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b = streamwise dimension of component, m d = streamwise component pitch, m n = streamwise component location (numbered from PCB leading edge) V = mean velocity based on minimum free flow area between PCBs, m/s

The units for the heat transfer coefficient and air density are W/(m2·°C) and kg/m3, respectively. • Without card guides at the PCB leading edges, the heat transfer coefficients are as follows:

 b 76.()[()]ρ V08.. d/b −1013 h.= 53  + (11.4)  d ()nd 02.

For convenience, Equation 11.3 can also be rewritten in terms of English units and it becomes [6]:

 b 0. 225()[()]ρ V08.. d/b −1 013 h.= 093  + (11.5)  d ()nd 036.

where the units of b and d are ft and V is in ft/min. The product ρV is often referred to as the mass velocity G. The units for the density and the heat transfer 3 2 coefficient are lbm/ft and Btu/(hr·ft ·°F), respectively. If only the high–velocity range data are considered, Equation 11.3 can be replaced by the following dimensionless equation:

 d 013. Nu=−019. Re07.  1 Pr13/ (11.6)  b 

The characteristic length used to define the Nusselt and Reynolds numbers is the distance from the PCB leading edge, x. Pr is the Prandtl number. Equation 11.6 yields values of the heat transfer coefficient typically twice those obtained from the usual relationship for turbulent flow over a flat plate. For plated through-hole (PTH) mounting technology, electronic components are typically mounted on one side of a PCB with the leads going through the holes and extending to the opposite (back) side of the PCB for soldering. There- fore, some of the heat dissipated from the components is conducted through the leads to the back side of the PCB and is convected to the airstream over the back side. This situation is especially true when there are a large number of copper circuit runs on the back side of the PCB to spread heat. In this case, the following heat transfer correlations for the parallel plates with uniform but different heat flux on each wall [7, 8] may be used:

• For fully developed laminar flow, use

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5. 385 Nu = i − ″ (11.7) 1 0.(/) 346 qqji″

• For fully developed turbulent flow, use

0.Re 0334 0. 755 Nu = (11.8) i − −01. ″ ″ 1049.Re(/)q jiq

where the quantities q″i and q″j are the heat fluxes on walls i and j, respectively. The Nusselt and Reynolds numbers are based on the hydraulic diameter. For parallel plates where the plate width is much larger than the gap, the hydraulic diameter is equal to twice the distance between the plates. If components are packaged close together, a new surface can be considered to form over the top of the components; therefore, the hydraulic diameter is equal to 2H.

11.3 PRESSURE DROP CORRELATIONS

Some pressure drop data are provided in references 1 and 3 for regular arrays with and without a tall module as well as with or without a barrier. For the case without a tall module or barrier, the correlations for parallel plates at fully turbulent flow under-predict the pressure drop over the array by just over 10%. In this case, the spacing between two plates is the distance H from the top of the modules to the upper plate, as defined following Equation 11.1, and the hydraulic diameter used to compute the friction factor is 2H. On the basis of experiments on ceramic DIPs, Wills [5] presented an empirical correlation for the pressure drop in the form

 0.L 033  ∆PV=+ρ 2 19.K∑  (11.9)  sV13.. 03  where

∆P = pressure drop, N/m2 K = loss coefficients L = PCB streamwise length, m s = PCB pitch, m V= mean velocity based on minimum free flow area, m/s ρ = air density, kg/m3

The loss factor for a card guide or turbulator is approximately as follows:

 y  2 K =   (11.10)  sy− 

where y is the height of the protuberance (e.g., card guide width).

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Equation 11.9 predicts a much higher pressure drop (about 4 times) than those obtained from the equations for parallel plates or the results of references 1 and 3 for the cases of an array with equal-height modules. This is because Equation 11.9 is obtained from tests with actual DIPs on the PCB. The leads of DIPs may cause additional turbulence and disturbance in the flow.

Example 11.1 DIPs, size 0.3 in × 1.2 in, are mounted on a PCB as shown in Figure 11.2 aligned in 17 columns pitched at 0.35 in and 5 rows pitched at 1.25 in. The DIPs protrude 0.2 in from the PCB. The dimensions of the PCB are 6 in × 6in× 0.1 in, and the spacing between PCBs is 0.8 in. Air entering the PCBs at 120°F with a mass flow rate of 51 lbm/h flows normal to the 1.2-in side of the DIPs. Determine the maximum and minimum heat transfer coefficients.

Solution Equation 11.5 will be used in the following calculations. The free flow area is defined as follows:

A = (0.8 – 0.1) × 6 – 5(0.2 × 1.2) = 3 in2

The mass flow velocity is:

m 51 ρVG== = =2448 lb /(ft2 ⋅ hr) =40. 8 lb /(ft2 ⋅ min) A/3 144 m m

From the dimensions in Figure 11.2,

 d 013.. 035. 013 b 03.  −1 =− 1 == 0. 792 and =085.  b   03.. d 035

From Equation 11.5, the heat transfer coefficient is

 b 0. 225()[()]ρ V08.. d/b −1 013 h.= 093  +  d ()nd 036.

The maximum heat transfer coefficient occurs at the first column (n = 1):

××08. =× +0. 225( 4 0.8) 0. 792 h.1 0 93 0. 857 (./)1× 0 35 12 036. =⋅13. 164 Btu/(hr ft2 ⋅° F)

The minimum heat transfer coefficient occurs at the 17th column (n = 17):

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FIGURE 11.2 Airflow over the PCB of Example 11.1.

××08. =× +0. 225( 4 0.8) 0. 792 h.17 0 93 0. 857 (./)17× 0 35 12 036. =⋅⋅526. Btu/(hr ft2 ° F)

If the nondimensional Equation 11.6 is used, the maximum and minimum heat transfer coefficients occur at x1 and x17, respectively. At the center of the first column, or x1 = 0.3/2 = 0.15 in = 0.0125 ft,

Gx 2448× 0. 0125 (Re ) ==1 = 637. 5 x 1 µ 0. 048

hx  d 013. ==Nu019 . Re0.7 − 1 Pr13/ k  b  =×0.... 19 91 862 × 0 792 × 0 892 = 12 . 33

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Figure 11.3 Local heat transfer coefficient over the DIPs of Exam- ple 11.1.

0016. h.=×12 33 =15. 78 Btu/(hr ⋅⋅° ft2 F) 1 0. 0125

where Pr = 0.715 and k = 0.016 Btu/(hr·ft2·°F). At the center of the 17th column, or x17 = 16 × 0.35 + 0.3/2 = 5.75 in = 0.479 ft,

2448× 0. 479 (Re ) = = 24437. 5 x 17 0. 048 Nu=×0 . 19 1179 . 342 × 0 . 792 × 0 . 892 = 158 . 3 0. 016 h.=×158 3 =528. Btu/(hr ⋅⋅° ft2 F) 17 0. 479

The results clearly indicate that Equation 11.5 and Equation 11.6 are in excellent agreement for the region with high Reynolds numbers. The local heat transfer coefficients based on Equation 11.5 are plotted in Figure 11.3. It appears that fully developed flow would begin at about the 20th column and the heat transfer coefficient at this condition would be 5 Btu/(hr·ft2·°F). It is of interest to compare the present results with those obtained from the correlation for parallel plates. Since the components are so closely packaged, a

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new surface is formed at the top of the components. The new spacing between plates becomes

H = 0.8 – 0.1 – 0.2 = 0.5 in

The free flow area is

A = (0.8 – 0.1 – 0.2) × 6 = 3 in2

and the mass velocity is

m 51 G == =2448 lb /(ft2 ⋅ hr) A/3 144 m

The hydraulic diameter for parallel plates is twice the distance between the plates, or Dh = 2 × 0.5 = 1 in. Thus,

GD 2448 × 1 Re ==h 12 =>⇒4250 2300 turbulent flow µ 0048.

For fully developed turbulent flow, Equation 11.8 can be applied. Assuming that only the bottom wall is heated,

Nu = 0.0334 Re0.755 = 0.0334(4250)0.755 = 18.331

and

0. 016 h.=×=18 331 352. Btu/(hr ⋅⋅° ft2 F) 112/

The results (h = 3.52 versus h = 5) clearly demonstrate that using the parallel plate assumption is too conservative.

11.4 HEAT TRANSFER ENHANCEMENT

To enhance thermal performance, individual small heat sinks are frequently added to hot components to maintain the component temperature at the desired limit. Three typical heat sinks used with PGAs are shown in Figure 11.4. Of these, the straight-finned heat sinks are also often applied to DIPs. The enhancement gained from most heat sinks is primarily due to the increased surface area for heat transfer. One way to increase the heat transfer area of a heat sink is to increase its number of fins. This action, however, reduces the spacing between fins and increases the flow resistance. The net result for a given fan or pump is to reduce the available flow through the heat sink, in turn decreasing the heat transfer coefficient.

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FIGURE 11.4 Diagrams of heat sinks applied to PGAs: (a) pin-fin, (b) disk-fin, and (c) straight-fin heat sinks.

Just as with flow over a solid, a boundary layer will develop in the space between fins. A general rule is that the fin spacing must be large enough to prevent boundary-layer interference at the lowest expected velocity. In other words, the space between fins must be greater than twice the boundary-layer thickness at the exit of the heat sink to avoid excessive flow resistance. The boundary-layer thickness over a flat plate is as follows:

δ = 5x 05. Rex (11.11)  µx 05. = 5   ρV 

where ρ and µ are density and viscosity of the fluid, V is the free-stream velocity, and x is the distance from the leading edge. To define the minimum fin spacing, x is set equal to the length of the heat sink in the flow direction. Figure 11.5 [9] illustrates the effect of the fin spacing on the flow pattern for heat sinks. The flow will be pushed away from the heat sink if boundary-layer interference occurs as shown in Figure 11.5c. A third surface, the fin base, must be accounted for in the analysis of any heat sink, complicating the examination of boundary-layer interference. Hannemann et al. [9] experimentally determined the critical space between fins as follows:

 µL 05. s = 341.   (11.12)  ρV 

where L is the length of the heat sink in the flow direction, and where s and L are in meters, V is in m/s, µ is in kg·m/s, and ρ is in kg/m3. For a given heat sink with

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FIGURE 11.5 Flow patterns in a heat sink: (a) free plates, s >> (L); (b)developing flow, s ≈ (L); (c) interfering, s << (L).

fixed dimensions, Equation 11.12 can be used to determine the critical velocity Vc required to avoid boundary-layer interference. The mean heat transfer coefficient experimentally determined over the total heat sink area is

05.  V  Af h = 437.  for ≥ 085 . (11.13)  L  A

2 where the units for h are W/(m ·°C), V is in m/s, and L is meters. Af /A is the ratio of the fin area to the total area of a heat sink. The above correlations

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are based on tests conducted for disk and straight fins with the following configurations:

Number of fins: 2–38 Fin length L: 1.52–8.38 cm Fin spacing s: 0.089–0.330 cm Velocity Vc: 0.5–11.3 m/s

A reduction in fin spacing decreases the flow through the heat sink, resulting in a decrease in the heat transfer coefficient. On the other hand, the number of fins on the heat sink is increased, and so is the total heat transfer surface. In optimizing thermal design, one should never focus on the heat transfer coefficient alone. The most important factor to consider is the product of the heat transfer coefficient, the heat transfer area, and the fin efficiency. In addition to the fin spacing, several factors such as the tip and side (lateral) clearances of a heat sink have a direct impact on the available flow through the heat sink. This bypass phenomenon of airflow over a heat sink is one of the most important factors affecting thermal performance. The larger the clearance, the less flow is available to the heat sink. This result follows because the fluid always flows through the least resistant path. The result of airflow bypass is to reduce the heat transfer coefficient. The subject of flow bypass has been discussed in detail [10–12]. The thermal resistance of a heat sink is defined as follows:

TT− θ = wai, (11.14) q

where q is the component heat dissipation, Tw is the average heat sink base temperature, and Ta, i is the bulk fluid temperature at the inlet of the heat sink. Equa- tion 11.14 can be rewritten in the following form:

θθ=+ θ ch − − TTao,, aiTT w ao , = + (11.15) q q =+11 η (11.16) mcpo hA

where Ta, o is the bulk fluid temperature at the exit of the heat sink, m is the mass flow rate and cp specific heat, ηο is the overall surface efficiency, h is the heat transfer coefficient, and A is the total heat transfer area.

11.5 FANS AND AIR-HANDLING SYSTEMS

The fan is the heart of any forced-air system. The fan propeller converts torque from the motor to increase the static pressure across the rotor of the fan and to energize the air particles. A fan is frequently employed in electronic equipment to

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provide the airflow required for cooling. Direct air cooling is a primary example of using a fan for thermal control of electronic systems.

11.5.1 Fan Types

Fans are generally classified into two types, namely, centrifugal (blower) and axial fans. The properties of the two categories are as follows:

1. Centrifugal fans • Receive air at their axis of rotation and exhaust air at their periphery in a direction normal to the rotation axis • Produce pressure from two sources: (1) centrifugal force created by the rotating air column, and (2) kinetic energy imparted to air as it leaves the impeller • Create a low flow rate against a high resistance (pressure) 2. Axial fans • Deliver air in a direction parallel to the fan blade axis • Produce pressure from the change in velocity passing through the impeller • Provide a high airflow rate but tend to work against low pressure

The axial fan can further be divided into three categories as follows:

1. Propeller fans are the simplest type and are designed to move large vol- umes of air at low velocity and develop low static pressure. 2. Tube axial fans are similar to the propeller fan but provide a higher total pressure capability than propeller fans. 3. Vane axial fans are the same as the tube axial fan but have vanes to straighten the swirling flow created as the air is accelerated, resulting in improved efficiency and noise characteristics.

Basically, tube axial and vane axial fans fill the gap for performance between centrifugal and propeller fans.

11.5.2 Fan Performance and System Resistance

The fan performance or characteristic curve as given in Figure 11.6 is generated through a series of tests. The fan is tested from shutoff conditions to nearly free delivery conditions. At the shutoff point, the duct connected to the fan outlet is completely blocked. On the other hand, the outlet resistance of the duct is reduced to zero at the free delivery point. Additional test points between these two limiting conditions are obtained when various flow restrictions are placed at the end of the duct to simulate various fan operating conditions. Fans designed to be used with a duct are generally tested with a length of duct between the fan and the measurement station. The purpose of this length of duct is to smooth out the flow as it exits the fan and to provide stable and uniform

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FIGURE 11.6 Typical fan curves and system operation point.

flow conditions at the measurement station. The measured pressures are then corrected back to the fan outlet conditions. Fans designed to be used without a duct are tested without the duct. As air moves through a system, the pressure is decreased because of pressure losses due to expansions, contractions, bends, other structure blockages, and friction. For a system under consideration, the pressure drop is a function of the flow rate; therefore, a system resistance curve can be generated. The intersection of the fan performance and system resistance curves is the operation point of the system, as illustrated in Figure 11.6 Figure 11.6 also shows the static efficiency η (= Qp/hp), which is defined as the ratio of the static-pressure power to the input power (in horsepower) of a fan. The maximum static efficiency identifies the ideal combination of the flow rate and pressure of the fan. Fans should be operated as close to this point as possible if they are to achieve optimum results as given in the figure. The fan laws relate the performance variables for any dynamically similar series of fans. The variables involved are fan size D, rotational speed N, gas density ρ, volumetric flow rate Q, pressure p, power (hp), and efficiency η. The relationships among the variables are as follows:

Q = ϕND3 (11.17a)

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p = ψρN2D2 (11.17b) hp = λρN3D5 (11.17c)

where ϕ, ψ, and λ are the constants for geometrically and dynamically similar operation and are also referred to as the flow coefficient, pressure coefficient, and power coefficient, respectively. The specific speed Ns is often used to determine the type of fan that is best suited for a given application, one with the highest efficiency. The specific speed is a dimensionless parameter created by combining the flow and pressure coefficients to eliminate the characteristic dimension D:

ϕ a N = s ψ b (11.18) b −− p = QNaba223  D ba  ρ 

In order to eliminate D,

2b – 3a = 0 (11.19)

Also, for simplicity and convenience of calculation, let N be raised to the first power, i.e.,

2b – a = 1 (11.20)

Combining Equations 11.19 and 11.20, we have

05. = NQ Ns (11.21) (/)p ρ 075.

where N is angular speed (rpm), Q is in cfm (ft3/min), and (p/ρ) is in inches of water. These are the units most commonly used by major fan manufacturers. Fig- ure 11.7 gives the range of specific speeds for each basic type of fan. Unless otherwise identified, fan performance data are based on dry air at stan- 3 dard conditions of 14.696 psia and 70°F with a density of 0.075 lbm/ft . Since most fans operate at a constant speed and provide a constant volumetric flow rate, one must be cautious about the operation of a fan at high altitudes or reduced density conditions. The electronic equipment may be overheated because the mass flow rate (m = ρQ) is reduced at the above-mentioned conditions. The effect of density (or altitude) on fan performance is illustrated in Fig- ure 11.8. Because of the reduction in air density at high altitudes, both fan performance and the system resistance curves are lowered. Sometimes, several fans are used simultaneously in a system. For two fans in series (push and pull), there is usually one fan at the inlet and another at the exit. The new fan characteristic curve for two fans in series is constructed by adding the values of pressure of individual fans at a given flow rate. Adding the flow rates

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FIGURE 11.7 Range of specific speed for various types of fan.

FIGURE 11.8 Effect of altitude or air density on system operation point.

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FIGURE 11.9 Performance of fans in series or parallel.

of individual fans at a given pressure yields the new characteristic curve for two fans in parallel. As compared with a single fan, two fans in series always increase the flow rate. In theory, however, it may not necessarily be true that two fans in parallel increase the flow rate. Figure 11.9 illustrates the performance of two fans in action. Thermally, the location of a fan in a system is also an important factor to consider. The fan located at the exit of a system is usually referred to as an exhaust fan; it draws air through the system. This type of installation reduces the internal system pressure (to less than the ambient pressure) and possibly draws dust or dirt into the system. The advantage of this method is that no heat from the fan motor is added to the system, and therefore there is no increase in the inlet air temperature. On the other hand, a blowing fan, which is located at the inlet of the system, creates a higher internal system pressure and keeps dust or dirt out of the system; however, the inlet air is heated by the heat dissipation from the fan motor, and thus the air temperature is increased prior to entering the system. An exhaust fan is generally less reliable because it operates at a higher air temperature.

11.5.3 Air-Delivery Systems

The system resistance is the sum of various pressure drops in the air-delivery system. The basic equations for the pressure drop as described in Chapter 3 are still applicable. In addition to the friction loss, the system pressure drop is caused by other losses such as inlet, exit, change of cross section (expansion and

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contraction), turns, and elbows. The pressure loss in the above items is typically expressed by the velocity head in the following form:

kVρ 2 ∆p = (11.22) 2gc

where k is the loss coefficient and can often be found in the Applied Fluid Dynam- 2 ics Handbook [13] and where gc is 32.2 lbm·ft/(lbf·s ). Frequently, the pressure drop in an airflow system is expressed in inches of water. For air at standard conditions, Equation 11.22 can be simplified as follows:

 V  2 ∆pk=   (11.23)  4005

where ∆p is in inches of water and V is in ft/min. The loss coefficients for various physical configurations are given in Table 11.1 [13]. A sharp bend in the tube is generally said to exist where the ratio of the radius of curvature to tube diameter is less than 2 and the bend angle is greater than 45°. A low-pressure region is created as illustrated in Figure 11.10 for turbulent flow over a sharp bend in the tube because of the flow separation from the inner wall. The flow separation and the low-pressure region along with the associated contraction of the axial flow beyond the bend are responsible for the high pressure loss in a sharp bend. Rounding the inner corner can significantly reduce the pressure drop in the bend by attenuating the separation zone, but the improvement is limited by rounding the outer corner of a bend. The existing test data suggest that the pressure loss for turbulent flow in a sharp bend is relatively independent of Reynolds numbers if the Reynolds number is greater than 2 × 105. For smaller Reynolds numbers, the pressure loss varies approximately as (Re)a, where a is between –0.17 and –0.2. Vanes are often installed in sharp bends as shown in Figure 11.11, to reduce pressure loss and provide more uniform flow at the exit from the bends. For sharp bends of 45° or greater, the reduction in pressure loss is very significant with vanes in place. For more gradual bends, however, the improvement is limited because the increase in skin friction and decrease in flow area due to the vanes offset the benefit of the improved flow patterns. Among the different types of vanes, thin concentric splitters (Figure 11.11a) are usually spaced so that all the vane channels have the same ratio of radius of curvature to duct width. This arrangement can be achieved if the radius of the individual splitters (rn) is chosen so that

+ r  R 11/(n ) n = i n = 012 ,, ,…  +  (11.24) rn+1  Rbi 

where n is the number of concentric splitters, r0 = Ri, rn +1 = Ri + b, and Ri and b are the inner radius of the bend and the duct width, respectively. Concentric

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Configuration Loss coefficient k Entrance flush with wall at right angle:

k = 0.5

Entrance flush with wall at arbitrary angle:

k = 0.5 + 0.3 cos θ + 0.2 cos2 θ

Round entrance in wall:

0. 0.0 0.0 0.0 R/D: 0 2 4 6 0.08 0.12 0.16 > 0.16 k: 0. 0.3 0.2 0.2 0.15 0.09 0.06 0.03 56 6 0

Round entrance in wall with neighboring wall: Values of k for various combinations of R/D and a/D: R/D a/D 0 0.1 0.1 0.200 0.30 0.5 0.800 10 25 50 0 0 0 0.2 — 0.8 0.4 0.190 0.09 0.05 0.050 00 50 0 0 0.3 — 0.5 0.3 0.170 0.07 0.04 0.040 00 20 0 0 0.5 0. 0.3 0.2 0.100 0.05 0.03 0.030 65 60 50 0 0 0

Abrupt contraction:

k = 0.5(1 – A2/A1); A2 < A1

Abrupt expansion:

2 k = (1 – A1/A2) ; A1 < A2

Exit:

k = 1.0

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FIGURE 11.10 Turbulent flow pattern over a sharp bend.

splitters are generally made of thin sheet metal with the thickness less than 1/40 of the inner bend radius. In the case of a sharp turn, short vanes such as circular arc and airfoil vanes (Figure 11.11b and c) can be used. The core length of the vane from the leading to the trailing edge is chosen so that

θ cR= 2 sin i 2 (11.25)

1/2 where θ is the bend angle. For example, c = 2 Ri for θ = 90°.

FIGURE 11.11 Flow in sharp bends with various types of vanes installed: (a) concentric splitters; (b) circular vanes; (c) airfoil vanes.

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FIGURE 11.12 Vanes in a diffuser.

For uniform spacing between short vanes, the recommended maximum number of vanes is

 b  nm=   −1 (11.26)  Ri 

where the value of m is between 2.1 and 2.5. Increasing the number of vanes beyond this recommended value may result in an increase in pressure loss because of the increased skin friction resistance. Flow separation can also occur in a diffuser. To minimize the pressure loss in the diffuser, the expansion angle between the diffuser walls should be limited to about 12° (2θ < 12°), as shown in Figure 11.12. Vanes should be applied to reduce the pressure drop in short, wide-angle diffusers when the expansion angles are greater than 12°. The design method is described as follows [13]:

1. The number of vanes is chosen so that the included angle of passage between vanes is between 7 and 10°, that is, β = 2θ/(n + 1) = 7 to 10°, where n is the number of vanes. The included angle of the wall passage, βw, is usually slightly larger than β.

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2. Equal spacing is used between the vanes so that a = w1/(n + 1), where a is the spacing between leading edges of the vanes and is measured perpendicular to the diffuser axis. 3. The leading edge of the vanes is set back slightly from the beginning of the diffuser divergence as 1 < b/a ≤ 1.2 or 0.1 < c/w1 ≤ 0.15. 4. The length of the vane can be shorter than the full length of the diffuser. Vanes with 0.25 < Lv/L < 0.6 have yielded very good performance (Lv and L are the length of the vane and diffuser, respectively). 5. The vanes are straight. The leading edges of the vanes are rounded and the trailing edges are tapered in at least a 5 to 1 taper to minimize the exit losses. 6. The thickness of the vanes is generally between 0.06w1 and 0.1w1. The thicker vanes require careful contouring of their leading and trailing edges.

Further improvements can also be made for two-dimensional diffusers having curved leading edges, as shown in Figure 11.13 [13]. The radius r is chosen so that

3w r = i (11.27a) 4θ *

where θ* is measured in radians. 2θ* is the sum of the diffuser included angle and a correction angle which depends only on the diffuser area ratio:

2θ* = 2θ + 2∆θ (11.27b)

where ∆θ in degrees is listed as follows for various values of A2/A1:

A2/A1: 3.0 3.5 4.0 4.5 5.0 ∆θ (degrees): 4.5 6.0 6.7 7.3 7.5

The number of vanes is chosen so that β = 2θ*/(n + 1) is between 7 and 10° and the spacing between vanes is a = w1/(n + 1). In a flow system, manifolds are frequently employed to provide proper flow rates to various branches. There are four types of manifolds, as shown in Figure 11.14. Two important factors involved in determining the pressure drop along the flow direction in a manifold are the inertia and friction terms as given in the following equation:

ρρdv()22fv () ds dP =− + (11.28) 2g D

where s is measured from the dead end and D is the manifold diameter. The differential (“d”) term can be replaced by a “∆” term in a numerical analysis. The sign of the first term that represents the change of velocity head is negative because an increase of pressure corresponds to a decrease of velocity. The

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FIGURE 11.13 Improvement on vanes in a diffuser.

friction loss represented by the second term of the equation, on the other hand, is always positive because of loss of fluid pressure along the length. Therefore, the relative magnitudes of the pressure regain due to the deceleration and the pressure loss due to friction determine whether the pressure rises or falls from the inlet end to the closed (or dead) end of the manifolds as illustrated in Figure 11.15. The branch flow rates are determined by the net pressure difference between the combined and divided manifold.

FIGURE 11.14 Four types of manifolds: (a) inlet manifold (divided manifold), (b) outlet manifold (combined manifold), (c) S-type manifold (parallel flow), and (d) U-type manifold (reverse flow).

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FIGURE 11.15 Pressure distribution in manifolds.

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No detailed analysis will be provided here. The reader should refer to Singhal et al. [14] for a detailed analysis. General features and characteristics of manifolds are described as follows:

1. The ratio of total lateral (branch) cross-sectional area to header (manifold) cross-sectional area should be less than 1 (as small as possible) in order to obtain a better uniform flow distribution in branches. 2. Large lateral (branch) resistance is desirable for uniform flow distribution in branches. 3. The effect of friction in the manifold may be neglected for small manifold length-to-diameter ratio. 4. The fully developed friction factor may be used if the spacing between laterals (branches) is large; however, the friction factor for developing flow should be considered when the spacing between the laterals is small. 5. Uniform flow distribution in the laterals is attained when the manifolds are large enough to act as infinite reservoirs. 6. A reverse (U-type) flow system has a better flow distribution than a parallel (S-type) flow system if other parameters remain the same. The total pressure in the former is also less than that in the latter.

Orifice plates of different sizes can be installed in the branches to regulate individual flows in order to achieve any desired flow distribution to each branch. In addition, the diameter or cross section of a manifold can also be varied to obtain the proper flow pattern.

REFERENCES

1. E. M. Sparrow, J. E. Niethammer, and A. Chaboki, “Heat Transfer and Pres- sure Drop Characteristics of Arrays of Rectangular Modules Encountered in Electronic Equipment,” Int. J. Heat and Mass Transfer 25(7), 1982. 2. E. M. Sparrow, A. A. Yanezmoreno, and D. R. Ottis, Jr., “Convective Heat Transfer Response to Height Differences in an Array of Blocks-Like Elec- tronic Components,” Int. J. Heat and Mass Transfer 27(3), 1984. 3. P. R. Souza Mendes and W. F. N. Santos, “Heat Transfer and Pressure Drop Experiments in Air-Cooled Electronic Component Arrays,” J. Thermophysics 1(4), 1987. 4. Y. P Gan, Q. J. Peng, C. F. Ma, Z. Y. Zhou, and D. Y. Cai, “Forced Convective Air Cooled from Electronic Component Arrays in a Parallel Channel,” in R. K Shah and A. Hashemi, eds., Aerospace Heat Exchanger Technology, Elsevier Science Publishers, Amsterdam, 1993. 5. M. Wills, “Thermal Analysis of Air-Cooled PCBs” (4 parts), Electronic Produc- tion, May–August, 1983. 6. F. E. Altoz, “Thermal Management,” Chapter 2 in C. A. Harper, ed., Electronic Packaging and Interconnection Handbook, McGraw-Hill, New York, 1991. 7. W. M. Kays and M. E. Crawford, Convective Heat and Mass Transfer, 2nd ed., McGraw-Hill, New York, 1980.

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8. R. J. Moffat and A. Ortega, “Direct Air-Cooling of Electronic Components,” in A. Bar-Cohen and A. D. Kraus, eds., Advances in Thermal Modeling of Elec- tronic Components and Systems, Vol. 1, Hemisphere, New York, 1988. 9. R. Hannemann, L. R. Fox, and M. Mahalingham, “Heat Transfer in Micro- electronic Components,” keynote paper, Proc. Int. Symp. Cooling Technol. Electronic Equipment, Honolulu, March 17–21, 1987. 10. K. S. Lau and R. P. Mahajan, “Effects of Tip Clearance and Fin Density on the Performance of Heat Sinks for VLSI Packages,” IEEE Trans. Components, Hybrids, and Manufacturing Technol. 12(4), 1989. 11. R. S. Lee, H. C. Huang, and W. Y. Chen, “A Thermal Characteristic Study of Extruded Type Heat Sinks in Considering Air Flow Bypass Phenomenon,” Proc. 6th Ann. IEEE Semiconductor Thermal and Temperature Measurement Symp., 1990. 12. C. L. Chapman, S. Lee, and B. L. Schmidt, “Thermal Performance of an Ellip- tical Pin Fin Heat Sink,” Proc. 10th Ann. IEEE Semiconductor Thermal and Temperature Measurement Symp., 1994. 13. R. D. Blevins, Applied Fluid Dynamics Handbook, Van Nostrand Reinhold, New York, 1984. 14. A. K. Singhal, L. W. Keeton, A. K. Majundar, T. Mukerjee, and R. S. Johnson, “An Improved Mathematical Formulation for the Computations of Flow Dis- tributions in Manifolds for Compact Heat Exchangers,” Adv. Heat Exchanger Des., ASME HTD 66, 1986.

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NATURAL AND MIXED CONVECTION

The basic theory of natural convection was discussed in Chapter 3. This chapter will focus on natural convection for special configurations such as in parallel plates, U-channels, pin fins, and enclosures. In addition, the problems of mixed convection in parallel plates will also be discussed here. For laminar flow with a moderate temperature gradient, the following simple correlation for natural convection with air over a vertical plate can be used:

= 025. Nu0.59 RaH (12.1a)

where

hH Nu = Nusselt number, k = RaH Rayleigh number, GrPr c µ Pr= Prandtl number, p k βρ23gH∆ T Gr ==Grashof number, BH3 ∆ T µ2 H = plate height ∆T = temperature difference between wall and ambient B = parameter from Table C.1 of Appendix C

This equation is often used as a base to develop other general correlations for more complicated configurations. For horizontal plates, the following equations can be applied:

Heated plate facing upward: =<025. 5 <×7 Nu054. RaL 10RaL 4 10 (laminar) (12.1b) =×13/ 7< Nu0162. RaLL 4 10 Ra (turbulent) (12.1c)

Heated plate facing downward: =×025. 5 <<×10 Nu0. 27 RaL 3 10RaL 3 10 (12.1d)

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FIGURE 12.1 Flow between parallel plates.

For a horizontal plate, the characteristic length L for a square plate is taken as the length of the side of the square; for a finite rectangular plate, as the arithmetic mean of the lengths of two sides; and for a circular disk, as 0.9 × disk diameter. For other shaped plates, L is defined as A/P, where A is the surface area and P is the perimeter.

12.1 PARALLEL PLATES Flow between parallel plates as illustrated in Figure 12.1 is frequently considered in heat transfer. For fully developed laminar flow, the pressure losses due to friction and due to buoyancy can be expressed by the following equations, respectively:

 dp 12µm   =− (12.2a)   ρ 3 dx F bw

and

 dp   = ()ρρ′ − g   a dx B (12.2b) =−ρβ′′− gT()fa T

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where m is the mass flow rate, w is the width of plate, ρ′ the average fluid density, β the coefficient of thermal expansion, and T′f and Ta the average fluid temperature and the ambient temperature, respectively. The mass flow rate m can be determined by relating Equations 12.2a and b as follows:

23 m ρβ′′gb() T− T = fa (12.3) w 12µ

For uniform heating on two plates, the local fluid temperature is found from:

dTf 2qw′′ = (12.4a) dx mcp

′′ =+2qxw TTfa (12.4b) mcp

where cp is the specific heat of the fluid and q″ is the constant heat flux. The average fluid temperature can be obtained by integration over the height of the plate and leads to

′′ ′ =+qH TfaT (12.5) mcp

Combining Equations 12.3 and 12.5, one obtains

12/ m  ρβ′′′23gbqH =   (12.6) µ w  12 cp 

Natural convection between vertical parallel plates was first studied experimentally by Elenbaas [1] in 1942, and the correlation for isothermal plates is as follows:

1  b  − Nu=   Ra[1− e 35Hb/( Ra)] 3 / 4 (12.7) 24  H 

where the Nusselt and Grashof numbers are based on the plate spacing b. All properties are evaluated at the wall temperature Tw, except that β is computed at the ambient (or free-stream) temperature Ta. Later, Sparrow and Bahrami [2] performed similar tests but extended the study to the cases with blockage along either one or both of the two edges. Comparisons between Elenbaas’ and

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Sparrow and Bahrami’s results are given in Figure 12.2. The differences as explained by Sparrow and Bahrami [2] are attributed to the following factors:

1. Some doubts persist about the appropriate property values used by Elenbaas in developing the correlation because of a large temperature difference (up to 330°C) in Elenbaas’ tests. 2. Elenbaas seemed to overestimate the experimental losses at the lower values of (b/H)Ra*.

In addition, one should also note that the plate sizes used in the tests in the two sources are different. Elenbaas used a larger plate (4.72 in × 4.72 in) as compared with a plate 3 in × 3 in used by Sparrow and Bahrami. As should be expected, better heat transfer occurs from the smaller plate. The results for blockage along the edges [2] are given in Figure 12.3. The results from Sparrow and Bahrami can be summarized as follows:

3. For (b/H)Ra* > 20, the results agree with Elenbaas’ data. 4. For (b/H)Ra* > 4, the results for edge blockage of one side are identical to those for the unblocked case. 5. For (b/H)Ra* > 10, the heat transfer results are independent of whether the edges are fully open or blocked along one or two lateral edges.

The curve fits for the data from Sparrow and Bahrami [2], as indicated in Fig- ure 12.2a, are as follows:

024. yy035. < 10 (12.8a)  −− Nu = 2.() 04ye0.// 135 1−≤ 35y 3 4 10 y≤ 100 (12.8b)  − 1 ()1−>ey35//y 3 4 100 (12.8c)  24

where y = (b/H)(Ra). Examining Figure 12.2b reveals that the heat transfer coefficient increases as the temperature difference Tw – Ta increases. For large plate spacing, data from both sources [1, 2] approach the results obtained from Equation 12.1. For vertical parallel-plate channels, the heat transfer rate from each plate decreases when the plate spacing is reduced. For a given envelope space, however, the number of plates, or total heat transfer surface area, is increased as the plate spacing decreases. Therefore, achieving the optimum plate spacing—the spacing at which the product of the heat transfer coefficient and the total heat transfer surface area is maximum—may maximize the total heat transfer rate. Elenbaas found that the optimum spacing between two vertical plates for laminar natural convection is

b GrPr ≈ 46 (12.9) H

This equation represents the case where optimum spacing refers to the spacing that permits the greatest convection heat transfer rate for a given plate temperature.

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FIGURE 12.2 Natural convection between vertical parallel plates: (a) nondimensional form; (b) dimensional form.

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FIGURE 12.3 Effects of edge blockage on heat transfer in parallel plates.

On the other hand, the spacing which minimizes the plate temperature for a given convection heat transfer rate can be determined from the following equation [3]:

b GrPr ≈ 600 (12.10) H

Here, optimum spacing means the spacing that allows the maximum heat transfer coefficient from each plate. Basically, achieving this optimization means separating the plates so that no boundary-layer interference occurs between plates. If the spacing between two plates is large enough as compared with the plate height, the pair can be treated as an isolated plate because the boundary layers associated with the individual plates do not interfere with each other. On the other hand, for very long plates, or small plate spacing, the boundary layers merge together near the entrance, and fully developed flow prevails along much of the channel. Based on these two limiting solutions, Bar-Cohen and Rohsenow [4] developed an analytical solution for each type of thermal boundary condition. A summary of their results is given in Table 12.1 Sparrow and Prakash [5, 6] proposed methods to improve heat transfer from vertical continuous-plate arrays by using discrete plates as shown in Figure 12.4.

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Table 12.1 Heat Transfer Correlations for Parallel Plates

Optimum Condition Nu Spacing, bo Optimum (Nu)o Isothermal plates  −05. 576 2. 873 –0.25 Symmetric Nu =  +  bo = 2.714P (Nu)o = 1.31 (Ra′ )205 (Ra′ ) . 

 −05. 144 2. 873 –0.25 Asymmetric Nu =  +  bo = 2.154P (Nu)o = 1.04 (Ra′ )205 (Ra′ ) . 

Isoflux plates  12 188. −05. = + –0.2 Symmetric (Nu)H / 2   bo = 1.472R (Nu)o, H/2 = 0.62 Ra′′ (Ra′ )04. 

 −05. = 6188+ . –0.2 Asymmetric (Nu)H / 2   bo = 1.169R (Nu)o, H/2 = 0.49 Ra′′ (Ra′ )04. 

where ρβ23∆Τ qb′′ cgHp Nu = Ra = − µ kT()wa T k ρβ2 ∆ ρβ24∆Τ cgT cgbp P = p Ra′ = Hkµ Hkµ ρβ25′′ ρβ25′′ cgbq cgbqp R = p Ra′′ = Hkµ Hkµ 2

Replacing continuous plates with discrete plates breaks up the boundary layer, resulting in an increased heat transfer rate; however, this enhancement technique will also increase the pressure drop and pumping power. For the case of forced convection, the heat transfer coefficient will always increase as long as the fan or pump can overcome the increased system pressure. The pumping power (or driving force) in natural convection is limited by the system height and the difference in densities, which is limited by the temperature gradient. An increase in the system pressure drop caused by heat transfer enhancement will reduce the mass flow rate. Under certain conditions, the heat transfer from discrete plates will actually be less than that for continuous plates because of a significant reduction in mass flow due to an increased system pressure drop. Comparisons between staggered discrete and continuous vertical plates are given in Figure 12.5a and b for heat transfer rate and inlet velocity, respectively. Sparrow and Prakash [5, 6] made three types of performance comparisons in their work. The first case was to maximize the heat transfer rate at fixed values of the temperature difference and surface area. The second case was to minimize the temperature difference for a fixed heat transfer rate and surface area. The final case was to minimize the array height for a fixed heat load and temperature difference. The results are presented in Figures 12.6 through 12.8.

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FIGURE 12.4 Continuous and discrete vertical plates: (a) continuous- plate array; (b) in-line discrete-plate array; (c) staggered discrete- plate array.

12.2 STRAIGHT-FIN ARRAYS

The straight-fin array as shown in Figure 12.9 consists of a number of U-shaped channels. The correlations for vertical parallel plates cannot be directly applied to a U-shaped channel because of the base involved. Aibara [7] indicated that U- shaped channels may be approximated by parallel plates within an error of 5% in the heat transfer rate only if the ratio of channel depth to channel width (L/S) is greater than 5. For many heat sinks in practical applications, however, the ratio L/S is often less than 5. In these cases, the base plate of the heat sink creates

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FIGURE 12.5 Comparisons between staggered discrete and continuous parallel plates: (a) ratio of heat transfer rates. Subscripts: *, continuous parallel-plate channel; s, staggered discrete plates.

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FIGURE 12.6 Comparisons of heat transfer rates at fixed ∆T and area: (a) qIL/q*; (b) qIL/qs. Subscripts: IL, in-line discrete plates; *, continuous parallel-plate channel; s, staggered discrete plates.

additional surface, where a third boundary layer is developed, and two corners that cause a significant reduction in the heat transfer. For vertical-fin arrays, several experimental data [8–10] are available. Among them, Izume and Nakamura [10] developed a mathematical relationship describing heat transfer from fin arrays; however, their equation does not hold

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FIGURE 12.7 Comparison of temperature difference at fixed q and area: (a) TIL/T*; (b) TIL/Ts. Subscripts: IL, in-line discrete plates; *, continuous parallel-plate channel; s, staggered discrete plates.

in the limiting cases of very large or very small ratios of channel depth to channel width. To overcome this problem, Van De Pol and Tierney [11] developed the following correlation applicable to any channel depth to width ratio:

   Ra*   05. 34/  Nu =−−1 exp ψ    (12.11) r ψ   *     Ra 

where

hr Nu = r k

FIGURE 12.8 Array height ratios vs. ˆq at fixed q and ∆T: (a) HIL/Hs; (b) HIL/H*. Subscripts: IL, in-line discrete plates; s, staggered discrete plates; *, continuous parallel-plate channel. Natural and Mixed Convection • 225

FIGURE 12.9 A vertical straight-fin array.

2LS r = 2LS+ r Ra* = Gr Pr H r  ρ  2 Gr= gTTrβ  (− ) 3 rw µ  a − −017./a ψ = 24(. 1 0 483e ) − {(1++− 0 . 5ae )[ 1 ( 1083..aV )( 9 . 14 ae 05 S− 0. 61 )]}3 S a = L − V =−11. 8 in 1

When r approaches S as L/S →∞, the Nusselt number in Equation 12.11 is reduced to the Elenbaas equation (12.1) for parallel plates. On the other hand, when the ratio L/S approaches zero, the resulting heat transfer coefficient (Nus- selt number) is reduced to Equation 12.1 for a single vertical plate. All physical properties are evaluated at the wall temperature, with the exception of β, which is computed at the fluid temperature.

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As in any natural convection, thermal radiation always plays an important role. Rea and West [12] proposed a method to estimate thermal radiation from finned heat sinks (U-channels). The thermal radiation from one U-channel can be expressed as

=−εσ 44 qSHTTra() wa (12.12)

where εa is the U-channel apparent emittance, σ the Stefan-Boltzmann constant, and S and H sink dimensions in Figure 12.9. The U-channel apparent emittance is a function of heat sink dimensions and surface emittance ε of the heat sink, and this apparent emittance can be greater than 1. Typical results for the U-channel emittance are given in Figure 12.10. Because the efficiency of a fin is generally less than 100%, the temperature of the fin at the tip is different from that at the base. Therefore, it is recommended that the average temperature be used to calculate the heat transfer rate from the heat sink to the ambient. If the base plate of a straight-fin array is oriented horizontally as shown in Fig- ure 12.11, the heat transfer coefficient for natural convection can be correlated [13] as follows:

−  GrPr Sn057.. H  0656 S 0412 . Nu=×522 . 10 3      (12.13a)  H   L   L

for

GrPr Sn 1067<<×2. 5 10 H

and

−  GrPr Sn0... 745 H  0 656 S 0 412 Nu=×2 . 787 10 3      (12.13b)  H   L   L

for

GrPr Sn 2. 5×< 1078 <×1. 5 10 H

where n is the number of spacings, H is the height, and 2L is the length of the fin array. Also, Nu and Gr are based on L (half-length of the fin array). All thermal properties are evaluated at the wall temperature.

Example 12.1 Determine the total heat loss from a vertical straight-fin array as shown in Figure 12.8. The temperatures of the heat sink and ambient are Tw and Ta,

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FIGURE 12.10 U-channel apparent emittance a for different values of .

respectively. It is assumed that the fin efficiency is 100% and the total number of fins, including two end fins, is n.

Solution In theory, there are four different heat transfer coefficients for natural convection:

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FIGURE 12.11 Horizontal fin array configuration.

hu: heat transfer coefficient from a U-channel, from Equation 12.11 hv: heat transfer coefficient from a vertical plate, from Equation 12.1a ht: heat transfer coefficient from a horizontal plate with heated surface facing up, from Equation 12.1b or 12.1c hb: heat transfer coefficient from a horizontal plate with heated surface facing down, from Equation 12.1d

Heat loss by natural convection is

=− + + + + qcu[( n12 )( LH HS ) h2 ( L B ) Hhv nHthv ++++ − Bw( htb h ) ntL ( h tb h )]( T wa T )

The components of this heat loss, represented by the terms inside the brackets, are the respective heat losses from (n – 1) U-channels, from the external surfaces of the two end fins, from n fin tips, and from the upward- and downward-facing surfaces of n base-plate thicknesses. Similarly, heat loss by radiation is

=−εεε + + + qr [( n12 ) HSa ( L B ) H nHt ++εεσ44 − 22Bw ntL]( Twa T )

where εa and ε are apparent emissivity (from Figure 12.10) and surface emissivity of the fin array. The absolute temperature must be applied to radiation heat transfer.

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In most practical design analyses, the heat losses from the tips, leading edges, and trailing edges of the fins might be ignored because the fin thickness t is generally small. The total heat loss from the array is the sum of the heat losses from natural convection and radiation. The dimension B as shown in Figure 12.9 is usually equal to zero because the base plate of the fin array is an integrated part of the chassis wall.

12.3 PIN-FIN ARRAYS

Pin-fin arrays that consist of many solid pins on a common base plate are frequently used to cool electronic equipment. The cross-section of individual fins can be any shape. Because of the difficulty of separating the radiation loss from the total heat loss of a pin-fin array, the heat loss is typically expressed by the combined natural convection and radiation. Sparrow and Vemuri [14, 15] performed experimental studies on a small pin-fin array (3 in × 3 in) with circular pins. Later Alessio and Kaminski [16] correlated all existing data, including their own results, into the following equation:

(aD− )cosθθ13.  H cos  −037. Nu= 1 . 035 (Ra )024.     (12.14) dd D   D 

where

a = fin pitch da= cosθ D = fin diameter H = fin length or height θ = angle of inclination from the horizontal hd Nu = d k gcTdβρ23∆ Ra = p d µk q h = AT∆

In the equation defining h, A is the total surface area of the array, including the sides of the fins, the tips of the fins, and the exposed area of the base plate. The surface emissivity is about 0.82 for the above test data. Since most of the data were based on the tests performed on a 3-in square base plate, the ratio of the edge fins to the total number of fins is very high. Because the edge fins have higher heat transfer rates as compared with inner fins, and also because they comprise a significant portion of the total number of fins, the results from Equation 12.14 will overestimate the heat transfer coefficient for larger pin-fin arrays. Conversely, smaller pin-fin arrays should perform better, and the results presented here would be underestimates.

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FIGURE 12.12 Heat transfer coefficients for pin-fin arrays: (a) arrays with vertical base plate; (b) arrays facing up; (c) arrays facing down.

It is difficult to correct the results from Equation 12.14 for various sizes of the array, because of a lack of available data. The only data available are the results of Taylor [17] for a large array with a base plate of 10 in by 20 in as shown in Fig- ure 12.12a. Based on these limited data, the following correction to Equation 12.14 is suggested:

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FIGURE 12.12 (Continued).

 R 08. hh′ =   (12.15)  04. 

where h is computed from Equation 12.14 and R is the ratio of edge fins to the total number of fins in an array under consideration. Fins used for comparison from Taylor [17] are small cones in a staggered arrangement with a fin pitch of 0.5 in. The fin major and minor diameters are

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FIGURE 12.12 (Continued).

0.25 and 0.12 in, respectively. The height of the fin is 1 in. The results for the pin- fin array facing up or down are also included in the same reference, and they are shown in Figure 12.12b and c, respectively. A pin-fin array is often made by the casting method. One should consult with the casting vendor about fin design limits. However, the following are basic design guidelines for casting pin fins with the configurations given in Figure 12.13:

Pin fins should be tapered at 0.25 inches per foot. The major diameter of a pin fin should not exceed its supporting wall thickness. The dimensional limits are St/D > 2, Sl/D > 2, and H/D ≤ 7. The minimum diameter D is 0.06 inches. H/D is limited to 2.5 when D is 0.06 inches.

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FIGURE 12.13 Pin-fin configurations.

Example 12.2 A pin fin array consists of 449 horizontal fins arrange in an equilateral configuration with 0.5 inch pitch. Determine the heat transfer coefficient of a pin-fin array with the following dimensions:

HDa===065... in 025 in 05 in

and

θ = 0 (horizontal fins) R = 0.25 (ratio of number of edge fins to total fins)

Solution Assume the ambient and the base-plate temperatures are 75 and 115°F, respectively. Then all properties are calculated at the average temperature of 95°F [= (75 + 115)/2]: ρ = 3 0. 0725 lbm /ft − β =°0. 001822 F1 µ =⋅ 0. 04595 lbm /(ft hr) k =⋅0. 0153 Btu/(hr ft⋅° F) =⋅° cp 024. Btu/(lbm F) d = 05. in ∆T =°40 F

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Finally, from Equation (12.14) =⇒= Radd39313..Nu 5 298

further implying

h=⋅⋅1. 945 Btu/(hr ft2 ° F)

and

 025. 08. h′ = 1. 945  =⋅⋅1. 335 Btu/(hr ft2 ° F)  04. 

12.4 ENCLOSURES

The following discussion deals with natural convection heat transfer between concentrically located isothermal spherical, cylindrical, and cubical inner bodies and their spherical and cubical enclosures [18, 19]. Typically, three possible heat transfer regimes can occur within the enclosure: (1) convection without the effect of enclosure—infinite-atmosphere solution for large gap (L/Ri); (2) convection with the effect of enclosure—enclosure solution for moderate gap (L/Ri); and (3) pure conduction—conduction solution for small gap (L/Ri). The following steps are used to determine the range of each regime so that a proper correlation can be applied:

1. The hypothetical radii for inner and outer bodies for spheres, cylinders, and cubes are defined as the corresponding radii of a sphere having the same volume as the body in question. a. Sphere:

== RDRDiioo05. 05 .

where Di and Do are actual diameters of inner and outer bodies, respectively. b. Cylinder:

13//13  3DH2   3DH2  =  ii =  oo Ri   Ro    16   16 

where H is the height of the cylinder. c. Cube:

13//13  3H3   3H3  =  i  =  o  Ri   Ro    4ππ  4 

where H is the side length of the cube.

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d. Rectangular block:

 3xyz13// 3xyz13 R =  iii R =  ooo i  4ππ o  4 

where x, y, and z are the sides of the rectangular block. 2. Gap width: L = Ro – Ri. 3. Boundary-layer length on the inner body:

a. Sphere: b = 0.5πDi. b. Cylinder: b = Hi + Di(0.5π – 1). c. Cube: b = 2Hi. 1/3 d. Rectangular block: b = 2Yi where Yi = (xiyizi) .   ==* L 4. Rabb Gr Pr and Ra b Ra b   Ri 

where b is used as the characteristic length for Grb and Rab. 5. The lower limit for an infinite region can be obtained by setting the solutions equal for regimes 1 (infinite region with no enclosure effect) and 2 (region with enclosure effect).  L  Sphere :   = 0. 978Ra0. 0593 a.   b Ri 1

 −025. 1/. 0 236 +  L  []DHii/(0 . 571 D )  Cylinder :   =   Ra0. 0593 b.   b Ri 1  0. 585   

 L  Cube :   = 126Ra. 0. 0593 c.   b Ri 1

6. Checking limits: a. If L/Ri > (L/Ri)1 (convection over a single inner body—no enclosure effect), then

hD 025. Nu ==i 052. Ra (12.16a) D k D

where Di is the inner diameter for spheres and cylinders, and the side length for cubes. b. If L/Ri < (L/Ri)1 (convection in enclosures—with enclosure effect), then

hb *.0 236 Nu ==0. 585(Ra ) (12.16b) bbk

c. If L/Ri < 0.45 (conduction may be important), then

qconv = *.0 239 02. (RaL ) (12.17) qcond

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FIGURE 12.14 Dimensionless shape factors for conduction.

where qconv and qcond are the heat transfer rates due to convection and conduction, respectively, and

 L3  L  Ra***=   Ra or Ra= Ra LbLL    b  Ri 

For very small values of L/Ri, heat transfer could be primarily due to conduction rather than convection. It is recommended that one use whichever predicts the higher value—convection or conduction—for heat transfer. For heat conduction in three-dimensional shells, Warrington et al. [20] provide graphical shape factors between two surfaces, defined as heat transfer divided by thermal conductivity and by temperature difference. Figure 12.14 charts for various configurations the nondimensional shape factor S* defined as follows:

S S* = (12.18) Si

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where Si = 2πDiDo/(Do – Di) and where Di and Do are equivalent diameters of the outer and inner surfaces, respectively. The equivalent diameter of a cube is its edge length, and the equivalent diameter of a cylinder or sphere is equal to its diameter. The heat transfer rate for conduction becomes as follows:

q = kS ∆T (12.19)

12.5 MIXED CONVECTION IN VERTICAL PLATES

In theory, natural and forced convection always occur together; however, in some cases, one may neglect either component depending on the flow condition. The following guideline can be applied to determine the nature of the problem:

Natural convection alone: Gr >> Re2 Forced convection alone: Gr << Re2 Mixed convection: Gr ≈ Re2

For mixed convection, the experimental data have revealed the existence of flow reversal in the fully developed region at a low Reynolds number, i.e., low forced-flow velocity. Mixed convection can take the form of either buoyancy- assisted (or aided) flow or buoyancy-resisted (or opposed) flow.

12.5.1 Buoyancy-Assisted Flow

Buoyancy-assisted flow refers to forced flow in the direction of buoyancy as shown in Figure 12.15. Cheng et al. [21] presented the solutions for three fundamental combinations of thermal boundary conditions on the respective wall surfaces, namely, isoflux-isoflux, isoflux-isothermal, and isothermal-isothermal. The flow reversal regions are given in graphical form in Figure 12.16. The expression of the Nusselt number for the isoflux-isoflux case is somewhat complicated, but the equations for the other two cases are relatively simple. For the isoflux-isothermal boundary condition,

720(Re/Gr) Nu = (12.20a) 1 − 360(Re/Gr) Q1

and

720(Re/Gr) Nu = 2 + (12.20b) 360(Re/Gr) Q1

where Q1 = q1b[k(T2 – T0)] and T0 is the ambient temperature. Equations 12.20a and b are valid only for cases where Q1 ≠ 0. For Q1 = 0, the Nusselt numbers at wall 1 and wall 2 become

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use FIGURE 12.15 Flow reversal of buoyancy-assisted flow in vertical plates.

FIGURE 12.16 Reversed-flow zone in buoyancy-assisted flow.

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Table 12.2 Nusselt Numbers for Isoflux-Isoflux

rH = q2/q1

0 0.4 0.5 0.6 1.0

Re/Gr Nu1 Nu2 Nu1 Nu2 Nu1 Nu2 Nu1 Nu2 Nu1 Nu2 0.001 5.312 0 5.319 9.026 5.403 7.371 5.506 6.695 6.081 6.081 0.01 3.176 0 3.534 13.984 3.644 7.746 3.764 6.034 4.353 4.353 0.1 2.745 0 3.172 19.078 3.300 8.611 3.440 6.315 4.141 4.141 1.0 2.698 0 3.130 19.903 3.260 8.736 3.402 6.359 4.120 4.120 ∞* 2.692 0 3.125 20.000 3.256 8.750 3.398 6.364 4.118 4.118

*Approach to pure forced convection.

Nu1 = 0 (12.21a)

and

Nu2 = 2.43 (12.21b)

For the isothermal-isothermal boundary condition with T1 ≠ T2,

720(Re/Gr) Nu = (12.22a) 1 +− 360(Re/Gr) rT 1

and

720(Re/Gr) Nu = 2 −+ (12.22b) 360(Re/Gr) rT 1

where rT = (T2 – T0)/(T1 – T0). For symmetric heating (T1 = T2),

Nu1 = Nu2 = 3.77 (12.23)

Nusselt numbers for the isoflux-isoflux case are given in Table 12.2. It should be noted that the Nusselt and Reynolds numbers are based on the spacing between two plates, which is one-half of the hydraulic diameter. Figure 12.16 reveals that there are two possible patterns of reversed-flow velocity profile for the isoflux-isoflux boundary condition. For the other two cases, only the single-peak pattern with a negative-velocity area adjacent to the colder wall could be found.

12.5.2 Buoyancy-Resisted Flow

Buoyancy-resisted (or opposed) flow refers to forced flow in the direction opposite to the buoyancy—e.g., the flow resulting when an upflow is cooled or a

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FIGURE 12.17 Nusselt numbers and flow reversal for isoflux-isoflux.

downflow is heated. Hamadah and Wirtz [22] obtained closed-form solutions for the fully developed flow with the following boundary conditions:

1. Both walls isothermal, with the right wall (Y = 1) at T2 and the left wall (Y = 0) at T1 where T2 > T1 2. Both walls isoflux, with the right wall at (q2)″ and the left wall at (q1)″ where (q2)″>(q1)″ 3. Isoflux-isothermal, with the right wall at (q2)″ and the left wall at T1 = T0 (the fluid inlet temperature)

For case 2, the flow reversal and the Nusselt number are given in Figure 12.17. On the other hand, simple equations are available for the other two cases:

Case 1: Isothermal-isothermal

4 Nu = R < 1 1 −− T (12.24a) 1Gr( 1RT )/( 1440 Re) 4 Nu = R < 1 2 +− T (12.24b) 1Gr( 1RT )/( 1440 Re)

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and the flow reversal occurs for

Gr > 288 − (12.25) Re 1 RT

where the Nusselt, Reynolds, and Grashof numbers are based on the hydraulic diameter (= 2b) and RT is defined as (T1 – T0)/(T2 – T0).

Case 3: Isoflux-isothermal

4 Nu = (12.26a) 1 1− Gr/( 2880 Re) 4 Nu = (12.26b) 2 1+ Gr/( 2880 Re)

and the flow reversal occurs for

Gr > 576 (12.27) Re

It should be noted that the heat transfer coefficients or Nusselt numbers in Equations 12.20 through 12.26 for both buoyancy-assisted and buoyancy- opposed flow are reduced to the values for fully developed forced convection if Gr/Re is zero.

REFERENCES

1. W. Elenbaas, “Heat Dissipation of Parallel Plates by Convection,” Physica 9(1), 1942. 2. E. M. Sparrow and P. A. Bahrami, “Experiments on Natural Convection from Vertical Parallel Plates with Either Open or Closed Edges,” J. Heat Transfer 102, 1980. 3. E. K. Levy, “Optimum Plate Spacing for Laminar Natural Convection from Vertical Isothermal Flat Plates,” J. Heat Transfer 93, 1971. 4. A. Bar-Cohen and W. M. Rohsenow, “Thermally Optimum Spacing of Vertical, Natural Convection Cooled Parallel Plates,” J. Heat Transfer 106, 1984. 5. E. M. Sparrow and C. Prakash, “Enhancement of Natural Convection Heat Transfer by a Staggered Array of Discrete Vertical Plates,” J. Heat Transfer 102, 1980. 6. E. M. Sparrow and C. Prakash, “Natural Convection Heat Transfer Perfor- mance Evaluation for Discrete- (In-Line or Staggered) and Continuous-Plate Arrays, Numerical Heat Transfer 3, 1980. 7. T. Aibara, “Natural Convective Heat Transfer in Vertical Parallel Fins of Rec- tangular Profiles,” Jap. Soc. Mech. Eng. 34, 1968.

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8. K. E. Starner and H. N. McManus, “An Experimental Investigation of Free Convection Heat Transfer from Rectangular Fin Arrays,” J. Heat Transfer 85, 1963. 9. J. R. Welling and C. B. Wooldridge, “Free Convection Heat Transfer Coeffi- cients from Rectangular Vertical Fins,” J. Heat Transfer 87, 1965. 10. K. Izume and H. Nakamura, “Heat Transfer by Convection on the Heated Surface with Parallel Fins,” Jap. Soc. Mech. Eng. 34, 1968. 11. D. W. Van De Pol and J. K. Tierney, “Free Convection Nusselt Number for Ver- tical U-Shaped Channels,” J. Heat Transfer 95, 1973. 12. S. N. Rea and S. E. West, “Thermal Radiation from Finned Heat Sinks,” IEEE Trans. Parts, Hybrids, and Packaging PHP-12(2), 1976. 13. F. Harahap and H. N. McManus Jr., “Natural Convection Heat Transfer from Horizontal Rectangular Fin Arrays,” J. Heat Transfer 89, 1967. 14. E. M. Sparrow and S. B. Vemuri, “Natural Convection/Radiation Heat Trans- fer from Highly Populated Pin Fin Arrays,” J. Heat Transfer 107, 1985. 15. E. M. Sparrow and S. B. Vemuri, “Orientation Effects on Natural Convec- tion/Radiation Heat Transfer from Pin Fin Arrays,” Int. J. Heat and Mass Transfer 29, 1986. 16. M. E. Alessio and D. A. Kaminski, “Natural Convection and Radiation Heat Transfer from an Array of Inclined Pin Fins,” J. Heat Transfer 111, 1989. 17. M. Taylor, “Experimental Comparison of Pin Fin Configurations for Extended Surface Heat Transfer in Space Applications,” Proc. 4th Int. Elec- tronics Packaging Conf., 1984. 18. R. O. Warrington, S. Smith, R. E. Powe, and R. L. Mussulman, “Boundary Effects on Natural Convection Heat Transfer for Cylinders and Cubes,” Heat Transfer in Enclosures, ASME HTD-39, 1984. 19. R. O. Warrington and R. E. Powe, “The Transfer of Heat by Natural Convec- tion between Bodies and Their Enclosures,” Int. J. Heat and Mass Transfer 28, 1985. 20. R. O. Warrington, R. E. Powe, and R. L. Mussulman, “Steady Conduction in Three Dimensional Shells,” J. Heat Transfer 104, 1982. 21. C. H. Cheng, H. S. Kou, and W. H. Huang, “Flow Reversal and Heat Transfer of Fully Developed Mixed Convection in Vertical Channels,” J. Thermophysics 4(3), 1990. 22. T. T. Hamadah and R. A. Wirtz, “Analysis of Laminar Fully Developed Mixed Convection in a Vertical Channel with Opposing Buoyancy,” J. Heat Transfer 113, 1991.

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HEAT EXCHANGERS AND COLD PLATES

A heat exchanger is a device that is used to transfer thermal energy from one fluid to another. The process involves two fluids. On the other hand, a cold plate involves only one fluid that is used to cool (or heat) a system. The general analysis and design principles for a cold plate are identical to those for a heat exchanger. Therefore, all discussion in this chapter will be confined to heat exchangers. Heat exchangers are normally used to cool the circulating fluid in electronic equipment.

13.1 COMPACT HEAT EXCHANGERS

The term compact heat exchanger (CHE) does not necessarily imply small bulk and weight but instead refers to a heat exchanger with a high ratio of heat transfer surface area to volume. Typically, the ratio of heat transfer surface area to volume (or area density) for a compact heat exchanger is greater than 213 ft2/ft3 (or 700 m2/m3). As compared with shell-and-tube heat exchangers, the distinguishing characteristics of compact heat exchangers are as follows:

1. They are very flexible, and it is easy to apply different area densities on the hot and cold sides separately as desired. 2. Fluids must be clean and relatively noncorrosive because of small flow passages used to achieve high surface area density, and one of the fluids is typically a gas. 3. Operating pressure and temperature are limited because of construction features of brazing joints. 4. The fluid pumping power (i.e., pressure drop) is often of equal importance to the heat transfer rate. 5. The header design of a compact heat exchanger is much more difficult and is also important for uniform flow distribution.

Compact heat exchanger design generally employs one of three fluid technologies: (1) gas to gas, (2) gas to liquid, and (3) gas to condensing or evapo- rating fluid.

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The reader should refer to Kays and London [1] for an in-depth discussion and additional information about compact heat exchangers. This source also contains much experimental data on heat transfer and friction factor for various types of fin.

13.2 FLOW ARRANGEMENT OF HEAT EXCHANGERS

Generally, a heat exchanger is configured in one of the following flow arrangements:

1. Parallel-flow heat exchanger. Both fluids flow in the same direction. The final temperature of the colder fluid can never reach the exit temperature of the hot fluid. 2. Counterflow heat exchanger. Two fluids flow in opposite directions. The exit temperature of the colder fluid may exceed the outlet temperature of the hot fluid. 3. Crossflow heat exchanger. The flow directions are normal to each other. This is the most common flow arrangement in compact heat exchangers.

In any heat exchanger, the fluids in the two streams (hot and cold streams) can be either fully mixed or unmixed.

13.3 OVERALL HEAT TRANSFER COEFFICIENT

When heat is transferred from one fluid to another through the separating wall of a heat exchanger, heat transfer occurs through conduction across the wall thickness and convection from both solid surfaces to fluids. The heat transfer rate can be expressed by the following equation:

q = UηoA(Th – Tc) (13.1)

where U is the overall heat transfer coefficient; A is the heat transfer surface area; Th and Tc are temperatures for hot and cold fluids, respectively; and ηo is the surface efficiency, defined as follows:

A η =−11f () −η (13.2) o A f

where A is the total surface area and Af and ηf are the surface area and efficiency of the fins under consideration. For a rectangular fin, ηf is approximately equal to (tanh ml)/ml, where the parameter m is (2h/kδ)0.5 and δ and l are fin thickness and height, respectively. (See Chapter 8 for details.) Various fin configurations result in different surface areas and efficiencies for the hot and cold sides of the heat exchanger; therefore, the overall heat transfer coefficient can be written based either on the hot side or on the cold side as follows. For the hot side,

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Table 13.1 Fouling Factors for Some Fluids

Fluids F, hr·ft2·°F/Btu Seawater below 125°F 0.0005 Seawater above 125°F 0.001 Treated boiler feedwater above 125°F 0.001 Fuel oil 0.005 Quenching oil 0.004 Alcohol 0.0005 Steam 0.0005 Industrial air 0.002 Refrigerating liquid 0.001

1 U = h 11a (13.3a) ++ +F ηη oh,,h hAA w/(/) h oc AAh c h c

and for the cold side,

1 U = c 11a (13.3b) ++ +F ηη oc,,h cAA w/(/) c oh AAh h c h

where a is the thickness of the plate separating the hot and cold fluids; the subscripts h and c represent the hot and cold fluids, respectively; Aw, Ah, and Ac are the heat transfer areas of the separating plate for conduction and of the finned surface areas for convection on the hot and cold sides of the heat exchanger, respectively; and F is the fouling factor (or resistance), defined as

=−11 F (13.4) UUdirty clean

The fouling factor represents the degree to which the heat transfer surfaces are corroded or coated with various deposits after some period of operation. The deposits on the surface create an additional thermal resistance. The average values of the fouling factors for some fluids are listed in Table 13.1 [2].

13.4 HEAT EXCHANGER EFFECTIVENESS

Heat exchanger effectiveness is defined as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate under the second law of thermodynamics:

q ε = (13.5) qmax

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where

=−=− qmcT()()()ph hi,, T ho CTT h hi ,, ho (13.6a) =−=− ()()()mcpc T co,, T ci C c T co ,, T ci (13.6b)

Ch and Cc are heat capacities for the hot and cold sides, respectively. qmax occurs in a counterflow heat exchanger with an infinite heat transfer area. In other words, if there is no external loss, the outlet temperature of the cold fluid equals the inlet temperature of the hot fluid when Cc < Ch. On the other hand, the temperature at the exit of the hot fluid is equal to the inlet temperature of the cold fluid when Ch < Cc. Thus,

qmax = Cmin(Th, i – Tc, i) (13.6c)

where Cmin is the minimum value of Ch and Cc. Combining Equations 13.5 and 13.6, one obtains

CT()− T ε = hhiho,, (13.7a) − CTTmin()hi,, ci CT()− T = ccoci,, − (13.7b) CTTmin()hi,, ci

The effectiveness ε is typically expressed by tabular or graphical values as a function of the following parameters:

 C  εε= NTU, min , flow arrangement (13.8)  Cmax 

where NTU is the number of heat transfer units and is defined as AU/Cmin. As described previously, the flow arrangement in a heat exchanger may be parallel flow, counterflow, or crossflow.

13.5 HEAT EXCHANGER ANALYSIS

Thermal analysis of a heat exchanger can be performed by either using the logarithmic mean temperature difference (LMTD) method or employing the heat exchanger effectiveness (ε-NTU) method. The former requires that the inlet and exit conditions of both fluids be specified. This approach is widely employed in the design of heat exchangers to given specifications. On the other hand, in many cases, the performance of a heat exchanger (i.e., U) is known, or can at least be estimated, but the fluid exit temperatures are unknown. This type of problem is often encountered in the selection of a heat exchanger or when the heat exchanger is tested at one flow rate while service conditions require different flow rates.

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The detailed derivations of both methods are available in References 1 through 4 and will not be repeated here. The heat transfer rate of a heat exchanger found by the LMTD method is

q = UA ∆Tm (13.9)

where

∆∆TT− ∆T = ab m ∆∆ ln (TTab / )

The subscripts a and b refer to the respective ends of the heat exchanger. As indicated previously, the effectiveness depends on the flow arrangement. For a parallel-flow system, the effectiveness is

−+ 1− e NTU(1 R ) ε = (13.10) 1+ R

where R is the capacity ratio Cmin/Cmax. Figure 13.1 shows the effectiveness as a function of NTU and of the capacity ratio for both parallel-flow and counterflow systems. The most common and most effective flow arrangement for a heat exchanger is the counterflow system, and its effectiveness is

− −−NTU()1 R ε = 1 e −− (13.11) 1− Re NTU()1 R

Equation 13.11 can be reduced to two special cases where R is equal to 0 and 1, respectively. When an exchanger serves as a condenser or boiler, the fluid temperature on the side where the boiling or condensation takes place is nearly constant. This case implies that the flow rate in the condenser or boiler is very large as compared with the fluid on the other side of the heat exchanger. In other words, R approaches zero because Cmax >> Cmin. Under this condition, Equation 13.11 becomes

ε = 1 – e–NTU (13.12a)

On the other hand, Equation 13.11 is reduced to the following expression when R = 1:

NTU ε = (13.12b) + NTU1

This case frequently occurs in the regenerative process of the thermodynamics cycle where the same flow rate is used. From Equations 13.11 and 13.12a and b, the effectiveness of a counterflow heat exchanger reaches 1 when NTU approaches infinity.

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FIGURE 13.1 Effectiveness of parallel- and counterflow heat exchangers.

The effectiveness of a crossflow exchanger with the two fluids unmixed is available in References 4 and 5 and is expressed by the following equation:

 m  m  ∞ −−n ()NTU ×n ()R × NTU ∑ 11− e NTU ∑  − e R NTU ∑  n=0  m=0 m!  m=0 m!  (13.13) ε =    R × NTU

This equation reduces to Equation 13.12a for R = 0. The greatest difference in performance among various flow systems occurs when the heat capacity ratio is equal to 1, i.e., when Cmin/Cmax = 1. A comparison of various flow arrangements’ effectiveness under this condition is given in Figure 13.2.

13.6 HEAT TRANSFER AND PRESSURE DROP

The heat transfer coefficient is typically expressed by a Colburn-type correlation, i.e.,

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FIGURE 13.2 Effectiveness of various flow arrangements at Cmin/Cmax = 1.

St = C Rem Prn (13.14)

where C is a constant and the dimensionless variables are defined as follows:

h St ==Stanton number Gcp DG Re ==Reynolds number h µ c µ Pr ==Prandtl number p k

where Dh is the hydraulic diameter and G is the mass velocity. All properties should be evaluated at the average fluid temperature of each stream (hot and cold fluids, respectively). Data are provided in Kays and London [1] for various types of fins in the form of the Colburn factor (j = St Pr2/3) versus the Reynolds numbers. The values of the

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friction factor are included in the database. The same reference also presents the following equation to calculate the pressure drop across a heat exchanger:

    ρ  ρ A ρ  ()Kf+−121σ 2 + 1 − +−−−1 () 1σ 2 K 1  G c  ρ  ρ e ρ ∆p =  2 mcA 2  (13.15) 2g ρ c 1  entrance flow core exit   effect acceleration friction effect   

where

A = total heat transfer area on one side = Ac minimum free flow area on one side A = L Achr f = friction factor = Kc contraction loss coefficient = Ke expansion loss coefficient L = flow length r==hydraulic radius1 hydraulic diameter D hh4 ρ = m mean density between “1” (inlet) and “2” (exit) 1 ρρ+ =∫ρ dA ≈12 A 2 σ = ratio of free flow area to frontal area

Generally, in the design of liquid-to-liquid heat exchangers, accurate knowledge of friction characteristics of the heat transfer surface is relatively unimportant because of the low power requirement for pumping high-density liquids (the ratio of pressure drop to heat transfer is low). On the other hand, the low gas density causes the friction power per unit mass flow rate to be relatively large in the case of gases.

13.7 GEOMETRIC FACTORS

Each heat exchanger includes two sides (fluids). Referring to Figure 13.3, the following relationships between geometric factors can be established.

Height: H = n1(b1 + a) + n2(b2 + a) + a (13.16a) Volume: V = HWD (13.16b) Frontal areas: Afr, 1 = HW and Afr, 2 = HD (13.16c) Free flow areas: Ac, j = σjAfr, j j =1 or 2 (13.16d) b β Surface per unit volume: α = jj j = 12 or (13.16e) j ++ bb122 a

Heat transfer area: Aj = αjVj= 1 or 2 (13.16f)

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FIGURE 13.3 Sketch of a heat exchanger.

where

== njj number of layers on side j12 or a = separation plate thickness == bjj spacing between plates on side j12 or σ = j ratio of free flow area to frontal area = α jh,jr β = j ratio of total surface area on one side to volume between plates of that side

The values of b, Dh, β, Af/A, and δ (fin thickness) for several types of fin are presented in Reference 1.

13.8 COLD-PLATE ANALYSIS

A finned cold plate as shown in Figure 13.4 cools electronic components. In this case, the coolant can be gases or liquids, depending on the heat load and other requirements. The following is a step-by-step analysis approach:

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FIGURE 13.4 Electronics on a finned cold plate.

1. Coolant temperature rise ∆Tc:

∆ = qe Tc (13.17a) mcp

where

= qe total heat load m = mass flow rate (either known or assumed initially) = cp specific heat of coolant

2. Average coolant temperature (Tf)ave:

∆T ()TT=+c (13.17b) f ave in 2

All fluid properties are evaluated at this temperature.

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3. Convection temperature rise ∆Tf:

q ∆T = e f η (13.17c) hA o

The group of terms hAηo is often referred to as the cold-plate conductance. Each value is determined as described in the following paragraphs: a. Frontal area of the cold plate:

Afr = WH (13.18a)

where

W = width of cold plate H = height of cold plate

b. Free flow area:

Ac = Afr – Afin – Aw (13.18b)

where

Afin = NfHδ Nf = number of fins δ = fin thickness

Ac can also be determined from Equation 13.16d if the surface geometry of the fin is known (from Reference 1 for several types of fin stock). Aw is the cross-sectional area due to the wall thickness. This item is required when the wall thickness is included in Equa- tion 13.18a. c. Mass velocity:

m G = (13.18c) Ac

d. Hydraulic diameter:

Dh = 4rh

The hydraulic diameter varies with the type of fin. This value is frequently provided in Reference 1; however, the hydraulic diameter can also be determined as follows:

2sH D = (13.18d) h Hs+

where s is the spacing between fins.

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e. Reynolds number Re and Prandtl number Pr:

GD Re = h µ

and

µc Pr = p k

The Prandtl number can also be found from the material property table or chart. f. Heat transfer coefficient h. For a given Re, the value of the group 2/3 (h/Gcp)Pr can be found from Reference 1 or other sources for various types of fins; therefore, the heat transfer coefficient can be readily determined. g. Fin and surface efficiency ηf and ηo. Once the heat transfer coefficient is known, the fin efficiency ηf can easily be calculated according to the equations described in the previous sections or Chapter 8. In calculating the efficiency, the fin length l should be set equal to the fin height for one side heating and to half the fin height for both sides heating. The ratio of the fin area to the total heat transfer area (Af/A) is constant for each fin configuration and is given in Reference 1; however, both areas can also be calculated. In addition, the ratio of the total heat transfer area to the volume of cold plate (β) is also provided in the same reference. Therefore, A and ηo can be calculated immediately. Consequently, the cold-plate conductance hAηo is known. 4. Cold-plate temperature Tc. The exit coolant temperature is

Tout = Tin + ∆Tc (13.18e)

and the cold-plate temperature is

Tc = Tf + ∆Tf (13.18f)

Tf is the local fluid temperature and is equal to Tout at the exit. For the uniformly heating case, the maximum cold-plate temperature occurs at the exit of the cold plate. To compute the actual (local) cold- plate temperature, heat conduction spreading must be included, and this often leads to a fully three-dimensional problem. In most cases, the calculated heat transfer coefficient and the surface efficiency are used as input to a thermal analyzer, which solves the temperature at various locations of the cold plate as well as electronic components. 5. Pressure drop ∆p. Equation 13.15 can be used to compute the cold- plate pressure drop. Kc and Ke are functions of σ and are given in Ref- erence 1. The friction factor f depends on the Reynolds number and is also available in the same reference.

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At this point, the cold-plate temperature and the pressure drop are known. Additional iterations are usually required to design a cold plate that meets the performance requirements. For example, the following steps may be taken if the pressure drop is too high:

1. Increasing the minimum free flow area by increasing the width and/or height. 2. Reducing the flow path length. However, this also decreases the heat transfer area and thus reduces the cold-plate conductance. The better way, if possible, is to increase the width and/or height of the cold plate, and at the same time to reduce the length of the cold plate. 3. Using different types of fins which give a lower friction factor at the same mass flow rate.

On the other hand, if the calculated cold-plate conductance is too low, the possible improvements are as follows:

1. Increasing the fin thickness. This approach leads to an increase in fin (and surface) efficiency. In addition, the heat transfer coefficient is also increased due to a reduction in the free flow area. 2. Increasing the fin density. This method results in an increase of the heat transfer coefficient and surface area. 3. Replacing the straight fin with a wavy or interrupted fin. Wavy surfaces create a high heat transfer coefficient by interrupting the boundary- layer development on the fin.

Generally, the methods for increasing heat transfer also increase pressure drop. A good design provides a good balance between the heat transfer performance and pressure drop.

13.9 CORRELATIONS

Empirical data on heat transfer and friction factor in graphical form for various types of fins are included in Kays and London [1]. Wieting [6] provided the following empirical correlations of experimental heat transfer and flow friction data for rectangular offset plate fins:

For laminar flow (Re ≤ 1000):

−0. 384  x  −− f.= 7 661  α 0. 092Re 0. 712 (13.19a)  Dh 

−0. 162  x  −− j.= 0 483  α 0. 184Re 0. 536 (13.19b)  Dh 

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FIGURE 13.5 Cross-sectional view of the heat exchanger of Example 13.1.

For turbulent flow (Re ≥ 2000):

−−0.. 781 0 534  x   t  − f = 1.R 136    e0198. (13.20a)  Dhh  D 

−0. 322 −0. 089  x   t  − j = 0.R 242    e0. 368 (13.20b)  Dhh  D 

where

x =≈individual fin length in flow direction 0.125 in w α ==aspect ratio p w = subchannel width p = subchannel height t = fin thickness

It should be noted that the empirical data listed in References 1 and 6 are limited to the case with air as the working fluid and they should not be used for liquid systems.

Example 13.1 The heat load for a counterflow air-to-air heat exchanger as shown in Figure 13.5 is 200 W. For simplicity, the same flow rate and fin configuration are assumed for both hot and cold sides. The fin thickness and height are 0.1 in and 1.25 in, respectively. The spacing between fins is 0.2 in, and there are 31 fins on each side. The depth (flow length) of the heat exchanger is 13 in. The flow rate is 400 lbm/hr,

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and the inlet temperature to the cold side is 120°F. Determine the exit temperatures for the hot and cold sides of the heat exchanger.

Solution The thick fins basically subdivide each side of the heat exchanger into many (32) subchannels. One may assume that the flow to subchannels is uniform.

Cmax = Cmin = 400 × 0.24 = 96 Btu/(hr·°F)

Cold Side Fluid temperature rise:

3. 414 ∆T =×200 =°71.F c 96

Fluid outlet temperature:

Tc, o = 120 + 7.1 = 127.1°F

Hydraulic diameter (subchannel):

202125××.. D = ==0.. 345 in 0 0287 ft h 02..+ 125

Width:

W = 0.2 × (31 + 1) + 31 × 0.1 = 9.5 in

Frontal area:

125. A =×95. =0. 0825 ft2 fr 144

Free flow area:

125. A =+××().31 1 0 2 =0. 0556 ft2 c 144 A 0. 0556 σ ==c =0. 673 Afr 0. 0825 2 σ = 0. 453

Mass velocity:

400 G ==7194 lb /(hr ⋅ ft2 ) 0. 0556 m

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Average fluid temperature:

120+ 127. 1 T = =°123.F 6 f 2

Fluid:

ρ =. 3 µ = ⋅ 0 0695 lbm /ft 0. 0478 lbm /(ft hr) k =⋅0.P 0159 Btu/(hr ft⋅°= F) r0. 715

Reynolds number:

7194× 0. 0287 Re = 0. 0478 =>4320 2300 () turbulent flow == K.ce037 K 0 (from Kays and London [1])

Nusselt number: (Index 6 of Table H.2 in Appendix H)

Nu = 5 + 0.012 Re0.83 (Pr + 0.29) = 17.55

Heat transfer coefficient:

17.. 55× 0 0159 h = =⋅⋅°972. Btu/(hr ft2 F) 0. 0287

Friction factor:

f = 0.079 Re–0.25 = 0.00974

Fin efficiency [assuming kfin = 100 Btu/(hr·ft·°F) for Aluminum]:

 2h 05. ml = l  δ   kfin   2972× . 05. 125. =    0./ 1× 10012 12 =⇒=η 05..f 092

Surface Areas:

2×× 1. 25 13 A =×31 = 6. 997 ft2 f 144 13 A =×++××[]0.( 2 31 1 ) 31 2 1 . 25 ×=7 . 574 ft2 144

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Surface efficiency:

A ηη=−11f (). − = 093 o A f

Pressure drop using Equation 13.15:

71942 ∆p = 2××× 0.. 0695 4 17 108   13   ××4 0. 00974  ++−−−−(.0 37 1 0 . 453 ) ( 1 0 . 453 0 )   0. 345  =+−==2 2 0.(.. 8929 1 468 0 917 0 .). 547 1 641 lbf /ft 0 . 0114 lbf /in = 0. 316 in of water

Hot Side In theory, we should repeat the above calculations for the hot side. Since we assume the same flow rate and fin configuration for both sides, we expect the same results if the effect due to the fluid properties (except for density related to pressure drop) is negligibly small.

NTU Neglecting fouling factors and heat conduction across the thickness of the separating plate, we find from Equation 13.3a or b:

11 1 = + ⇒=U 452. Btu/(h ⋅ ft2 ⋅° F) U 093..× 972 093 ..× 972 AU 452. NTU ==×=7. 574 0. 357 Cmin 96

Effectiveness The effectiveness can generally be found from the charts in Kays and Lon- don[1]. For the present case, it can be computed from Equation 13.12b as follows:

0. 357 CT()− T ε = ==0. 263 ccoci,, + − 1 0. 357 CTTmin()hi,, ci 71. T =+120 =147.. 0 °F and T =147 0 −= 7. 1 139. 9 °F hi,,0. 263 ho

For the present example, the pressure drop can approximately be related to the pressure drop of the cold side as follows:

ρ 0. 0695 ()∆∆pp==×=c () 0.. 316 0 327 in of water h ρ c h 0. 0671

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REFERENCES

1. W. M. Kays and A. L. London, Compact Heat Exchangers, 2nd ed., McGraw- Hill, New York, 1955. 2. J. H. Seely and R. C. Chu, Heat Transfer in Microelectronic Equipment: A Prac- tical Guide, Chapter IV, Marcel Dekker, New York, 1972. 3. F. Kreith, Chapter 11, Principles of Heat Transfer, 7th ed., International Text- book, Scranton, PA, 1963. 4. A. D. Kraus and A. Bar-Cohen, Chapter 10, Thermal Analysis and Control of Electronic Equipment, Hemisphere, Washington, DC, 1983. 5. J. L. Mason, “Heat Transfer in Cross Flow,” Proc. 2nd U.S. Natl. Congr. Applied Mechanics, ASME, Ann Arbor, MI, 1955. 6. A. R. Wieting, “Empirical Correlations for Heat Transfer and Flow Friction Characteristics of Rectangular Offset-Fin Plate-Fin Heat Exchangers,” J. Heat Transfer, August 1975.

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ADVANCED COOLING TECHNOLOGIES I: SINGLE-PHASE LIQUID COOLING

In the cooling of electronic equipment, liquid cooling is frequently applied to high–heat dissipation electronic equipment. The reason is that the liquid has better heat transfer properties than that of the gas. Liquid cooling can further be divided into single- and two-phase systems. The latter involves change-of-phase processes such as boiling or condensation. Selection of a coolant for a liquid cooling system becomes a very important design consideration.

14.1 COOLANT SELECTION

A number of factors must be considered when selecting a coolant for a given system. The first step is to establish the important system characteristics, such as weight/volume limitations, power consumption requirements, and leaktightness. In general, a fluid may be very good in some respects and quite deficient in others. Therefore, there is no single, ideal fluid that is good for all systems and applications. Consequently, various fluid properties of coolants should be evaluated against the particular system applications under consideration. The important properties that affect the selection of a coolant are discussed in detail in Knight [1]. Coolants must be chemically compatible with the materials that they will contact within the coolant loop, and protected from leakage. Also, particular attention must be paid to the selection of seals, gaskets, and adhesives used in the cooling system. Kelly [2] has rated the compatibility of a number of common compounds and some of the general classes of fluids that might be used as coolants. For cooling of electronics, the dielectric property of a liquid is also very important, especially for direct liquid cooling. The reader is encouraged to read both references. The fluid properties that are directly related to thermal and hydrodynamic performance of a coolant are specific heat cp, thermal conductivity k, viscosity µ, and density ρ. Specific heat is an indicator of the coolant’s ability to store thermal energy within a given mass. Thus, it is desired that the fluid have a high specific heat. Thermal conductivity defines the coolant’s ability to transfer heat and should be high. Fluid viscosity increases the fluid resistance, which results in a high system pressure drop and required pumping power, and also reduces the effectiveness of heat transfer; therefore, the viscosity of the fluid should be as low

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as possible. The density of the coolant relates to the mass of the fluids. The higher the density, the more mass is in a given volume. The density, unlike specific heat and thermal conductivity, also has some negative attributes. Fluid weight and pumping power increase with the density. Thus, it may be desirable to have either a large or a small density, depending on the specific application. The important properties affecting the boiling process for a given fluid are surface tension σ and latent heat of vaporization hfg. A coolant with a low value of surface tension will result in a small contact angle and wet the heat transfer surface. This high wetting nature of the fluid deactivates many potential nucleation sites on the surface and can thus lead to a significant surface temperature overshoot at boiling incipience. Care must be exercised because a fluid having a low surface tension is more likely to cause leakage problems at seals, gaskets, cracks, connectors, etc. Coolants may be rated on the basis of heat transfer characteristics as they affect mass flow rate, pumping power, volume, and weight. From the viewpoint of thermohydraulic performance, fluids with a high heat transfer coefficient always have a higher flow resistance that requires a higher pumping power to overcome the higher system pressure drop. Depending on the system requirement, a careful balance between the heat transfer coefficient and pressure drop is needed to achieve an acceptable design for thermal control of the system. For convenience, a figure of merit (FOM) is frequently used to compare various fluids. The FOM represents the relationship among key properties or characteristics of a fluid under prescribed flow conditions and is used to guide the selection and optimization of fluids. FOMs based on heat transfer conditions alone are derived for single-phase (liquid) flow in both forced and free convection [3]. For two phase flow, FOMs may be obtained from the equations of boiling incipience; and of the critical heat flux for pool and flow boiling [3]. For example, if the FOM for a fluid is based on formulas for natural convection, it can be defined as follows:

  1   n  −1  βρ2  n   ckp (14.1) FOM =    µ 

where n is 0.25 for laminar flow and 0.33 for turbulent flow. For fully developed laminar forced convection in a channel where Nu is a constant, the heat transfer coefficient h depends only on the thermal conductivity k and the channel hydraulic diameter d, i.e., on h ∝ k/d. Based on this definition, the FOM does not include the specific heat of fluids in some cases of forced convection. Therefore, this definition for FOM appears to be incomplete. Any good heat transfer fluid should possess the ability to transfer heat between the wall and the fluid and should also have an ability to transport heat in the fluid that is related to the specific heat of the fluid. Furthermore, in practical engineering applications, it is beneficial to maximize heat transfer performance while minimizing the required pump power to save power consumption, system weight, and cost. It is proposed, therefore, that the FOM for single-phase flow under forced convection be defined as follows:

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ch FOM = p (14.2) P

where P is the required pump power. If the pump power is not a factor in need of consideration, e.g., under conditions of natural convection, let P = 1 in Equation 14.2. The pump power for an incompressible fluid can be expressed as follows:

mp∆ P = ρζ (14.3)

where

m = mass flow rate ∆p = fluid pressure drop ζ = pump efficiency

If the gravitational head is neglected, the pressure drop in a channel is mainly due to the frictional loss, i.e.,

 L ρV 2 ∆pf= 4   (14.4a)   d 2gc

where

−a −  ρVd fa= ′ Re a = a′  (14.4b)  µ 

and where a and a′ are constant, depending on the flow condition, and L and d are the length and the diameter of the channel, respectively. Combining Equations 14.2 through 14.4, one obtains

a hc b ρ hc p = p (14.5) P m µa

where

b = constant ζ ()a+1 = 2gdc −− 4aLV′ ()2 a

There are two ways to prescribe the mass flow rate m. One is when the mass flow rate is given, i.e., m = constant, while another is the case with a constant volumetric flow rate, VA = constant. In the latter case,

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m = ρVA (14.6)

where A is the free flow area that is considered to be constant for a given system. Therefore, the FOM can be summarized in the following form. For constant mass flow,

ρahc FOM = p (14.7a) m µa

and for constant volumetric flow rate,

− ρa 1hc FOM = p (14.7b) v µa

The heat transfer coefficients in forced internal flow under the fully developed condition are given in the following expressions:

k  h =×constant  d  for laminar flow (14.8a) af=×=1(Reconstant)

and

 k  h = 0.RePr 023  08.. 033   d  for turbulent flow (14.8b) −  af==0.(.Re) 25 0 079 025. 

For the purpose of illustration, substituting Equations 14.8a and b into Equation 14.7a leads to

 ρ  kcp FOM = F′  for laminar flow (14.9a) m  µ 

and

 kc067133..ρ 105 . = ′′ p  FOMm F   for turbulent flow (14.9b)  µ072. 

where F′ and F″ are the constants for the laminar and turbulent flow, respectively. Numerical values of FOM, based on Equations 14.9a and b, for various fluids at 104°F (40°C) are given in Table 14.1 (F′ and F″ are assumed to be 1 in the equations). Table 14.1 indicates that water is the best coolant and is followed by

Table 14.1 Coolant Figures of Merit

Fluid Laminar Turbulent Water 14.207 27.688 FC-43 0.155 0.633 FC-75 0.405 1.249 FC-78 0.666 1.751 Coolanol-25 0.250 0.980 Coolanol-45 0.071 0.397 Glycol/water 1.382 4.482 PAO (2CST) 0.222 0.950

Notes: 1. Constant mass flow rate. 2. Based on quantities in English units: k, Btu/(h·ft·°F); cp, 3 Btu/(lbm·°F); ρ, lbm/ft ; µ, lbm/(h·ft).

ethylene glycol/water (62.5%/37.5%). Since thermal properties of fluids are a function of temperature, the temperature may greatly affect numerical values of FOM. One needs to consider the operation temperature range of the system in determining the FOM. Silicate esters, such as Coolanol, are among the most widely used coolants in military products. In the past, however, fluid decomposition in the presence of moisture and electrical stress led to the formation of highly flammable alcohol that caused system failures, and detrimental deposits of silica gel that clogged fil- ters and reduced cooling capability. Polyalphaolefin (PAO), a synthetic fluid, is a leading candidate to replace silicate esters. Extensive tests were performed on PAO to study its compatibility, lubricity, and heat transfer characteristics. The results were largely favorable, with the exception of higher viscosities at extremely low temperatures, which resulted in longer system warmup time as compared with Coolanol.

14.2 NATURAL CONVECTION

The simplest method of cooling electronic equipment is to immerse it in a pool of liquid. Because the liquid then comes into direct contact with the electronics, a dielectric fluid is a must. For low heat dissipation systems, the natural convection that takes place in liquids can be sufficient. The direct immersion of parallel vertical plates was studied by Bar-Cohen and Schweitzer [4]. Their correlations are included in Table H.4 of Appendix H. Park and Bergles [5] performed a series of experiments to investigate the effect of the heater size on the heat transfer coefficient in the case of natural convection. For flush heaters, the heat transfer coefficient increases as the heater width decreases. The width effect is more pronounced in R113 than in water. The data indicate that the heat transfer coefficient is more than 20% higher than those predicted by conventional correlations. This is attributed to leading-edge effects. The experimental results show that the width effect is negligible at about 70 mm. The results for R113 and water are correlated into the following equation:

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= b Nuxxa Ra (14.10)

where

hx Nu = x k = Raxx Gr Pr  ρ  2 qx′′ 4 Gr = gβ  x  µ  k x = the vertical space coordinate

The constants a and b are as follows. For R113,

 0. 00111 0. 2745 a.=+0 906 1   (/ww′ )3. 965 

− −  264. × 10 5  0. 0362 b.=+0184 1   (/ww′ )9. 248 

and for water,

 0. 09886 0. 04654 a.=+0 906 1   (/ww′ )408. 

− −  2. 219× 10 9  0. 003966 b.=+0184 1   (/ww′ )9. 834  w′ = 70 mm

where all properties are evaluated at the film temperature, (Tw + Tb)/2. The authors also found that the heat transfer coefficient for a single, protruding heater is about 14% higher than that for a flush heater. The increase is probably caused by increased flow disturbance at the leading edge and trailing edge. An experimental study of arrays of small (5 mm × 5 mm) heaters was also performed with two or three heaters in-line or staggered. With in-line flush heaters, the heat transfer coefficients for the upper heaters are lower than that of the bottom one. When the distance between heaters increases, the heat transfer coefficients for the upper heaters also increase. In a staggered array, and at a small ratio of the vertical heater pitch to heater height, the heat transfer coefficient for the top heater increases as the transverse distance between heaters increases; however, the opposite effect is found at a large ratio. The average heat transfer coefficient over a 3 × 3 array of small heated blocks in an enclosure containing FC-75 is also available in reference 5. The dimensions of the block are 24 mm × 6 mm × 8 mm (protuberance). The data were correlated

FIGURE 14.1 Sketch of a counterflow system for electronic cooling.

into the same form as in Equation 14.10 with a = 0.279 and b = 0.224. Nu and Re are based on the heater length, L.

14.3 FORCED CONVECTION

14.3.1 Channel Flow

Forced convection through a cold plate is frequently used to cool electronic equipment. In most cases, flow is unidirectional, which leads to a significant temperature gradient in the cold plate. A counterflow arrangement as shown in Figure 14.1 [6] may be applied to overcome the large temperature gradient. Typi- cal temperature profiles of the cold plate and fluid for unidirectional flow and counterflow systems are presented in Figure 14.2. The advantage of using counterflow rather than a unidirectional flow system is a lowering of the cold-plate wall temperature and the temperature gradient. Yeh [6] provides an analytical solution, as follows, for the cases where the wall heat dissipation and the heat transfer coefficients are functions of the space:

–Bx –Bx θx = T2(x) – T1(x) = θ0e + Q(x)e (14.11a) θ =− 02TT()00 1 () 12 hxW()   hxhxW() ()  =  2 ∫L qxdx′′()  −  12 ∫Lθ dx (14.11b) + 0 + 0 x C2  hx12() h () x   hx12() h () x 

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FIGURE 14.2 Cold-wall temperature distribution for unidirectional and counterflow.

hxW() Tx()−= T ()021 ∫ x[()()]qx′′ + hxθ dx 11 + 2 x (14.11c) Chx11[() hx 2 ()] 2qx′′()++ hxTx () () hxTx () () Tx()= 11 2 2 w + (14.11d) hx12() h () x

where

2WCh[()()] x+ Ch x A = 21 12 + CC12[() h 1 x h 2 ()] x Wh() x h ()( x C− C ) B = 12 21 + CC12[() h 1 x h 2 ()] x =− x ′′ ′−Bx′ ′ q() x∫0 Aq ( x ) e dx == Cmciipi()112 ,

This approach was further extended to a multiple-channel counterflow system [7, 8] and has been applied to a solid-state phased radar array [9]. Forced convection with liquid flowing over a discrete heat source was studied by Incropera et al. [10]. The authors obtained correlations for a smooth heat source (12.7 mm × 12.7 mm), flush-mounted in a rectangular channel. The correlation based on experiments on water and FC-77 is

 µ 025. Nu= 0 . 13 Re064.. Pr 038 i  1000<

FIGURE 14.3 Schematic of heat source array with finned pins.

where NuL is based on heat source length, ReD is based on channel hydraulic diameter, and µi and µh are the dynamic viscosity evaluated at liquid inlet and heater surface temperatures, respectively. A similar study in a smaller heater (0.25 mm × 3.0 mm) was made by Samant and Simon [11]. Because of the short length with a thin thermal boundary layer, the heat transfer coefficient is much higher than that of Equation 14.12 and is as follows:

= 058.. 05 NuHH047 . Re Pr (14.13)

where both the Nusselt and Reynolds numbers are based on the height of the channel perpendicular to the heater surface. Additional tests were performed by Ramadhyani and Incropera [12] for water and FC-77 over multiple heat sources with or without surface enhancement. Their work as shown in Figure 14.3 included: (1) unaugmented (bare) surface, (2) augmentation with a 4 × 4 in-line array of copper pins silver-soldered to each heat source (the pin is 11.2 mm in height and 2.03 mm in diameter), and (3) augmentation with square fins integrally attached to each pin. For the unaugmented surface, the data presented in Figure 14.4 suggest a transition from laminar to turbulent flow over the range of 2000 < ReL < 3000. The fully developed flow is approached after the third row. For turbulent flow (ReL > 3000), the data of Fig- ure 14.5 are correlated by an expression of the form

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FIGURE 14.4 Average Nusselt number for unaugmented heat source array.

 µ 011. Nu= C Rem Pr038.  i  LL µ (14.14)  h 

where C and m are also given in Figure 14.4. Maddox and Mudawar [13] performed a series of studies for a vertically upward flow with FC-72 over smooth and enhanced surfaces. The results for single-phase flow are given in Figure 14.6. For a smooth surface, the correlation is as follows:

 <<×5 = 0.. 608 0 33 2800ReL 1 . 5 10 NuLL0 . 237 Re Pr  (14.15a) 8.Pr. 9<< 12 6

Three low-profile microstuds of height varying from 0.25 mm to 1.02 mm, each having a base area of 12.7 mm × 12.7 mm, were also included in the figure. The

FIGURE 14.5 Average Nusselt number for each row of pin-fin heat source.

single-phase correlation for a microstud surface that is oriented diagonally with respect to the flow is

 <<×5 = 069.. 033 7700ReL 1 . 6 10 NuLL076 . Re Pr  (14.15b) 10.Pr. 1<< 12 4

where the Nusselt and Reynolds numbers in both equations are based on the heater length. For a smooth surface, Equation 14.15a can also be correlated based on ReD as in Equation 14.12 for a horizontal flow. A better heat transfer coefficient was found for a vertically upward flow where the constant in Equation 14.12 was 5 increased from 0.13 to 0.18 in the range 4200 < ReD < 2.25 × 10 . As in air cooling, extended surfaces such as offset fins are frequently used in liquids to enhance heat transfer. Mandruiak and Carey [14, 15] tested various sizes of thick offset fins as shown in Figure 14.7; however, attempts to correlate

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FIGURE 14.6 Heat transfer data for smooth and microstud surfaces.

the single-phase heat transfer data for all surfaces using a single curve were not very successful. The fins were machined from a copper block, and they are an integral part of the test block. The results indicate that the transition from the laminar to the turbulent flow occurs at about Reynolds number 700. Note that the Reynolds number is based on the heated perimeter. The heat transfer correlations for all surfaces listed in Figure 14.7 are as follows:

Surface 1

 −082. < 23/ = 3.Re 831 Re700 (14.16a) StPr  − 0.Re 215 036. Re≥ 700 (14.16b)

Surface 2 St Pr2/3 = 0.141 Re–0.3 Re > 700 (14.16c)

Surface 3 St Pr2/3 = 0.046 Re–0.19 Re > 700 (14.16d)

Yeh [16] conducted an experiment to study the heat transfer and flow friction of offset fins in a narrow passage, as shown in Figure 14.8, with various types of

FIGURE 14.7 Fin dimensions and cutaway view of thick offset fins.

liquid. The channel height under consideration is 0.04 in. The results, given in Figure 14.9, were correlated into the following equations:

−−  x  0.. 384 w 0 092 − fC=     Re n (14.17a)  D  H  D

FIGURE 14.8 Rectangular (thin) offset fins.

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FIGURE 14.9 Friction factor and Nusselt number for offset fins in a narrow passage.

and

 074./ 13 01.Re Pr Re ≥ 90 (14.17b) Nu =  DD  043./ 13 < 04.ReDD Pr Re 90 (14.17c)

where C and n are 8.591 and 0.66 for water, respectively. The values for air are listed in Chapter 13.

14.3.2 Jet Impingement

One effective way of cooling a surface is to use impinging fluid jets because of the high heat transfer coefficients that are attainable. The impinging jet can be gaseous or liquid. For liquid-jet impingement, the jet can be either submerged or free. In submerged-jet impingement, the system is filled with the same fluid as that of the jet. On the other hand, the liquid jet is exposed to a gaseous environment in free-jet impingement. Submerged-jet impingement is greatly affected by the distance between the jet orifice and the target plate because of viscous hydrodynamic interaction. While the effect of this distance is insignificant in free-jet impingement, the liquid layer flow on the solid surface is exposed to an ambient gas environment.

An experimental investigation of liquid multiple-jet impingement was conducted by Jiji and Dagan [17]. A single correlation for the average heat transfer coefficient with Prandtl numbers between 5 and 27 (for water and FC-77) was obtained as follows:

 0. 008nL  Nu=+384 .  1 Re12// Pr 13 (14.18) L  d 

where

Re = jet Reynolds number d = jet diameter L = heat source size n = number of jets per heat source = NuL average Nusselt number

In their study, the data for 1, 4, and 9 jets per heat source at flow rates smaller than 20 cm3/s indicate that thermal resistance is on the order of 1°C/W for FC-77 and 0.1°C/W for water. Reduction of thermal resistance for a given flow rate can also be achieved by reducing the jet diameter and increasing the number of jets per heat source. Similar experiments were performed by Wadsworth and Mudawar [18] to investigate single-phase heat transfer from a smooth 12.7 mm × 12.7 mm simulated chip to a two-dimensional jet of dielectric Fluorinert FC-72 liquid issuing from a thin rectangular slot into a channel confined between the chip surface and nozzle plate. The test data were correlated by the following equation:

Nu  LW− 0. 664 L =+3.Re. 0605.. 0 099 Re0664  (14.19) Pr13/  W 

where the Reynolds number is based on the nozzle hydraulic diameter (2W) and L is the heater length. All properties are evaluated at the mean liquid temperature, which is the average temperature of liquid at the nozzle inlet and the heater surface.

14.3.3 Microchannel Flow

Another effective forced convection cooling approach, using microscopic heat transfer structures, was first proposed by Tuckerman and Pease [19, 20]. The general features of heat transfer and flow friction of microchannels can be summarized as follows:

1. Conversion from laminar (Re < 300–400) to turbulent (Re > 1000–1500) regimes occurs at much smaller Reynolds numbers, and the transition region is limited in the smaller Reynolds number zone in comparison

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FIGURE 14.10 Friction factor vs. Reynolds number in microchannels.

with conventional channels. The transition to turbulent flow is also influenced by liquid temperature, velocity, and microchannel sizes. 2. The friction factors and heat transfer correlations often deviate from the conventional ones for large tubes. Generally, Nu may decrease with an increase of Re. In the transition region (300–400 < Re < 1000–1500), the Nu-Re relationship diverges from a single straight-line configuration. In the turbulent flow region, however, all data fall along a fairly narrow band that can be represented by a single straight line in which Nu increases as Re increases. 3. The geometric parameters of microchannels, especially the hydraulic diameter and the aspect ratio of the channel, have a critical effect not only on the flow transition the Reynolds number, but also on the friction factor and heat transfer coefficient. 4. Liquid properties change dramatically over the channel length because of the small amount of liquid in the microchannels.

As the characteristic diameter approaches the molecular or substructural level, an analysis based on classical continuum mechanics and the Navier-Stokes equations is in question. This may be one of the reasons that the results for microchannels differ significantly from the classical results. Harley et al. [21] performed tests on liquid (n-propanol) flow in rectangular channels with a width of 100 µm but with varying channel depth. The friction factor is given in Figure 14.10. The results indicate that the product f × Re is no longer constant for fully developed laminar flow. Physical phenomena of heat transfer in microchannels are discussed in detail in References 22 through 27. Figure 14.11 [22] is used as an example to illustrate the typical experimental results for the heat transfer coefficient versus the wall temperature in single-phase forced convection. The variation in heat transfer coefficient can be divided into three distinct regions: the region on the left of the graph in which the heat transfer coefficient slightly increases or decreases; the middle region, where great variation emerges; and a region where the heat transfer coefficient increases monotonically.

FIGURE 14.11 Variations of heat transfer coefficient with wall temperature.

Wang and Peng [23] investigated single-phase forced convection on water and methanol flowing through four to six parallel rectangular channels with three sides heated (the top was insulated). The height of the microchannel was 0.7 mm and the channel width varied from 2.4 to 4.0 mm. Fully developed turbulent flow was initiated at about Re = 1000–1500, and the heat transfer correlation was found to be as follows:

Nu = 0.00805 Re4/5 Pr1/3 (14.20)

where Nu and Re are based on the channel hydraulic (or effective) diameter. The heat transfer coefficient is defined as

q′′ h = − (14.21) TTwf

″ where q is based on all heated surfaces, Tw is the wall temperature at the end of the channel, and Tf is the liquid inlet temperature. Peng et al. [24] conducted experimental studies of forced convection of water through parallel rectangular microchannels having channel hydraulic diameters Dh of 0.133 to 0.367 mm and aspect ratios H/W between 0.333 and 1. The experimental data were correlated by the following equations for the laminar and turbulent flow, respectively:

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Table 14.2 Empirical Constants for Heat Transfer

Dh, mm L, mm W, mm H, mm H/WCH, l CH, t 0.343 50 0.4 0.3 0.750 0.0580 0.01340 0.300 50 0.3 0.3 1.000 0.0384 0.00726 0.267 50 0.4 0.2 0.500 0.0426 0.01660 0.240 50 0.3 0.2 0.667 0.0472 0.09260 0.200 50 0.2 0.2 1.000 0.0468 0.00696 0.150 50 0.3 0.1 0.333 0.0104 0.00483 0.133 50 0.2 0.1 0.500 0.0285 0.00939

0.62 1/3 Laminar flow: Nu = CH, l Re Pr (14.22)

4/5 1/3 Turbulent flow: Nu = CH, t Re Pr (14.23)

where CH, l and CH, t are empirical constants that depend on the geometric configuration of the microchannels. Numerical values of CH, l and CH, t are listed in Table 14.2. The geometric parameters, especially the hydraulic diameter and aspect ratio, are important variables that have a significant effect on the flow and heat transfer. The experimental results indicated that the laminar convection heat transfer had the maximum value when the aspect ratio was approximately equal to 0.75. On the other hand, the turbulent heat transfer approached the maximum value when the aspect ratio was in the range between 0.5 and 0.75. According to Peng et al. [28, 29], the empirical correlations for the friction factors for single-phase liquid flow in microchannels are as follows:

Laminar flow:

Cfl, f = (14.24) Re198.

Turbulent flow:

Cft, f = (14.25) Re172.

where Cf, l and Cf, t are empirical constants. These values are also dependent on the microchannel configuration and geometrical parameters, and are listed in Table 14.3, where Wc is the center-to-center distance between microchannels; L, H, and W are the length, height, and width of the microchannel, respectively; and Z = min (H, W)/max (H, W). On the basis of the 12 different microchannel structures in Table 14.3, Peng and Peterson [29] provide the following general correlations for laminar and turbulent convection heat transfer with water:

Table 14.3 Empirical Constants for Friction Factor

Dh, mm Wc, mm L, mm W, mm H, mm H/WZ Cf, l Cf, t 0.267 4.5 45 0.4 0.2 0.500 0.500 28,600 40,400 0.343 2.8 45 0.4 0.3 0.750 0.750 44,800 34,200 0.343 2.0 45 0.4 0.3 0.750 0.750 44,800 34,200 0.240 4.6 45 0.3 0.2 0.667 0.667 42,600 18,200 0.300 2.8 45 0.3 0.3 1.000 1.000 109,000 38,600 0.300 2.0 45 0.3 0.3 1.000 1.000 109,000 38,600 0.200 4.5 45 0.2 0.2 1.000 1.000 32,400 20,100 0.240 2.8 45 0.2 0.3 1.500 0.667 42,600 18,200 0.240 2.0 45 0.2 0.3 1.500 0.667 42,600 18,200 0.133 4.5 45 0.1 0.2 2.000 0.500 5,200 1,820 0.150 2.8 45 0.1 0.3 3.000 0.333 24,200 6,920 0.150 2.0 45 0.1 0.3 3.000 0.333 24,200 6,920

Laminar flow:

−079.  D 081.  H  Nu= 0 . 1165 h    Re062./ Pr 13 (14.26)    Wc  W

Turbulent flow:

 D 115. Nu= 0 . 072 h  [ 1−− 2 . 421 (Z 0 . 5 )20813 ]Re./ Pr (14.27)  Wc 

The test data indicate that there exists an optimum value of Z that maximizes the turbulent convection heat transfer, and that this value, approximately Z = 0.5, is constant regardless of the ratio H/W or W/H. For gas flow in microchannels, References 30 and 31 provide equations that can be used to predict the heat transfer coefficient. From Wu and Little [30],

Nu = 0.00222 Re1.09 Pr0.4 Re > 3000 (14.28)

where Nu and Re are based on the hydraulic diameter. From Choi et al. [31],

1. Laminar flow, Re < 2000: Nu = 0.000972 Re1.17 Pr1/3 (14.29a) 2. Turbulent flow, 2500 < Re < 20,000: Nu = 3.82 × 10–6 Re1.96 Pr1/3 (14.29b)

REFERENCES

1. A. F. Knight, “Choice of Fluids for Cooling Electronic Equipment,” Electro- Technology, June 1963.

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2. D. J. Kelly, “Resistance of Materials to Hydraulic Fluids,” Machine Design, January 21, 1972. 3. T. Y. Lee, J. Saylor, T. Simon, W. Tong, P. S. Wu, and A. Bar-Cohen, “Fluid Selection and Property Effects in Single- and Two-phase Immersion Cooling,” Proc. Intersoc. Conf. Thermal Phenomena in Electronic Components, May 1988. 4. A. Bar-Cohen, and H. Schweitzer, “Convective Immersion of Parallel Plates,” Proc. 4th Int. Electronic Packaging Conf., 1984. 5. K. A. Park and A. E. Bergles, “Natural Convection Heat Transfer Characteris- tics of Simulated Chips,” Heat Transfer in Electronics Equipment, ASME HTD- 48, 1985. 6. L. T. Yeh, “Analytical Solutions for a Counterflow Heat Exchanger with Space Dependent Wall Heat Dissipation,” Heat Transfer in Electronics Equipment, ASME HTD-48, 1985. 7. L. T. Yeh and W. K. Gingrich, “Numerical Solutions for a Multiple-Channel Counterflow Heat Exchanger with Space Dependent Wall Heat Dissipations,” Proc. 8th Int. Heat Transfer Conf. 6, San Francisco, 1986. 8. W. K. Gingrich, L. T. Yeh, and T. D. Fuhr, “Transient Temperature Distribution for a Multiple-Channel Counterflow Heat Exchanger with Space Dependent Wall Heat Dissipations,” ASME Winter Annual Meeting, San Francisco, 1989. 9. L. T. Yeh, “Thermal Design of a Multiple-Channel Bidirectional Flow Cold Plate for Solid State Phased Array Radars,” in W. Aung, ed., Cooling Technol- ogy for Electronic Equipment, Hemisphere, New York, 1988. 10. F. P. Incropera, J. S. Kerby, D. F. Moffat, and S. Ramadhyani, “Convection Heat Transfer from Discrete Heat Sources in a Rectangular Channel,” Int. J. Heat and Mass Transfer 29, 1986. 11. K. R. Samant and T. W. Simon, “Heat Transfer from a Small High-Heat Flux Patch to a Subcooled Turbulent Flow,” Proc. AIAA/ASME Heat Transfer and Thermophysics Conf., 1986. 12. S. Ramadhyani and F. P. Incropera, “Forced Convection Cooling of Discrete Heat Sources with and without Surface Enhancement,” in W. Aung, ed., Cooling Technology for Electronic Equipment, Hemisphere, New York, 1988. 13. D. E. Maddox and I. Mudawar, “Single- and Two-Phase Convection Heat Transfer from Smooth and Enhanced Microelectronic Heat Sources in a Rec- tangular Channel,” ASME HTD-96, 1, 1988. 14. G. D. Mandruiak, V. P. Carey, and X. Xu, “An Experimental Study of Convec- tive Boiling in a Partially Heated Horizontal Channel with Offset Strip Fin,” J. Heat Transfer 110, 1988. 15. G. D. Mandruiak and V. P. Carey, “Convective Boiling in Vertical Channels with Different Offset Strip Fin Geometries,” J. Heat Transfer 111, 1989. 16. L. T. Yeh, “An Experimental Study of Offset Fins in a Narrow Channel,” Proc. 1st Int. Conf. Aerospace Heat Exchanger Technol., Palo Alto, CA, 1993. 17. L. M. Jiji and Z. Dagan, “Experimental Investigation of Single-Phase Multijet Impingement Cooling of an Array of Microelectronic Heat Sources,” in W. Aung, ed., Cooling Technology for Electronic Equipment, Hemisphere, New York, 1988. 18. D. C. Wadsworth I. and Mudawar, “Cooling of a Multichip Electronic Module by Means of Confined Two Dimensional Jets of Dielectric Liquid,” J. Heat Transfer 112, 1990.

19. D. B. Tuckerman, “Heat Transfer Microstructures for Integrated Circuits,” Doctoral Thesis, Stanford University, 1980. 20. D. B. Tuckerman and F. F. Pease, “High Performance Heat Sinking for VLSI,” IEEE Electron Devices Letter EDL-2, 1981. 21. J. Harley, J. Pfahler, H. Bau, and J. Zemel, “Transport Process in Micro and Submicro Channels,” ASME HTD-116, 1989. 22. X. F. Peng and G. P. Peterson, “The Effect of Thermofluid and Geometrical Parameters on Convection of Liquids through Rectangular Microchannels,” Int. J. Heat Mass Transfer 38(4), 1994. 23. B. X. Wang and X. F. Peng, “Experimental Investigation on Liquid Forced Convection Heat Transfer through Microchannels,” Int. J. Heat Mass Transfer 37, Suppl. 1, 1994. 24. X. F. Peng, G. P. Peterson, and B. X. Wang, “Heat Transfer Characteristics of Water Flowing through Microchannels,” Experimental Heat Transfer 7, 1994. 25. X. F. Peng and B. X. Wang, “Cooling Characteristics with Microchanneled Structures,” Enhanced Heat Transfer 1(4), 1994. 26. X. F. Peng, B. X. Wang, G. P. Peterson, and H. B. Ma, “Experimental Investi- gation of Heat Transfer in Flat Plates with Rectangular Microchannels,” Int. J. Heat Mass Transfer 38(1), 1995. 27. X. F. Peng and B. X. Wang, “Forced Convection and Flow Boiling Heat Trans- fer for Liquid Flowing through Microchannels,” Int. J. Heat Mass Transfer 36(14), 1993. 28. X. F. Peng, G. P. Peterson, and B. X. Wang, “Friction Flow Characteristics of Water Flowing in Microchannel Structures,” Int. J. Heat Mass Transfer 39(12), 1996. 29. X. F. Peng and G. P. Peterson, “Convective Heat Transfer and Flow Friction for Water Flowing through Microchannels,” Experimental Heat Transfer 7, 1994. 30. P. Wu and W. A. Little, “Measurement of the Heat Transfer Characteristics of Gas Flow in Fin Channel Heat Exchangers Used for Microminiature Refrig- erators,” Cryogenics 24, 1984. 31. S. B. Choi, R. F. Barron, and R. O. Warrington, “Fluid Flow and Heat Trans- fer in Microtubes,” Micromechanical Sensors, Actuators, and Systems, ASME DSC-32, 1991.

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ADVANCED COOLING TECHNOLOGIES II: TWO-PHASE FLOW COOLING

As mentioned in Chapter 14, dielectric fluids are generally required for cooling of electronics, especially in direct-immersion cooling. Dielectric liquids such as Freon from Dupont and Fluorinert (FC series) from 3M Company are the most popular fluids used in direct cooling of electronics. The basic characteristics of those fluids are as follows:

1. Excellent electrical properties: high dielectric strength, low dielectric constant, and high electrical resistivity 2. High thermal stability 3. Low chemical reactivity 4. Poor thermal properties: low thermal conductivity, specific heat, and heat of vaporization (latent heat) 5. Extremely low surface tension 6. High solubility for air

As they can for the liquid phase, fluid figures of merit (FOMs) can be obtained and used as a guide to select the working fluid for a two-phase system.

15.1 FIGURE OF MERIT The boiling incipience and critical heat flux (CHF) are the most important factors to be considered in developing an FOM for pool boiling. The CHF represents the maximum allowable design heat flux of a system under consideration. For boiling incipience, Marto and Lepere [1] defined the FOM as follows:

σT FOM = s i ρ (15.1) hfg g

where Ts is saturation temperature and σ, hfg, and ρg are surface tension, heat of vaporization, and vapor density, respectively. The classic CHF correlation derived by Zuber et al. [2] for a large horizontal surface under conditions of saturated pool boiling is expressed in the following form:

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π 12//14 CHF =−hgρσρρ[( )] (15.2) Z 24 fg g f g

According to this equation, an FOM for the pool-boiling CHF was proposed in Saylor et al. [3] as follows:

=−ρσρρ12//14 FOMCHF, PB, sat hgfg g[( f g )] (15.3)

Generally, CHF increases when heater size decreases. Equation 15.2 was modified by Lienhard and Dhir [4] for a small heater as follows:

14. CHF  Z L′ < 6 (15.4a) = 14/ CHFL&D  ()L′  09. L′ > 6 (15.4b)

where

 σ 12/ LL′ =   ; L = heater length ρρ−  g()fg

By combining all properties together, the FOM for a small heater becomes

=−ρσ12// 38 ρ ρ 18/ FOMCHF, PB, sat hgfg g [(f g )] (15.5)

For subcooled boiling, where the liquid temperature is below its saturation temperature, the CHF correlation developed by Ivey and Morris [5] is as follows:

   ρ 14/    g TT−  CHF =+CHF101 .   c ρ  sat bulk  (15.6) I&M Z   ρ  pf, f h ρ   f fg g 

Similarly, the FOM for subcooling can be expressed by the following equation:

   ρ 14/ ρ  g cpf, f FOM =−+hgρσρρ12//[ ( )]14 101 .   (15.7) CHF, PB, sub fg g f g   ρ  h ρ   f fg g 

Again, following Saylor et al. [3], the CHF for flow boiling over a small heater as presented by Lee et al. [6] is

027. = U ρρσ0... 239 0 396 0 365 CHFLs0 . 0742 (hFfg g f ) ub(15.8) L0. 365

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where

 ρ 0. 118 1. 414 f cpf, F =+1 0. 952    ∆T1. 414 (15.9) sub ρ sub  g   hfg 

It should be noted that U is the bulk velocity. In addition, ∆T represents the degree of subcooling, and it becomes zero at the saturation temperature. FOMs for CHF under saturated and subcooled boiling conditions are obtained from Equations 15.8 and 15.9, respectively, as follows:

= ρρσ0... 239 0 396 0 365 FOMCHF, FB, sat hfg g f (15.10)

and

 ρ 0. 118 1. 414 f cpf, FOM =     CHF, FB, sub ρ (15.11)  g   hfg 

All properties are functions of the pressure alone in the saturated condition and are evaluated at the system pressure; therefore, a fluid FOM actually depends on the pressure alone in two-phase flow.

15.2 DIRECT-IMMERSION COOLING

Liquid immersion remains a primary candidate for cooling of high-power electronic systems because it can accommodate relatively high heat flux without the need for mechanical devices such as pumps or other enhancement techniques. Marto and Lepere [1] conducted pool-boiling heat transfer experiments with R-113 and FC-72 on a plain copper tube with an outside diameter of 15.8 mm. Their results are given in Figure 15.1. As shown in the figure, a greater incipient boiling superheat is required for R-113 than for FC-72. Danielson et al. [7] provided pool-boiling characteristics for a number of dielectric fluids, such as FC-43, 72, 75, 77, 84, and 87 and R-113. Data were obtained with a classical 0.25-mm platinum wire. It was found that nucleate boiling characteristics of highly wetting inert fluids can be predicted with the following correlation proposed by Rohsenow [8] for water and other fluids:

   cT∆  q′′ (/)σ g 12/ a pf, = C    Prb (15.12) h sf µ h  ρρ− 12/  fg  ffg()fg

where a is generally taken as 1/3 and b is taken equal to 1 for water and 1.7 for other fluids. The surface/fluid coefficient Csf had ranged from 0.002 to 0.016 in all previously published data. For modest superheat variations where fluid

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FIGURE 15.1 Pool boiling data for a plain tube.

properties can be assumed to be invariant with temperature, Equation 15.12 can be further simplified as follows:

q″ = C (∆T)n (15.13)

Most empirical boiling curves, either pool or flow boiling, exhibit such a relationship for most, or often all, of the nucleate boiling range. The CHF data of Danielson et al. [7] compare reasonably well with the prediction from Equation 15.4. Among various fluids tested, FC-43 displays the

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lowest CHF value. The experimental data also indicated that superheat excursions (temperature overshoot) at boiling incipience always occurred but their extent could vary. In addition, the thermal transport in the nucleate boiling region is very efficient so that the corresponding thermal resistance is about 1.2 to 2.1°C/W for a 1-cm2 surface. A heat transfer system should be designed to operate in the nucleate boiling region because of the extremely effective cooling that takes place in this region. Park and Bergles [9] studied the effects of heater size on boiling in R-113. They obtained results for varying height (1 mm to 80 mm) and width (2.5 mm to 70 mm). They found that the size of the heater had no effect on established boiling, in contrast to results reported previously in the literature. As previously reported, a temperature overshoot was observed, and on some occasions, a double overshoot was also found. A typical boiling curve is shown in Figure 15.2. Park and Bergles found that the CHF for wide heaters increased with decreasing heater height. The CHF also increased with decreasing heater width. Com- bining this with other available data, including those in Reference 4, a general correlation of CHF for a wide (W ≥ 20 mm) heater is expressed as follows:

 014. =+152 CHFP&B 086 . CHFZ  1  (15.14)  L′329. 

where L′ is the nondimensional height as defined in Equation 15.4. The effects of the width and the height on CHF for the 5-mm-high and the 5-mm-wide heaters, respectively, are given in the following equations:

 37. =+52 CHFP&B 093 . CHFZ  1  (15.15a)  I102. 

and

 0. 087 =+241, 300 CHFP&B 07 . CHFZ  1  (15.15b)  L′403. 

2 0.5 where I is the nondimensional width, defined as I = (ρfWσ/µ ) ; and W is the width of the heaters.

15.3 ENHANCEMENT OF POOL BOILING

As stated earlier, most dielectric fluids have an extremely low surface tension which often results in a large temperature overshoot in the case of pool boiling. The large temperature overshoot is undesirable and unacceptable for electronic equipment. The following methods can be applied to eliminate or minimize the temperature overshoot at the incipience of boiling. These methods may also enhance pool boiling in general.

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FIGURE 15.2 Typical boiling for small heaters immersed in R113.

15.3.1 Surface Enhancement Surface enhancement may reduce the temperature overshoot and increase the values of nucleate boiling and CHF. However, these improvements may not always be achieved simultaneously. Anderson and Mudawar [10] studied pool boiling of FC-72 with various surface enhancement schemes. The surface treatments included surface roughness, cavities (horizontal, vertical, and inclined arrays of holes), vertical microfins, horizontal microstuds, and inclined microgrooves. The basic findings from this study are as follows:

1. Common incipience theories based on cavity sizes do not apply to highly wetting fluids. Incipience excursion (temperature overshoot) of

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FIGURE 15.3 Pool-boiling curves for plain and ABM surfaces.

such fluids actually depends on the radius of vapor embryos existing within the surface. 2. Increasing the nonboiling waiting period increases the incipience temperature and the probability of incipient hysteresis. 3. Increasing surface roughness promotes earlier incipience and therefore reduces the probability of significant incipience excursion. 4. Artificial cavities on the order of 0.3 mm diameter are ineffective in lowering the incipience temperature, enhancing nucleate boiling, or increasing CHF. 5. Microstructure significantly enhances nucleate boiling while increasing the probability of incipience excursion.

More recently, Chang and You [11] studied the effects of surface micro- and macrogeometry on boiling heat transfer in saturated FC-87 and R-123. Among the coatings studied was ABM, which was developed by these authors and stands for aluminum–brushable ceramic–MEK. Selected results are presented in Figure 15.3.

15.3.2 Bubble Generation Another method to reduce temperature overshoot for pool boiling in highly wetting fluids was proposed by Bergles and Kim [12] using bubble generation. The

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FIGURE 15.4 Bubble generation heater and bubble paths for various spacings.

scheme as shown in Figure 15.4 involves the generation of bubbles below the heating element (chip). The incipient boiling superheats for plain and sintered copper surfaces with R-113 were reduced from 30°C to as low as 8°C and from 22°C to as low as 7°C, respectively. A reasonable explanation for the effectiveness of this technique is that the impacting bubbles activate temporarily dormant cavities that in turn activate large neighboring cavities.

15.3.3 Falling Liquid Film

Mudawar et al. [13] studied microelectronic cooling by boiling heat transfer from a vertical surface to an inert fluorocarbon liquid (FC-72). A sketch of the system is given in Figure 15.5. The results of this work indicate that the surface temperature overshoot is less than 2°C as compared with more than 10°C for a typical pool boiling. The possible reason for reducing the temperature overshoot is the strong influence of liquid motion on the growth and departure of bubbles. As compared with pool boiling on a vertical plate (Figure 15.6a), a large vapor embryo, as shown in Figure 15.6b, is trapped in the cavity by liquid drag forces, which resist the effect of buoyancy forces pulling the embryo from the cavity. Fig- ure 15.7 shows the boiling curve for a falling liquid film. The correlations for the heat flux in the fully developed nucleate boiling region and for the CHF are expressed, respectively, as follows:

″ ∆ 2.62 q = 114.1( Tsat) (15.16)

and

 ρ  23/  042. f σ ()qU′′ = 0 .( 121 ρ h )    (15.17) CHF sat gfg ρ  ρ 2   g   fUL

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FIGURE 15.5 Falling liquid film.

where q″ is expressed in W/m2 and ∆T (wall superheat) in °C, L is the heater length, and U is the velocity (m/s). The ratio of CHF values corresponding to subcooled and saturated conditions is as follows:

 ∆∆13/  ρ  23/ ()q′′ cTpf, sub 016. fpfcT, sub CHF sub =+11  +  (15.18) ′′ ρ ()qCHF sat  hfg   gfgh 

FIGURE 15.6 Liquid-vapor exchange at a nucleation site. (a) Pool boiling on a vertical surface. (b) Boiling in a falling liquid film.

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FIGURE 15.7 Boiling curve of a falling liquid film.

The results also indicated that increasing the velocity or subcooling (∆Tsub) increases the value of CHF. On the other hand, increasing the heater length decreases CHF. Grimley et al. [14, 15] applied several surface enhancement schemes as shown in Figure 15.8 to improve boiling in a falling liquid film. The results of using microfins and microstuds are presented in Figure 15.9, while the data with deflectors are given in Figure 15.10. The key findings of this work can be summarized as follows:

1. As compared with a smooth surface, microfins aligned in the flow direction enhance nucleate boiling and increase CHF. On the other hand, microstuds provide superior performance in the nucleate regime but do not increase CHF relative to a smooth surface. 2. The effect of subcooling on CHF is much more pronounced in falling films than in pool boiling. The increase in CHF can approximately be related to subcooling by the following expression:

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FIGURE 15.8 Surface enhancement for falling liquid film.

FIGURE 15.9 Boiling data for microfin and microstud surfaces.

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FIGURE 15.10 Boiling data for a microfin surface with and without a deflector.

′′ qCHF ∝ ∆ exp(Tsub ) (15.19) ()q′′ ∆ = CHF Tsub 0

Also, the effect of subcooling is less pronounced for the 12.7-mm-long heater than for a 62.5-mm-long heater.

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3. Although the louvered flow deflector enhances CHF for saturated conditions, no improvement is found when subcooling is used in combination with the deflector. 4. CHF may be improved by a combination of reduced heater length, subcooling, and flow deflector, but the effects are not independent and superposition does not apply.

15.3.4 Jet-Impingement and Spray Cooling

Both jet-impingement cooling and spray cooling are created by forcing liquid through a small nozzle (or orifice). As described in the discussion of single-phase jet impingement in Chapter 14, liquid jets may be subdivided into two types, namely, submerged jet and free jet. In the case of sprays, liquid is shattered into a dispersion of fine droplets prior to impact. A spray can be formed either by high pressure (as a plain orifice spray) or atomized with the aid of pressurized air (as an atomized spray). Ma and Bergles [16] studied the heat transfer characteristics of a chip-size electrically heated test surface with normally impinging circular submerged jets of saturated or subcooled R-113. Typical results are given in Figure 15.11. The temperature overshoot is much less than that observed with pool boiling. Estes and Mudawar [17] compared the performance of free jets and sprays in cooling a 12.7-mm × 12.7-mm simulated chip. The fluid used in the experiment was FC-72. For jet cooling, three different sizes of nozzle were used, with desig- nations 1, 2, and 3 in the order of increasing orifice diameter—0.66, 0.86, and 1.14 mm, respectively. Figure 15.12 shows the boiling curve of various orifice diameters. The proposed CHF correlation is given by the following expression:

−  ρ 0. 645  0. 343 0. 364 ()q′′ f 2σ  L′ CHF sub = 0. 221    1+  ρ  ρ   ρ 2 ′ −    vjfguh v  fjuL() d d   ρ 05.   f 2  ×+1117.   H   ρ  M  (15.20)  v 

where

d = jet diameter L = heater (chip) length = uj jet velocity LL′ = 205. cT∆ = pf, sub HM hfg

The boiling data for spray cooling as given in Figure 15.12 are provided in reference 17. The CHF correlation is expressed as follows:

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FIGURE 15.11 Jet impingement boiling curve for R-113.

 2  −035. (4 /π )(q′′ )  ρ 03. ρ Qd′′ CHF sub = 23. f  f 32  ρ ′′  ρ   σ  vfgQh  v    (15.21)   ρ 05.   f  ×+1 0. 0019  H   ρ  M   v 

where d32 is the Sauter mean diameter and Q″ is the volumetric spray flux at the outer edge of the spray impact area. d32 and Q″ are defined as follows:

1/2 –0.259 d32 = 3.67do(We Re) (15.22a)

and

Q′′  θθ =+051. cos cos (15.22b) QL/(π 2 /4 )  22

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297

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where

ρρ(/)2∆pd We ==Weber number vfo σ ρρ(/)2∆pd12/ Re ==Reynolds number ffo µ f = do orifice diameter ∆p = pressure drop across spray nozzle Q = mean spray volumetric flow rate L = heater (or chip) length θ = spray cone angle

Estes and Mudawar [17] indicated that dryout occurs on the chip (heater) surface prematurely in the case of jet impingement when chip power is increased too fast, but this phenomenon has not been observed in the case of spray cooling. In fact, CHF is very repeatable with spray cooling even during chip power fluctu- ations. For both jet impingement and spray cooling, single-phase heat transfer and CHF are enhanced with increasing jet diameter, increasing jet velocity or spray flow rate, and increasing subcooling. However, the effect of subcooling on CHF is less pronounced in spray cooling than in jet impingement. Another study using an FC-72 impinging jet to cool a wafer was conducted by Goodling et al. [18]. The paper describes the setup and prototypes used but it does not present any significant results. Yao [19] studied the effects of droplet size, subcooling, and mass flux in spray cooling with FC-72. It was concluded that spray cooling offers special advantages of high CHF, uniform heat flux, and very small or no temperature overshoot. Selected results are presented in Figure 15.13. Ghodbane and Holman [20] obtained heat transfer data for horizontal sprays on vertical constant–heat flux surfaces with subcooled R-113. Using both full- cone circular and square hydraulic spray nozzles with three different orifice diameters (d = 0.238, 0.159, 0.063 cm), the flow rates vary from 0.5 to 126 cm3/s. A CHF correlation was developed by the authors. With additional test data, this correlation was later modified by Holman and Kendall [21] as follows:

 ρ 2 06. x fjpud ()q′′ = 95 .  H15. (15.23) CHF sub µ  σ  M ffgh  

where x is the distance from the nozzle to the heat source and dp is the droplet diameter. Other characteristics on which jet-impingement and spray cooling differ are listed in Table 15.1

15.3.5 Mixture Of Dielectric Liquids

Another method to reduce the temperature overshoot was proposed by Norm- ington et al. [22], involving mixing of different dielectric liquids. The results of

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FIGURE 15.13 Variation of peak heat flux with liquid mass flux in spray cooling.

Table 15.1 Differences between Jet-Impingement and Spray Cooling

Attribute Jet-impingement cooling Spray cooling Speed High speed Low speed Surface Possible target surface erosion No surface erosion Cooling area Small area per jet Large area Heat transfer Very high value at stagnation point; Very uniform over sharp decline away from this point a large area Heating surface Nonuniform surface temperature Uniform surface temperature Pressure drop High pressure drop due to high Small pressure drop speed and small orifices Temperature overshoot Small Very small Nozzles Very small nozzles; easily plugged up Relatively large

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Table 15.2 Summary Results of Experimental Data on Mixing of Dielectric Fluids

Subcooling, Liquid composition Saturation Temperature °C (by volume) temperature, °C overshoot, °C CHF, W/cm2 50 100% D80 81 19–29 29.2 50 95% D80, 5% HT100 83 3–6 37.7 50 85% D80, 15% HT100 84 0–6 32.5 50 50% D80, 50% HT100 92 5 — 50 20% D80, 80% HT100 100 3–6 — 50 100% HT100 109 25–32 — 10 100% D80 81 0–4 19.3 10 95% D80, 5% HT100 83 0–7 20.9 10 85% D80, 15% HT100 84 0 21.9 10 20% D80, 80% HT100 103 2–6 — 10 100% HT100 110 2–4 —

their work are summarized in Table 15.2. As can be seen from the table, the effect of liquid mixing in reducing the temperature overshoot is pronounced only in the high-subcooling case. Similarly, the value of CHF was extended in the mixtures at 50°C subcooling. The effect of the mixture on CHF was insignificant at 10°C subcooling.

15.4 FLOW BOILING

Flow boiling has significant advantages over pool boiling, such as enhancing nucleate boiling, increasing CHF, and reducing the temperature overshoot. For a typical rectangular-channel flow, the saturated boiling critical heat flux can be expressed as follows [23]:

0. 133 13/ ()q′′  ρ   σρ  1 CHF sat = 015.  v   v  (15.24) ρ  2  + Ghfg  f  GL 1 0./ 0077LDe

where G is mass velocity, L is heated length, and De is heated equivalent diameter (= 4 × flow area/heated perimeter). An ultrahigh-flux heat sink as shown in Figure 15.14 using a curved surface with subcooled nucleate boiling in FC-77 was developed by Iversen and Whitaker [24]. The radial acceleration v2/r can be utilized to enhance flow boiling. The correlation for CHF is given by

 14/  −  a qv′′ =×.36 10 412/ ∆T1 +   (15.25) CHF sub      g 

2 where (q″)CHF is in kW/cm , v in cm/s, and subcooling ∆Tsub in °C, and where a and g are acceleration and gravitational acceleration (m/s2), respectively.

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FIGURE 15.14 Curved-surface semiconductor cooling system.

When the radial acceleration is negligible, e.g., a = 0, Equation 15.25 reduces to the following form:

′′ ∝ 12/ qv CHF (15.26a)

On the other hand, when a/g is large compared to 1, we have

′′ ∝ qv CHF (15.26b)

An experimental study was performed by Samant and Simon [25] for heat transfer from a small heated patch to a subcooled fully developed turbulent flow (R-113 and FC-72). The test setup and the typical boiling curve are shown in Fig- ure 15.15. The summary results are listed in Table 15.3. As can be seen from the table, temperature excursion decreases with increasing subcooling and/or velocity. The maximum temperature excursion in this investigation is 27°C, which is much higher than previously observed. The large temperature excursions observed here are presumed to be due to the very small heater. Based on data from Samant and Simon [25], Lee and Simon [26] derived a generalized correlation for CHF. The form of correlation is based on an analogy with different flow situations, such as channel flows, flow across a cylinder, and

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FIGURE 15.15 Flow boiling over a small heater.

impinging free and submerged jets. With additional test data from Lee and Simon [26], the generalized CHF correlation was later revised as follows:

   0. 994 2  −033.  ρ 0. 779 ()q′′ ρ GL  f 0. 424  CHF sub = 0.. 0397 v    1+ 3 026  H (15.27) Gh  ρσρ     ρ  M  fg ff v 

The heated length in Lee and Simon [26] varies over 0.25 mm, 0.5 mm, and 0.75 mm for a Nichrome heater patch and 1 mm and 3 mm for a platinum patch. Their results indicated that CHF increases with increasing velocity and subcool-

Table 15.3 Heat Transfer from a Small Heated Patch: Test Conditions and Key Boiling Characteristics

Test condition Result Bulk Bulk Pressure Temperature velocity, temperature, at patch, Subcooling overshoot, 2 m/s °C kPa ∆Tsub, °C °C CHF, mW/m 2.05 48.1 118.8 13.4 26.7 1.22 4.11 44.7 103.5 12.6 10.1 1.19 4.11 47.3 375.8 54.4 0 2.10 7.25 47.4 394.8 56.2 0 2.69 7.25 50.3 165.7 21.7 1.7 1.71 16.86 51.8 338.1 45.7 0 3.28

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FIGURE 15.16 Boiling curves of smooth and microstud surfaces.

ing (∆Tsub = Tsat – Tb). The CHF is a weak function of pressure and it decreases approximately as the 0.3 power of the heat length. Maddox and Mudawar [27] performed single- and two-phase experiments with FC-72 over a single heat source flush-mounted to one wall of a vertical rectangular channel. The heat sources tested in this study included a smooth surface and three low-profile microstud surfaces (as shown in Chapter 14) of varying height, each having a base of 12.7 mm × 12.7 mm. The boiling curves are presented in Figure 15.16. As shown in Figure 15.16, the higher velocities result in significant enhancement in the single-phase and nucleate boiling regions. For example, at a surface temperature of 85°C, the heat flux is increased by a factor of 2.6 when the velocity is increased from 0.4 to 2.25 m/s. The same figure also shows the effect of fin height on the boiling curve. All fin heights provide substantial enhancement in the single-phase region; however, the degree of enhancement is then decreased in the nucleate boiling region. The experimental data indicate that the worst case of temperature overshoot in the study is 7.5°C, occurring with the smooth surface at a very low velocity (0.75 m/s) and almost zero subcooling. Under similar conditions, however, the maximum temperature overshoot is 4°C in the case of microstud surfaces.

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FIGURE 15.17 Boiling curves for mini- and microchannel heat sinks.

FIGURE 15.18 Fluid exit temperature for mini- and microchannels.

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Peng, Wang, Peterson, and Ma [28–30] have jointly or separately performed a series of tests of flow boiling in microchannels. Their work is an extension of studies in single-phase flow described in Chapter 14. Their findings are as follows:

1. No bubbles are observed in the microchannels. This implies that there exists a minimum liquid bulk size (or space) at which bubbles can grow in liquid. 2. With a sufficiently high superheat, flow boiling is initiated at once and immediately progresses to fully developed nucleate boiling. Hence, although the liquid subcooling is sufficiently great, no apparent partial nucleate boiling regime exists. This is quite different from the well- known case of subcooled flow boiling in conventional-sized channels or tubes. 3. The velocity and subcooling appear to have no obvious effect on the flow nucleate boiling; however, they may significantly alter the onset of flow nucleate boiling.

Bowers and Mudawar [31] also performed an experimental study of flow boiling with R-113 in minichannels (D = 2.54 mm) and microchannels (D = 510 µm). The test channel length was 1 cm. The test conditions included inlet subcooling ranging from 10 to 32°C, and a range of flow rates up to a maximum of 95 ml/min. For the microchannels, the following CHF correlation was proposed:

− q′′ −  L  054. CHF,P = 016.We019.     (15.28) Ghfg D

where qCHF, P is CHF based on the heated channel inside area. The Weber number 2 We is defined as G L/(σ P f) and G and L are the mass velocity and tube heated length, respectively. The selected results are shown in Figures 15.17 and 15.18, and the key findings are as follows:

1. The CHF for mini- and microchannel heat sinks is not a function of inlet subcooling at low flow rates (see Figure 15.17), because the fluid reaches the saturation temperature at a short distance into the heated section of the channel. 2. The pressure drop in the microchannel is much higher than that of the minichannel. A reduction in the exit fluid temperature (see Figure 15.18) for the microchannel sink with increasing heat fluxes occurs because the fluid near the exit is at the saturation temperature as a function of the exit pressure. 3. The flow-boiling performance of both the mini- and microchannel heat sinks is superior to comparable single-phase systems. However, the minichannel clearly has the practical advantages over the microchannel because of a much lower system pressure drop (see Figure 15.19).

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FIGURE 15.19 Comparison of performance between mini- and microchannel heat sinks.

For example, the microchannel yielded a 28% increase in CHF as compared with the minichannel. A significant increase in the pressure drop, however, is required, from 0.008 bar for the minichannel to 0.32 bar for the microchannel.

REFERENCES

1. P. J. Marto and V. J. Lepere, “Pool Boiling Heat Transfer from Enhanced Sur- faces to Dielectric Fluids,” J. Heat Transfer 104, 1982. 2. N. Zuber, M. Tribus, and J. W. Westwater, “The Hydrodynamic Crisis in Pool Boiling of Saturated and Subcooling Liquids,” Int. Dev. Heat Transfer, Part II, ASME, 1961. 3. J. R. Saylor, A. Bar-Cohen, T. Y. Lee, T. W. Simon, W. Tong, and P. S. Wu, “Fluid Selection and Property Effects in Single- and Two-Phase Immersion Cooling,” IEEE Trans. Components, Hybrids, and Mfg. Technol. 11(4), 1988. 4. J. H. Lienhard and V. K. Dhir, “Hydrodynamic Prediction of Peak Pool Boiling Heat Fluxes from Finite Bodies,” J. Heat Transfer 95, 1973. 5. H. J. Ivey and D. J. Morris, “On the Relevance of the Vapor-Liquid Exchange Mechanism for Subcooled Boiling Heat Transfer at High Pressure,” U.K. Rep. AEEW-R-137, 1962. 6. T. Y. Lee, T. W. Simon, and A. Bar-Cohen, “An Investigation of Short-Heating- Length Effects on Flow Boiling Critical Heat Flux in a Subcooled Turbulent Flow,” in W. Aung, ed., Proc. Intl. Symp. on Cooling Technology for Electronic Equipment, Honolulu, HI, March 17-21, 1987.

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7. R. D. Danielson, L. Tousignant, and A. Bar-Cohen, “Saturated Pool Boiling Characteristics of Commercially Available Perfluorinated Insert Liquids,” ASME-JSME Thermal Eng. Joint Conf. 3, 1987. 8. W. M. Rohsenow, “A Method of Correlating Heat Transfer Data for Surface Boiling Liquids,” Trans. ASME 74, 1952. 9. K. A. Park and A. E. Bergles, “Effects of Size of Simulated Microelectronic Chips on Boiling and Critical Heat Flux,” J. Heat Transfer 110, 1988. 10. T. M. Anderson and I. Mudawar, “Microelectronic Cooling by Enhanced Boil- ing of a Dielectric Fluorocarbon Liquid,” ASME HTD-96 1, 1988. 11. J. L. Chang and S. M. You, “Enhanced Boiling Heat Transfer from Micro- Porous Cylindrical Surfaces in Saturated FC-87 and R-123,” J. Heat Transfer 119, 1997. 12. A. E. Bergles and C. J. Kim, “A Method to Reduce Temperature Overshoots in Immersion Cooling of Microelectronic Devices,” Proc. Intersoc. Conf. Ther- mal Phenomena in Electronic Components, May 1988. 13. I. A. Mudawar, T. A. Incropera, and F. P. Incropera, “Microelectronic Cool- ing by Fluorocarbon Liquid Films,” in W. Aung, ed., Proc. Intl. Symp. on Cooling Technology for Electronic Equipment, Honolulu, HI, March 17-21, 1987. 14. T. A. Grimley, I. A. Mudawar, and F. P. Incropera, “Enhancement of Boiling Heat Transfer in Falling Films,” Proc. ASME-JSME Thermal Eng. Joint Conf. 3, 1987. 15. T. A. Grimley, I. A. Mudawar, and F. P. Incropera, “Limits to Critical Heat Flux Enhancement in a Liquid Film Falling over a Structured Surface That Simu- lates a Microelectronic Chip,” J. Heat Transfer 110, 1988. 16. C. F. Ma and A. E. Bergles, “Boiling Jet Impingement Cooling of Simulated Microelectronic Chips,” Heat Transfer in Electronic Equipment, ASME HTD- 28, 1983. 17. K. A. Estes and I. Mudawar, “Comparison of Two-Phase Electronic Cooling Using Free Jets and Sprays,” J. Electronic Packaging 117, 1995. 18. J. S. Goodling, R. C. Jaeger, N. V. Williamson, C. D. Ellis, and T. D. Slagh, “Wafer Scale Cooling Using Jet Impingement Boiling Heat Transfer,” ASME Paper No. 87-WA/EEP-3, 1987. 19. S. C. Yao, “Boiling Heat Transfer for Electronic System Thermal Control,” 3rd International Symposium on Transport Phenomena in Thermal Control, Taipei, 1988. 20. M. Ghodbane and J. P. Holman, “Experimental Study of Spray Cooling with Freon-113,” Int. J. Heat Mass Transfer 34(4/5), 1991. 21. J. P. Holman and C. M. Kendall, “Extended Studies of Spray Cooling with Freon-113,” Int. J. Heat Mass Transfer 36(8), 1993. 22. P. J. C. Normington, M. Mahalingam, and T. Y. T. Lee, “Thermal Management Control without Overshoot Using Combinations of Boiling Liquids,” IEEE Trans. Components, Hybrids, and Mfg. Technol. 15(5), 1992. 23. Y. Katto, “General Features of CHF of Forced Convection Boiling in Uni- formly Heated Rectangular Channels,” Int. J. Heat Mass Transfer 24, 1981. 24. A. H. Iversen and S. Whitaker, “Uniform Temperature, Ultrahigh Flux Heat Sinks Using Curved Surface Subcooled Nucleation Boiling,” Proc. 5th IEEE SEM-THERM Symp., 1989.

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25. K. R. Samant and T. W. Simon, “Heat Transfer from a Small Heated Region to R-113 and FC-72,” J. Heat Transfer 111, 1989. 26. T. Y. Lee and T. W. Simon, “Critical Heat Flux in Forced Convection Boiling from Small Regions,” ASME HTD-119, 1989. 27. D. E. Maddox and I. Mudawar, “Single And Two-Phase Convective Heat Transfer from Smooth and Enhanced Microelectronic Heat Sources in a Rec- tangular Channel,” ASME HTD-96 1, 1988. 28. X. F. Peng and B. X. Wang, “Cooling Characteristics with Microchanneled Structures,” Enhanced Heat Transfer 1(4), 1994. 29. X. F. Peng and B. X. Wang, “Forced Convection and Flow Boiling Heat Trans- fer for Liquid Flowing through Microchannels,” Int. J. Heat Mass Transfer 36(14), 1993. 30. X. F. Peng, B. X. Wang, G. P. Peterson, and H. B. Ma, “Experimental Investi- gation of Heat Transfer in Flat Plates with Rectangular Microchannels,” Int. J Heat Mass Transfer 38(1), 1995. 31. M. B. Bowers and I. Mudawar, “High Flux Boiling in Low Flow Rate, Low Pressure Drop Mini-Channel and Micro-Channel Heat Sinks,” Int. J. Heat Mass Transfer 37(2), 1994.

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HEAT PIPES

A heat pipe is a self-contained closed system that transfers heat through boiling and condensation. The heat pipe is a passive device that provides a means to transfer large amounts of heat with relatively small temperature gradients from a single source or multiple sources to heat sinks. However, heat pipes alone cannot cool equipment. In fact, effective cooling of the condenser section of a heat pipe is required to maintain proper heat pipe performance. A similar device is known as a two-phase thermosyphon. It has the same basic heat transfer mechanism as a heat pipe, but an important difference exists in the mechanism of condensate return in these devices. A heat pipe relies on the capillary action of an internal wick and a working fluid, whereas a thermosyphon employs an external force field such as gravity or centrifugal force for condensate return to the evaporator. Since there is no wick involved, a two-phase thermosyphon is often referred to as a wickless heat pipe.

16.1 OPERATION PRINCIPLES

Functionally, a heat pipe consists of three sections, namely, evaporator, adiabatic section, and condenser. The adiabatic section may be omitted in some cases. The principle of operation of a heat pipe is shown in Figure 16.1. Heat that is added to the evaporator section of the heat pipe causes the liquid in the wick structure to vaporize. Thermal energy is carried by the high-pressure vapor from the evaporator (hot) end to the cold end of the heat pipe, where the vapor is condensed. The condensed liquid returns to the evaporator end by capillary action through a wick structure. The capillary action is the force upon the wick structure resulting from the relationship of viscosity, surface tension, and wetting ability of the liquid. The main driving force for the vapor is created by a higher vapor pressure in the evaporator, and the vapor flow is dependent on the geometry of the vapor passages and the viscosity of the vapor. Basic theories and applications of heat pipes are presented in Chi [1]. Thermal control of electronic equipment through use of heat pipes is presented in Reference 2.

16.2 USEFUL CHARACTERISTICS

Heat pipes have significant advantages over solid conduction due to several unique features:

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FIGURE 16.1 Heat pipe structure.

1. High heat transfer capacity. As a closed two-phase system, the heat pipe utilizes the latent heat of fluids. 2. Precise isothermal control. Increasing heat input to the evaporator results in only an increase in the rate of evaporation without any significant increase in operating temperature. 3. Functional independence of evaporator and condenser. The evaporator and condenser operate independently. This permits high heat fluxes over a small area of the evaporator to be dissipated over a larger area on the condenser section with reduced heat fluxes. 4. Quick thermal response. Because of the two-phase system, the thermal response time is much faster than that of a solid. 5. Remote applications. Heat pipes can be made in various shapes to serve remote heat sources. 6. High reliability. Since there are no moving parts, the heat pipe can provide silent, reliable, long-life operation. 7. Small size and light weight. Heat pipes are relatively compact in volume and weight.

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16.3 CONSTRUCTION

A heat pipe physically consists of a container, a wick structure, and a working fluid. The outer shell of a heat pipe is often referred to as the container, as it is utilized to enclose all functioning parts, and to lend it structural rigidity. The internal pressure is generally different from the environmental pressure, and the container must be capable of withstanding this pressure differential in order to maintain the structural integrity. The materials for the container must also be compatible with the working fluid. Possible candidate materials are glass, copper, aluminum, stainless steel, and tantalum. The container provides radial heat conduction paths from heat sources and sinks to the working fluid. The wall thickness of the container should be as thin as possible to minimize the thermal resistance. Heat conduction also takes place between the evaporator and the condenser through the container wall. If the majority of heat is transferred by conduction along the container wall, a significant temperature gradient will be created between the evaporator and the condenser. This thermal path should be insulated to force heat to go to the working fluid, where heat can be transferred by evaporation and condensation with a small temperature gradient along the length of the heat pipe. Therefore, a thin but strong container wall is desirable. The wick structure is usually a porous material filled with small, randomly interconnected capillary channels. The possible types of wick structures are (1) porous wicks, such as open cell foams, screens, and sintered powders; (2) longitudinal or circumferential grooves; (3) composite wicks, such as conventional artery and spiral artery; and (4) packed beads. The basic requirements of a wick structure are as follows:

• Pumping power. Small pores, especially in the evaporator section, lead to a large liquid pumping force developed by surface tension of fluid in the wick at the liquid-vapor interface. • Liquid flow path. Large and smooth wall channels reduce the viscous losses when liquid returns from the evaporator to the condenser. • Radial thermal path. High thermal conductivity of both liquid and wick material results in a small resistance across the liquid-wick composite structures at both evaporator and condenser sections. • Liquid-vapor flow path. Physical separation between vapor and liquid paths is desirable in reducing the counterflow shear, particularly in high vapor velocity conditions.

The selection of working fluids for a heat pipe is determined by the physical properties of the fluid and by the chemical compatibility of the fluid with the container wall and wick structure. The heat pipe should operate at a pressure that is located at a steep-slope section of the vapor pressure-temperature curve. Then, any pressure drop in the vapor core between the evaporator and condenser will generate only a very small temperature drop along the length of the heat pipe. Other important fluid properties are as follows:

1. Latent heat hfg. A high latent heat is desirable in order to transfer the maximum amount of heat with the minimum mass flow rate.

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2. Thermal conductivity k. High thermal conductivity of the liquid minimizes the radial temperature gradient across the liquid layer or liquid- wick structure. 3. Surface tension σ. Increasing the surface tension increases the capillary pumping force; however, a decrease in surface tension increases the wetting ability of a fluid. 4. Viscosity µ. Low viscosity for both liquid and vapor reduces the flow resistance. 5. Boiling point. The operating temperature is a function of the fluid boiling point.

Possible candidate fluids are liquid nitrogen, alcohol, water, Freon, potassium, sodium, cesium, lithium, and silver. Based upon the importance of fluid properties on the operation of the heat pipe, the liquid transport factor, defined in the following equation, is used to evaluate various working fluids at specific operating temperatures:

ρσh N = lfg (16.1) l µ l

The liquid transport factor, which is given in units of W/m2, is often also referred to as the figure of merit. This factor is usually used to select the working fluid for a given heat pipe.

16.4 OPERATION LIMITS

The proper functioning of heat pipes depends on continuous circulation of the working fluid. Consequently, virtually all of the limitations for successful heat pipe operation are associated with interruption of fluid circulation. For a steady- state condition, the heat pipe is subjected to operation limits discussed in the following subsections.

16.4.1 Capillary Wicking Limit

In order for the fluid to circulate in a heat pipe, the following condition must be met:

∆pca ≥∆pn + ∆pa + ∆pl + ∆pv (16.2)

where individual components in the equation are described in the following (to aid in the description, heat pipe nomenclature is illustrated in Figure 16.2):

1. ∆pca is the net capillary pressure difference between the evaporator and the condenser:

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FIGURE 16.2 Heat pipe nomenclature.

∆ =− −− pppppca()() v l e v l c σθ σθ =−22cose cos c (16.3) rre c

where θe and re , and θc and rc are the contact angles and radii of curvature in the evaporator and condenser, respectively. For most heat pipe wick structures, the maximum capillary pressure may be expressed in terms of a single radius of curvature by letting θ be zero in both evaporator and condenser. Due to vaporization, the liquid in the evaporator recedes into the wick, thereby reducing the capillary radius re. On the other hand, because of condensation, the liquid in the condenser floods the wick, increasing the capillary radius rc. The difference in the two capillary radii of curvature causes a difference in pressure, which pumps the liquid from the condenser to the evaporator. The maximum heat transfer occurs when the evaporation and condensation rates are at maximum. The maximum liquid flow rate exists when the meniscus radius of curvature in the evaporator is at minimum (θe =0°) and that in the condenser is at maximum (θc =90°). The maximum capillary pumping becomes

σ ∆ = 2 pc, max (16.4) re

Values for the effective capillary radius (re or rc) in some common wick structures are listed in Table 16.1 [1].

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Table 16.1 Expressions for Effective Capillary Radius

Structure re or rc = Definitions Circular cylinder rr= radius of liquid flow passage Rectangular groove ww= groove width Triangular groove w/cos β w = groove width β = half-included angle Parallel wires ww= wire spacing Wire screens (w + dw)/2 w = wire spacing = 1/2Ndw = wire diameter N = screen mesh number Packed spheres 0.41rr= sphere radius

2. ∆pn is the normal hydrostatic pressure drop. This pressure drop occurs only in a heat pipe in which circumferential connection of the liquid in the wick is possible, and it thus equals zero for the rectangular axial groove structure:

∆pn = ρlgdv cos φ (16.5)

where dv is the vapor core diameter and φ is measured from the horizontal for an inclined heat pipe. 3. ∆pa is the axial hydrostatic pressure drop:

∆pa = ±ρlgL sin φ (16.6)

where L is the length of the heat pipe. Depending on the relative positions of the evaporator and condenser, this pressure drop may either assist or resist the capillary pumping process in the gravitational environment. The plus sign in Equation 16.6 is applied when the evaporator is above the condenser, while the minus sign is used when the condenser is above the evaporator. The axial hydrostatic pressure drop will enhance the heat pipe performance in the latter case. ∆ 4. pl is the viscous pressure drop in the liquid. Feldman [3] analyzed the viscous losses in liquid flowing through wick structures. Since the velocity encountered in wicks is typically small, the flow can generally be assumed to be laminar and relatively free from inertial effects. The pressure drop in liquid through porous media gives the following equation:

µ Lm′ ∆p = ll l ρ (16.7) lwKA

where

′ L = effective flow channel length =++ 05.Lea L 05 .L c = = q  ml liquid mass flow rate  hfg  K = wick permeability = Aw total cross - sectional area of the wick

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Table 16.2 Expressions for Wick Permeability

Structure Wick permeability K= Definitions Circular artery r2/8 r = radius 2 Rectangular grooves 2εr /(f Re)l ε = porosity = w/s w = groove width s = groove pitch r = 2wd/(w +2d) δ = groove depth * 2 Circular annular wick 2r /(f Re)l r = r1 – r2 Wrapped screen wick d2ε3/122(1 – ε)2 d = wire diameter ε = 1 – 1.05πNd/4 N = mesh number Packed spheres r2ε3/37.5(1 – ε)2 r = sphere radius ε = porosity

* (f Re)l is a product of friction factor and Reynolds number in a liquid flow.

and where Le, La, and Lc are the lengths of the evaporator, adiabatic section, and condenser, respectively. Table 16.2 lists expressions for wick permeability K. 5. ∆pv is the viscous pressure drop in the vapor core. For incompressible flow, this pressure drop can be computed by the following equation, i.e.,

′ 4fLDm(/)(/) A2 ∆p = v v ρ (16.8a) 2 v

where f depends on laminar or turbulent flow. The flow may be treated as a com- pressible flow if the Mach number is greater than 0.3. Chi [1] considered a com- pressible flow with the Mach number greater than 0.2 and proposed the following equation to calculate the vapor viscous pressure drop:

Cf(Re)µ Lm′ ∆p = vvvv (16.8b) v 2ρ 2rAvvv

where = Av total cross - sectional area of vapor core = rv vapor core hydraulic radius == mvfgvapor mass flow rate (q/h ) = fv friction factor 2rq Re = v v µ vvfgAh q = heat rate

and where C is a function of the Mach number Rev. According to Chi [1], C can be estimated from the following equations.

C = 1 for Mv < 0.2 (16.9a)

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−12/  M (γ −1)  + v  CM = 1 for v > 0.2 (16.9b)  2 

where

q M = v γρ12/ (16.9c) ()RTvvv v Ah vfg

The friction factor is expressed as follows:

fvRev = 16 for Rev < 2300 (16.9d)

and

-1/4 fv = 0.038 (Rev) for Rev > 2300 (16.9e)

where Rv and γv are the gas constant and the ratio of specific heats (cp/cv), the latter of which is equal to 1.67, 1.4, and 1.33 for monatomic, diatomic, and poly- atomic gases, respectively. Mv is the local Mach number, and Tv is the vapor temperature. If the sum of all pressure drops on the right-hand side of Equation 16.2 is greater than the net capillary force between the evaporator and condenser, the liquid cannot be returned to the evaporator fast enough to compensate for its loss by evaporation. Eventually, the wick structure becomes starved of liquid and dries out. This condition is known as the capillary wicking limit, and the heat transfer rate is

qc = mlhfg (16.10)

With the aid of Equations 16.5 through 16.9, one can solve for ml (=Mv) by equat- ing both sides of Equation (16.2).

16.4.2 Sonic Limit

The sonic limit is a condition found in heat pipes in which the vapor flow chokes at the evaporator exit. Lowering the condenser temperature results in a decrease in the evaporator temperature that in turn reduces the vapor density and increases the vapor velocity. Any further reduction in the condenser temperature does not continue to reduce the evaporator temperature when the vapor Mach number becomes 1 at the evaporator exit. This condition is called the sonic limit of a heat pipe. The sonic limit heat transfer rate can be determined from the definition given in the following equation:

05.(RTγ )12/ qAh= ρ  vvv svvfg + γ (16.11)  1 v 

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Increasing the heat rejection rate at the condenser beyond the sonic limit creates very large axial temperature gradients along the heat pipe, but it does not increase the heat transfer rate.

16.4.3 Entrainment Limit

Due to the interaction of the vapor flowing in one direction and the liquid flowing in the opposite direction along the heat pipe, liquid droplets may be picked up, or entrained, into the vapor flow. At high-velocity operation conditions, this interaction is sufficient to draw a large amount of liquid out of the wick, in turn reducing the amount of liquid flow to the evaporator, and causes dryout of the evaporator. The onset of entrainment generally occurs when the Weber number is equal to 1. The Weber number, We, which is defined as the ratio of the viscous shear force to the force resulting from the liquid surface tension, is expressed as follows:

2rVρ 2 We = wvv (16.12) σ

where the vapor velocity is

q V = v ρ (16.13) Ahvvfg

To avoid reaching the entrainment limit, We must be less than 1. The heat transport capacity at the entrainment limit is

 σρ 12/ = v qAhevfg  (16.14)  2rw 

where rw is the hydraulic radius of the wick structure and is defined as twice the area of the wick pore at the wick-vapor interface divided by the wetted perimeter at this interface.

16.4.4 Boiling Limit

With increasing radial heat flux at the evaporator, vapor bubbles may be formed in the wick. The vapor bubbles in the wick structure may cause hot spots and block the circulation of liquid. Hence, there is a heat flux limit for the evaporator, and this limit is called the boiling limit, defined in the following equation:

22πLkT′  σ  q = ev − ∆p b ρ  ca (16.15) hrrrfg vln ( i / n )  n 

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Table 16.3 Effective Thermal Conductivity of Liquid and Wick

Wick structure ke =

kk Wick and liquid in series lw εε+− kkwl()1

Wick and liquid in parallel εkw + (1 – ε)kl +−−−ε kkll[( k w ) (1 )( k l k w )] Wrapped screen ++−−ε ()()()kklw1 kk lw +−−−ε kkkllw[(221 ) ( )( kk lw )] Packed spheres ++−−ε ()()()21kklw kk lw δδ++ wkkflw wk l(.0 185 wk fw k l) Rectangular grooves ++δ ()(.wwfffl0 185 wk k )

where kl = liquid thermal conductivity w = groove thickness kw = wick thermal conductivity wf = groove fin thickness ε = wick porosity δ = groove depth

where ke is the effective thermal conductivity of the liquid-wick combination, for which values are given in Table 16.3 [1]; ri is the radius of the heat pipe inner wall; and rn is the nucleate site radius, which can be assumed to be from 2.54 × 10–5 to 2.54 × 10–7 according to Dunn and Reay [4].

16.4.5 Viscous Vapor Flow Limit

At very low operating temperatures, the vapor pressure difference between the evaporator and the condenser may be so small that the viscous forces within the vapor core become dominant and limit the vapor flowing from the evaporator to the condenser. The following criterion is suggested as a limit in Feldman [3]:

∆pv < 0.1pv (16.16)

where pv is the system vapor pressure.

Combining all the limits defined in this section, a heat pipe can operate only under the curve ABCDEF as shown in Figure 16.3.

16.5 MATERIALS COMPATIBILITY

The limits discussed in Section 16.4 define the operation boundary of heat pipes. From a manufacturing viewpoint, the heat pipe is also subjected to many potential failure modes due to contamination, leaks, corrosion, and erosion. To

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FIGURE 16.3 Heat pipe operating limits.

Table 16.4 Heat Pipe Material Compatibility

Working fluid Wick Freon Freon material H2O Acetone NH3 Methanol Dow-A Dow-E 11 13 Copper RU RU NR RU RU RU RU RU Aluminum GNC RL RU NR UK NR RU RU Stainless steel GNT PC RU GNT RU RU RU RU Nickel PC PC RU RL RU RL UK UK Refrasil RU RU RU RU RU RU UK UK

RU Recommended by past successful use RL Recommended by the literature PC Probably compatible NR Not recommended UK Unknown GNC Generation of noncondensable gas at all temperatures GNT Generation of noncondensable gas at elevated temperature when oxide is present

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FIGURE 16.4 Heat pipe operating temperatures.

insure long-life operation of heat pipes, materials compatibility between the working fluid, wick structure, and container material is required. Basiulis and his associates [5, 6] performed extensive tests to assess material compatibility over a long period, and their results are given in Table 16.4.

16.6 OPERATING TEMPERATURES

Based on the operating temperature, heat pipes can be classified into three categories, namely, low-, moderate-, and high-temperature heat pipes. Low-temperature heat pipes are applied to cryogenic cooling and are also called cryogenic heat pipes. On the other hand, high-temperature heat pipes utilize liquid metals as the working fluid; hence, they are also referred to as liquid metal heat pipes. Moderate-temperature heat pipes use organic and inorganic fluids. The operating temperature ranges, which depend on the boiling points of working fluids for the above three types of heat pipes, are given in Figure 16.4. As stated by Kraus and Bar-Cohen [7], most applications of heat pipes for cooling of electronics require the boiling point of the working fluids to be between 250 and 375 ˚K. These fluids include ammonia; Freon 11, 21, and 113; acetone; methanol; and water.

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The effect of the various operating limits on heat pipe operating temperature ranges varies. The most significant operating limits for the individual heat pipe types are as follows:

1. High-temperature heat pipes: sonic and entrainment limits 2. Moderate-temperature heat pipes: capillary wicking limit 3. Low-temperature heat pipes: viscous vapor flow and capillary limits

16.7 OPERATION METHODS

Most heat pipes automatically adjust the operating temperature to match the heat source and sink conditions in order to satisfy the law of conservation of energy. There are applications in which a specific operating temperature range must be maintained in spite of the variations in the source or sink conditions. In these applications, it is necessary to actively or passively control the heat pipe to maintain the desired temperature ranges. Marcus and Fleischman [8] presented three fundamental approaches to controlling heat pipes: (1) control of the vapor flow, (2) control of the liquid flow, and (3) control of the heat transfer into or out of the heat pipe. One example of a method of controlling heat transfer out of the heat pipe is the use of noncondensable gases. The presence of noncondensable gases reduces the heat transfer area in the condenser and creates a serious problem for a heat pipe. This characteristic, which is normally undesirable, can be put to advantage to control the amount of heat transfer out of the heat pipe as shown in Figure 16.5a [1]. A heat pipe which is intentionally filled with a noncondensable gas is called a gas-controlled heat pipe. The controlling gas effectively shuts off a certain portion of the condenser. As the operating temperature increases, the vapor pressure increases, compressing the noncondensable gas into a smaller volume and pushing it toward the gas reservoir. This action increases the active condenser area. On the other hand, as the operating temperature falls, the vapor pressure decreases, causing the noncondensable gas to expand to a greater volume. This in turn reduces the active condenser area. The net effect is to provide the heat pipe with a passively controlled variable thermal conductance. With the addition of sensors at the evaporator and heaters at the gas reservoir, the heat pipe becomes an actively controlled variable-conductance device, as indicated in Figure 16.5b [1]. The temperature sensor at the evaporator provides a signal to turn the reservoir heaters on or off in order to control the volume of the noncondensable gas. Figure 16.5c [1] illustrates the vapor-modulated variable-conductance heat pipe, which utilizes a throttling valve to control the amount of vapor flow from the evaporator to condenser. As the evaporator temperature increases, the fluid in the bellows chamber expands. This leads the throttling valve to close down and reduces the flow of vapor to condenser. This type of heat pipe is typically applied to the case where the evaporator temperature varies and a constant condenser temperature is desired. The liquid-modulated heat pipe shown in Figure 16.5d [1] has two separate wick structures. The second wick structure on the opposite end of the condenser serves as a liquid trap. In the normal mode of operation, the trap wick is dry. As

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FIGURE 16.5 Control techniques for heat pipe operations.

the heat flux increases or the environment of the heat pipe changes, the liquid trap section becomes the cold end of the heat pipe. Condensation begins to occur in the trap. As the liquid accumulates in the trap end, the main heat pipe wick becomes starved, and this results in a rapid reduction in capillary pumping capacity. Eventually, the main wick dries out completely and the heat pipe ceases to function.

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FIGURE 16.6 Electrothermal analog for a heat pipe.

16.8 THERMAL RESISTANCES

The thermal resistance between the evaporator and condenser is the most important information required in applying a heat pipe to any thermal control system. This thermal resistance can be estimated by using an analogous electrothermal network presented by Marto and Peterson and shown in Figure 16.6 [2]. The overall thermal resistance consists of nine different resistances arranged in a series-parallel combination. The individual thermal resistances shown in Figure 16.6 are defined as follows:

Rpe radial conduction resistance of the heat pipe wall at the evaporator Rwe resistance of the liquid-wick combination at the evaporator Rie resistance of the liquid-vapor interface at the evaporator Rva resistance of the adiabatic vapor section Rpa axial resistance of the heat pipe wall Rwa axial resistance of the liquid-wick combination Ric resistance of the liquid-vapor interface at the condenser Rwc resistance of the liquid-wick combination at the condenser Rpc radial conduction resistance of the heat pipe wall at the condenser

The values of individual thermal resistances calculated by Asselman and Green [9] are listed in Table 16.5. In comparing these values, it appears that the

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Table 16.5 Comparative Values of Heat Pipe Resistances

Individual resistance Value, °C/W

–1 Rpe and Rpc 10 +1 Rwe and Rwc 10 –5 Rie and Ric 10 –8 Rva 10 +2 Rpa 10 +4 Rwa 10

network can be simplified by omitting the two axial thermal resistances Rpa and Rwa. The overall or total thermal resistance Rt then becomes the sum of a series of resistances, i.e.,

Rt = Rpe + Rwe + Rie + Rva + Ric + Rwc + Rpc (16.17)

Depending on the heat pipe configuration, rectangular or round, the wall conduction resistance is

 t  for a rectangular or flat heat pipe (16.18a)  kA RR (or ) =  pp pe pc ln (dd / )  oi for a round heat pipe (16.18a)  π  2 Lkpp

where kp is the thermal conductivity of the heat pipe container wall and t, do, and di are the thickness and outer and inner diameters of the container wall, respectively. Ap and Lp are the heat transfer area and length, respectively, for the evaporator or condenser. Similarly, the equivalent thermal resistance of the liquid/wick combination for the evaporator or condenser is

ln (dd/ ) RR (or ) = oi we wc π (16.19) 2 Lkpe

where ke is the effective thermal conductivity listed in Table 16.3. The adiabatic section vapor resistance can be expressed by the following equation:

Tp()− p R = vveve,, va ρ (16.20) vfghq

where Tv is the saturated vapor temperature and pv, e and pv, c are the vapor pressures at the evaporator and condenser, respectively.

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FIGURE 16.7 Effect of orientation on heat pipe performance.

16.9 APPLICATIONS

The operation of a heat pipe is not dependent on gravity; however, gravity does have a significant effect on heat pipe performance, depending on the orientation of the heat pipe. Heat pipe performance is improved if the condenser is located above the evaporator. Figure 16.7 [10] clearly illustrates the effect of orientation on the performance of heat pipes. The minus sign indicates a heat pipe orientation where the condenser is above the evaporator. In this case, the liquid is assisted by gravity in flowing from the condenser to the evaporator, and the result is higher heat transfer rates. As the inclined angle increases from negative to positive values, the heat transfer rate decreases because of an increase in liquid flow resistance from the condenser to the evaporator. Figure 16.8 shows the performance of a heat pipe with two condensers, one on each end. The overall dimensions of this heat pipe are identical to those shown in Figure 16.7. Examining both figures in detail reveals that the heat transfer rate is doubled as compared with the previous single-condenser heat pipe. The degree of the inclined angle is still important to heat pipe performance, but the direction of the inclination becomes irrelevant due to the symmetrical configuration of the heat pipe.

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FIGURE 16.8 Performance of double-condenser heat pipe.

The application of heat pipes to electronic equipment has become more popular in recent years. Sekhon [11] described circuit-card (or board) heat pipes and standard electronic module (SEM) heat pipes developed by Hughes. The experimental results of testing circuit-card heat pipes with 30 W heat load are presented in Figure 16.9. As can be seen from the figure, the temperature rise over the heat pipe is much smaller than the rises over copper and aluminum plates used for the same purpose. Similar trends were also found by Feldmanis [12], who made comparisons of temperature distribution along a circuit board between applications involving both conventional conduction paths and heat pipes. As shown in Figure 16.10 [11] for SEM heat pipes, heat dissipated by electronic components is transferred by heat pipes to the edges of modules (or cards) and is then rejected from the heat pipes’ condenser to the coolant or the cold walls. Basiulis et al. [13] employed heat pipes for cooling the printed circuit or wiring boards as shown in Figure 16.11. By utilizing the above concepts, Token et al. [14] summarized various system studies using heat pipes for avionics thermal control. Nelson et al. [15] proposed direct heat pipe cooling of semiconductor devices as illustrated in Figure 16.12. The circuit, including the open transistor cases and the entire inside surface of the module housing, is coated with a compatible wick

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FIGURE 16.9 Performance of circuit-card heat pipes.

FIGURE 16.10 Cutaway view of heat pipe–cooled SEM.

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FIGURE 16.11 Edge-cooled heat pipe printed wiring board.

FIGURE 16.12 Direct heat pipe cooling of a semiconductor.

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FIGURE 16.13 Temperatures of a PCB rack with and without heat pipes.

material. Heat is absorbed by the evaporation of heat pipe fluid and carried to the alumina substrate, where it is released by condensing fluid and transferred through the module housing to the heat sink. With this approach, the thermal resistance from junction to case can be reduced as much as 33%. An approach using heat pipes for cooling a rack which contains numerous conduction-cooled printed circuit boards (PCBs) is shown in Figure 16.13. The PCB rack is enclosed in a large electronic box. All the PCBs are mechanically clamped to the aluminum structures in the rack. Heat generated from electronic components is transferred to the edges of the individual PCB, and is further carried away by conduction through three aluminum structures to the base plate. Heat is finally rejected to the ambient by a finned heat sink which is bonded to the base plate. Thermally, the base plate provides a means to spread the heat prior to the finned heat sink. In the system, the finned heat sink is the only surface exposed to the ambient. Due to heat conduction, a large temperature gradient exists in the structure between the first and the last PCBs. To improve the thermal design, heat pipes are embedded in the aluminum structures. The horizontal section of the heat pipe is the evaporator, while the vertical section serves as the condenser. There are 10, 10, and 6 heat pipes in the top, middle, and bottom structures, respectively. The middle structure carries almost twice as much heat as compared with the top or the bottom structure. Therefore, the number of heat pipes required is

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FIGURE 16.14 Heat pipe heat exchanger for cooling enclosed electronics.

larger than for the other two structures. The performance of heat pipes in the top structure is less effective than that in the middle or the bottom structure because the evaporator is above the condenser. For this reason, 10 heat pipes are also used in the top structures. Heat dissipation for each PCB is also listed in Figure 16.13. With the application of heat pipes, the temperature gradient along the aluminum structure is reduced from about 20 to 1°C. The same amount of temperature reduction is expected for the electronic components on those PCBs located far away from the base plate. The temperature at two major locations with and without heat pipes is also included in the figure for comparison. Heat pipes can also be used as heat exchangers. Figure 16.14 shows an air-to- air heat exchanger consisting of many parallel heat pipes. In this case, two airstreams are completely separated. This heat pipe heat exchanger as shown in the figure is utilized to prevent electronic components in the cabinet from being contaminated by the ambient air.

16.10 MICRO HEAT PIPES

Cotter [16] first proposed using micro heat pipes for cooling microelectronic devices. He defines a micro heat pipe as “one so small that the mean curvature of

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the vapor liquid interface is necessarily comparable in magnitude to the reciprocal of the hydraulic radius of the total flow channel.” Mathematically, this can be expressed by the following equation [17]:

∝ 1 rc (16.21a) rh

rc ≥ 1 (16.21b) rh

The latter equation better defines a micro heat pipe and helps to differentiate between small conventional heat pipes and the micro heat pipes. In practical applications, a micro heat pipe is a wickless, noncircular channel with an approximate diameter of 100 to 1000 µm and a length of about 10 to 20 mm. Figure 16.15 shows two different configurations of the micro heat pipe [18–20]. The fundamental principles of micro heat pipes are the same as those in larger, conventional heat pipes. Heat that is added to one end of the heat pipe vaporizes the liquid in the evaporator. The vapor then flows to the cold end, where it condenses and gives up the latent heat of vaporization. The vaporization and condensation process causes the liquid-vapor interface in the liquid arteries to change continually along the length of the heat pipe as indicated in Figure 16.15 and results in a capillary pressure difference between the evaporator and the condenser. The corner regions serve as liquid arteries, and thus no wicking structure is required. The steady-state condition is dealt with in Reference 18, while Reference 19 considers the transient condition. However, most references consider the cases with a single micro heat pipe. Mallik et al. [21] developed a transient three- dimensional numerical model of an array of triangular micro heat pipes fabricated as an integral part of silicon wafers. Later, Peterson et al. [22] performed an experimental investigation to determine the thermal behavior of arrays of micro heat pipes fabricated in silicon wafers. In their work, two types of micro heat pipe arrays were evaluated. One used machined rectangular channels 45 µm wide and 80 µm deep, and the other employed an anisotropic etching process to produce triangular channels 120 µm wide and 80 µm deep. The experimental data were then compared with those for a plain silicon wafer. The experimental data indicate that embedding micro heat pipes in the wafers significantly increases the effective thermal conductivity of the wafers. This leads to a great reduction in the wafer temperature and temperature gradient. As shown in Figure 16.16, at a power input of 4 W, increases in the effective thermal conductivity of 31 and 81% and reductions in the maximum chip temperature of 14.1°C and 24.9°C are found for the machined rectangular and etched triangular heat pipe arrays, respectively.

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FIGURE 16.15 Micro heat pipes.

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FIGURE 16.16 Data for rectangular and triangular heat pipe arrays.

REFERENCES

1. W. S. Chi, Heat Pipe Theory and Practice: A Sourcebook, McGraw-Hill, New York, 1976. 2. P. J. Marto and G. P. Peterson, “Application of Heat Pipes to Electronics Cool- ing,” in A. Bar-Cohen and A. D. Kraus, eds., Advances in Thermal Modeling of Electronic Components and Systems, Vol. 1, Hemisphere, New York, 1988. 3. K. T. Feldman, “Analysis and Design of Heat Pipes,” University of New Mex- ico, Albuquerque, 1970. 4. P. D. Dunn and D. A. Reay, Heat Pipes, 3rd ed., Pergamon, New York, 1982. 5. A. Basiulis and M. Filler, “Operating Characteristics and Long Life Capabili- ties of Organic Fluid Heat Pipes,” AIAA Paper 71-408, 1971. 6. A. Basiulis, R. C. Prager, and T. R. Lamp, “Compatibility and Reliability of Heat Pipe Materials,” Proc. 2nd Int. Heat Pipe Conf., Bologna, Italy, 1976. 7. A. D. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment, Hemisphere, Washington, DC, 1983.

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8. B. D. Marcus and G. L. Fleischman, “Steady State and Transient Performance of Hot Reservoir Gas Control Heat Pipes,” NASA CR-73420, 1971. 9. G. A. A. Asselman and D. B. Green, “Heat Pipes,” Phillips Tech. Rev. 16, 1973. 10. T. Murase, K. Yoshida, J. Fujikake, T. Koizumi, and N. Ishida, “Heat Pipe Heat Sink Heat Kicker for Cooling of Semiconductors,” Furukawa Rev. 2, 1982. 11. K. S. Sekhon, “Applications of Heat Pipes to Control Electronics Tempera- ture in Radars,” IEEE Mechanical Engineering in Radar Symposium, 1977. 12. C. Feldmanis, “Cooling Techniques and Thermal Analysis of Circuit Board Mounted Electronic Equipment,” Heat Transfer in Electronic Equipment, ASME HTD-20, 1982. 13. A. Basiulis, H. Tanzer, and S. McCabe, “Heat Pipes for Cooling of High Den- sity Printed Wiring Boards,” Proc. 6th Int. Heat Pipe Conf., Grenoble, France, 1987. 14. K. H. Token, E. C. Garner, R. R. Cook, and J. E. Stone, “Heat Pipe Avionic Thermal Control,” AIAA-80-1511, 1980. 15. L. A. Nelson, K. S. Sekhon, and J. E. Fritz, “Direct Heat Pipe Cooling of Semiconductor Devices,” Proc. 3rd Int. Heat Pipe Conf., Palo Alto, CA, 1978. 16. T. P. Cotter, “Principles and Prospects of Micro Heat Pipes,” Proc. 5th Int. Heat Pipes Conf., Tsukuba, Japan, 1984. 17. G. P. Peterson, “Investigation of Miniature Heat Pipes,” Final Report, Wright Patterson AFB Contract No. F33615-86-C-2733, Task 9, 1988. 18. B. R. Babin, G. P. Peterson, and D. Wu, ”Analysis and Testing of a Micro Heat Pipe during Steady-State Operation,” J. Heat Transfer 112, 1990. 19. D. Wu and G. P. Peterson, “Investigation of the Transient Characteristics of a Micro Heat Pipe,” J. Thermophysics 5, 1991. 20. G. P. Peterson, “Overview of Micro Heat Pipe Research and Development,” Appl. Mech. Rev. 45, 1992.21. 21. A. K. Mallik, G. P. Peterson, and M. H. Weichold, “On the Use of Micro Heat Pipes as an Integral Part of Semiconductors,” J. Heat Transfer 114, 1992. 22. G. P. Peterson, A. B. Ducan, and M. H. Weichold, “Experimental Investiga- tion of Micro Heat Pipes Fabricated in Silicon Wafers,” J. Heat Transfer 115, 1993.

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THERMOELECTRIC COOLERS

The principals of thermoelectricity were first discovered during the first half of the 19th century, but no practical applications of the principles have been applied until in the second half of the twentieth century. Two major classes of application of thermoelectric devices are heat pumps and power generators. In the former, electrical energy is supplied to transport thermal energy from one location to other locations; while in the latter, thermal energy is converted into electrical energy. The following characteristics of thermoelectric heat pumps are particularly useful in thermal management of electronic packaging:

1. Heating or cooling can be achieved by controlling the direction of the electrical current. 2. The heat transfer capacity depends on the current. 3. The system is very reliable and quiet because it has no moving parts. 4. The device is independent of gravity and system orientation.

On the other hand, thermoelectric heat pumps have the disadvantage of a low coefficient of performance (COP). In other words, it requires a large electrical power input to pump a small amount of heat.

17.1 BASIC THEORIES OF THERMOELECTRICITY

The net thermoelectric effect consists of several distinct individual effects that are described in this section. Seebeck Effect. An electromotive force (emf) or voltage will be generated if two junctions of two different materials are held at different temperatures. This emf is proportional to the temperature difference between the two junctions as follows

dV = α12 dT (17.1)

where the proportional constant α is called the Seebeck coefficient and has the units of V/°C. The constant α12, defined in the next equation, can be either positive or negative:

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α12 = α1 – α2 (17.2)

Generally, α12 is positive if material 1 is electrically positive with respect to material 2 when the couple is opened at the cold junction. Peltier Effect. Heat will be generated or rejected at the junction of two different materials when a direct current is applied. The amount of heat is found to be proportional to the current:

qP = π12I (17.3)

where the proportional constant π is called the Peltier coefficient and can be positive or negative depending on the relationship between materials 1 and 2 as in the case of the Seebeck coefficient. For a positive value of π12, material 1 is thermoelectrically positive with respect to material 2. Then, qP is produced (i.e., has a positive value) if the current I from 1 to 2 is considered to be positive. Based on a thermodynamic analysis, William Thomson [1,2] developed the following relationship:

π = αT (17.4)

where T is the absolute temperature. Therefore, the heat absorbed or rejected per unit time at a junction between two different materials can be expressed as follows:

qP = α12IT (17.5)

Thomson Effect. Based on experimental studies, Thomson discovered that heat absorbed or rejected per unit length could be expressed in the following form:

dT qI= τ (17.6) T dx

where τ is called the Thomson coefficient. The above three effects represent reversible interchange of electrical and thermal energies. Two other effects, which are irreversible effects of heat evolution, are described in the following paragraphs. Joule Effect. The Joule effect states that the heat rate generated by an electrical current is equal to the product of the electrical resistance of the conductor and the square of the current, i.e.,

2 qJ = I R (17.7)

where the resistance R = ρ(L/A) and ρ is the electrical resistivity, in Ω·cm. Fourier Effect. The Fourier law states that the heat rate is proportional to the area normal to heat flow and the temperature gradient along the conduction path:

dT qkA=− (17.8) F dx

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where k, A, and x are thermal conductivity of the material, conduction area, and length, respectively. Since Thomson heating usually plays a secondary role in the design of thermoelectric devices, it is often neglected to simplify the analysis.

17.2 NET THERMOELECTRIC EFFECT

In a single thermoelectric medium with two ends maintained at fixed temperatures of Th and Tc, the net heat transfer for a small element dx under the steady- state condition can be expressed in the following form:

qF, x – qF, x + dx + qJ = 0 (17.9)

With the aid of Equations 17.7 and 17.8, Equation 17.9 can be rewritten into a differential form as follows:

dT2 I2ρ kA +=0 (17.10) dx2 A

where the boundary conditions are

T = Th x = 0 (17.11a) T = Tc x = L (17.11b)

and where Th > Tc. The general solution to Equations 17.10 and 17.11 is

 x   x  2 Tx()=− T [( T − T ) − P ]  − P  (17.12) hhc  L  L

where

 IL 2 ρ L  ρL P = 05.   == 05. (IR2 ) R   Ak kA  A

To find the location of the maximum temperature, denoted by x*, let dT/dx be zero. Hence

LTTL()− x* =−hc (17.13) 22P

This clearly indicates that heat generated due to the Joule effect will flow to both * ends of the material because the temperature at x is greater than Th and Tc. The fractional amount of energy transferred from x* to the cold end (junction) is

x* f =−1 (17.14) L

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The conductive heat flow to the cold end is

= 2 qfIRc 2 − (17.15) =+IR() Thc T kA 2 L

Finally, if both ends are junctions of two different materials forming a thermoelectric couple, then the net cooling capability, including the Peltier effect, at the cold junction becomes

=− qqqnet P c IR2 () T− T kA (17.16) =−()ααTI − −hc 12c 2 L

17.3 FIGURE OF MERIT

From the preceding discussion as well as Equation 17.16, the following summary may be obtained in regard to the desired properties of thermoelectric materials:

1. The Seebeck coefficient must be high in order to generate a large thermoelectric power. 2. The electric resistivity must be low to reduce the Joule heating effect. 3. The thermal conductivity must be low to maintain a large temperature gradient between two ends of the material.

For convenience, these three properties are often combined into a figure of merit Z, which is a measurement of the usefulness of a material in a thermoelectric device:

α 2 Z = (17.17) kρ

where the unit for the figure of merit is °C–1. Since most thermoelectric devices consist of a pair of dissimilar materials, it will be more practical to define the figure of merit in the following way:

 αα− 2 Z =  12  (17.18) ρρ05..+ 05 ()kk11 (22 )

A pure metal has a very low thermoelectric power, typically a few µV/°C. Another disadvantage of pure metals as thermoelectric elements is that the product of thermal conductivity and electrical resistivity is a constant. The figure of merit cannot be improved by changing k and ρ, because they cannot be changed individually. On the other hand, the thermal conductivity and electrical resistivity of semiconductors can be adjusted individually by appropriate doping.

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17.4 OPERATION PRINCIPLES

Free electrons contained in metals can be set in motion when heat or an electrical field is applied to the conductor. For example, the kinetic energy of the free electrons will be increased when heat is added to one end of a metal rod. This will result in a net electron flow to the cold end. A charge is associated with each electron, and the direction of the electrical current is opposite to the flow of electrons. The Seebeck effect governs the relationship between the temperature gradient and the change of voltage in a thermoelectric material. A difference in potential energy of the charge carriers exists between the two materials when an electric current passes through a thermoelectric couple. On the basis of the law of conservation of energy and the Peltier effect, energy is generated or rejected to the surroundings at the junction when the charge carriers move across the junction. This phenomenon can be used to transport heat from the cold junction to the hot junction. Two types of material are employed in a thermoelectric device: n-type material, which has an excess of electrons and is generally classified as having negative thermoelectric power; and p-type material, which has positive thermoelectric power because it has a deficiency of electrons. When an electron moves across the junction from p-type to n-type, it rises to a higher energy level at the expense of the junction, where the temperature is then reduced. On the other hand, the junction temperature will be increased when an electron passes across the junction from n-type to p-type.

17.5 SYSTEM CONFIGURATIONS

Figure 17.1 shows the basic configuration of a thermoelectric device called a Peltier couple. In a thermoelectric device, p-type and n-type semiconductor materials are soldered in series with copper bars to form the cold and hot junctions. The copper bars between p-type and n-type materials have no effect on the junction characteristics. This is because the presence of a third material in a thermoelectric circuit does not change the Seebeck effect as long as the junction is maintained at the same temperature. Due to the extremely high thermal conductivity and low electrical resistivity of copper bars, the junctions of the p-type and n-type materials can be held at constant temperatures. Generally, a single-couple device has a relatively small cooling capacity. In order to improve the performance, individual couples are assembled together as shown in Figure 17.2. Thermal insulation material is packed between the p- and n-type materials and copper bars to reduce thermal leakage in the system. Another configuration that is particularly useful for a gas stream is shown in Fig- ure 17.3. The gas flow may be heated or cooled according to the direction of the electrical current. Since there are limitations on the achievable temperature difference across a given thermoelectric couple, a multiple-stage thermoelectric module or cascaded module as shown in Figure 17.4 can be used to improve performance. For practical application, the number of stages is generally limited to about three or four. In this design, the sectional areas of the semiconductors may be varied in order to maintain a constant current density throughout the entire system.

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FIGURE 17.1 Base configuration of a Peltier couple.

17.6 PERFORMANCE ANALYSIS

To simplify analysis of a thermoelectric cooler, the following assumptions are made:

1. The thermal and electrical properties of the materials are independent of temperature. 2. A linear temperature gradient exists between the hot and cold junctions. 3. The Thomson effect is neglected. 4. The hot and cold junctions have no effect on the electric circuits. 5. Convection and radiation heat transfer within the system are neglected. 6. Geometrical similarity exists between the p-type and n-type materials.

FIGURE 17.2 Interconnected thermoelectric couples.

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FIGURE 17.3 Thermoelectric device for gas flow systems.

For a single-stage thermoelectric cooler as shown in Figure 17.1, the following equation represents the net cooling capacity under the steady-state condition when heat is pumped from the cold to the hot junction with temperatures at Tc and Th, respectively:

IR2 qT=−()αα I − −KT () − T (17.19) net 12ch2 c

FIGURE 17.4 Two-stage cascaded thermoelectric module.

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where

kA kA K =+11 22 (17.20a) L1 L2 ρρL L R =+11 22 (17.20b) A1 A2

On the other hand, to power the device, electrical energy must be supplied. The total power input including the resistive voltages and the Seebeck voltages in the couple is

Pe = I[IR + (α1 – α2)(Th – Tc)] (17.21)

where the subscripts 1 and 2 represent the p-type and n-type materials, respectively. The coefficient of performance (COP) for a refrigerator is generally defined as the ratio of the cooling capability to the required power input, i.e.,

q COP = net (17.22) Pe

It is frequently assumed that the conductive heat transfer rate, the electrical resistance, and the geometry are equal for each leg of the couple, i.e.,

kA kA 11= 22 (17.23a) L1 L2 ρρL L 11= 22 (17.23b) A1 A2

and

L1 = L2 (17.23c)

With the aid of Equations 17.23a–c, the coefficient of performance can be rewritten as follows.

()αα−−−TI I2 R /[()()]()/2 kρ05.. + k ρ05 2 T − T R COP = 12ch11 22 c(17.24) 2 +−αα − IR I()()12 Thc T

Figure 17.5 shows a typical plot for qnet (Equation 17.19) and COP (Equation 17.24) versus current. The figure further indicates that the operating current can be optimized for either maximum COP or maximum cooling capacity. To obtain the maximum cooling capacity, let d(qnet)/dI = 0, and we have

05.(αα− )22T ()q = 12c −−KT() T (17.25a) net max R hc

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FIGURE 17.5 Cooling capacity and COP of a typical thermoelectric device.

with

()αα− T I = 12c (17.25b) max R

Similarly, d(COP)/dI = 0 leads to

T ()/1+−ZT05. T T  COP = c  ave hc (17.26a) max − ++05. TThc ()11ZTave 

with

()()αα−−TT I = 12hc max +−05. (17.26b) RZT()11ave

where Tave is defined as (Th + Tc)/2 and Z is the figure of merit defined in Equation 17.18. From Equation 17.26a, the maximum COP approaches Tc/(Th – Tc) as Z → ∞. Tc/(Th – Tc) is the COP of the Carnot refrigeration cycle. On the other hand, COPmax is independent of Th or Tc as Z → 0.

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Under the maximum current condition, the maximum temperature difference between the cold and hot junctions can be achieved when the cold junction is thermally insulated. In this case, Equation 17.25a becomes zero; i.e.,

()αα− 22T ()TT−=12c hcmax 2RK = 2 (17.27) 05. ZTc

The total heat rejection at the hot junction that must be removed by some external means is the heat pumped plus the power required for the pumping as expressed below

qhot = qnet + Pe (17.28)

Substituting Equations 17.19 and 17.2 into Equation 17.28 leads to

IR2 [( kρρ )05..+− ( k )05 ] 2 ( T T ) qT=−()αα I + −11 22 hc (17.29) hot 12h 2 R

17.7 PRACTICAL DESIGN PROCEDURE

Thermoelectric modules have wide applications and can be incorporated into the cooling of electronics in a variety of ways. For example, a thermoelectric cooler may act as an air-conditioning unit to maintain the air temperature inside an equipment chamber or enclosure below the outside ambient temperature, or may function as a cold plate where the electronic system is mounted to the cold junction of a thermoelectric couple. To enhance heat transfer, finned heat sinks may be bonded to both cold and hot surfaces of a thermoelectric module. However, electrical insulation is required between the finned heat sinks and the copper bars connecting the p-type and n-type materials. The design procedure must consider three distinct phases. First, heat is applied to the cold junction of a thermoelectric module. Second, the energy must be pumped from the cold to the hot junction of the thermoelectric cooler. Finally, the total energy is then rejected from the hot junction to the ambient. The first and the last phases require the application of the laws of heat transfer, while the second phase involves application of the principles of thermoelectricity. Figure 17.6, showing some electronics inside an enclosure, is used as an example to illustrate the design procedure. The objective of using a thermoelectric module is to keep the internal temperature Ti of the enclosure below the outside ambient temperature Ta to maintain proper thermal control over the electronics. The design procedure is as follows:

1. Determine the maximum allowable internal temperature Ti of the enclosure to maintain the electronics at an acceptable temperature. This allowable internal temperature depends on the cooling schemes

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FIGURE 17.6 Thermoelectric module for cooling electronics inside an enclosure.

employed inside the enclosure. This step also establishes the maximum allowable temperature difference ∆Tmax between the ambient and the enclosure. 2. Compute the heat load ql upon the cold junction of the thermoelectric module, which may include the solar flux or any additional heating on the enclosure surfaces:

qnet = qe + ql (17.30)

where

= qnet total heat load on cold junction that must be pumped thermoelectrically to hot junction = qe heat load from electronics = ql heat load leakage from ambient to enclosure

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FIGURE 17.7 Temperature difference vs. net heat transfer rate.

3. Calculate ∆Tc and Tc. As usual, extended surfaces (finned heat sinks) can be employed at the cold junction to reduce the temperature difference ∆Tc between the cold junction of the thermoelectric module and the internal temperature of the enclosure. This temperature difference is dependent on the cooling method—e.g., free convection or forced convection due to air circulation by a fan. Included in ∆Tc are the temperature gradients across the heat sink base thickness and the bonding material between the cold heat sink and cold junction. Once ∆Tc is known, Tc can be readily determined. 4. Calculate ∆Th-c and Th. The temperature difference ∆Th-c between the hot and the cold junctions can be computed from qnet according to Equa- tion 17.19 for a given thermoelectric module and operation current. 5. Determine the total heat load on the hot junction of the thermoelectric couple based on Equation 17.28:

qhot = qnet + Pe

6. Calculate the temperature difference ∆Th between the ambient and the hot junction of the thermoelectric cooler. Again, ∆Th may include the temperature gradients across the heat sink base thickness and the bonding material between the hot heat sink and the hot junction. 7. Check Ti against (Ti)′ (= Ta + ∆Th – ∆Th-c + ∆Tc). 8. Obtain a design if (Ti)′ < Ti. Otherwise, repeat the calculations by using a different thermoelectric module.

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To facilitate the computations, Equation 17.19 is frequently presented in a graphical form provided by manufacturers. Generally, the temperature difference Th – Tc is plotted against the net heat transfer rate with current as a parameter as illustrated in Figure 17.7

REFERENCES

1. J. H. Seely and R. C. Chu, Chapter VIII, Heat Transfer in Microelectronic Equip- ment: A Practical Guide, Marcel Dekker, New York, 1972. 2. A. D. Kraus and A. Bar-Cohen, Chapter 18, Thermal Analysis and Control of Electronic Equipment, Hemisphere, Washington, DC, 1983.

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MATERIAL THERMAL PROPERTIES

Thermal Specific conductivity, heat, Density, 3 Material Btu/(hr·ft·°F) Btu/(lbm·°F) lbm/ft ABS (plastics) 0.081–0.179 66.8 Acrylic (conformal coating) 0.087 Alloy 42 (42% Ni/58% Fe) 6.0 0.12 508.03 Alumina, 96% pure 17.0 0.282 230.98 Alumina, 94% pure 10.41 241.91 Aluminum 2024 80.0 0.23 172.8 Aluminum 6063 T6 116.0 0.23 168.56 Aluminum 6061 T6 90.0 0.23 168.56 Aluminum 7075 76.0 0.23 174.53 A-40 (60% Al/40% Si) 72.8 157.9

Beryllia, 99.5% pure 140.0 0.24 Brass (70% Cu/30% Zn) 58.0 0.092 532.0

Copper 220.0 0.09 558.0 Cu/Invar/Cu (12.5%/75%/12.5%) 64.0 (kxy), 11.0 (kz) 0.112 518.0 Cu/Invar/Cu (20%/60%/20%) 97.0 (kxy), 12.7 (kz) 0.106 527.0

Epoxy (thermally conductive) 0.2–0.5 0.3–0.35 156.04

Fiberglass (G-10) 0.17 0.2 112.32 FR-4 0.173 0.21 120.99

Gallium arsenide (GaAs) 26.0 0.077 332.0 Gold 170.0 0.03 1203.0

Invar (36% Ni/64% Fe) 8.0 0.123 499.39 IR glass 0.128 0.066 292.03 Iron, pure 36.0 0.104 491.0 Cast iron, alloy 29.0 0.1 455.0 Cast iron, plain 32.0 0.11 474.0

Kovar (54% Fe, 29% N, 8.0 0.11 511.49 17% Co, Mn, C) (continues)

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Thermal Specific conductivity, heat, Density, 3 Material Btu/(hr·ft·°F) Btu/(lbm·°F) lbm/ft Lead 20.0 0.03 705.0

Magnesium 91.0 0.232 109.0 Mica 0.34–0.41 0.21 Molybdenum (Mo) 81.0 0.067 638.01 Mo/Cu (85% Mo/15% Cu) 106.4 0.066 624.9

Nickel 34.0 0.103 555.0

Phenolic (asbestos-filled) 0.2 0.28–0.32 106.0 Polyimide/glass 0.3 138.22 Polyimide/Kevlar 0.15 108.69 Polyimide/quartz 0.3 120.99

RTV (thermally conductive) 0.3 0.25 133.6–141.9

Silicon 86.7 0.19 145.5 Silicon rubber 0.11 Silver 240.0 0.056 655.0 Solder (63% tin/37% lead) 29.6 0.04 553.0 Solder (Au/Sn) 146.9 Stainless steel (302–304, 347) 8.0 0.11 448.0 Steel, SAE 1020 32.0 0.116 490.6 Styrofoam 0.02

Teflon 0.15 Thermal grease (Wakefield 120) 0.43 143.4 Thermal grease (Emerson TC4) 0.68 131.3 Thermal grease (Dow Corning 0.12 162.4 DC 340) Titanium 4.2 0.12 276.5 Tungsten (W) 94.3 0.032 1204.9 W/Cu (90% W/10% Cu) 121.00 0.039 1061.3 W/Cu (80% W/20% Cu) 142.7 0.044 977.0

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THERMAL CONDUCTIVITY OF SILICON AND GALLIUM ARSENIDE

Temperature, Silicon (Si) Gallium arsenide (GaAs) ˚K conductivity, W/(cm·˚K) conductivity, W/(cm·˚K) 200 2.65 0.65 250 1.95 0.53 300 1.58 0.44 350 1.28 0.37 400 1.07 0.32 500 0.79 0.24 600 0.63 0.22 700 0.51 0.20 800 0.44 0.19 900 0.38 0.18

Thermal conductivity of silicon (Si) vs. temperature.

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Thermal conductivity of gallium arsenide (GaAs) vs. temperature.

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PROPERTIES OF AIR, WATER, AND DIELECTRIC FLUIDS

Table number, fluid, and applicable temperature range, °F: C. 1 Air, –100 < T < 1000 C. 2 Water, 32 < T < 600 C. 3 FC-43, –80 < T < 345 C. 4 FC-75, –80 < T < 215 C. 5 FC-77, –80 < T < 207 C. 6 FC-78, –80 < T < 122 C. 7 Coolanol-20, –100 < T < 440 C. 8 Coolanol-25, –100 < T < 440 C. 9 Coolanol-35, –100 < T < 440 C.10 Coolanol-45, –100 < T < 440 C.11 Freon E2, –100 < T < 220 C.12 Glycol/water, –50 < T < 233 C.13 PAO (2 CST), –60 < T < 400 C.14 PAO (4 CST), –60 < T < 400 C.15 PAO (6 CST), –60 < T < 400 C.16 PAO (8 CST), –60 < T < 400

Properties in all tables and their units: Temperature T, °F 3 Density ρ, lbm/ft Viscosity µ, lbm/(hr·ft) Specific heat cp, Btu/(lbm·°F) Thermal conductivity k, Btu/(hr·ft·°F) Thermal expansion coefficient β, °F–1 2 2 –1 –3 Parameter B = gcβρ /µ , °F ·ft Prandtl number Pr Atmosphere pressure or saturation pressure Pam, psi

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use am BP β Pr k ρµ p 0.00 0.2380E+00 0.8585E-01 0.3983E-01 0.1320E-01 0.7181E+00 0.2147E-02 0.4160E+07 0.1470E+02 20.0040.00 0.2381E+0060.00 0.2383E+0080.00 0.2384E+00 0.8278E-01 0.2386E+00 0.7984E-01 0.7702E-01 0.4116E-01 0.7433E-01 0.4246E-01 0.4375E-01 0.1364E-01 0.4501E-01 0.1408E-01 0.1453E-01 0.7183E+00 0.1497E-01 0.7183E+00 0.2073E-02 0.7180E+00 0.2002E-02 0.7176E+00 0.1933E-02 0.3496E+07 0.1868E-02 0.2951E+07 0.2499E+07 0.1470E+02 0.2125E+07 0.1470E+02 0.1470E+02 0.1470E+02 –20.00 0.2380E+00 0.9078E-01 0.3827E-01 0.1265E-01 0.7198E+00 0.2260E-02 0.5302E+07 0.1470E+02 620.00 0.2497E+00 0.3632E-01 0.7336E-01 0.2637E-01 0.6974E+00 0.9200E-03 0.9404E+06 0.1470E+02 100.00120.00 0.2388E+00140.00 0.2390E+00160.00 0.2393E+00180.00 0.7177E-01 0.2395E+00200.00 0.6932E-01 0.2398E+00220.00 0.6698E-01 0.4626E-01 0.2401E+00240.00 0.6475E-01 0.4749E-01 0.2404E+00260.00 0.6264E-01 0.4870E-01 0.1541E-01 0.2408E+00280.00 0.6062E-01 0.4990E-01 0.1584E-01 0.2411E+00300.00 0.5871E-01 0.5107E-01 0.1628E-01 0.2415E+00320.00 0.7171E+00 0.5690E-01 0.5223E-01 0.1672E-01 0.2419E+00340.00 0.7164E+00 0.5517E-01 0.5338E-01 0.1715E-01 0.2423E+00360.00 0.1806E-02 0.7157E+00 0.5354E-01 0.5451E-01 0.1759E-01 0.2427E+00380.00 0.1746E-02 0.7149E+00 0.5200E-01 0.5562E-01 0.1802E-01 0.2431E+00400.00 0.1689E-02 0.7141E+00 0.1812E+07 0.5054E-01 0.5672E-01 0.1845E-01 0.2435E+00420.00 0.1635E-02 0.7132E+00 0.1551E+07 0.4916E-01 0.5780E-01 0.1888E-01 0.2440E+00440.00 0.1583E-02 0.7122E+00 0.1332E+07 0.4786E-01 0.1470E+02 0.5887E-01 0.1931E-01 0.2445E+00460.00 0.1533E-02 0.7112E+00 0.1148E+07 0.4662E-01 0.1470E+02 0.5992E-01 0.1974E-01 0.2449E+00480.00 0.1486E-02 0.7103E+00 0.9927E+06 0.4546E-01 0.1470E+02 0.6096E-01 0.2016E-01 0.2454E+00500.00 0.1441E-02 0.7093E+00 0.8612E+06 0.4437E-01 0.1470E+02 0.6199E-01 0.2059E-01 0.2459E+00520.00 0.1399E-02 0.7083E+00 0.7497E+06 0.4334E-01 0.1470E+02 0.6300E-01 0.2101E-01 0.2464E+00540.00 0.1358E-02 0.7072E+00 0.6548E+06 0.4237E-01 0.1470E+02 0.6400E-01 0.2144E-01 0.2470E+00560.00 0.1320E-02 0.7062E+00 0.5739E+06 0.4145E-01 0.1470E+02 0.6499E-01 0.2186E-01 0.2475E+00 0.1283E-02 0.7053E+00 0.5047E+06 0.4059E-01 0.1470E+02 0.6596E-01 0.2228E-01 0.2480E+00 0.1249E-02 0.7043E+00 0.4454E+06 0.3977E-01 0.1470E+02 0.6693E-01 0.2269E-01 0.1216E-02 0.7033E+00 0.3944E+06 0.3901E-01 0.1470E+02 0.6788E-01 0.2311E-01 0.1185E-02 0.7024E+00 0.3504E+06 0.3828E-01 0.1470E+02 0.6882E-01 0.2352E-01 0.1156E-02 0.7015E+00 0.3124E+06 0.1470E+02 0.6975E-01 0.2393E-01 0.1128E-02 0.7006E+00 0.2795E+06 0.1470E+02 0.7067E-01 0.2435E-01 0.1101E-02 0.6997E+00 0.2509E+06 0.1470E+02 0.2475E-01 0.1077E-02 0.6989E+00 0.2260E+06 0.1470E+02 0.2516E-01 0.1053E-02 0.6981E+00 0.2043E+06 0.1470E+02 0.1031E-02 0.6973E+00 0.1852E+06 0.1470E+02 0.1010E-02 0.6966E+00 0.1685E+06 0.1470E+02 0.9900E-03 0.1537E+06 0.1470E+02 0.9711E-03 0.1407E+06 0.1470E+02 0.1291E+06 0.1470E+02 0.1188E+06 0.1470E+02 0.1470E+02 0.1470E+02 Table C.1Table of Air Properties Part 1: At Sea Level (Altitude H = 0 ft) Tc

354

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use am BP β Pr k ρµ p 0.00 0.2380E+00 0.5798E-01 0.3983E-01 0.1320E-01 0.7181E+00 0.2147E-02 0.1897E+07 0.9925E+01 20.0040.00 0.2381E+0060.00 0.2383E+0080.00 0.2384E+00 0.5591E-01 0.2386E+00 0.5392E-01 0.5202E-01 0.4116E-01 0.5020E-01 0.4246E-01 0.4375E-01 0.1364E-01 0.4501E-01 0.1408E-01 0.1453E-01 0.7183E+00 0.1497E-01 0.7183E+00 0.2073E-02 0.7180E+00 0.2002E-02 0.7176E+00 0.1933E-02 0.1595E+07 0.1868E-02 0.1346E+07 0.1140E+07 0.9925E+01 0.9690E+06 0.9925E+01 0.9925E+01 0.9925E+01 –20.00 0.2380E+00 0.6131E-01 0.3827E-01 0.1265E-01 0.7198E+00 0.2260E-02 0.2418E+07 0.9925E+01 620.00 0.2497E+00 0.2453E-01 0.7336E-01 0.2637E-01 0.6974E+00 0.9200E-03 0.9404E+06 0.1470E+02 100.00120.00 0.2388E+00140.00 0.2390E+00160.00 0.2393E+00180.00 0.4847E-01 0.2395E+00200.00 0.4681E-01 0.2398E+00220.00 0.4523E-01 0.4626E-01 0.2401E+00240.00 0.4373E-01 0.4749E-01 0.2404E+00260.00 0.4230E-01 0.4870E-01 0.1541E-01 0.2408E+00280.00 0.4094E-01 0.4990E-01 0.1584E-01 0.2411E+00300.00 0.3965E-01 0.5107E-01 0.1628E-01 0.2415E+00320.00 0.7171E+00 0.3843E-01 0.5223E-01 0.1672E-01 0.2419E+00340.00 0.7164E+00 0.3726E-01 0.5338E-01 0.1715E-01 0.2423E+00360.00 0.1806E-02 0.7157E+00 0.3616E-01 0.5451E-01 0.1759E-01 0.2427E+00380.00 0.1746E-02 0.7149E+00 0.3512E-01 0.5562E-01 0.1802E-01 0.2431E+00400.00 0.1689E-02 0.7141E+00 0.8266E+06 0.3413E-01 0.5672E-01 0.1845E-01 0.2435E+00420.00 0.1635E-02 0.7132E+00 0.7075E+06 0.3320E-01 0.5780E-01 0.1888E-01 0.2440E+00440.00 0.1583E-02 0.7122E+00 0.6076E+06 0.3232E-01 0.9925E+01 0.5887E-01 0.1931E-01 0.2445E+00460.00 0.1533E-02 0.7112E+00 0.5236E+06 0.3149E-01 0.9925E+01 0.5992E-01 0.1974E-01 0.2449E+00480.00 0.1486E-02 0.7103E+00 0.4528E+06 0.3070E-01 0.9925E+01 0.6096E-01 0.2016E-01 0.2454E+00500.00 0.1441E-02 0.7093E+00 0.3928E+06 0.2997E-01 0.9925E+01 0.6199E-01 0.2059E-01 0.2459E+00520.00 0.1399E-02 0.7083E+00 0.3419E+06 0.2927E-01 0.9925E+01 0.6300E-01 0.2101E-01 0.2464E+00540.00 0.1358E-02 0.7072E+00 0.2987E+06 0.2861E-01 0.9925E+01 0.6400E-01 0.2144E-01 0.2470E+00560.00 0.1320E-02 0.7062E+00 0.2618E+06 0.2799E-01 0.9925E+01 0.6499E-01 0.2186E-01 0.2475E+00 0.1283E-02 0.7053E+00 0.2302E+06 0.2741E-01 0.9925E+01 0.6596E-01 0.2228E-01 0.2480E+00 0.1249E-02 0.7043E+00 0.2031E+06 0.2686E-01 0.9925E+01 0.6693E-01 0.2269E-01 0.1216E-02 0.7033E+00 0.1799E+06 0.2634E-01 0.9925E+01 0.6788E-01 0.2311E-01 0.1185E-02 0.7024E+00 0.1598E+06 0.2585E-01 0.9925E+01 0.6882E-01 0.2352E-01 0.1156E-02 0.7015E+00 0.1425E+06 0.9925E+01 0.6975E-01 0.2393E-01 0.1128E-02 0.7006E+00 0.1275E+06 0.9925E+01 0.7067E-01 0.2435E-01 0.1101E-02 0.6997E+00 0.1144E+06 0.9925E+01 0.2475E-01 0.1077E-02 0.6989E+00 0.1031E+06 0.9925E+01 0.2516E-01 0.1053E-02 0.6981E+00 0.9316E+05 0.9925E+01 0.1031E-02 0.6973E+00 0.8447E+05 0.9925E+01 0.1010E-02 0.6966E+00 0.7683E+05 0.9925E+01 0.9900E-03 0.7011E+05 0.9925E+01 0.9711E-03 0.6416E+05 0.9925E+01 0.5889E+05 0.9925E+01 0.5420E+05 0.9925E+01 0.9925E+01 0.9925E+01 Table C.1Table of Air Properties Part 2: At Altitude H = 10,000 ft Tc

355

Table C.2 Properties of Water

Tcp ρµ k Pr β BPam 40.00 0.1005E+01 0.6260E+02 0.3762E+01 0.3254E+00 0.1162E+02 0.1953E-04 0.2254E+07 0.1215E+00 50.00 0.1004E+01 0.6252E+02 0.3195E+01 0.3326E+00 0.9642E+01 0.5034E-04 0.8036E+07 0.1779E+00 60.00 0.1002E+01 0.6243E+02 0.2740E+01 0.3393E+00 0.8094E+01 0.8430E-04 0.1826E+08 0.2563E+00 70.00 0.1001E+01 0.6234E+02 0.2375E+01 0.3455E+00 0.6883E+01 0.1189E-03 0.3413E+08 0.3631E+00 80.00 0.9998E+00 0.6223E+02 0.2082E+01 0.3512E+00 0.5928E+01 0.1515E-03 0.5641E+08 0.5069E+00 90.00 0.9985E+00 0.6212E+02 0.1844E+01 0.3564E+00 0.5165E+01 0.1796E-03 0.8500E+08 0.6978E+00 100.00 0.9972E+00 0.6200E+02 0.1647E+01 0.3613E+00 0.4546E+01 0.2017E-03 0.1192E+09 0.9486E+00 110.00 0.9949E+00 0.6187E+02 0.1483E+01 0.3657E+00 0.4035E+01 0.2212E-03 0.1605E+09 0.1275E+01 120.00 0.9983E+00 0.6173E+02 0.1345E+01 0.3697E+00 0.3632E+01 0.2406E-03 0.2113E+09 0.1693E+01 130.00 0.1001E+01 0.6158E+02 0.1228E+01 0.3734E+00 0.3292E+01 0.2602E-03 0.2730E+09 0.2219E+01 140.00 0.1004E+01 0.6142E+02 0.1127E+01 0.3767E+00 0.3002E+01 0.2797E-03 0.3466E+09 0.2883E+01 150.00 0.1006E+01 0.6125E+02 0.1039E+01 0.3797E+00 0.2754E+01 0.2993E-03 0.4333E+09 0.3711E+01 160.00 0.1008E+01 0.6107E+02 0.9633E+00 0.3823E+00 0.2539E+01 0.3188E-03 0.5344E+09 0.4735E+01 356 170.00 0.1009E+01 0.6088E+02 0.8965E+00 0.3847E+00 0.2352E+01 0.3384E-03 0.6509E+09 0.5989E+01 180.00 0.1011E+01 0.6069E+02 0.8375E+00 0.3869E+00 0.2188E+01 0.3581E-03 0.7840E+09 0.7511E+01 190.00 0.1012E+01 0.6048E+02 0.7850E+00 0.3888E+00 0.2043E+01 0.3777E-03 0.9348E+09 0.9344E+01 200.00 0.1013E+01 0.6026E+02 0.7383E+00 0.3905E+00 0.1915E+01 0.3974E-03 0.1104E+10 0.1154E+02 210.00 0.1014E+01 0.6004E+02 0.6963E+00 0.3920E+00 0.1801E+01 0.4171E-03 0.1293E+10 0.1414E+02 220.00 0.1015E+01 0.5980E+02 0.6585E+00 0.3933E+00 0.1699E+01 0.4368E-03 0.1502E+10 0.1720E+02 230.00 0.1016E+01 0.5956E+02 0.6242E+00 0.3945E+00 0.1607E+01 0.4565E-03 0.1733E+10 0.2080E+02 240.00 0.1017E+01 0.5930E+02 0.5932E+00 0.3956E+00 0.1525E+01 0.4763E-03 0.1985E+10 0.2499E+02 250.00 0.1018E+01 0.5903E+02 0.5648E+00 0.3960E+00 0.1452E+01 0.4961E-03 0.2259E+10 0.2985E+02 260.00 0.1019E+01 0.5876E+02 0.5390E+00 0.3957E+00 0.1388E+01 0.5159E-03 0.2557E+10 0.3544E+02 270.00 0.1020E+01 0.5847E+02 0.5152E+00 0.3954E+00 0.1329E+01 0.5357E-03 0.2877E+10 0.4187E+02 280.00 0.1022E+01 0.5818E+02 0.4934E+00 0.3950E+00 0.1276E+01 0.5556E-03 0.3221E+10 0.4920E+02 290.00 0.1023E+01 0.5787E+02 0.4732E+00 0.3946E+00 0.1227E+01 0.5754E-03 0.3588E+10 0.5754E+02 300.00 0.1025E+01 0.5755E+02 0.4546E+00 0.3941E+00 0.1183E+01 0.5953E-03 0.3979E+10 0.6698E+02 320.00 0.1030E+01 0.5689E+02 0.4213E+00 0.3929E+00 0.1105E+01 0.6352E-03 0.4829E+10 0.8959E+02 340.00 0.1037E+01 0.5618E+02 0.3924E+00 0.3913E+00 0.1040E+01 0.6752E-03 0.5769E+10 0.1179E+03 360.00 0.1046E+01 0.5543E+02 0.3673E+00 0.3892E+00 0.9874E+00 0.7152E-03 0.6793E+10 0.1529E+03 425.00 0.1094E+01 0.5269E+02 0.3040E+00 0.3779E+00 0.8804E+00 0.8460E-03 0.1060E+11 0.3264E+03 am BP β Pr k ρµ p 0.00 0.2310E+00 0.1218E+03 0.5049E+02 0.3915E-01 0.2979E+03 0.6204E-03 0.1506E+07 0.1110E-02 10.0020.00 0.2331E+0030.00 0.2352E+0040.00 0.1211E+03 0.2373E+0050.00 0.1203E+03 0.2394E+0060.00 0.4090E+02 0.1196E+03 0.2415E+0070.00 0.3330E+02 0.1188E+03 0.2436E+0080.00 0.2725E+02 0.1181E+03 0.2457E+0090.00 0.3913E-01 0.2242E+02 0.1173E+03 0.2478E+00 0.3911E-01 0.1853E+02 0.1166E+03 0.2499E+00 0.3909E-01 0.2436E+03 0.1540E+02 0.1158E+03 0.3907E-01 0.2003E+03 0.1287E+02 0.1150E+03 0.3904E-01 0.6243E-03 0.1654E+03 0.1081E+02 0.3902E-01 0.6282E-03 0.1374E+03 0.9123E+01 0.3900E-01 0.2282E+07 0.6322E-03 0.1146E+03 0.3898E-01 0.3421E+07 0.6362E-03 0.9616E+02 0.3896E-01 0.5076E+07 0.6403E-03 0.8108E+02 0.1777E-02 0.7455E+07 0.6444E-03 0.6870E+02 0.2789E-02 0.1083E+08 0.6486E-03 0.5852E+02 0.4297E-02 0.1558E+08 0.6529E-03 0.6507E-02 0.2218E+08 0.6571E-03 0.9694E-02 0.3125E+08 0.1422E-01 0.4357E+08 0.2057E-01 0.2934E-01 0.4132E-01 –30.00–20.00 0.2247E+00–10.00 0.2268E+00 0.1241E+03 0.2289E+00 0.1234E+03 0.9802E+02 0.1226E+03 0.7817E+02 0.6267E+02 0.3922E-01 0.3919E-01 0.3917E-01 0.5616E+03 0.4523E+03 0.6091E-03 0.3662E+03 0.6128E-03 0.4072E+06 0.6166E-03 0.6364E+06 0.9842E+06 0.2376E-03 0.4066E-03 0.6795E-03 340.00 0.3024E+00 0.9614E+02 0.7061E+00 0.3842E-01 0.5557E+01 0.7863E-03 0.6080E+10 0.1333E+02 100.00110.00 0.2520E+00120.00 0.2541E+00130.00 0.1143E+03 0.2562E+00140.00 0.1135E+03 0.2583E+00150.00 0.7741E+01 0.1128E+03 0.2604E+00160.00 0.6602E+01 0.1120E+03 0.2625E+00170.00 0.5660E+01 0.1113E+03 0.2646E+00180.00 0.3894E-01 0.4877E+01 0.1105E+03 0.2667E+00190.00 0.3892E-01 0.4224E+01 0.1098E+03 0.2688E+00200.00 0.3889E-01 0.5010E+02 0.3678E+01 0.1090E+03 0.2709E+00210.00 0.3887E-01 0.4310E+02 0.3219E+01 0.1082E+03 0.2730E+00220.00 0.3885E-01 0.6615E-03 0.3728E+02 0.2832E+01 0.1075E+03 0.2751E+00240.00 0.3883E-01 0.6659E-03 0.3241E+02 0.2504E+01 0.1067E+03 0.2772E+00260.00 0.3881E-01 0.6013E+08 0.6704E-03 0.2831E+02 0.2226E+01 0.1060E+03 0.2814E+00280.00 0.3879E-01 0.8212E+08 0.6749E-03 0.2486E+02 0.1988E+01 0.1052E+03 0.2856E+00300.00 0.3877E-01 0.1110E+09 0.6795E-03 0.2195E+02 0.5748E-01 0.1786E+01 0.1037E+03 0.2898E+00 0.3875E-01 0.1485E+09 0.6841E-03 0.1947E+02 0.7904E-01 0.1612E+01 0.1022E+03 0.2940E+00 0.3872E-01 0.1965E+09 0.6888E-03 0.1075E+00 0.1736E+02 0.1334E+01 0.1007E+03 0.3870E-01 0.2575E+09 0.6936E-03 0.1447E+00 0.1556E+02 0.1127E+01 0.9917E+02 0.3868E-01 0.3339E+09 0.6985E-03 0.1928E+00 0.1402E+02 0.9722E+00 0.3864E-01 0.4285E+09 0.7034E-03 0.2546E+00 0.1269E+02 0.8560E+00 0.3860E-01 0.5442E+09 0.7083E-03 0.3331E+00 0.1155E+02 0.3855E-01 0.6840E+09 0.7134E-03 0.4321E+00 0.9716E+01 0.3851E-01 0.8509E+09 0.7185E-03 0.5560E+00 0.8341E+01 0.1048E+10 0.7290E-03 0.7100E+00 0.7308E+01 0.1276E+10 0.7398E-03 0.8998E+00 0.6535E+01 0.1837E+10 0.7509E-03 0.1132E+01 0.2536E+10 0.7623E-03 0.1416E+01 0.3358E+10 0.2170E+01 0.4266E+10 0.3248E+01 0.4758E+01 0.6830E+01 Table C.3Table of FC-43 Properties Tc

357

Table C.4 Properties of FC-75

Tcp ρµ k Pr β BPam –40.00 0.2200E+00 0.1201E+03 0.2089E+02 0.3997E-01 0.1150E+03 0.7105E-03 0.9793E+07 0.7712E-02 –30.00 0.2224E+00 0.1192E+03 0.1711E+02 0.3972E-01 0.9580E+02 0.7156E-03 0.1448E+08 0.1226E-01 –20.00 0.2247E+00 0.1184E+03 0.1413E+02 0.3947E-01 0.8042E+02 0.7208E-03 0.2110E+08 0.1907E-01 –10.00 0.2271E+00 0.1175E+03 0.1175E+02 0.3922E-01 0.6803E+02 0.7260E-03 0.3028E+08 0.2909E-01 0.00 0.2294E+00 0.1167E+03 0.9849E+01 0.3897E-01 0.5798E+02 0.7313E-03 0.4279E+08 0.4358E-01 10.00 0.2317E+00 0.1158E+03 0.8317E+01 0.3872E-01 0.4978E+02 0.7367E-03 0.5955E+08 0.6416E-01 20.00 0.2341E+00 0.1150E+03 0.7077E+01 0.3847E-01 0.4307E+02 0.7422E-03 0.8165E+08 0.9295E-01 30.00 0.2364E+00 0.1141E+03 0.6068E+01 0.3822E-01 0.3754E+02 0.7477E-03 0.1102E+09 0.1326E+00 40.00 0.2388E+00 0.1132E+03 0.5242E+01 0.3797E-01 0.3297E+02 0.7534E-03 0.1466E+09 0.1866E+00 50.00 0.2411E+00 0.1124E+03 0.4563E+01 0.3772E-01 0.2917E+02 0.7591E-03 0.1920E+09 0.2591E+00

358 60.00 0.2435E+00 0.1115E+03 0.4002E+01 0.3747E-01 0.2600E+02 0.7649E-03 0.2478E+09 0.3551E+00 70.00 0.2458E+00 0.1107E+03 0.3537E+01 0.3722E-01 0.2336E+02 0.7708E-03 0.3148E+09 0.4810E+00 80.00 0.2482E+00 0.1098E+03 0.3149E+01 0.3697E-01 0.2114E+02 0.7768E-03 0.3940E+09 0.6443E+00 90.00 0.2505E+00 0.1090E+03 0.2825E+01 0.3672E-01 0.1928E+02 0.7829E-03 0.4857E+09 0.8539E+00 100.00 0.2529E+00 0.1081E+03 0.2554E+01 0.3647E-01 0.1771E+02 0.7890E-03 0.5897E+09 0.1120E+01 110.00 0.2552E+00 0.1073E+03 0.2327E+01 0.3622E-01 0.1639E+02 0.7953E-03 0.7051E+09 0.1456E+01 120.00 0.2576E+00 0.1064E+03 0.2135E+01 0.3597E-01 0.1529E+02 0.8017E-03 0.8304E+09 0.1875E+01 130.00 0.2599E+00 0.1056E+03 0.1975E+01 0.3572E-01 0.1437E+02 0.8082E-03 0.9632E+09 0.2394E+01 140.00 0.2623E+00 0.1047E+03 0.1840E+01 0.3547E-01 0.1360E+02 0.8147E-03 0.1100E+10 0.3032E+01 150.00 0.2646E+00 0.1039E+03 0.1727E+01 0.3522E-01 0.1298E+02 0.8214E-03 0.1238E+10 0.3811E+01 160.00 0.2670E+00 0.1030E+03 0.1634E+01 0.3497E-01 0.1247E+02 0.8282E-03 0.1372E+10 0.4754E+01 170.00 0.2693E+00 0.1022E+03 0.1558E+01 0.3472E-01 0.1208E+02 0.8352E-03 0.1498E+10 0.5889E+01 180.00 0.2716E+00 0.1013E+03 0.1496E+01 0.3447E-01 0.1179E+02 0.8422E-03 0.1610E+10 0.7247E+01 190.00 0.2740E+00 0.1005E+03 0.1448E+01 0.3422E-01 0.1159E+02 0.8493E-03 0.1705E+10 0.8860E+01 200.00 0.2763E+00 0.9960E+02 0.1412E+01 0.3397E-01 0.1148E+02 0.8566E-03 0.1778E+10 0.1077E+02 210.00 0.2787E+00 0.9874E+02 0.1387E+01 0.3372E-01 0.1146E+02 0.8640E-03 0.1826E+10 0.1301E+02 am BP β Pr k ρµ p 0.00 0.2294E+00 0.1175E+03 0.9714E+01 0.3904E-01 0.5708E+02 0.7227E-03 0.4408E+08 0.6803E-01 10.0020.0030.00 0.2317E+0040.00 0.2341E+00 0.1166E+0350.00 0.2364E+00 0.1158E+0360.00 0.2388E+00 0.8213E+01 0.1149E+0370.00 0.2411E+00 0.6992E+01 0.1141E+0380.00 0.2435E+00 0.5995E+01 0.1132E+0390.00 0.2458E+00 0.3876E-01 0.5177E+01 0.1124E+03 0.2482E+00 0.3848E-01 0.4502E+01 0.1115E+03 0.2505E+00 0.3820E-01 0.4910E+02 0.3943E+01 0.1107E+03 0.3792E-01 0.4253E+02 0.3478E+01 0.1098E+03 0.3765E-01 0.7279E-03 0.3711E+02 0.3089E+01 0.3737E-01 0.7333E-03 0.3260E+02 0.2764E+01 0.3709E-01 0.6122E+08 0.7387E-03 0.2884E+02 0.3681E-01 0.8384E+08 0.7442E-03 0.2569E+02 0.3653E-01 0.1132E+09 0.7498E-03 0.2305E+02 0.9845E-01 0.1507E+09 0.7554E-03 0.1403E+00 0.2083E+02 0.1978E+09 0.7612E-03 0.1971E+00 0.1895E+02 0.2559E+09 0.7670E-03 0.2731E+00 0.3264E+09 0.7729E-03 0.3736E+00 0.4106E+09 0.5049E+00 0.5091E+09 0.6747E+00 0.8921E+00 0.1167E+01 Table C.5Table of FC-77 Properties Tc –40.00–30.00–20.00 0.2200E+00–10.00 0.2224E+00 0.1209E+03 0.2247E+00 0.1200E+03 0.2271E+00 0.2041E+02 0.1192E+03 0.1677E+02 0.1183E+03 0.1388E+02 0.4016E-01 0.1157E+02 0.3988E-01 0.3960E-01 0.1118E+03 0.3932E-01 0.9352E+02 0.7024E-03 0.7878E+02 0.7073E-03 0.6682E+02100.00 0.1028E+08 0.7124E-03110.00 0.1511E+08 0.7175E-03120.00 0.2529E+00 0.2189E+08 0.1301E-01 130.00 0.2552E+00 0.3129E+08 0.2025E-01 0.1090E+03140.00 0.2576E+00 0.3089E-01 0.1081E+03150.00 0.2599E+00 0.4624E-01 0.2490E+01 0.1073E+03160.00 0.2623E+00 0.2259E+01 0.1064E+03170.00 0.2646E+00 0.2064E+01 0.1056E+03180.00 0.2670E+00 0.3625E-01 0.1900E+01 0.1047E+03200.00 0.2693E+00 0.3597E-01 0.1760E+01 0.1039E+03 0.2716E+00 0.3569E-01 0.1737E+02 0.1643E+01 0.1031E+03 0.2763E+00 0.3541E-01 0.1603E+02 0.1544E+01 0.1022E+03 0.3514E-01 0.7790E-03 0.1490E+02 0.1462E+01 0.1005E+03 0.3486E-01 0.7851E-03 0.1394E+02 0.1393E+01 0.3458E-01 0.6224E+09 0.7913E-03 0.1314E+02 0.1293E+01 0.3430E-01 0.7502E+09 0.7976E-03 0.1247E+02 0.3402E-01 0.8915E+09 0.8040E-03 0.1513E+01 0.1192E+02 0.3346E-01 0.1044E+10 0.8105E-03 0.1944E+01 0.1148E+02 0.1206E+10 0.8171E-03 0.2475E+01 0.1113E+02 0.1374E+10 0.8239E-03 0.3126E+01 0.1068E+02 0.1542E+10 0.8307E-03 0.3918E+01 0.1707E+10 0.8448E-03 0.4874E+01 0.1863E+10 0.6021E+01 0.2127E+10 0.7387E+01 0.9007E+01 0.1315E+02

359

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use am BP β Pr k ρµ p 0.005.00 0.2345E+00 0.2352E+00 0.1133E+03 0.1128E+03 0.3568E+01 0.3371E+01 0.3869E-01 0.3852E-01 0.2162E+02 0.2058E+02 0.8081E-03 0.8113E-03 0.3398E+09 0.3792E+09 0.6409E+00 0.7541E+00 –5.00 0.2337E+00 0.1138E+03 0.3781E+01 0.3887E-01 0.2274E+02 0.8048E-03 0.3038E+09 0.5428E+00 10.0015.00 0.2360E+0020.00 0.2367E+0025.00 0.1124E+03 0.2374E+0030.00 0.1119E+03 0.2382E+0035.00 0.3188E+01 0.1115E+03 0.2389E+0040.00 0.3018E+01 0.1110E+03 0.2397E+0050.00 0.2860E+01 0.1106E+03 0.2404E+0060.00 0.3835E-01 0.2714E+01 0.1101E+03 0.2419E+0070.00 0.3818E-01 0.2578E+01 0.1096E+03 0.2434E+0080.00 0.3801E-01 0.1961E+02 0.2451E+01 0.1087E+03 0.2448E+0090.00 0.3784E-01 0.1871E+02 0.2333E+01 0.1078E+03 0.2463E+00 0.3766E-01 0.8147E-03 0.1787E+02 0.2122E+01 0.1069E+03 0.2478E+00 0.3749E-01 0.8180E-03 0.1708E+02 0.1937E+01 0.1060E+03 0.3732E-01 0.4223E+09 0.8213E-03 0.1635E+02 0.1777E+01 0.1051E+03 0.3698E-01 0.4692E+09 0.8247E-03 0.1567E+02 0.1637E+01 0.3663E-01 0.5202E+09 0.8281E-03 0.8841E+00 0.1503E+02 0.1515E+01 0.3629E-01 0.5755E+09 0.8316E-03 0.1033E+01 0.1388E+02 0.3595E-01 0.6352E+09 0.8351E-03 0.1203E+01 0.1287E+02 0.3560E-01 0.6996E+09 0.8421E-03 0.1397E+01 0.1199E+02 0.7688E+09 0.8492E-03 0.1617E+01 0.1122E+02 0.9223E+09 0.8565E-03 0.1867E+01 0.1054E+02 0.1097E+10 0.8639E-03 0.2148E+01 0.1293E+10 0.8714E-03 0.2822E+01 0.1510E+10 0.3668E+01 0.1749E+10 0.4721E+01 0.6020E+01 0.7608E+01 –40.00–30.00 0.2286E+00–25.00 0.2300E+00–20.00 0.1170E+03 0.2308E+00–15.00 0.1161E+03 0.2315E+00–10.00 0.5852E+01 0.1156E+03 0.2323E+00 0.5137E+01 0.1151E+03 0.2330E+00 0.4821E+01 0.1147E+03 0.4007E-01 0.4530E+01 0.1142E+03 0.3973E-01 0.4260E+01 0.3955E-01 0.3398E+02 0.4011E+01 0.3938E-01 0.2975E+02 0.3921E-01 0.7828E-03 0.2813E+02 0.3904E-01 0.7889E-03 0.2663E+02 0.1304E+09 0.7921E-03 0.2524E+02 0.1679E+09 0.7952E-03 0.2394E+02 0.1899E+09 0.7984E-03 0.1519E+00 0.2142E+09 0.8016E-03 0.2232E+00 0.2412E+09 0.2688E+00 0.2710E+09 0.3224E+00 0.3850E+00 0.4580E+00 Table C.6Table of FC-78 Properties Tc

360

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use am BP β Pr k ρµ p 0.00 0.4287E+00 0.5673E+02 0.1697E+02 0.7276E-01 0.1000E+03 0.2853E-03 0.1329E+07 0.4562E+00 10.0020.00 0.4320E+0030.00 0.4354E+0040.00 0.5656E+02 0.4388E+0050.00 0.5640E+02 0.4421E+0060.00 0.1382E+02 0.5624E+02 0.4455E+0070.00 0.1143E+02 0.5608E+02 0.4488E+0080.00 0.9579E+01 0.5592E+02 0.4522E+0090.00 0.7235E-01 0.8138E+01 0.5576E+02 0.4556E+00 0.7193E-01 0.7003E+01 0.5559E+02 0.4589E+00 0.7152E-01 0.8255E+02 0.6098E+01 0.5543E+02 0.7111E-01 0.6916E+02 0.5371E+01 0.5527E+02 0.7069E-01 0.2862E-03 0.5876E+02 0.4780E+01 0.7028E-01 0.2870E-03 0.5060E+02 0.4296E+01 0.6986E-01 0.1998E+07 0.2878E-03 0.4413E+02 0.6945E-01 0.2916E+07 0.2886E-03 0.3895E+02 0.6903E-01 0.4137E+07 0.2895E-03 0.4839E+00 0.3476E+02 0.5714E+07 0.2903E-03 0.5133E+00 0.3136E+02 0.7696E+07 0.2911E-03 0.5444E+00 0.2856E+02 0.1012E+08 0.2920E-03 0.5774E+00 0.1301E+08 0.2929E-03 0.6125E+00 0.1637E+08 0.6496E+00 0.2021E+08 0.6890E+00 0.7309E+00 0.7752E+00 –30.00–20.00 0.4186E+00–10.00 0.4220E+00 0.5721E+02 0.4253E+00 0.5705E+02 0.3459E+02 0.5689E+02 0.2683E+02 0.2117E+02 0.7401E-01 0.7359E-01 0.7318E-01 0.1957E+03 0.1538E+03 0.2829E-03 0.1230E+03 0.2837E-03 0.3226E+06 0.2845E-03 0.5348E+06 0.8569E+06 0.3823E+00 0.4055E+00 0.4301E+00 330.00 0.5396E+00 0.5139E+02 0.9714E+01 0.5907E-01 0.8873E+01 0.3150E-03 0.3675E+09 0.3187E+01 100.00110.00 0.4623E+00120.00 0.4656E+00130.00 0.5511E+02 0.4690E+00140.00 0.5495E+02 0.4724E+00150.00 0.3896E+01 0.5478E+02 0.4757E+00160.00 0.3564E+01 0.5462E+02 0.4791E+00170.00 0.3284E+01 0.5446E+02 0.4824E+00180.00 0.6862E-01 0.3048E+01 0.5430E+02 0.4858E+00190.00 0.6820E-01 0.2846E+01 0.5414E+02 0.4892E+00200.00 0.6779E-01 0.2625E+02 0.2673E+01 0.5397E+02 0.4925E+00210.00 0.6737E-01 0.2433E+02 0.2521E+01 0.5381E+02 0.4959E+00220.00 0.6696E-01 0.2937E-03 0.2272E+02 0.2389E+01 0.5365E+02 0.4992E+00240.00 0.6654E-01 0.2946E-03 0.2137E+02 0.2270E+01 0.5349E+02 0.5026E+00260.00 0.6613E-01 0.2450E+08 0.2955E-03 0.2022E+02 0.2164E+01 0.5333E+02 0.5093E+00280.00 0.6571E-01 0.2920E+08 0.2963E-03 0.1924E+02 0.2066E+01 0.5317E+02 0.5160E+00300.00 0.6530E-01 0.3428E+08 0.2972E-03 0.8222E+00 0.1840E+02 0.1975E+01 0.5284E+02 0.5228E+00 0.6488E-01 0.3968E+08 0.2981E-03 0.8721E+00 0.1766E+02 0.1889E+01 0.5252E+02 0.5295E+00 0.6447E-01 0.4537E+08 0.2990E-03 0.9250E+00 0.1701E+02 0.1725E+01 0.5219E+02 0.6405E-01 0.5131E+08 0.2999E-03 0.9811E+00 0.1642E+02 0.1565E+01 0.5187E+02 0.6364E-01 0.5747E+08 0.3008E-03 0.1041E+01 0.1589E+02 0.1402E+01 0.6281E-01 0.6385E+08 0.3017E-03 0.1104E+01 0.1539E+02 0.1234E+01 0.6198E-01 0.7046E+08 0.3026E-03 0.1171E+01 0.1492E+02 0.6115E-01 0.7735E+08 0.3035E-03 0.1242E+01 0.1399E+02 0.6032E-01 0.8459E+08 0.3044E-03 0.1317E+01 0.1303E+02 0.9229E+08 0.3063E-03 0.1397E+01 0.1199E+02 0.1006E+09 0.3082E-03 0.1482E+01 0.1083E+02 0.1198E+09 0.3101E-03 0.1572E+01 0.1447E+09 0.3120E-03 0.1667E+01 0.1792E+09 0.1875E+01 0.2300E+09 0.2110E+01 0.2374E+01 0.2670E+01 Table C.7Table of Coolanol-20 Properties Tc

361

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use am BP β Pr k ρµ p 0.00 0.3980E+00 0.5786E+02 0.6064E+02 0.7806E-01 0.3091E+03 0.4682E-03 0.1777E+06 0.2837E-03 10.0020.00 0.4039E+0030.00 0.4099E+0040.00 0.5759E+02 0.4158E+0050.00 0.5732E+02 0.4218E+0060.00 0.4619E+02 0.5705E+02 0.4277E+0070.00 0.3579E+02 0.5678E+02 0.4337E+0080.00 0.2820E+02 0.5651E+02 0.4396E+0090.00 0.7781E-01 0.2258E+02 0.5624E+02 0.4455E+00 0.7755E-01 0.1836E+02 0.5597E+02 0.4515E+00 0.7729E-01 0.2398E+03 0.1515E+02 0.5570E+02 0.7703E-01 0.1892E+03 0.1268E+02 0.5543E+02 0.7676E-01 0.4704E-03 0.1517E+03 0.1076E+02 0.7649E-01 0.4726E-03 0.1237E+03 0.9241E+01 0.7622E-01 0.3050E+06 0.4748E-03 0.1023E+03 0.7594E-01 0.5054E+06 0.4771E-03 0.8592E+02 0.7566E-01 0.8101E+06 0.4794E-03 0.7315E+02 0.3119E-03 0.1257E+07 0.4817E-03 0.6312E+02 0.3430E-03 0.1893E+07 0.4840E-03 0.5515E+02 0.3771E-03 0.2766E+07 0.4864E-03 0.4147E-03 0.3930E+07 0.4888E-03 0.4560E-03 0.5437E+07 0.5014E-03 0.7332E+07 0.5513E-03 0.6062E-03 0.6665E-03 –30.00–20.00 0.3801E+00–10.00 0.3861E+00 0.5868E+02 0.3920E+00 0.5841E+02 0.1533E+03 0.5813E+02 0.1104E+03 0.8107E+02 0.7880E-01 0.7856E-01 0.7831E-01 0.7397E+03 0.5427E+03 0.4617E-03 0.4058E+03 0.4638E-03 0.2819E+05 0.4660E-03 0.5411E+05 0.9992E+05 0.2134E-03 0.2346E-03 0.2580E-03 330.00 0.5942E+00 0.4892E+02 0.1965E+01 0.6782E-01 0.1722E+02 0.5537E-03 0.1431E+09 0.6503E-02 100.00110.00 0.4574E+00120.00 0.4634E+00130.00 0.5515E+02 0.4693E+00140.00 0.5488E+02 0.4753E+00150.00 0.8034E+01 0.5461E+02 0.4812E+00160.00 0.7066E+01 0.5434E+02 0.4872E+00170.00 0.6281E+01 0.5407E+02 0.4931E+00180.00 0.7537E-01 0.5640E+01 0.5380E+02 0.4991E+00190.00 0.7508E-01 0.5112E+01 0.5353E+02 0.5050E+00200.00 0.7479E-01 0.4876E+02 0.4675E+01 0.5326E+02 0.5110E+00210.00 0.7449E-01 0.4361E+02 0.4310E+01 0.5299E+02 0.5169E+00220.00 0.7419E-01 0.4912E-03 0.3942E+02 0.4002E+01 0.5272E+02 0.5229E+00240.00 0.7389E-01 0.4936E-03 0.3598E+02 0.3742E+01 0.5245E+02 0.5288E+00260.00 0.7358E-01 0.9651E+07 0.4960E-03 0.3316E+02 0.3521E+01 0.5217E+02 0.5407E+00280.00 0.7327E-01 0.1242E+08 0.4985E-03 0.3082E+02 0.3330E+01 0.5190E+02 0.5526E+00300.00 0.7296E-01 0.1564E+08 0.5010E-03 0.2888E+02 0.7329E-03 0.3166E+01 0.5136E+02 0.5645E+00 0.7264E-01 0.1930E+08 0.5035E-03 0.2726E+02 0.8059E-03 0.3021E+01 0.5082E+02 0.5764E+00 0.7232E-01 0.2337E+08 0.5061E-03 0.2590E+02 0.8861E-03 0.2779E+01 0.5028E+02 0.7200E-01 0.2781E+08 0.5087E-03 0.2477E+02 0.9743E-03 0.2578E+01 0.4974E+02 0.7167E-01 0.3256E+08 0.5113E-03 0.2380E+02 0.1071E-02 0.2400E+01 0.7100E-01 0.3755E+08 0.5139E-03 0.2299E+02 0.1178E-02 0.2229E+01 0.7032E-01 0.4273E+08 0.5165E-03 0.2229E+02 0.1295E-02 0.6963E-01 0.4804E+08 0.5192E-03 0.2116E+02 0.1424E-02 0.6892E-01 0.5341E+08 0.5219E-03 0.2026E+02 0.1566E-02 0.5882E+08 0.5274E-03 0.1945E+02 0.1722E-02 0.6423E+08 0.5331E-03 0.1864E+02 0.1893E-02 0.7513E+08 0.5388E-03 0.2082E-02 0.8637E+08 0.5447E-03 0.2289E-02 0.9863E+08 0.2768E-02 0.1131E+09 0.3346E-02 0.4046E-02 0.4891E-02 Table C.8Table of Coolanol-25 Properties Tc

362

Table C.9 Properties of Coolanol-35

Tcp ρµ k Pr β BPam –30.00 0.3942E+00 0.5796E+02 0.6622E+03 0.8064E-01 0.3237E+04 0.4488E-03 0.1433E+04 0.2344E-04 –20.00 0.3992E+00 0.5770E+02 0.4354E+03 0.8031E-01 0.2164E+04 0.4508E-03 0.3300E+04 0.2817E-04 –10.00 0.4041E+00 0.5744E+02 0.2933E+03 0.7998E-01 0.1482E+04 0.4528E-03 0.7244E+04 0.3385E-04 0.00 0.4091E+00 0.5718E+02 0.2021E+03 0.7963E-01 0.1038E+04 0.4549E-03 0.1518E+05 0.4067E-04 10.00 0.4141E+00 0.5692E+02 0.1425E+03 0.7928E-01 0.7441E+03 0.4570E-03 0.3041E+05 0.4888E-04 20.00 0.4190E+00 0.5666E+02 0.1026E+03 0.7893E-01 0.5448E+03 0.4591E-03 0.5834E+05 0.5874E-04 30.00 0.4240E+00 0.5640E+02 0.7547E+02 0.7856E-01 0.4073E+03 0.4612E-03 0.1074E+06 0.7058E-04 40.00 0.4290E+00 0.5614E+02 0.5661E+02 0.7819E-01 0.3106E+03 0.4633E-03 0.1899E+06 0.8482E-04 50.00 0.4339E+00 0.5588E+02 0.4329E+02 0.7781E-01 0.2414E+03 0.4655E-03 0.3233E+06 0.1019E-03 60.00 0.4389E+00 0.5562E+02 0.3372E+02 0.7743E-01 0.1911E+03 0.4677E-03 0.5306E+06 0.1225E-03 70.00 0.4439E+00 0.5536E+02 0.2672E+02 0.7703E-01 0.1540E+03 0.4699E-03 0.8406E+06 0.1472E-03 80.00 0.4488E+00 0.5510E+02 0.2154E+02 0.7663E-01 0.1262E+03 0.4721E-03 0.1288E+07 0.1769E-03 90.00 0.4538E+00 0.5484E+02 0.1764E+02 0.7622E-01 0.1050E+03 0.4743E-03 0.1912E+07 0.2126E-03 363 100.00 0.4588E+00 0.5458E+02 0.1466E+02 0.7581E-01 0.8874E+02 0.4766E-03 0.2753E+07 0.2555E-03 110.00 0.4638E+00 0.5432E+02 0.1237E+02 0.7538E-01 0.7607E+02 0.4789E-03 0.3853E+07 0.3070E-03 120.00 0.4687E+00 0.5406E+02 0.1057E+02 0.7495E-01 0.6609E+02 0.4812E-03 0.5248E+07 0.3689E-03 130.00 0.4737E+00 0.5380E+02 0.9149E+01 0.7451E-01 0.5816E+02 0.4835E-03 0.6970E+07 0.4433E-03 140.00 0.4787E+00 0.5353E+02 0.8014E+01 0.7407E-01 0.5179E+02 0.4859E-03 0.9041E+07 0.5327E-03 150.00 0.4836E+00 0.5327E+02 0.7098E+01 0.7361E-01 0.4663E+02 0.4882E-03 0.1147E+08 0.6402E-03 160.00 0.4886E+00 0.5301E+02 0.6351E+01 0.7315E-01 0.4242E+02 0.4906E-03 0.1426E+08 0.7693E-03 170.00 0.4936E+00 0.5275E+02 0.5736E+01 0.7268E-01 0.3895E+02 0.4930E-03 0.1739E+08 0.9245E-03 180.00 0.4985E+00 0.5249E+02 0.5226E+01 0.7221E-01 0.3608E+02 0.4955E-03 0.2084E+08 0.1111E-02 190.00 0.5035E+00 0.5223E+02 0.4799E+01 0.7172E-01 0.3369E+02 0.4979E-03 0.2460E+08 0.1335E-02 200.00 0.5085E+00 0.5197E+02 0.4437E+01 0.7123E-01 0.3167E+02 0.5004E-03 0.2863E+08 0.1604E-02 210.00 0.5134E+00 0.5171E+02 0.4128E+01 0.7074E-01 0.2996E+02 0.5030E-03 0.3291E+08 0.1928E-02 220.00 0.5184E+00 0.5145E+02 0.3861E+01 0.7023E-01 0.2850E+02 0.5055E-03 0.3743E+08 0.2317E-02 240.00 0.5283E+00 0.5093E+02 0.3422E+01 0.6920E-01 0.2613E+02 0.5107E-03 0.4716E+08 0.3346E-02 260.00 0.5383E+00 0.5041E+02 0.3069E+01 0.6813E-01 0.2425E+02 0.5159E-03 0.5805E+08 0.4832E-02 280.00 0.5482E+00 0.4989E+02 0.2767E+01 0.6704E-01 0.2263E+02 0.5213E-03 0.7068E+08 0.6978E-02 300.00 0.5582E+00 0.4937E+02 0.2491E+01 0.6591E-01 0.2110E+02 0.5268E-03 0.8627E+08 0.1008E-01 330.00 0.5731E+00 0.4859E+02 0.2094E+01 0.6417E-01 0.1870E+02 0.5353E-03 0.1202E+09 0.1749E-01 am BP β Pr k ρµ p 0.00 0.4091E+00 0.5799E+02 0.3334E+03 0.8088E-01 0.1687E+04 0.4665E-03 0.5883E+04 0.8983E-05 10.0020.00 0.4141E+0030.00 0.4190E+00 0.5772E+0240.00 0.4240E+00 0.5745E+0250.00 0.2416E+03 0.4290E+00 0.5718E+0260.00 0.1785E+03 0.4339E+00 0.5691E+0270.00 0.8053E-01 0.1345E+03 0.4389E+00 0.5664E+0280.00 0.8018E-01 0.1031E+03 0.4439E+00 0.1242E+04 0.5637E+0290.00 0.7982E-01 0.8045E+02 0.4488E+00 0.9331E+03 0.5610E+02 0.7946E-01 0.4686E-03 0.6382E+02 0.4538E+00 0.7142E+03 0.5583E+02 0.7908E-01 0.4709E-03 0.5144E+02 0.1115E+05 0.5566E+03 0.5556E+02 0.7871E-01 0.4731E-03 0.4209E+02 0.2033E+05 0.4414E+03 0.7832E-01 0.4753E-03 0.1130E-04 0.3494E+02 0.3567E+05 0.3559E+03 0.7793E-01 0.4776E-03 0.1421E-04 0.6038E+05 0.2915E+03 0.7753E-01 0.4799E-03 0.1787E-04 0.9871E+05 0.2424E+03 0.4822E-03 0.2248E-04 0.1561E+06 0.2045E+03 0.4845E-03 0.2828E-04 0.2391E+06 0.4869E-03 0.3556E-04 0.3554E+06 0.4473E-04 0.5133E+06 0.5626E-04 0.7076E-04 Table C.10Table of Coolanol-45 Properties Tc –30.00–20.00 0.3942E+00–10.00 0.3992E+00 0.5880E+02 0.4041E+00 0.5853E+02 0.9935E+03 0.5826E+02 0.6756E+03 0.8187E-01 0.4696E+03 0.8155E-01 0.4783E+04 0.8122E-01 0.3307E+04 0.4600E-03 0.2337E+04 0.4621E-03 0.6719E+03 0.4643E-03 0.1446E+04 0.4515E-05 0.2979E+04100.00 0.5678E-05 110.00 0.7142E-05 0.4588E+00120.00 0.4638E+00 0.5528E+02130.00 0.4687E+00 0.5501E+02140.00 0.2940E+02 0.4737E+00 0.5474E+02150.00 0.2505E+02 0.4787E+00 0.5447E+02 0.7712E-01160.00 0.2161E+02 0.4836E+00 0.5420E+02 0.7671E-01170.00 0.1885E+02 0.4886E+00 0.1749E+03 0.5393E+02 0.7629E-01180.00 0.1662E+02 0.4936E+00 0.1515E+03 0.5366E+02 0.7587E-01190.00 0.4893E-03 0.1480E+02 0.4985E+00 0.1328E+03 0.5339E+02 0.7544E-01200.00 0.4917E-03 0.1330E+02 0.5035E+00 0.7216E+06 0.1177E+03 0.5312E+02 0.7500E-01210.00 0.4941E-03 0.1204E+02 0.5085E+00 0.9885E+06 0.1055E+03 0.5285E+02 0.7455E-01220.00 0.4966E-03 0.8900E-04 0.1099E+02 0.5134E+00 0.1322E+07 0.9544E+02 0.5258E+02 0.7410E-01240.00 0.4990E-03 0.1119E-03 0.1010E+02 0.5184E+00 0.1728E+07 0.8714E+02 0.5231E+02 0.7364E-01260.00 0.5016E-03 0.1408E-03 0.9337E+01 0.5283E+00 0.2213E+07 0.8022E+02 0.5204E+02 0.7317E-01280.00 0.5041E-03 0.1771E-03 0.8676E+01 0.5383E+00 0.2777E+07 0.7442E+02 0.5150E+02 0.7270E-01300.00 0.5066E-03 0.2227E-03 0.8098E+01 0.5482E+00 0.3424E+07 0.6950E+02 0.5096E+02 0.7222E-01300.00 0.5092E-03 0.2802E-03 0.7129E+01 0.5582E+00 0.4152E+07 0.6530E+02 0.5042E+02 0.7174E-01 0.5118E-03 0.3524E-03 0.6332E+01 0.5731E+00 0.4959E+07 0.6168E+02 0.4988E+02 0.7074E-01 0.5145E-03 0.4432E-03 0.5639E+01 0.5843E+07 0.5852E+02 0.4906E+02 0.6972E-01 0.5171E-03 0.5575E-03 0.5006E+01 0.6803E+07 0.5325E+02 0.6868E-01 0.5198E-03 0.7011E-03 0.4106E+01 0.7838E+07 0.4888E+02 0.6761E-01 0.5253E-03 0.8819E-03 0.8950E+07 0.4502E+02 0.6595E-01 0.5308E-03 0.1109E-02 0.1143E+08 0.4133E+02 0.5365E-03 0.1395E-02 0.1433E+08 0.3568E+02 0.5424E-03 0.2207E-02 0.1788E+08 0.5513E-03 0.3491E-02 0.2245E+08 0.5523E-02 0.3282E+08 0.8738E-02 0.1739E-01

364

Table C.11 Properties of Freon E2 Tcp ρµ k Pr β BPam –40.00 0.2198E+00 0.1137E+03 0.1329E+02 0.4235E-01 0.6894E+02 0.7778E-03 0.2377E+08 0.6620E-02 –30.00 0.2215E+00 0.1129E+03 0.1107E+02 0.4190E-01 0.5852E+02 0.7839E-03 0.3397E+08 0.1054E-01 –20.00 0.2233E+00 0.1120E+03 0.9290E+01 0.4146E-01 0.5004E+02 0.7900E-03 0.4785E+08 0.1649E-01 –10.00 0.2252E+00 0.1111E+03 0.7854E+01 0.4101E-01 0.4312E+02 0.7963E-03 0.6642E+08 0.2540E-01 0.00 0.2270E+00 0.1102E+03 0.6689E+01 0.4057E-01 0.3743E+02 0.8027E-03 0.9085E+08 0.3849E-01 10.00 0.2289E+00 0.1093E+03 0.5738E+01 0.4012E-01 0.3274E+02 0.8092E-03 0.1225E+09 0.5745E-01 20.00 0.2309E+00 0.1084E+03 0.4958E+01 0.3968E-01 0.2885E+02 0.8158E-03 0.1627E+09 0.8446E-01 30.00 0.2328E+00 0.1075E+03 0.4316E+01 0.3923E-01 0.2562E+02 0.8225E-03 0.2129E+09 0.1224E+00 40.00 0.2348E+00 0.1067E+03 0.3784E+01 0.3878E-01 0.2291E+02 0.8294E-03 0.2747E+09 0.1748E+00 50.00 0.2369E+00 0.1058E+03 0.3343E+01 0.3834E-01 0.2065E+02 0.8363E-03 0.3492E+09 0.2462E+00 60.00 0.2389E+00 0.1049E+03 0.2974E+01 0.3789E-01 0.1875E+02 0.8434E-03 0.4374E+09 0.3422E+00

365 70.00 0.2410E+00 0.1040E+03 0.2665E+01 0.3745E-01 0.1716E+02 0.8505E-03 0.5400E+09 0.4696E+00 80.00 0.2432E+00 0.1031E+03 0.2406E+01 0.3700E-01 0.1581E+02 0.8578E-03 0.6569E+09 0.6362E+00 90.00 0.2453E+00 0.1022E+03 0.2188E+01 0.3656E-01 0.1469E+02 0.8652E-03 0.7875E+09 0.8515E+00 100.00 0.2475E+00 0.1014E+03 0.2005E+01 0.3611E-01 0.1374E+02 0.8728E-03 0.9303E+09 0.1126E+01 110.00 0.2498E+00 0.1005E+03 0.1850E+01 0.3567E-01 0.1295E+02 0.8805E-03 0.1083E+10 0.1473E+01 120.00 0.2520E+00 0.9958E+02 0.1719E+01 0.3522E-01 0.1230E+02 0.8883E-03 0.1242E+10 0.1905E+01 130.00 0.2543E+00 0.9870E+02 0.1610E+01 0.3477E-01 0.1177E+02 0.8963E-03 0.1405E+10 0.2439E+01 140.00 0.2567E+00 0.9781E+02 0.1518E+01 0.3433E-01 0.1135E+02 0.9044E-03 0.1565E+10 0.3090E+01 150.00 0.2590E+00 0.9693E+02 0.1443E+01 0.3388E-01 0.1103E+02 0.9126E-03 0.1718E+10 0.3878E+01 160.00 0.2614E+00 0.9604E+02 0.1381E+01 0.3344E-01 0.1079E+02 0.9210E-03 0.1858E+10 0.4821E+01 170.00 0.2638E+00 0.9516E+02 0.1331E+01 0.3299E-01 0.1064E+02 0.9296E-03 0.1981E+10 0.5940E+01 180.00 0.2663E+00 0.9428E+02 0.1293E+01 0.3255E-01 0.1058E+02 0.9383E-03 0.2081E+10 0.7257E+01 190.00 0.2688E+00 0.9339E+02 0.1264E+01 0.3210E-01 0.1059E+02 0.9472E-03 0.2155E+10 0.8794E+01 200.00 0.2713E+00 0.9251E+02 0.1246E+01 0.3165E-01 0.1068E+02 0.9563E-03 0.2198E+10 0.1058E+02 210.00 0.2739E+00 0.9162E+02 0.1237E+01 0.3121E-01 0.1085E+02 0.9655E-03 0.2210E+10 0.1263E+02 220.00 0.2764E+00 0.9074E+02 0.1237E+01 0.3076E-01 0.1111E+02 0.9749E-03 0.2189E+10 0.1497E+02 am BP β Pr k ρµ p 0.00 0.6881E+00 0.6617E+02 0.9576E+02 0.2198E+00 0.2997E+03 0.3803E-03 0.7570E+05 0.0000E+00 10.0020.00 0.6936E+0030.00 0.6992E+0040.00 0.6591E+02 0.7048E+0050.00 0.6566E+02 0.7104E+0060.00 0.7131E+02 0.6541E+02 0.7159E+0070.00 0.5368E+02 0.6516E+02 0.7215E+0080.00 0.4083E+02 0.6491E+02 0.2204E+00 0.7271E+0090.00 0.3140E+02 0.6466E+02 0.2209E+00 0.7326E+00 0.2440E+02 0.6441E+02 0.2213E+00 0.7382E+00 0.2244E+03 0.1917E+02 0.6415E+02 0.2214E+00 0.1699E+03 0.1522E+02 0.6390E+02 0.2215E+00 0.3817E-03 0.1301E+03 0.1221E+02 0.2214E+00 0.3832E-03 0.1007E+03 0.9905E+01 0.2212E+00 0.1360E+06 0.3846E-03 0.7887E+02 0.2208E+00 0.2391E+06 0.3861E-03 0.6247E+02 0.2202E+00 0.4116E+06 0.3876E-03 0.0000E+00 0.5003E+02 0.6934E+06 0.3891E-03 0.0000E+00 0.4053E+02 0.1144E+07 0.3907E-03 0.0000E+00 0.3320E+02 0.1846E+07 0.3922E-03 0.0000E+00 0.2918E+07 0.3937E-03 0.0000E+00 0.4513E+07 0.0000E+00 0.6833E+07 0.0000E+00 0.0000E+00 0.0000E+00 –40.00–30.00 0.6657E+00–20.00 0.6713E+00–10.00 0.6717E+02 0.6769E+00 0.6692E+02 0.6825E+00 0.3465E+03 0.6667E+02 0.2472E+03 0.6642E+02 0.1783E+03 0.2159E+00 0.1300E+03 0.2171E+00 0.2182E+00 0.1068E+04 0.2191E+00 0.7644E+03 0.3746E-03 0.5532E+03 0.3760E-03 0.4049E+03 0.5871E+04 0.3774E-03 0.1149E+05 0.3788E-03 0.2200E+05 0.0000E+00 0.4125E+05 0.0000E+00 0.0000E+00 0.0000E+00 230.00 0.8159E+00 0.6038E+02 0.1620E+01 0.1977E+00 0.6684E+01 0.4157E-03 0.2414E+09 0.0000E+00 100.00110.00 0.7438E+00120.00 0.7493E+00130.00 0.6365E+02 0.7549E+00140.00 0.6340E+02 0.7605E+00150.00 0.8120E+01 0.6315E+02 0.7660E+00160.00 0.6728E+01 0.6290E+02 0.7716E+00170.00 0.5634E+01 0.6264E+02 0.2196E+00 0.7771E+00180.00 0.4769E+01 0.6239E+02 0.2188E+00 0.7827E+00190.00 0.4080E+01 0.6214E+02 0.2178E+00 0.7882E+00200.00 0.2751E+02 0.3528E+01 0.6189E+02 0.2167E+00 0.7938E+00 0.2305E+02 0.3084E+01 0.6164E+02 0.2154E+00 0.7993E+00 0.3953E-03 0.1953E+02 0.2724E+01 0.6139E+02 0.2140E+00 0.3969E-03 0.1674E+02 0.2432E+01 0.6113E+02 0.2125E+00 0.1013E+08 0.3984E-03 0.1451E+02 0.2195E+01 0.2108E+00 0.1469E+08 0.4000E-03 0.1272E+02 0.2002E+01 0.2090E+00 0.2087E+08 0.4016E-03 0.0000E+00 0.1128E+02 0.2070E+00 0.2901E+08 0.4033E-03 0.0000E+00 0.1011E+02 0.2049E+00 0.3947E+08 0.4049E-03 0.0000E+00 0.9173E+01 0.5258E+08 0.4065E-03 0.0000E+00 0.8416E+01 0.6855E+08 0.4082E-03 0.0000E+00 0.7809E+01 0.8749E+08 0.4099E-03 0.0000E+00 0.1093E+09 0.4116E-03 0.0000E+00 0.1337E+09 0.0000E+00 0.1600E+09 0.0000E+00 0.0000E+00 0.0000E+00 Table C.12Table Mixture of Glycol-Water Properties Tc

366

Table C.13 Properties of PAO (2 CST)

Tcp ρµ k Pr β BPam –40.00 0.4749E+00 0.5201E+02 0.5473E+03 0.8599E-01 0.3023E+04 0.4300E-03 0.1619E+04 0.1267E-05 –30.00 0.4795E+00 0.5177E+02 0.3317E+03 0.8567E-01 0.1857E+04 0.4300E-03 0.4367E+04 0.2373E-05 –20.00 0.4841E+00 0.5154E+02 0.2106E+03 0.8535E-01 0.1195E+04 0.4300E-03 0.1073E+05 0.4320E-05 –10.00 0.4887E+00 0.5130E+02 0.1396E+03 0.8503E-01 0.8021E+03 0.4300E-03 0.2423E+05 0.7658E-05 0.00 0.4933E+00 0.5107E+02 0.9616E+02 0.8471E-01 0.5600E+03 0.4300E-03 0.5057E+05 0.1324E-04 10.00 0.4979E+00 0.5083E+02 0.6866E+02 0.8440E-01 0.4051E+03 0.4300E-03 0.9826E+05 0.2237E-04 20.00 0.5025E+00 0.5060E+02 0.5065E+02 0.8408E-01 0.3027E+03 0.4300E-03 0.1789E+06 0.3697E-04 30.00 0.5071E+00 0.5036E+02 0.3848E+02 0.8376E-01 0.2330E+03 0.4300E-03 0.3072E+06 0.5987E-04 40.00 0.5117E+00 0.5013E+02 0.3001E+02 0.8344E-01 0.1840E+03 0.4600E-03 0.5351E+06 0.9509E-04 50.00 0.5163E+00 0.4989E+02 0.2397E+02 0.8312E-01 0.1489E+03 0.4600E-03 0.8311E+06 0.1483E-03 60.00 0.5209E+00 0.4966E+02 0.1955E+02 0.8280E-01 0.1230E+03 0.4600E-03 0.1238E+07 0.2274E-03 70.00 0.5255E+00 0.4942E+02 0.1624E+02 0.8249E-01 0.1035E+03 0.4600E-03 0.1776E+07 0.3431E-03

367 80.00 0.5301E+00 0.4919E+02 0.1372E+02 0.8217E-01 0.8849E+02 0.4600E-03 0.2466E+07 0.5099E-03 90.00 0.5347E+00 0.4895E+02 0.1175E+02 0.8185E-01 0.7674E+02 0.4600E-03 0.3330E+07 0.7468E-03 100.00 0.5392E+00 0.4872E+02 0.1018E+02 0.8153E-01 0.6736E+02 0.4600E-03 0.4389E+07 0.1079E-02 110.00 0.5438E+00 0.4848E+02 0.8919E+01 0.8121E-01 0.5972E+02 0.4600E-03 0.5668E+07 0.1539E-02 120.00 0.5484E+00 0.4825E+02 0.7877E+01 0.8089E-01 0.5340E+02 0.4600E-03 0.7196E+07 0.2169E-02 130.00 0.5530E+00 0.4801E+02 0.7006E+01 0.8058E-01 0.4808E+02 0.5100E-03 0.9988E+07 0.3020E-02 140.00 0.5576E+00 0.4778E+02 0.6266E+01 0.8026E-01 0.4353E+02 0.5100E-03 0.1237E+08 0.4160E-02 150.00 0.5621E+00 0.4754E+02 0.5628E+01 0.7994E-01 0.3958E+02 0.5100E-03 0.1517E+08 0.5671E-02 160.00 0.5667E+00 0.4731E+02 0.5072E+01 0.7962E-01 0.3610E+02 0.5100E-03 0.1850E+08 0.7653E-02 170.00 0.5713E+00 0.4707E+02 0.4582E+01 0.7930E-01 0.3301E+02 0.5100E-03 0.2245E+08 0.1023E-01 180.00 0.5759E+00 0.4684E+02 0.4146E+01 0.7899E-01 0.3022E+02 0.5100E-03 0.2715E+08 0.1355E-01 190.00 0.5804E+00 0.4660E+02 0.3754E+01 0.7867E-01 0.2770E+02 0.5100E-03 0.3277E+08 0.1779E-01 200.00 0.5850E+00 0.4637E+02 0.3402E+01 0.7835E-01 0.2540E+02 0.5100E-03 0.3951E+08 0.2318E-01 220.00 0.5941E+00 0.4590E+02 0.2794E+01 0.7771E-01 0.2136E+02 0.5700E-03 0.6412E+08 0.3841E-01 240.00 0.6033E+00 0.4543E+02 0.2297E+01 0.7708E-01 0.1798E+02 0.5700E-03 0.9301E+08 0.6184E-01 260.00 0.6124E+00 0.4496E+02 0.1892E+01 0.7644E-01 0.1516E+02 0.5700E-03 0.1342E+09 0.9697E-01 280.00 0.6215E+00 0.4449E+02 0.1569E+01 0.7580E-01 0.1287E+02 0.5700E-03 0.1910E+09 0.1484E+00 300.00 0.6306E+00 0.4402E+02 0.1320E+01 0.7517E-01 0.1107E+02 0.5700E-03 0.2645E+09 0.2221E+00 am BP β Pr k ρµ p 0.00 0.5100E+00 0.5241E+02 0.6523E+03 0.8969E-01 0.3709E+04 0.4200E-03 0.1131E+04 0.0000E+00 10.0020.00 0.5140E+0030.00 0.5180E+0040.00 0.5219E+02 0.5220E+0050.00 0.5196E+02 0.5260E+0060.00 0.4313E+03 0.5174E+02 0.5300E+0070.00 0.2950E+03 0.5152E+02 0.5340E+0080.00 0.2082E+03 0.5129E+02 0.5380E+0090.00 0.8935E-01 0.1513E+03 0.5107E+02 0.5420E+00 0.8902E-01 0.1128E+03 0.5084E+02 0.5460E+00 0.8868E-01 0.2481E+04 0.8609E+02 0.5062E+02 0.8835E-01 0.1717E+04 0.6714E+02 0.5039E+02 0.8801E-01 0.4200E-03 0.1226E+04 0.5337E+02 0.8767E-01 0.4200E-03 0.9006E+03 0.4316E+02 0.8734E-01 0.2565E+04 0.4200E-03 0.6791E+03 0.8700E-01 0.5434E+04 0.4400E-03 0.5243E+03 0.8667E-01 0.1081E+05 0.4400E-03 0.0000E+00 0.4136E+03 0.2128E+05 0.4400E-03 0.0000E+00 0.3325E+03 0.3795E+05 0.4400E-03 0.0000E+00 0.2719E+03 0.6456E+05 0.4400E-03 0.0000E+00 0.1052E+06 0.4400E-03 0.0000E+00 0.1650E+06 0.0000E+00 0.2501E+06 0.0000E+00 0.0000E+00 0.0000E+00 –40.00–30.00 0.4940E+00–20.00 0.4980E+00–10.00 0.5331E+02 0.5020E+00 0.5309E+02 0.5060E+00 0.5111E+04 0.5286E+02 0.2857E+04 0.5264E+02 0.1674E+04 0.9103E-01 0.1024E+04 0.9069E-01 0.9036E-01 0.2774E+05 0.9002E-01 0.1569E+05 0.4200E-03 0.9299E+04 0.4200E-03 0.5756E+04 0.1905E+02 0.4200E-03 0.6046E+02 0.4200E-03 0.1747E+03 0.0000E+00 0.4626E+03 0.0000E+00 0.0000E+00 0.0000E+00 300.00 0.6300E+00 0.4568E+02 0.2828E+01 0.7963E-01 0.2238E+02 0.5300E-03 0.5765E+08 0.0000E+00 100.00110.00 0.5500E+00120.00 0.5540E+00130.00 0.5017E+02 0.5580E+00140.00 0.4994E+02 0.5620E+00150.00 0.3543E+02 0.4972E+02 0.5660E+00160.00 0.2949E+02 0.4950E+02 0.5700E+00170.00 0.2483E+02 0.4927E+02 0.5740E+00180.00 0.8633E-01 0.2112E+02 0.4905E+02 0.5780E+00190.00 0.8600E-01 0.1813E+02 0.4882E+02 0.5820E+00200.00 0.8566E-01 0.2257E+03 0.1568E+02 0.4860E+02 0.5860E+00220.00 0.8533E-01 0.1900E+03 0.1364E+02 0.4837E+02 0.5900E+00240.00 0.8499E-01 0.4400E-03 0.1617E+03 0.1194E+02 0.4815E+02 0.5980E+00260.00 0.8466E-01 0.4400E-03 0.1391E+03 0.1050E+02 0.4792E+02 0.6060E+00280.00 0.8432E-01 0.3678E+06 0.4400E-03 0.1207E+03 0.9271E+01 0.4748E+02 0.6140E+00 0.8399E-01 0.5264E+06 0.4800E-03 0.1055E+03 0.8212E+01 0.4703E+02 0.6220E+00 0.8365E-01 0.7358E+06 0.4800E-03 0.0000E+00 0.9289E+02 0.6497E+01 0.4658E+02 0.8332E-01 0.1099E+07 0.4800E-03 0.0000E+00 0.8219E+02 0.5189E+01 0.4613E+02 0.8298E-01 0.1479E+07 0.4800E-03 0.0000E+00 0.7307E+02 0.4184E+01 0.8231E-01 0.1959E+07 0.4800E-03 0.0000E+00 0.6521E+02 0.3413E+01 0.8164E-01 0.2562E+07 0.4800E-03 0.0000E+00 0.5839E+02 0.8097E-01 0.3314E+07 0.4800E-03 0.0000E+00 0.4720E+02 0.8030E-01 0.4246E+07 0.4800E-03 0.0000E+00 0.3852E+02 0.5398E+07 0.5300E-03 0.0000E+00 0.3173E+02 0.6816E+07 0.5300E-03 0.0000E+00 0.2644E+02 0.1180E+08 0.5300E-03 0.0000E+00 0.1815E+08 0.5300E-03 0.0000E+00 0.2739E+08 0.0000E+00 0.4038E+08 0.0000E+00 0.0000E+00 0.0000E+00 Table C.14Table (4 CST) of PAO Properties Tc

368

Table C.15 Properties of PAO (6 CST)

Tcp ρµ k Pr β BPam –40.00 0.4675E+00 0.5370E+02 0.1636E+05 0.9256E-01 0.8265E+05 0.4100E-03 0.1841E+01 0.0000E+00 –30.00 0.4718E+00 0.5348E+02 0.8460E+04 0.9234E-01 0.4322E+05 0.4100E-03 0.6832E+01 0.0000E+00 –20.00 0.4762E+00 0.5326E+02 0.4624E+04 0.9212E-01 0.2390E+05 0.4100E-03 0.2268E+02 0.0000E+00 –10.00 0.4805E+00 0.5305E+02 0.2661E+04 0.9191E-01 0.1391E+05 0.4100E-03 0.6793E+02 0.0000E+00 0.00 0.4848E+00 0.5283E+02 0.1606E+04 0.9169E-01 0.8493E+04 0.4100E-03 0.1850E+03 0.0000E+00 10.00 0.4892E+00 0.5261E+02 0.1013E+04 0.9147E-01 0.5415E+04 0.4100E-03 0.4616E+03 0.0000E+00 20.00 0.4935E+00 0.5240E+02 0.6643E+03 0.9125E-01 0.3593E+04 0.4100E-03 0.1063E+04 0.0000E+00 30.00 0.4978E+00 0.5218E+02 0.4521E+03 0.9103E-01 0.2472E+04 0.4100E-03 0.2278E+04 0.0000E+00 40.00 0.5022E+00 0.5196E+02 0.3179E+03 0.9081E-01 0.1758E+04 0.4300E-03 0.4789E+04 0.0000E+00 50.00 0.5065E+00 0.5175E+02 0.2304E+03 0.9059E-01 0.1288E+04 0.4300E-03 0.9045E+04 0.0000E+00 60.00 0.5108E+00 0.5153E+02 0.1715E+03 0.9038E-01 0.9694E+03 0.4300E-03 0.1619E+05 0.0000E+00 70.00 0.5152E+00 0.5131E+02 0.1308E+03 0.9016E-01 0.7472E+03 0.4300E-03 0.2761E+05 0.0000E+00

369 80.00 0.5195E+00 0.5110E+02 0.1018E+03 0.8994E-01 0.5883E+03 0.4300E-03 0.4513E+05 0.0000E+00 90.00 0.5238E+00 0.5088E+02 0.8083E+02 0.8972E-01 0.4719E+03 0.4300E-03 0.7104E+05 0.0000E+00 100.00 0.5282E+00 0.5066E+02 0.6522E+02 0.8950E-01 0.3849E+03 0.4300E-03 0.1082E+06 0.0000E+00 110.00 0.5325E+00 0.5045E+02 0.5339E+02 0.8928E-01 0.3184E+03 0.4300E-03 0.1601E+06 0.0000E+00 120.00 0.5368E+00 0.5023E+02 0.4425E+02 0.8907E-01 0.2667E+03 0.4300E-03 0.2311E+06 0.0000E+00 130.00 0.5411E+00 0.5001E+02 0.3706E+02 0.8885E-01 0.2257E+03 0.4600E-03 0.3493E+06 0.0000E+00 140.00 0.5455E+00 0.4980E+02 0.3132E+02 0.8863E-01 0.1927E+03 0.4600E-03 0.4849E+06 0.0000E+00 150.00 0.5498E+00 0.4958E+02 0.2666E+02 0.8841E-01 0.1658E+03 0.4600E-03 0.6632E+06 0.0000E+00 160.00 0.5541E+00 0.4936E+02 0.2284E+02 0.8819E-01 0.1435E+03 0.4600E-03 0.8959E+06 0.0000E+00 170.00 0.5584E+00 0.4915E+02 0.1967E+02 0.8797E-01 0.1248E+03 0.4600E-03 0.1198E+07 0.0000E+00 180.00 0.5627E+00 0.4893E+02 0.1700E+02 0.8776E-01 0.1090E+03 0.4600E-03 0.1589E+07 0.0000E+00 190.00 0.5670E+00 0.4872E+02 0.1475E+02 0.8754E-01 0.9552E+02 0.4600E-03 0.2093E+07 0.0000E+00 200.00 0.5713E+00 0.4850E+02 0.1283E+02 0.8732E-01 0.8392E+02 0.4600E-03 0.2743E+07 0.0000E+00 220.00 0.5800E+00 0.4807E+02 0.9765E+01 0.8688E-01 0.6518E+02 0.5000E-03 0.5052E+07 0.0000E+00 240.00 0.5886E+00 0.4763E+02 0.7493E+01 0.8644E-01 0.5102E+02 0.5000E-03 0.8426E+07 0.0000E+00 260.00 0.5972E+00 0.4720E+02 0.5802E+01 0.8601E-01 0.4029E+02 0.5000E-03 0.1380E+08 0.0000E+00 280.00 0.6058E+00 0.4677E+02 0.4553E+01 0.8557E-01 0.3223E+02 0.5000E-03 0.2199E+08 0.0000E+00 300.00 0.6144E+00 0.4633E+02 0.3645E+01 0.8513E-01 0.2631E+02 0.5000E-03 0.3368E+08 0.0000E+00 Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table C.16 Properties of PAO (8 CST)

Tcp ρµ k Pr β BPam –40.00 0.4945E+00 0.5412E+02 0.4030E+05 0.9393E-01 0.2121E+06 0.4000E-03 0.3009E+00 0.0000E+00 –30.00 0.4980E+00 0.5391E+02 0.1929E+05 0.9373E-01 0.1025E+06 0.4000E-03 0.1302E+01 0.0000E+00 –20.00 0.5014E+00 0.5369E+02 0.9865E+04 0.9353E-01 0.5289E+05 0.4000E-03 0.4941E+01 0.0000E+00 –10.00 0.5049E+00 0.5347E+02 0.5360E+04 0.9332E-01 0.2900E+05 0.4000E-03 0.1660E+02 0.0000E+00 0.00 0.5084E+00 0.5326E+02 0.3079E+04 0.9312E-01 0.1681E+05 0.4000E-03 0.4991E+02 0.0000E+00 10.00 0.5119E+00 0.5304E+02 0.1861E+04 0.9292E-01 0.1025E+05 0.4000E-03 0.1355E+03 0.0000E+00 20.00 0.5154E+00 0.5283E+02 0.1178E+04 0.9271E-01 0.6548E+04 0.4000E-03 0.3355E+03 0.0000E+00 30.00 0.5189E+00 0.5261E+02 0.7776E+03 0.9251E-01 0.4361E+04 0.4000E-03 0.7635E+03 0.0000E+00 40.00 0.5224E+00 0.5239E+02 0.5330E+03 0.9231E-01 0.3017E+04 0.4200E-03 0.1692E+04 0.0000E+00 50.00 0.5259E+00 0.5218E+02 0.3780E+03 0.9211E-01 0.2159E+04 0.4200E-03 0.3336E+04 0.0000E+00 60.00 0.5295E+00 0.5196E+02 0.2763E+03 0.9190E-01 0.1592E+04 0.4200E-03 0.6194E+04 0.0000E+00 70.00 0.5330E+00 0.5174E+02 0.2074E+03 0.9170E-01 0.1206E+04 0.4200E-03 0.1090E+05 0.0000E+00

370 80.00 0.5365E+00 0.5153E+02 0.1594E+03 0.9150E-01 0.9347E+03 0.4200E-03 0.1830E+05 0.0000E+00 90.00 0.5400E+00 0.5131E+02 0.1250E+03 0.9129E-01 0.7396E+03 0.4200E-03 0.2949E+05 0.0000E+00 100.00 0.5436E+00 0.5110E+02 0.9982E+02 0.9109E-01 0.5957E+03 0.4200E-03 0.4589E+05 0.0000E+00 110.00 0.5471E+00 0.5088E+02 0.8090E+02 0.9089E-01 0.4870E+03 0.4200E-03 0.6928E+05 0.0000E+00 120.00 0.5507E+00 0.5066E+02 0.6640E+02 0.9069E-01 0.4032E+03 0.4200E-03 0.1020E+06 0.0000E+00 130.00 0.5543E+00 0.5045E+02 0.5507E+02 0.9048E-01 0.3373E+03 0.4500E-03 0.1574E+06 0.0000E+00 140.00 0.5578E+00 0.5023E+02 0.4607E+02 0.9028E-01 0.2847E+03 0.4500E-03 0.2230E+06 0.0000E+00 150.00 0.5614E+00 0.5001E+02 0.3881E+02 0.9008E-01 0.2419E+03 0.4500E-03 0.3116E+06 0.0000E+00 160.00 0.5650E+00 0.4980E+02 0.3287E+02 0.8987E-01 0.2066E+03 0.4500E-03 0.4307E+06 0.0000E+00 170.00 0.5686E+00 0.4958E+02 0.2795E+02 0.8967E-01 0.1772E+03 0.4500E-03 0.5904E+06 0.0000E+00 180.00 0.5721E+00 0.4937E+02 0.2384E+02 0.8947E-01 0.1525E+03 0.4500E-03 0.8044E+06 0.0000E+00 190.00 0.5757E+00 0.4915E+02 0.2038E+02 0.8927E-01 0.1315E+03 0.4500E-03 0.1091E+07 0.0000E+00 200.00 0.5793E+00 0.4893E+02 0.1746E+02 0.8906E-01 0.1135E+03 0.4500E-03 0.1474E+07 0.0000E+00 220.00 0.5865E+00 0.4850E+02 0.1285E+02 0.8866E-01 0.8500E+02 0.4900E-03 0.2912E+07 0.0000E+00 240.00 0.5938E+00 0.4807E+02 0.9502E+01 0.8825E-01 0.6393E+02 0.4900E-03 0.5229E+07 0.0000E+00 260.00 0.6010E+00 0.4764E+02 0.7083E+01 0.8785E-01 0.4846E+02 0.4900E-03 0.9242E+07 0.0000E+00 280.00 0.6083E+00 0.4720E+02 0.5357E+01 0.8744E-01 0.3727E+02 0.4900E-03 0.1586E+08 0.0000E+00 300.00 0.6156E+00 0.4677E+02 0.4153E+01 0.8704E-01 0.2938E+02 0.4900E-03 0.2591E+08 0.0000E+00 Appendix D

TYPICAL EMISSIVITIES OF MATERIALS

Material Emissivity

Dielectric and semiconducting materials Acrylic (conformal coating) 0.9 Alumina, 96% pure 0.9 Alumina, 94% pure 0.9 Fiberglass (G-10) 0.9 FR-4 0.9 Polyimide/glass 0.9 Polyimide/Kevlar 0.9 Polyimide/quartz 0.9 Silicon 0.94

Metals Aluminum: Commercial sheet 0.09 Polished 0.06 Anodized 0.25–0.86* Heavily oxidized 0.21 Brass: polished 0.06 Copper: Polished 0.05 Heavily oxidized 0.78 Gold: Polished 0.02 Plated (0.0001 in) 0.11 Inconel: polished Type B, 0.19; Type X, 0.21 Iron: Polished 0.05 Rusted 0.65 Cast iron, polished 0.21 Cast iron, oxidized 0.95 Wrought iron, smooth 0.33 Wrought iron, dull 0.94 Wrought iron, polished 0.28 *Some references indicate that anodizing and similar surface treatments sometimes are difficult to control and are very irregular in many given surfaces. (continues)

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Material Emissivity Lead: Pure polished 0.06 Gray polished 0.28 Oxidized 0.63 Magnesium 0.06 Molybdenum 0.07 Nickel: Polished 0.05 Unpolished 0.11 Oxidized 0.34 Platinum 0.05 Silver: polished 0.05 Steel: Polished 0.08 Oxidized 0.8 Stainless 0.15 Tin 0.07

Various nonmetals Glass: Smooth 0.94 Conrex D 0.8 Nonex 0.82 White or black lacquer 0.8–0.95 Aluminum paint and lacquer 0.52 Rubber: Hard black 0.94 Soft gray 0.86 Water: Liquid water or ice 0.95–0.96 Snow 0.86

Ground surfaces Asphalt pavement 0.93 Concrete 0.94 Sand 0.76 Soil Dry, 0.92; wet, 0.95

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use Appendix E

SOLAR ABSORPTIVITIES AND EMISSIVITIES OF COMMON SURFACES

Surface finish Solar absorptivity α Emissivity εα/ε Metallic Aluminum: Vapor-deposited 0.08–0.17 0.04 2–4.25 Bare 0.09–0.17 0.03–0.1 3–1.7 Anodized* 0.25–0.86 0.04–0.88 6.25–0.98 Gold: Vapor-deposited 0.19–0.3 0.03 6.33–1 Bare 0.16 0.02 8 Molybdenum 0.43 0.03 14 Silver 0.07 0.01 7

Nonmetallic

Aluminum Kapton: 0.5 mil 0.34 0.55 0.62 1.0 mil 0.38 0.67 0.57 2.0 mil 0.41 0.75 0.55 5.0 mil 0.46 0.86 0.53 Black paints: Chemglaze Z306 0.92–0.98 0.89 1.03–1.10 3M black velvet 0.97 0.84 1.15 White paints: S13G-LO 0.20–0.25 0.85 0.24–0.29 Z93 0.17–0.2 0.92 0.18–0.22 ZOT 0.18–0.2 0.91 0.20–0.22 Chemglaze A276 0.22–0.28 0.88 0.25–0.32 Camouflage paint: Tan 0.7 0.86 0.81 Black 0.94 0.89 1.06 Green 0.66 0.89 0.74

*Some references indicate that anodizing and similar surface treatments sometimes are difficult to control and are very irregular in many given surfaces. (continues)

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use 374 • THERMAL MANAGEMENT OF MICROELECTRONIC EQUIPMENT

Surface finish Solar absorptivity a Emissivity e a/e Mylar: 0.5-mil aluminized Mylar Aluminum side 0.13 0.04 3.25 Mylar side 0.18 0.9 0.2 Beta cloth 0.32 0.86 0.37 Astro quartz 0.22 0.8 0.28 MAXORB 0.9 0.1 9 Optical solar reflector: 8-mil quartz mirrors 0.05–0.08 0.8 0.063–0.1 2-mil silver Teflon 0.05–0.09 0.66 0.076–0.12 5-mil silver Teflon 0.05–0.09 0.78 0.064–0.12 2-mil aluminized Teflon 0.10–0.16 0.66 0.15–0.24 5-mil aluminized Teflon 0.10–0.16 0.78 0.13–0.21

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use Appendix F

PROPERTIES OF PHASE-CHANGE MATERIALS

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use — — — — — — — — — — — — — — — — — — — — — — 0.6 0.2 0.1 19.0 — — — — — — — — — — — — — — — — — — — — — — — kJ/(kg·°C) W/(m·°C) 3 91 8600 33 9400 0.17 — 8900 301 2180 754 — 242 1160 2.8 302 — 190 1212 232 — 174 1449 272 1720 281285 836 846242 1.5–2.2 1.3–1.9 1800 0.2–0.3 0.2–0.3 236244167249256 — 254 — 266 — 271 — 274 — 803279 809 815 820 1.9–2.4 831 1.9–2.5 1.9–2.5 1.9–2.5 0.2–0.3 1.6–2.3 0.2–0.3 0.2–0.3 0.2–0.3 0.2–0.3 320 — 344 860 227 — 211 — 214 — 164 — 279 — 216 1280 332 1000 2.0–4.2 5.6 0.0 –9.6 57.8 78.0 27.8 81.0 30.0 66.0 15.6 70.0 77.0 64.3 18.1 28.1 35.0 44.0 56.3 61.2 69.7 79.0 86.4 99.0 30.0 82.0 31.0 23.0 37.2 58.0 105.5 115.3 102.0 Melting Latent Density, Specific heat, conductivity, conductivity, heat, Specific Density, Melting Latent point, °C heat, kJ/kg kg/m 4 O O 2 2 O 2 2 O O ) 2 2 Chemical form _3H 3 + 2B(Othe) O 3 _12H 2 _H 4 6 4 CH _10H CH 2 OBr 4 2 202 26 30 34 38 42 46 54 58 66 78 90 122 142 15 ON 5 H 5 Cl HPO CrO SO H H H H H H H H H H H H H H 4 2 2 2 O O H 2 2 12 14 16 18 20 22 26 28 32 38 44 60 70 100 10 3 En+2NH Na N PH C C 8%Sn-23%Pb-45%Bi-19%In-5%Cd12%Sn-18%Pb-49%Bi-21%In 13%Sn-27%Pb-50%Bi-10%Cd 47.2 C Na C C C C C C Na C C C C C C Ba(OH)_8H NaOH_H NaCO (CHCO H C Ethylene diammonium ditetranethoxyborate Sodium chromate Nitrogen pentoxide Methyl fumarate Phosphonium chloride Transit-Heat 86 Transit-Heat Cerrolow-136 Organic paraffins: Dodecane Organic nonparaffins: Polyglycol Bromcamphor Inorganic: 60 Transit-Heat Eutectic alloy (soft solder): Cerrolow-117 Cerrobend Acetamide Sodium sulfate Hectane Octadecane eicosane Technical Docosane Hexacosane Octacosane Dibasic sodium phosphate Water Dotriacontane Octatriacontane Tetrateracontane Hexacontane Heptacontane Hexadecane Tetradecane Barium hydroxide octahydrate Sodium hydroxide monohydrate Sodium acetate trihydrate

376

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FRICTION FACTOR CORRELATIONS

Physical condition Range of validity Index Correlation Ref. Smooth circular tube Fully developed laminar flow Re < 2300 1 f·Re = 16 1 Fully developed turbulent flow 5000 < Re < 300,000 2a f = 0.079 Re–0.25 1 300,000 < Re < 1,000,000 2b f = 0.046 Re–0.2 1

Rough tube Fully turbulent zone, ε = absolute roughness ε 1  D D/ 3 = 174.ln  + 228 . 2 > 001.  ε  fRe f Transition turbulent zone (Implicit) 4 3 1  ε/D 251.  =−2log +  f  37. Re f  Transition turbulent zone (Explicit) 5   4  6  13/  20, 000ε 10  f ≈+0. 0055 1  +   D Re     378 Rectangular tubes Fully developed laminar flow 6 f ⋅=Re23 . 922 − 30 . 201α 5 +−32.. 897αα23 12 439 a b αα==≤≤aspect ratio 01 b a 2ab D = h ab+

For parallel plates:Dh = 2a; a = 0 Rectangular n-sided Fully developed laminar flow 3 ≤ n ≤ 15 7 6 ⋅= + polygonal tubes fARe8 . 1394 14 . 9123 −+9.. 0574AA23 1 6533 = Anlog10

Isosceles triangular tubes Fully developed laminar flow 8 f ⋅=Re12 . 0408 + 0 . 09824θ 5 θ b = 30° for equilateral triangular duct − 0. 002168θ 2 2θ = apex angle 2θ + 0. 00001189θ 3 b Appendix G • 379

REFERENCES

1. W. M. Kays, Convective Heat and Mass Transfer, McGraw-Hill, New York, 1966. 2. J. G. Knudsen and D. L. Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958. 3. L. F. Moody, “Friction Factors for Pipe Flow,” Trans. ASME 66, 1944. 4. L. F. Moody, “An Approximate Formula for Pipe Friction Factors,” Mech. Eng., 1950. 5. Developed by L. T. Yeh, according to Reference 7. 6. L. T. Yeh, “Validity of One Heat Transfer Correlation to Various Shaped Ducts via Hydraulic Diameter Concept,” Proc. 4th Int. Electronics Packaging Conf., Baltimore, 1984. 7. R. K. Shah and A. L. London, “Laminar Flow Forced Convection in Ducts,” Advances in Heat Transfer, Supplement 1, Academic Press, New York, 1978.

HEAT TRANSFER CORRELATIONS

Table H.1 Common Correlations for External Forced Convection

Type Fluid properties Physical situation of fluid Range of validity Index Heat transfer correlation Ref evaluated at: Forced convection over Gas or Local value a flat plate heated over liquid Rex = ρVx/µ Nux = hx/k the entire length L Constant wall temp Pr > 0.6 1 1 Film temp Tf = 5 1/2 1/3 Laminar flow Rex < 5 × 10 (Nux)T = 0.332(Rex) Pr (T∞ + Tw)/2 where 0.8 1/3 Turbulent flow (Nux)T = 0.0295(Rex) Pr T∞ = fluid temp, Tw = wall temp Constant heat flux 2 1 Film temp 5 1/2 1/3 Laminar flow Rex < 5 × 10 (Nux)H = 0.453(Rex) Pr 5 0.8 1/3 Turbulent flow Rex > 5 × 10 (Nux)H = 0.0307(Rex) Pr 382 Mean value ReL = ρVL/µ NuL = hL/k Constant wall temp Pr > 0.6 3 1 Film temp 5 1/2 1/3 Laminar flow ReL < 5 × 10 (NuL)T = 0.664(Rex) Pr 5 0.8 1/3 Turbulent flow ReL > 5 × 10 (NuL)T = 0.0369(Rex) Pr Constant heat flux 4 1 Film temp 5 1/2 1/3 Laminar flow ReL < 5 × 10 (Nux)H = 0.906(Rex) Pr 5 0.8 1/3 Turbulent flow ReL > 5 × 10 (NuL)H = 0.0384(Rex) Pr

Forced convection over Gas or Combined laminar and turbulent NuL = hL/k a flat plate heated over liquid flow over plate: Pr > 0.6, 5 entire length L ReL = ρVL/µ > 5 × 10

Constant wall temp 5 =−13/ 08. 2 Film temp (NuLT )0. 0369 Pr (ReL 23, 515)

Constant heat flux 6 =−13/ 08. 2 Film temp (NuLH )0. 0384 Pr (ReL 19, 557) Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Forced convection over Gas Pr = 0.7 7 =+ +× −41612. / 3 Film temp ≤ × 5 Nudd2 ( 0. 25 Re 3 10 Re ) smooth spheres 0 Red < 3 10 d 3 × 105 < Re < 6 × 106 =+×* −3 d Nudd430 0. 5 10 Re +×−−9 2 −×17 3 025.. 10Redd 31 10 Re

Red = ρVD/µ Nud = hD/k

Liquid 1 < Red < 200,000 8  µ  025. 4 Free-stream temp; Nu=+ (12 . 053 . Re054..)Pr 03  µ d d µ w at wall temp  w 

4 5 0.5 Liquid 3.56 × 10 ≤ Red < 1.6 × 10 9Nud = 2 + 0.386(Red Pr) 5 metal

Forced convection over Gas, liquid, Pr Red < 0.2 10 = 1 6 Nud smooth cylinders or liquid Nud = hD/k − 12/ 0.ln 8237 (Pr Red ) metal Red = ρVD/µ 383 Pr Re > 0.2 11 A 7 Film temp d Nu =+03.. 062Re12/ Pr 13/ d d B    Re  58/ 45/ A =+1  d        282, 000     04.  23/ 14/ B =+1         Pr 

′ Pr Red > 0.2 =+ 12/ 13/ A 7 Film temp 4 5 Nud 03.. 062Red Pr 7 × 10 < Red < 4 × 10 B  Re  05. A′ =+1  d   282, 000

*5 × 10–3 in the original paper, but 0.5 × 10–3 appears to give better results compared with reference 3. (continues) Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table H.1 (Continued)

Type Fluid properties Physical situation of fluid Range of validity Index Heat transfer correlation Ref evaluated at:

Forced convection Gas or Re = U∞Deρ/µ Film temp Tf crossflow over liquid noncircular cylinders** Nu = C Ren Pr1/3

384 nC 5000–100,000 12 0.588 0.222 8

2500–15,000 13 0.612 0.224 8

5000–100,000 14 0.638 0.138 8

5000–19,500 15 0.638 0.144 8

19,500–100,000 16 0.782 0.035 8 Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

3000–15,000 17 0.804 0.085 8

5000–100,000 18 0.675 0.092 8

4000–15,000 19 0.731 0.205 8

20,000–90,000 20 0.5 0.86 9

**This is for a rectangular or square plate with finite dimension. The angle of attack α is varied from 90° (normal incidence) to 25°. The Reynolds num-

385 ber Re is based on characteristic length De, which is defined as follows: De = 4A/C, where A = surface area of plate and C = circumference of plate. Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table H.2 Common Correlations for Internal Forced Convection

Type Fluid properties Physical situation of fluid Range of validity Index Heat transfer correlation Ref evaluated at:

Forced convection in Gas or liquid Laminar flow Nux = hD/k smooth circular tubes Re = ρVD/µ < 2300 x*= (x/D)/(Re Pr)

Local value

Constant wall temperature 1 10 Bulk temp x* →∞ (Nux)T = 3.657 (fully developed flow) 3 –0.488 –57.2x* x* > 0.01 (Nux)T = 3.657 + 6.874(10 x*) e –1/3 x* < 0.01 (Nux)T = 1.077(x*) – 0.7 386 Constant wall temperature 2 10 Bulk temp x* →∞ (Nux)H = 4.364 (fully developed flow) 3 –0.506 –41x* x* > 0.0015 (Nux)H = 4.364 + 8.68(10 x*) e –1/3 0.00005 < x* < 0.0015 (Nux)H = 1.32(x*) – 0.5 –1/3 x* ≤ 0.0015 (Nux)H = 1.32(x*) – 1.0 Mean value Constant wall temperature 3 11 Bulk temp =+0./ 0668 x* (NumT ) 3. 657 − 1004+ . (*)x 23/

Constant heat flux 4 10 Bulk temp x* > 0.03 (Nux)H = 4.364 + 0.0722/x* –1/3 x* ≤ 0.03 (Nux)H = 1.953(x*) Forced convection in Gas or liquid Turbulent flow 5 smooth circular tubes Re = rVD/m > 4000 Fully developed turbulent flow Nu = hD/k Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

4 0.1 < Pr < 10 Nu =+5015. Reaa12 Pr 12 Mean bulk temp 104 ≤ Re ≤ 106 024. a =−088. L/D > 30 1 4 + Pr =+1 −06. Pr a2 3 05. e

Gas 104 ≤ Re ≤ 106 6 Nu = 5 + 0.012 Re0.83 (Pr + 0.29) 12 Mean bulk temp L/D > 30

Forced convection in Gas or liquid Variable-property case (high =+ aa12 12 b: bulk temp Nub 5015. Ref Prw smooth circular tubes heating/cooling rate) 7 f: film temp 024. 0.01 < Pr < 104 a =−088. w: wall temp 1 + 104 ≤ Re ≤ 106 4 Prw − =+1 06. Prw ae2 3 05.

Gas Variable-property case (high =+ 0.83 + 12 b: bulk temp Nub 5 0. 012Ref (Prw 0.29) 387 heating/cooling rate) 8 f: film temp 104 ≤ Re ≤ 106; Pr < 0.9 w: wall temp

3 5 0.827 Liquid metal 3.6 × 10 ≤ Re ≤ 9 × 10 9 (Nu)H = 4.82 + 0.0185(Re Pr) 13 Mean bulk temp 2 4 0.85 0.93 10 ≤ Re Pr ≤ 10 10 (Nu)T = 4.8 + 0.0156 Re Pr 14 Mean bulk temp Liquid Turbulent thermal entrance: 11 Nu = 0.036 Re0.8 Pr1/3 (d/L)0.055 15 Mean bulk temp Pr > 0.9 (mean value) 10 < L/d < 400

Gas Turbulent thermal entrance 12 Nu/Nu∞ = 1 + y/(x/d) (mean value) 16 Mean bulk temp Pr = 0.7 y = 2.0 – 4.6 × 10–6(Re – 50,000) 104 < Re < 2 × 105 Nu∞ = 5 + 0.012 Re0.83 (Pr + 0.29)

Liquid metal Turbulent thermal entrance; 13 (Nux /Nu∞)H = 1 + 2/(x/d) 17 Mean bulk temp 0.85 0.93 x/d > 4 (Nu∞)H = 6.3 + 0.0167 Re Pr

14 (Nux /Nu∞)T = 1 + 2/(x/d) 17 Mean bulk temp 0.85 0.93 (Nu∞)T = 4.8 + 0.0156 Re Pr (continues) Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table H.2 (Continued)

Type Fluid properties Physical situation of fluid Range of validity Index Heat transfer correlation Ref evaluated at: Forced convection in Gas or liquid Laminar flow Fully developed laminar flow smooth noncircular Re = ρVDh/µ < 2300 Nu = hDh/k tubes Dh = hydraulic diameter = 4Ac/P P = perimeter Ac = cross-sectional area

2 Rectangular n-sided Gas or liquid Constant heat flux 15 (Nu)H = 0.4079 + 7.9281A – 5.0087A 18 Mean bulk temp ducts, 3 ≤ n ≤ 15 (n > 15, + 0.9352A3 considered round tubes) A = log10 n

2 Constant wall temperature 16 (Nu)T = 0.1065 + 7.6126A – 4.8028A 18 Mean bulk temp + 0.8945A3 388 2 Rectangular ducts Gas or liquid Constant heat flux 17 (Nu)H = 8.236 – 16.183α + 20.168α 18 Mean bulk temp a (aspect ratio) = 2b/2a <1 – 8.628α3 a = 0 for parallel plates 2 Constant wall temperature 18 (Nu)T = 7.548 – 18.136α + 25.648α 18 Mean bulk temp Dh = 4b – 12.094α3

2 Elliptical ducts Gas or liquid Constant heat flux 19 (Nu)H = 5.269 – 1.8748α + 0.9643α 18 Mean bulk temp α = 2b/2a Constant wall temperature 20 3.. 4922+ 2 6658α 18 Mean bulk temp   −<αα2 = 5.;. 885 0 25 (Nu)T  3.. 658−− 0 1787();α 1   α > 025. Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

2 Circular ducts with Gas or liquid Constant heat flux 21 (Nu)H = 8.2007 – 15.0708α + 18.9437α 18 Mean bulk temp diametrically opposite – 7.7285α3 flat sides α = 2b/2a

2 Rectangular ducts with Gas or liquid Constant heat flux 22 (Nu)H = 8.2766 – 13.698α + 16.6645α 18 Mean bulk temp semicircular ends – 6.8788α3 α = 2b/2a

389 Rectangular ducts Gas or liquid Local Value x* = (x/Dh)/(Re Pr) Thermally developing laminar flow Constant heat flux 23 (Nux)H = c1 + c2/x* 18 Mean bulk temp 2 3 x* ≥ 0.005 c1 = 8.15 – 15.173α + 19.918α – 8.745α 2 3 c2 = 0.0085 + 0.0492α – 0.0894α – 0.0499α Constant wall temperature 24 (Nux)T = d1 + d2/x* 2 3 x* ≥ 0.005 d1 = 7.518 – 17.74α + 24.819α – 11.606α 2 3 d2 = 0.0047 + 0.03α – 0.0385α – 0.0166α 18 Mean bulk temp Mean value Constant heat flux 25 18 Mean bulk temp xt = 0.01 + 0.02α –1/3 xt ≥ x* ≥ 0.005 (Num)H = f1(x*) 2 3 f1 = 2.43 – 2.737α – 4.301α – 2.214α x* > xt (Num)H = f2 + f3/x* 2 3 f2 = 8.24 – 15.809α + 19.577α – 8.398α 2 3 f3 = 0.0352 + 0.0559α + 0.0228α – 0.0389α (continues) Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table H.2 (Continued)

Type Fluid properties Physical situation of fluid Range of validity Index Heat transfer correlation Ref evaluated at: Constant wall temperature 26 –1/3 0.005 ≤ x* ≤ 0.01 (Num)T = g1(x*) 2 3 g1 = 2.038 – 2.206α + 2.738α – 1.181α 18 Mean bulk temp x* > 0.01 (Num)T = g2 + g3 /x* 2 3 g2 = 7.554 – 17.287α + 23.533α – 10.79α 2 3 g3 = 0.0229 + 0.0546α – 0.0417α – 0.0141α 18 Mean bulk temp Parallel plates Gas or liquid Thermally developing x* = (x/ Dh)/(Re Pr) laminar flow Local value Constant wall temperature 27 10 Mean bulk temp –1/3 x* ≤ 0.001 (Nux)T = 1.233(x*) + 0.4 390 3 –0.488 –245x* x* > 0.001 (Nux)T = 7.541 + 6.874(10 x*) e Constant heat flux 28 10 Mean bulk temp –1/3 x* ≤ 0.0002 (Nux)H = 1.49(x*) –1/3 † 0.0002 < x* ≤ 0.001 (Nux)H = 1.49(x*) + 0.4 3 –0.506 –164x* x* > 0.001 (Nux)H = 8.235 + 8.68(10 x*) e Mean value Constant wall temperature 29 10 Mean bulk temp –1/3 x* ≤ 0.0005 (Num)T = 1.849(x*) –1/3 0.0005 < x* ≤ 0.006 (Num)T = 1.849(x*) + 0.6 x* > 0.006 (Num)T = 7.541 + 0.0235/x* Constant heat flux 30 10 Mean bulk temp –1/3 x* ≤ 0.001 (Num)H = 2.236(x*) –1/3 0.001 < x* ≤ 0.01 (Num)H = 2.236(x*) + 0.9 x* > 0.01 (Num)H = 8.235 + 0.0364/x* Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Parallel plates Constant temperature at two surfaces

Gas or liquid Fully developed turbulent 31 Nu =+12 0. 03Reaa32 Pr 19 Mean bulk temp flow − ae=+1 05. 06. Pr 104 ≤ Re ≤ 106 2 3 0.1 < Pr < 104 024. a =−088. 3 36. + Pr

Liquid metal 0.04 < Pr < 0.1 32 Nu =+83.. 002Re082. Pr a4 19 Mean bulk temp 104 ≤ Re ≤ 106 0. 0096 a =+052. 4 002. + Pr

Constant heat flux and insulated surfaces

Gas 104 < Re < 106 33 Nu = 0.0196 Re0.8 Pr1/3 20 Mean bulk temp

Rectangular ducts Liquid Fully developed turbulent flow 34 Nu = 0.078 Re0.738 Pr0.33 21 Mean bulk temp × 3 ≤ ≤ × 5 391 3 10 Re 2 10 5 < Pr < 15 (1/α) > 4

Forced convection Liquid 40 < Reg < 1100 35 = 0. 375 23 Nu 4. 207Reg in spherical annulus Reg = ρVg/µ Mean bulk temp

V = average velocity at θ =90° Nu = hg/k where g = ro – ri and ro and ri are outer and inner sphere radius

3 4 0.5 2 Gas 10 < Rea <4× 10 36 Nu = 0.41(Rea ) (1 + ri /ro) 24 Mean bulk temp Rea = (2m)R Nu = 2hri /k where outer sphere is hot d = diameter of inlet and inner sphere is cold and ro and ri 2 2 R = [0.8ri /(ro – ri ) + 0.4/d] are outer and inner sphere radius m = G(πd 2/4) G = mass velocity

†+0.4 appears to be better than the original value of –0.4 in Reference 10. Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table H.3 Common Correlations for External Free Convection Type Fluid properties Physical situation of fluid Range of validity Index Heat transfer correlation Ref evaluated at: Free convection from Gas, liquid, Constant wall temperature 1 Nu = hL/k 25 Film temp vertical flat plates or liquid RaL = Gr Pr or vertical cylinders metal Gr = gρ 2β ∆TL3/µ 2 9 Laminar flow: RaL < 10 ; 14/ =+ 067. RaL Pr > 0 (NuLT ) 068. [1+ ( 0./ 492 Pr)916// ] 49 θ < 60° ⇒ Ra is replaced by Ra cos θ Turbulent flow: Ra > 109  16/ 2 L  0. 387Ra  (Nu ) =+0. 825 L  LT 916827//  [1+ ( 0 . 492 /Pr) ] 

Constant heat flux 2 Nu = hL/k 25 Film temp RaL = Gr Pr

392 Gr = gρ 2β ∆TL3/µ 2 9 Laminar flow: RaL < 10 ; 14/ =+0. 745RaL Pr > 0 (NuLH ) 068. [1+ ( 0./ 437 Pr)9164// ] 9 θ < 60° ⇒ Ra is replaced by Ra cos θ Turbulent flow: Ra > 109  16/ 2 L  0. 387Ra  (Nu ) =+0. 825 L  LH 916827//  [(./Pr)]1+ 0 437  Free convection from Gas or Constant wall temperature 3 26 Film temp 5 7 1/4 heated plates facing liquid Laminar: 10 < ReL < 4 × 10 (NuL)T = 0.622(RaL) 5 7 1/3 upward Turbulent: 10 < ReL < 4 × 10 (NuL)T = 0.162(RaL) Free convection from heated Gas or Constant wall temperature 4 27 Film temp 5 10 1/3 plates facing downward liquid 3 × 10 < ReL < 3 × 10 (NuL)T = 0.27(RaL) Free convection from Gas or Constant heat flux 5 28 Film temp (b is × 8 vertical or inclined liquid Laminar: RaL < 2 10 =− θ 103. a based on (NuLH ) [0 . 6 488 (sin ) ]Ra L cylinders 0° ≤ θ ≤ 90° ambient temp) 1 (sinθ )175. 0.5 < Pr < 30 a =+ θ = angle from vertical 412 Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Gas Constant wall temperature 6 29 Film temp 4 9 0.25 Laminar: 10 < Rad < 10 [(NuL)T]a = (cos θ) [(Nud)T]horiz a = 90°– θ < 90° Free convection from Gas, liquid, Constant wall temperature 7 Nu = hd/k 30 Film temp horizontal cylinders or liquid Rad = Gr Pr metal Gr = gρ 2β ∆Td3/µ 2 Pr > 0 9 14/ Laminar flow: Rad < 10 0. 518Ra =+ d (NudT ) 036. [1+ ( 0./ 559 Pr)91649// ]

9  16/ 2 Turbulent flow: Rad > 10  0. 387Ra  (Nu ) =+06. d  dT 9168// 27  [(./Pr)]1+ 0 559  Constant heat flux 8 Nu = hd/k 30 Film temp Rad = Gr Pr Gr = gρ 2β ∆Td3/µ 2 393 Pr > 0 9 14/ Laminar flow: Rad < 10 0. 579Ra =+ d (NudH ) 036. [1+ ( 0./ 442 Pr)91649// ]

9  16/ 2 Turbulent flow: Rad > 10  0. 387Ra  (Nu ) =+06. d  dH 916827//  [(./Pr)]1+ 0 559 

1/4 Free convection from Gas Pr = 1 9 Nud = 2 + 0.43(Rad) 31 Film temp spheres Rad = Gr Pr Gr = gρ 2β ∆Td3/µ 2 < Gr < 105 Notes: –0.24 1/4 1. The general criterion for a vertical cylinder to be treated as a vertical flat plate is (d/L) ≥ 47 Pr /GrL . 2. For short cylinders and blocks, the characteristic length dimension is defined as (1/L) = (1/vertical dimension) + (1/horizontal dimension). 3. For inclined plates, the correlations for vertical plates in the laminar flow region can be replaced by Gr cos θ if θ <60°. 4. For a horizontal plate, the characteristic length L for a square plate is taken as the length of side of the square, for a finite rectangular plate as the arithmetic mean of the lengths of two dimensions, and for a circular disk as 0.9 × disk diameter. For other shaped plates, L is defined as L = A/P, where A is the surface area and P is the perimeter. Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table H.4 Common Correlations for Free Convection in Enclosures or from Parallel Vertical Plates

Type Fluid properties Physical situation of fluid Range of validity Index Heat transfer correlation Ref evaluated at:

Free convection in horizontal Gas Isothermal conditions 1 Nub = hb/k 8 Average temp Tave = 2 3 2 enclosed spaces (lower Gr = gρ β(T1 – T2)b /µ (T1 + T2)/2 where plate is hotter; if upper b = gap distance T1 = surface 1 temp, plate is hotter, no Grb < 1700 Nub = 1 T2 = surface 2 temp 4 5 1/4 convection occurs) 10 < Grb < 4 × 10 Nub = 0.195(Grb) 5 1/ 3 Nub = 1 Grb > 4 × 10 Nub = 0.068(Grb ) 1/ 3 0.407 Liquid or Rab = Grb Pr 2 Nub = 0.069(Grb) Pr 32 Average temp 5 9 liquid 3 × 10 < Grb Pr < 7 × 10 metal 0.2 < Pr < 8750

Free convection in vertical Gas Isothermal conditions Nub = hb/k Average temp Tave = enclosed spaces A = height to gap ratio H/b (T1 + T2)/2

Grb < 2000 3 Nub = 1 8 394 4 5 0.344 2 × 10 < Grb <10 ; A = 1 4 Nub = 0.082(Grb) 33 4 5 0.315 –0.204 2.5 ≤ A ≤ 5; 10 < Grb < 3 × 10 5Nub = 0.135(Grb) A 33 3 0.209 –0.131 5 < A ≤ 79; 1.5 × 10

Liquid Isothermal conditions NuH = hH/k Average temp Tave = 1/4 0.051 –0.11 (T + T )/2 Laminar flow: 7 NuH = 0.36(RaH) Pr A 35 1 2 7 9 10 < RaH <4× 10 RaH = GrH Pr 2 3 2 5 ≤ A ≤ 48 GrH = gρ β(T1 – T2)H /µ 3 < Pr < 40,000 0.3 0.051 Transition flow: 8 NuH = 0.084(RaH) Pr 35 9 12 4 × 10 < ReH <4× 10 1 < Pr < 200 5 < A <48 1/ 3 Turbulent flow: 9 NuH = 0.039(RaH) 35 12 ReH > 4 × 10 1 < Pr < 4 5 < A < 48 Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Free convection in vertical Gas Mixing conditions: constant Nub = hb/k Average temp Tave = enclosed spaces heat flux on hot wall, constant (T1 + T2 )/2 temperature on cold wall A = height to gap ratio H/b Laminar flow Pr = 0.7; 1.5 × 103 4

Free convection in inclined Gas or 9 ≤ A ≤ 36 Nub = hb/k Average temp Tave = 3 5 2 rectangular enclosures liquid 4 × 10 < Grb < 3.1 × 10 14 Nub = 0.118[Grb Pr cos 38 (T1 + T2 )/2 45° ≤ φ ≤ 90° (φ – 45°)]0.29 φ : enclosure tilt from horizontal

Free convection in concentric Gas or 2 ≤ A ≤ 35 Nub = hb/k Average temp Tave =

cylindrical enclosures liquid b = Ro – Ri ; Ro = outer radius, (T1 + T2 )/2 Ri: = inner radius 10 4 < Gr <3× 10 5 15 = 0.. 315 0 027 33 b Nubb0 . 149 Ra Pr 0. 459 −  R  A 0. 204 o   Ri  (continues) Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table H.4 (Continued) Type Fluid properties Physical situation of fluid Range of validity Index Heat transfer correlation Ref evaluated at:

Free convection a body and Gas or 16 Nub = hb/k 39 Tave = (Ti + To)/2 where its spherical enclosures liquid 2 × 106 < (Ra )* < 1010 = *.0 222 T = inner temp, b Nubb0664 . (Ra ) i ≤ × 3 0.7 < Pr 4.3 10  RR−  0. 165 To = outer temp Pr0. 018 oi  Ri  Definitions: (Rab)* = Rab (Ro – Ri )/Ri where Ro = outer body radius, Ri = inner body radius for spheres; it is a hypothetical radius of spheres with a volume equal to that of cylinders or cubes b = π Di /2, spheres; H + Di ( π /2 – 1), cylinders; 2H, cubes H = height of cylinders or length of cubes

Free convection from vertical Gas Isothermal condition 17 Nub = hb/kTave = (Tw + Ta)/2 where parallel plates (all edges H = plate height Tw = wall temp, open) b = plate spacing Ta = air temp

396 Rab = Grb Pr 2 3 2 Grb = ρ gβ(Tw – Ta)b /µ : 0.35 y = (b/H)Rab < 10 Nub = 0.24y 40 –0.135 –35/y 3/4 10 ≤ y ≤ 200 Nub = 2.04y [( y/24)(1 – e ) ]40 –35/y 3/4 y > 200 Nub = ( y/24)(1 – e ) 42 Free convection from vertical Gas Isoflux heating 18 Nub = hb/kTave = (TH + Ta)/2 where 5 parallel plates (all edges Rab = ρgβqb /µα Hk TH = outlet air temp, open) q = heat flux Ta = inlet air temp H = plate height b = plate spacing Ra ≤ 10 4  x  −m 43 b (Nu )=   (Nu ) bx  H  bH 0.Ra 144 (Nu ) = b bH + 09.. 033 (1 0 . 0156 Rab ) (Nu ) 19 (Nu ) = bH 43 bm 1− m Ra where m =−04. 01 . log b 10 1000 Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Free convection from vertical Gas Symmetric isothermal plates: 20 Nub = hb/kTave = (Tw + Tf)/2 where parallel plates (all edges Rab = Grb Pr Tw = wall temp, 2 3 2 open) Grb = ρ gβ (Tw – Tf)b /µ Tf = inlet temp 2 0.5 –0.5 y = (b/H)Rab Nub = (576/y + 2.873/y ) 44 2 0.5 –0.5 Asymmetric isothermal plates 21 Nub = (144/y + 2.873/y ) 44 (isothermal and insulated plates) 0.4 –0.5 1 Gas Symmetric isoflux plates 22 NuH/2 = [12/Rab + 1.88/(Rab) ] 44 Tave = ⁄2 (Tw, H/2 + Ta ) qA/  b Nu =   H/2 −   ()TTwH/2 a k 5 Rab = rgbqb /maHk 0.4 –0.5 Asymmetric isoflux plates 23 NuH/2 = [6/Rab + 1.88/(Rab) ] 44 Tave = (Tw, H/2 + Ta ) (isoflux and insulated plates)

Free convection from vertical Liquid Symmetric isothermal plates 24 Nub = hb/kTave = (Tw + Tf)/2 where parallel plates (all edges Pr = 5 Rab = Grb Pr Tw = wall temp, 2 3 2 open) Grb = ρ gβ(Tw – Tf )b /µ Tf = inlet temp 2 0.5 –0.5 y = (b/H)Rab Nub = (576/y + 2.1/y ) 45

397 2 0.5 –0.5 Asymmetric isothermal plates 25 Nub = (144/y + 2.1/y ) 45 (isothermal and insulated plates) 0.4 –0.5 Symmetric isoflux plates 26 NuH/2 = [12/Rab + 2.1/(Rab ) ] 44 Tave = (Tw, H/2 + Ta ) qA/  b Nu =   H/2 −   ()TTwH/2 a k 5 Rab = ρgβqb /µαHk 0.4 –0.5 Asymmetric isoflux plates 27 NuH/2 = [6/Rab + 2.1/(Rab ) ] 44 Tave = (Tw, H/2 + Ta ) (isoflux and insulated plates) 4 Free convection a body Gas or 0.7 < Pr ≤ 1.4 × 10 28 Nub = hb/k 48 Tave = (Ti + To)/2 where 6 10 0.236 and its enclosures liquid 2.9 × 10 < (Rab)* < 3 × 10 Nub = 0.585[(Rab)*)] Ti = inner temp, 6 10 4.6 × 10 < Rab ≤ 4 × 10 To = outer temp Definitions: (Rab)* = Rab (Ro – Ri )/Ri. b = πDi /2, spheres; Hi + Di (π /2 – 1), cylinders; 2Hi , cubes; 2Yi , rectangular blocks. Ri = hypothetical sphere radius for inner body; Ro = hypothetical sphere radius for outer body. 1/ 3 H = height of cylinders or side of cubes; Yi = (length × width × height) ; h is based on inner body surface area. 0.059 0.059 L /Ri = (Ro – Ri )/Ri ; ≤ 1.26(Rab) , cubes; ≤ 0.978(Rab) , spheres. If L/Ri is small (< 0.45), h is based on the higher value of convection or conduction. Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table H.5 Common Correlations for Forced Convection in Tube Banks Terms used: N = number of longitudinal rows; ST = transverse pitch; SL = longitudinal pitch; V = free-stream velocity; CZ = correction for number of rows; CΦ = correction for angle of attack.

Type Fluid properties Physical situation of fluid Range of validity Index Heat transfer correlation Ref evaluated at: Crossflow in tube banks Gas or 0.7 < Pr < 500   025. m 036. Pr Nu= CC CΦ Re Pr   liquid DZ D,max  Prw 

Prw at wall temperature NuD = hD/k (ReD)max = ρVmax D/µ

(1) In-line banks Vmax = STV/(ST – D) 1 49 Free-stream temperature CCZ Cφ m 398 2 10 ≤ Remax ≤ 10 0.80 1.0 (b) 0.40 2 3 10 ≤ Remax ≤ 10 0.37 1.0 (b) 0.54 3 5 10 ≤ Remax ≤ 2 × 10 0.27 (a) (b) 0.63 5 6 2 × 10 ≤ Remax ≤ 7 × 10 0.021 (a) (b) 0.84  23− 3 = 0.. 6089+− 0 1092NN 0 . 01069 +× 0 . 3439 10 NNfor <20 (a) CZ  120for N ≥ −−− 0.. 4278−×+× 0 8718 1023ΦΦΦΦ 0 . 9481 1024 −× 0 . 159 10 3for <°80 (b) CΦ =  180for Φ ≥°

(2) Staggered banks Vmax = STV/(ST – D), or 2 51 Free-stream temp Vmax = 0.5STV/(SD – D) CZ = 1 for N ≥ 20 For SD < 0.5(ST + D): CCZ Cφ m 2 2 0.5 * SD =[SL + (ST /2) ] 1.0 (c) (b) 0.40 2 3 10 ≤ Remax ≤ 10 0.5 (c) (b) 0.54 3 5 10 ≤ Remax ≤ 2 × 10 (e) (d) (b) 0.6 Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 01/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

For ST /SL < 2: 3 5 10 ≤ Remax ≤ 2 × 10 0.4 (d) (b) 0.6 For ST /SL < 2: 5 2 × 10 ≤ Remax 0.022 (d) (b) 0.84 51 ≤ 7 × 106 52 =+×−×+×−−1223 −3 < (c) CNNNZ 0.. 7817 0 5705 10 0 . 5432 10 0 . 1729 10 for N20 =+ − ×−−12 + ×33 < (d) CNNNNZ 0.. 52 0 1389 0 . 1377 10 0. 4439 10 for 20 = 0.2 (e) C 0.35(SSTL / ) Parallel flow in tube banks Gas or Constant axial heat flux Nu = hD/k 50 liquid Constant circumferential wall D = tube diameter temperature 2s = tube central spacing Fully developed flow r = 2s/D Laminar flow Equilateral triangular r ≤ 1.14 3 Nu = –14.7623 – 63.048r + 175.506r2 – 86.1054r 3 399 2 3 arrays r > 1.14 log10 Nu = 1.31053 – 2.87379A + 2.5908A – 0.96909A A = log10 r Square arrays 4 log10 Nu = 0.6853 + 1.199A – 8.2421A2 + 11.6131A3 – 5.2249A4 Turbulent flow Nu = HDe /k Re = rVDe /m De = hydraulic diameter =(D/2)[2.2(2s/D)2 – 2] Equilateral triangular Gas 1.15 ≤ r ≤ 1.25 5 Nu = 0.04 Re0.772 Pr0.6 51 arrays 7000 ≤ Re ≤ 2 × 105 Water 400 ≤ Re ≤ 50,000 r = 1.0 6 Nu = 0.01 Re0.8 Pr0.43 52 1.1 ≤ r ≤ 1.2 Nu = 0.025 Re0.8 Pr0.43 1.2 < r < 1.4 Nu = 0.028 Re0.8 Pr0.4 1.4 ≤ r ≤ 1.5 Nu = 0.032 Re0.8 Pr0.43

*1.0 is better than the original value of 0.9 in reference 51 400 • THERMAL MANAGEMENT OF MICROELECTRONIC EQUIPMENT

REFERENCES

1. W. M. Kays, Convective Heat Transfer, McGraw-Hill, New York, 1966. 2. Developed by L. T. Yeh, according to reference 1. 3. E. Achenbach, “Heat Transfer from Spheres up to Re = 6 × 106,” Proc. 6th Int. Heat Transfer Conf. 5, Hemisphere, Washington, DC, 1978. 4. G. C. Vliet and G. Leppert, ”Forced Convection Heat Transfer for Flow in Pipes, Past Flat Plates, Single Cylinders, Single Spheres and Flow in Packed Beds and Tube Bundles,” AIChE J. 18, 1972. 5. L. C. Whitte, “An Experimental Study of Forced Convection Heat Transfer from a Sphere to Liquid Sodium,” J. Heat Transfer 90, 1968. 6. S. Nakai and T. Okazaki, “Heat Transfer from a Horizontal Circular Wire at Small Reynolds and Grashof Numbers: I, Pure Conduction,” Int. J. Heat Mass Transfer 18, 1975. 7. S. W. Churchill and M. Bernstein, “A Correlation Equation for Forced Con- vection from Gases and Liquids to a Circular Cylinder in Crossflow,” J. Heat Transfer 99, 1977. 8. M. Jakob, Heat Transfer, Vol. 1, Wiley, New York, 1949. 9. E. M. Sparrow, J. W. Ramsey, and E. A. Mass, “Effect of Finite Width on Heat Transfer and Fluid Flow about an Inclined Rectangular Plate,” J. Heat Trans- fer 101, 1979. 10. R. K. Shah and A. L. London, “Laminar Flow Forced Convection in Ducts,” Advances in Heat Transfer, Supplement 1, Academic Press, New York, 1978. 11. H. Husen, “Darstellung des Warmeübergangszahl in Rohre,” durch verallge- meinerte, VDI-Z. 4, 1942. 12. C. H. Sleicher and M. W. Rouse, “A Convective Correlation for Heat Transfer to Constant and Variable Properties Fluids in Turbulent Pipe Flow,” Int. J. Heat Mass Transfer 18, 1975. 13. E. S. Skupinski, J. Tortel, and L. Vautrey, “Détermination des Coefficients de Convection d’un Alliage Sodium-Potassium dans un Tube Circulaire,” Int. J. Heat Mass Transfer 8, 1965. 14. R. H. Notter and C. A. Sleicher, “A Solution to the Turbulent Graetz Prob- lem—III, Fully Developed and Thermal Entry Region Heat Transfer Rate,” Chem. Eng. Sci. 23, 1972. 15. W. Nusselt, Der Warmeaustausch zwischen Wand und Wasser in Rohre,” Forsch. Gebiete Ingenieurw. 2, 1931. 16. Developed by L. T. Yeh, according to reference 1. 17. N. Z. Azer and B. T. Cha, “Turbulent Heat Transfer in Liquid Metals—Fully Developed Pipe Flow with Constant Wall Temperature,” Int. J. Heat Mass Transfer 3, 1961. 18. Developed by L. T. Yeh, according to reference 10. 19. A. A. Shibani and M. N. Ozisik, “A Solution to Heat Transfer in Turbulent Flow between Parallel Plates,” Int. J. Heat Mass Transfer 20, 1977. 20. Developed by L. T. Yeh, according to Reference 1. 21. Developed by L. T. Yeh, according to Reference 22. 22. F. D. Haynes and G. D. Ashton, “Turbulent Heat Transfer in Large Aspect Channels,” J. Heat Transfer 102, 1980. 23. D. B. Tuft and H. Brandt, “Forced-Convection Heat Transfer in a Spherical Annulus Heat Exchanger,” J. Heat Transfer 104, 1982.

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24. K. N. Astill, P. J. Kerney, and R. L. Newton, “An Experimental Investigation of Forced Convection between Concentric Spheres at Low Reynolds Numbers,” 6th Int. Heat Transfer Conf. FC(6):1, 1978. 25. S. W. Churchill and H. S. Chu, “Correlating Equations for Laminar and Turbu- lent Free Convection from a Vertical Plate,” Int. J. Heat Mass Transfer 18, 1975. 26. W. W. Yousef, J. D. Tarasuk, and W. J. McKeen, “Convection Heat Transfer from Upward-Facing Isothermal Horizontal Surfaces,” J. Heat Transfer 104, 1982. 27. W. H. McAdams, Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954. 28. M. Al-Arabi and Y. K. Saleman, “Laminar Natural Convection Heat Transfer from an Inclined Cylinder,” Int. J. Heat Mass Transfer 23, 1980. 29. V. T. Morgan, “The Overall Convective Heat Transfer from Smooth Circular Cylinders,” in Advances in Heat Transfer, Vol. 11, Academic Press, New York, 1975. 30. S. W. Churchill and H. S. Chu, “Correlating Equations for Laminar and Tur- bulent Free Convection from a Horizontal Cylinder,” Int. J. Heat Mass Trans- fer 18, 1975. 31. M. N. Ozisik, Basic Heat Transfer, McGraw-Hill, New York, 1981. 32. S. Glob and D. Dropkin, “Natural Convection Heat Transfer in Liquids Con- fined by Two Horizontal Plates from Below,” J. Heat Transfer 81C, 1959. 33. M. E. Newell and F. W. Schmidt, “Heat Transfer by Laminar Natural Convec- tion within Rectangular Enclosures,” J. Heat Transfer 92, 1970. 34. S. H. Yin, Y. T. Wung, and K. Chen, “Natural Convection in an Air Layer Enclosed with Rectangular Cavities,” Int. J. Heat Mass Transfer 21, 1978. 35. N. Seki, S. Fukusako, and H. Inaba, “Heat Transfer of Natural Convection in a Rectangular Cavity with Vertical Walls of Different Temperatures,” Bull. JSME 21, 1978. 36. M. N. A. Said and A. C. Trupp, “Laminar Free Convection in Vertical Air- Filled Cavities with Mixed Boundary Conditions,” ASME Paper 79-HT-110. 37. R. K. MacGregor and A. F. Emery, “Free Convection through Vertical Plane Layers—Moderate and High Prandtl Number Fluids,” J. Heat Transfer 91, 1969. 38. K. R. Randall, J. W. Mitchell, and M. W. Ei-Wakil, “Natural Convection Heat Transfer Characteristic of Flat Plate Enclosures,” J. Heat Transfer 101, 1979. 39. R. O. Warrington and R. E. Powe, “Natural Convection Heat Transfer between Bodies and Their Spherical Enclosure,” presented at 19th National Heat Transfer Conference, Orlando, FL, July 27–30, 1980. 40. Developed by L. T. Yeh, according to reference 41. 41. E. M. Sparrow and P. A. Bahrami, “Experiments on Natural Convection from Vertical Parallel Plates with Either Open or Closed Edges,” J. Heat Transfer 102, 1980. 42. W. Elenbaas, “Heat Dissipation of Parallel Plates by Free Convection,” Phys- ica 9, 1942. 43. R. A. Wirtz and R. J. Stutzman, “Experiments on Free Convection between Vertical Plates with Symmetric Heating,” J. Heat Transfer 104, 1982. 44. A. Bar-Cohen and W. M. Rohsenow, “Thermally Optimum Spacing of Verti- cal, Natural Convection Cooled Parallel Plates,” J. Heat Transfer 106, 1984. 45. A. Bar-Cohen and H. Schweitzer, “Convective Immersion of Parallel Vertical Plates,” Proc. 4th Int. Packaging Conf., 1984.

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46. M. Molki and E. M. Sparrow, “An Empirical Correlation for Average Heat Transfer Coefficient in Circular Tubes,” J. Heat Transfer 108, 1986. 47. M. J. Chamberlain, K. G. T. Hollands, and G. B. Raithby, “Experiments and Theory on Natural Convection Heat Transfer from Bodies of Complex Shapes,” J. Heat Transfer 107, 1985. 48. R. O. Warrington, S. Smith, R. E. Powe, and R. Mussulman, “Boundary Effects on Natural Convection Heat Transfer for Cylinders and Cubes,” Heat Transfer in Enclosures, ASME HTD 39, 1984. 49. A. Zukanskas, “Heat Transfer from Tubes in Crossflow,” in Advances in Heat Transfer, Vol. 8, Academic Press, New York, 1972. 50. B. T. F. Chung, F. Lin, M. M. Kermani, and L. T. Yeh, “Heat Transfer from Par- allel Flow through Isosceles Triangular and Rectangular Tube Bundles,” Advances in Heat Exchanger Design, ASME HTD 66, 1986. 51. W. A. Sutherland and W. M. Kays, “Heat Transfer in Parallel Rod Arrays,” J. Heat Transfer 88, 1988. 52. V. I. Subbotin, P. A. Ushakov, P. L. Kirillov, M. K. Ibragimov, M. N. Ivanovsky, E. V. Nomophilov, D. M. Ovechkin, D. N. Sorokin, and V. P. Sorokin, “Heat Removal from the Reactor Fuel Elements Cooled by Liquid Metals,” 3rd United Nations Int. Conf. Peaceful Uses of Atomic Energy, A/CONF 28/P/328, May 1964.

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UNITS CONVERSION TABLE

Density 3 3 1 lbm/ft = 16.018 kg/m

Energy 1 Btu = 1054.4 joule (J) = 0.252 kcal (kilocalorie) = 3.9308 × 10–4 hp = 2.9291 × 10–4 W·hr

Force 1 lbf = 4.4482 N = 0.4536 kgf

Heat flux 1 Btu/(hr·ft2) = 3.1525 W/m2 = 0.2712 cal/(hr·cm2)

Heat transfer coefficient 1 Btu/(hr·ft2·˚F) = 5.6475 W/(m2·˚C) = 0.003661 W/(in2·˚C)

Length 1 ft = 0.3048 m 1 in = 2.54 cm

Mass 1 lbm = 0.45359 kg

Power 1 Btu/hr = 0.29288 W =3.9275 × 10–4 hp

Pressure 1 psi = 6894.8 Pa = 3.3227 dyn/cm2 = 27.7 in of water

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Specific heat 1 Btu/(lbm·˚F) = 4.184 kK/kg = 1 cal/(g·˚C)

Temperature 1°F = 0.5556°C

Thermal conductivity 1 Btu/(hr·ft·˚F) = 1.7296 W/(m·˚C) = 0.04393 W/(in·˚C)

Velocity 1 ft/s = 0.3048 m/s

Viscosity –4 1 lbm/(hr·ft) = 4.1338 × 10 Pa·s = 4.1338 × 10–1 cp (centipois e)

Volume 1 ft3 = 0.028317 m3 = 7.4794 gallon (U.S.)

Volumetric flow rate 1 ft3/min = 4.7195 × 10–4 m3/s

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Ablation, 28 Bulk temperature, 43 Absolute surface roughness, 45 Buoyancy-assisted flow, 237–239 Absorptivity, 54 Buoyancy-resisted flow, 239–241 Ahmad parameter, 107 Burnout, 98 Air, properties of, 354–355 Burnout heat flux, 77 Air cooling, direct, see Direct air cooling Chen correlation, 102, 103 Air-delivery systems, 202–210 Chip package, 169 Air-handling systems, 197–202 Chip package attachment, 173–176 Air temperature and solar radiation, Chip package thermal resistance, hot dry daily cycle of, 63 172–173 Airflow system, pressure drop in, Clausius-Clapeyron relation, 70 203 Coefficient of thermal expansion Annular flow, 95, 125–126 (CTE), 4, 173 Aspect ratio, 40 Colburn factor, 249 Cold-plate analysis, 251–255 Baker flow map, 123–124 Cold-plate conductance, 253 Ball grid array (BGA), 171 Cold-plate temperature, 254 Biot number, 20, 138 Compact heat exchangers (CHE), Bluff body, flow over, 32–33 243–244 Boiling, 67 Component-level resistance, 4–5 incipient, at heating surfaces, 72–76 Condensation, 115–129 Boiling crisis, 98–99 direct-contact, 116 Boiling curve dropwise, 115 for jet impingement cooling, 297 filmwise, see Filmwise for microchannel heat sinks, 304 condensation for microstud surfaces, 303 homogeneous, 116 for minichannel heat sinks, 304 inside horizontal tube, 123–126 for pool boiling, 67–70 modes of, 115–116 for smooth surfaces, 303 noncondensable gas in, 127–129 for spray cooling, 297 Conduction, 9–28, 18 Boiling incipience, 283 Fourier law, 9 Boundary conditions, effect of, 38–41 fundamental law of, 9–10 Boundary layer, thermal, 32 general differential equations for, Bubbly flow, 95 10–16

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Conduction (continued) pressure drop correlations for, lumped-system transient analysis, 190–194 20–24 previous work, 185–187 one-dimensional, 16–17 Direct-contact condensation, 116 with phase change, 25–28 Direct-immersion cooling, 285–287 steady-state condition, 16 Dittus-Boelter equation, 99 thermal/electrical analogy, 17–20 Dropwise condensation, 115 transient condition, 16–17 Dryout, 97, 98 Configuration factor, 55–58 Dual in-line package (DIP), 169–172 Contact angle, 72 Contact pressure, apparent, 144 Emissive power, 53 Convection, 18, 31–51 Emissivities, 54 external forced, common of materials, 371–372 correlations for, 382–385 of surfaces, 373–374 external free, common correlations Enclosures for, 392–393 common correlations for free forced, see Forced convection convection in, 394–397 free, see also Natural convection natural convection in, 234–237 common correlations for, in Epoxy joints, 159–160 enclosures or from parallel Exhaust fan, 202 vertical plates, 394–397 Extended surfaces, 131–140 internal forced, common fin efficiency, 134–137 correlations for, 386–391 selection and design of fins, mixed, in vertical plates, 237–241 137–140 natural, see Natural convection External thermal resistance, 5 single-phase natural, 67 Extreme climatic conditions, Coolanol-20, properties of, 361 radiation in, 61–64 Coolanol-25, properties of, 362 Coolanol-35, properties of, 363 Falling liquid film, 290–295 Coolanol-45, properties of, 364 Fan performance and system Coolant selection, 261–265 resistance, 198–202 Coolers, thermoelectric, see Fan types, 198 Thermoelectric coolers axial fans, 198 Cooling, direct-immersion, 285–287 blowing fan, 202 Counterflow heat exchanger, 244 centrifugal fans, 198 Critical heat flux (CHF), 69, 77–79, propeller fans, 198 98, 283–285 tube axial fans, 198 Crossflow heat exchanger, 244 vane axial fans, 198 Fans, 197–202 Density of materials, 349–350 specific speed for, 201 Departure from nucleate boiling Fast burnout, 99 (DNB), 98 FC-43, properties of, 357 Diffusers, 206–207 FC-75, properties of, 358 Direct air cooling, 185–197 FC-77, properties of, 359 heat transfer correlations for, FC-78, properties of, 360 187–190 Figure of merit (FOM), 262, 312 heat transfer enhancement and, numerical values of, 264–265 194–197 for pool boiling, 283–285

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for single-phase flow, 262 Gap conductance, 145 for thermoelectric coolers, 338 Gas-controlled heat pipes, 321 for two-phase flow, 262 Glycol-water mixture, properties of, Filler materials, 158–159 366 Film boiling, 69 Grashof number, 37 Film temperature, 43 effect of, 37–38 Filmwise condensation, 115–116 Gravity, effect of, on pool boiling, heat transfer in, 116–121 87–91 Fins, 131 Groeneveld correlation, 108 efficiency of, 134–137 selection and design of, 137–140 H condition, 39 Flow Heat exchangers, 243–251 buoyancy-assisted, 237–239 compact, 243–244 buoyancy-resisted, 239–241 effectiveness of, 245–246 Flow boiling, 67, 95–113, 300–306 flow arrangement of, 244 boiling crisis, 98–99 geometric factors in, 250–251 burnout, 98 heat transfer coefficient of, dryout, 97, 98 244–245 flow patterns, 95 heat transfer in, 248–249 heat transfer equations for, 99–109 pressure drop in, 248–249 heat transfer mechanisms in, 95–98 shell-and-tube, 243 pressure drop in, 109–113 thermal analysis of, 246–248 thermal enhancement in, 109 Heat flux correlations, minimum, Flow cooling, two-phase, 283–306 79–81 Flow patterns, 95 Heat pipes, 309–333 Forced-air cooling, 185 applications of, 325–330 Forced convection, 31 boiling limit of, 317–318 common correlations for, in tube capillary action, 158 banks, 398–399 effective capillary radius, 313–314 through cold plate, 267–274 capillary wicking limit, 312–316, Form drag, 34 316 Forster-Zuber equation, 102 construction of, 311–312 Fouling factors, 245 control techniques for, 322 Fourier effect, 336–337 entrainment limit of, 317 Fourier equation, 10 materials compatibility in, Fourier law, 9 318–320 Free-jet impingement, 274 micro, 330–333 Freezing, 25–26 nomenclature for, 313 Freon E2, properties of, 365 operating temperatures in, 320–321 Friction factor operation limits of, 312–318 in fully developed laminar flow, 41 operation methods for, 321–322 in laminar flow, 40 operation principles, 309 pressure drop and, 43–46 sonic limit of, 316–317 Friction factor correlations, 377–378 structure of, 310 Friedel correlation, 111–113 thermal resistances of, 323–324 useful characteristics of, 309–310 Gallium arsenide, thermal Heat sink, 177, 194–197 conductivity of, 351, 352 thermal resistance of, 197

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Heat transfer Isothermal surfaces in filmwise condensation, 116–121 black, radiation between, 55–58 in heat exchangers, 248–249 gray, radiation between, 58–61 parameter effects on, 35–43 post-burnout, 108–109 Jet impingement, 274–275 Heat transfer coefficient, 34–35 Jet-impingement boiling curve, 296 of heat exchangers, 244–245 Jet-impingement cooling, 295–298 Heat transfer correlations, 381–399 boiling curve for, 297 for direct air cooling, 187–190 differences between spray cooling for parallel plates, 219 and, 299 Heat transfer enhancement, direct air Joint thermal contact resistance, cooling and, 194–197 145–147 Heat transfer equations for flow Joule effect, 336 boiling, 99–109 Heat transfer mechanisms in flow Kirchhoff’s law, emissivity and, 54 boiling, 95–98 Heating surfaces, incipient boiling at, Laminar flow, 31 72–76 friction factor in, 40 Heterogeneous nucleation, 70 fully developed friction factor in, 41 High-temperature heat pipes, Nusselt number in, 41, 42, 43 320–321 Laplace equation, 11 Homogeneous condensation, 116 Latent heat of vaporization, 67 Homogeneous nucleation, 74 Leidenforst point, 79–81 Horizontal tube, condensation inside, Liquid cooling, single-phase, 261–279 123–126 Liquid-deficient region, 108 Hot dry daily cycle of air temperature Liquid film, falling, 290–295 and solar radiation, 63 Liquid immersion, 285–287 Hottel crossed-string method, 58 Liquid transport factor, 312 Hydraulic diameter, 32, 40, 253 Logarithmic mean temperature Hydrodynamic entry length, 31 difference (LMTD) method, Hydrodynamic instability, 77 246–247 Hydrostatic pressure drop, 314 Low-temperature heat pipes, 320–321 Hysteresis, thermal, 87 Lumped-system transient analysis for conduction, 20–24 Impinging fluid jets, 274–275 Infrared (IR) energy, 64 Mach number, 315 Integrated circuits (IC), 169 Manifolds, 207–210 large-scale integration (LSI), 1 pressure distribution in, 209 medium-scale integration (MSI), 1 Martinelli parameter, 102 small-scale integration (SSI), 1 Material hardness, 143 ultra-large-scale integration (ULSI), Materials 1 density of, 349–350 very-large-scale integration (VLSI), emissivities of, 371–372 1 specific heat of, 349–350 Internal thermal resistance, 4–5, 172 thermal conductivity of, 349–350 Interstitial filler, 148–152 thermal properties of, 349–350 Isoflux-isoflux, Nusselt number for, Materials compatibility in heat pipes, 239 318–320

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Mean time between failures (MTBF), 3 Nusselt analysis, improvements over, Melting, 25–26 121–123 Metallic coatings, soft, 152–156 Nusselt model, 116–120 Micro heat pipes, 330–333 Nusselt number, 35, 36 Microchannel flow, 275–279 in fully developed laminar flow, 41, Microchannel heat sinks 42, 43 boiling curve for, 304 for isoflux-isoflux, 239 comparison of performance between mini- and, 306 Offset fins, 271–274 Microstud surfaces, boiling curve for, Ohm’s law, 17 303 Opaque bodies, 54 Minichannel heat sinks Optimum plate spacing, 216–218 boiling curve for, 304 Overshoot, 75 comparison of performance between micro- and, 306 Packaging-level resistance, 5 Minimum heat flux correlations, PAO (2 CST), properties of, 367 79–81 PAO (4 CST), properties of, 368 Moderate-temperature heat pipes, PAO (6 CST), properties of, 369 320–321 PAO (8 CST), properties of, 370 Modulus of elasticity, 143 Parallel-flow heat exchanger, 244 Molecular diffusivity of heat, 37 Parallel plates Molecular diffusivity of momentum, heat transfer correlations for, 219 37 natural convection in, 214–220 Moody chart, 44, 45 Parallel vertical plates, common Multichip module (MCM), 169 correlations for free convection Multiple-jet impingement, 275 in, 394–397 Parameter effects Natural convection, 31, 213–237, on heat transfer, 35–43 265–267 on pool boiling, 81–87 in enclosures, 234–237 Partial film boiling, 69 in parallel plates, 214–220 Partial nucleate boiling, 96 in pin-fin arrays, 229–234 Peak heat flux, 77 single-phase, 67 Peltier couple, 339, 340 in straight-fin arrays, 220–229 Peltier effect, 336 Net thermoelectric effect, 337–338 Phase change, conduction with, Noncondensable gas in condensation, 25–28 127–129 Phase-change materials (PCM), Nucleate boiling, 69, 70–72, 96, 25–28, 156–158 99–101 properties of, 375–376 departure from (DNB), 98 Pin-fin arrays, natural convection in, partial, 96 229–234 Nucleate boiling correlations, 76 Pin-grid array (PGA), 169–172 Nucleation Plate spacing, optimum, 216–218 heterogeneous, 70 Plated through-hole (PTH) mounting homogeneous, 74 technology, 189 in isothermal surroundings, 72 Plug flow, 95, 124–125 Number of heat transfer units (NTU), Poisson equation, 10 246 Poisson’s ratio, 143

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Polyalphaolefin (PAO), 265 Rayleigh number, 38 Pool boiling, 67–91 Rectangular offset plate fins, 255–259 boiling curve, 67–70 Reference temperature approach, 43 dissolved gases and, 84, 85 Reflectivity, 54 effect of gravity on, 87–91 Reliability, 3 enhancement of, 287–300 temperature and, 3–4 figure of merit (FOM) for, 283–285 Reynolds number, 32, 36 parameter effects on, 81–87 effects of, 36 subcooling and, 81–82, 83 surface conditions and, 82–84 Seebeck effect, 335–336 wettability and, 84–85 Separation point, 33 Post-burnout heat transfer, 108–109 Shape factor, 55–58, 236 Prandtl number, 32, 36 Shell-and-tube heat exchangers, 243 effect of, 36–37 Silicate esters, 265 Pressure distribution in manifolds, Silicon, thermal conductivity of, 351 209 Single-chip package (SCP), 169 Pressure drag, 34 Single-phase liquid cooling, 261–279 Pressure drop Single-phase natural convection, 67 in airflow system, 203 Sky temperature, 64 in flow boiling, 109–113 Slow burnout, 99 friction factor and, 43–46 Slug flow, 95, 124–125 in heat exchangers, 248–249 Smooth surfaces, boiling curve for, hydrostatic, 314 303 total, 110 Soft metallic coatings, 152–156 viscous, 314 Solar absorptivities of surfaces, Pressure drop correlations for direct 373–374 air cooling, 190–194 Solar constant, 61 Printed circuit boards (PCB), Solar flux, 61 169–182, 329–330 Solar radiation and air temperature, cooling methods for, 176–177 hot dry daily cycle of, 63 thermal analysis of, 177–178 Solder joints, 159–160 Property ratio method, 43 Solidification, 25–26 Solids Radiation, 18, 53–64 mechanical properties of, 143 absorptivity, 54 thermal properties of, 143 between black isothermal surfaces, Sonic limit of heat pipes, 316–317 55–58 Specific heat, 42, 261 between gray isothermal surfaces, of materials, 349–350 58–61 Specific speed for fans, 201 black body, 54 Spray cooling, 295–298 emissivity, see Emissivity boiling curve for, 297 in extreme climatic conditions, differences between 61–64 jet-impingement cooling and, reflectivity, 54 299 shape or configuration or view Standard electronic module (SEM) factor, 55–58 heat pipes, 326 transmissivity, 54 Stanton number, 249 Radiosity, 58 Stefan-Boltzmann law, 53

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Straight-fin arrays, natural Thermal enhancement in flow convection in, 220–229 boiling, 109 Stratified flow, 124 Thermal greases, 148 Streamlined body, flow over, 32–33 Thermal hysteresis, 87 Subcooling Thermal interface resistance, 141–165 degree of, 69 factors affecting, 141–145 effect of, 121 methods of reducing, 147–159 pool boiling and, 81–82, 83 practical design data for, 160–165 Submerged-jet impingement, 274 typical values of, 161 Suppression factor, 104 for various materials, 163 Surface enhancement, 288–289 Thermal properties Surface/fluid coefficient, 285 of fluids, 46–47 Surface-fluid parameters, 76 of materials, 349–350 Surface roughness, 141 Thermal resistance absolute, 45 chip package, 172–173 and waviness, 142–143 of heat pipes, 323–324 Surface tension, 71, 262 of heat sink, 197 Surfaces levels of, 4–5 emissivities of, 373–374 total, 4 solar absorptivities of, 373–374 Thermal resistance factor, 13, 14 System-level resistance, 5 Thermoelectric coolers, 335–347 characteristics of, 335 T condition, 39 coefficient of performance (COP), Temperature 335, 342–343 cold-plate, 254 figure of merit (FOM) for, 338 reliability and, 3–4 operation principles for, 339 Temperature cycling, 3, 4 performance analysis of, 340–344 Thermal analysis of heat exchangers, practical design procedure for, 246–248 344–347 Thermal boundary layer, 32 system configurations for, 339 Thermal compounds, properties of, Thermoelectric couples, 161 interconnected, 340 Thermal conductivity, 9–10, 261 Thermoelectric effects, 335–337 equivalent, 178–182 Fourier effect, 336–337 of gallium arsenide, 351, 352 Joule effect, 336 of materials, 349–350 net, 337–338 normal, 179–180 Peltier effect, 336 planar, 178–179 Seebeck effect, 335–336 of silicon, 351 Thomson effect, 336 Thermal contact conductance, 141 Thermoelectricity, basic theories of, Thermal contact resistance, joint, 335–337 145–147 Thermosyphon, two-phase, 309 Thermal design considerations, 5–6 Thomson coefficient, 336 Thermal diffusivity, 10 Thomson effect, 336 Thermal/electrical analogy for Transmission coefficient, 61 conduction, 17–20 Transmissivity, 54 Thermal-energy storage (TES) Tube banks, common correlations for systems, 25 forced convection in, 398–399

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Turbulence, effect of, 122 Viscous pressure drop, 314 Turbulent flow, 31 Viscous vapor flow limit, 318 Two-phase flow cooling, 283–306 Voids, 173 Two-phase thermosyphon, 309 Wall superheat, 71 U-channel apparent emittance, 226, Wall superheat excursion, 75 227 Wall temperature, 43 U-shaped channels, 220 Water, properties of, 356 Units conversion table, 403–404 Wavy flow, 95, 124–125 Weber number, 298, 317 Vanes, 203, 205 Wettability, pool boiling and, 84–85 Vaporization, latent heat of, 67 Wick permeability, expressions for, Vertical plates, mixed convection in, 315 237–241 Wick structure, 311 View factor, 55–58 Wicking limit, capillary, 312–316, 316 for two finite surfaces, 57 Wickless heat pipes, 309

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Lian-Tuu Yeh received his B.S. in mechanical engineering from National Cheng Kung University, Taiwan, in 1967 and his M.S. and Ph.D. in mechanical engineering from the University of Akron in Ohio in 1972 and 1976, respectively. Dr. Yeh has been practicing in thermal sciences within diverse industries for more than 26 years, with such major U.S. companies as Babcock & Wilcox, Texas Instruments, Compaq Computer, Lockheed Martin, and Chiaro Networks. Since 1982, Dr. Yeh’s work has been concentrated in thermal management of electronic equipment, and he has made contributions in the development of advanced cooling technologies for high-power electronic systems. His experience includes but is not limited to liquid-cooled solid-state phased radar arrays, avionics, desktop and notebook computers, and various other liquid- or air-cooled electronic systems. His work extends to thermal management of electronics in vehicles and missiles, and also intensively in telecommunications systems. As an adjunct pro- fessor in the Department of Aerospace and Mechanical Engineering at the Uni- versity of Texas at Arlington, Dr. Yeh taught a course in thermal analysis and design of electronic equipment. Dr. Yeh has been an active participant in and session organizer of national and international heat transfer conferences. He has served as a technical reviewer for various heat transfer journals and conferences, and also for the National Science Foundation in reviewing proposals for research grants. Dr. Yeh is a practicing engineer and is also a researcher. He has published over 43 technical papers in the field of heat transfer and received one U.S. patent for cooling of electronic equipment. Dr. Yeh was elected a Fellow of the American Society of Mechanical Engineers (ASME) in 1990. He is a member of Sigma Xi and a senior member of American Institute of Aeronautics and Astronautics, Inc. (AIAA). Currently, he is a member of the ASME Division K-16 Committee on Heat Transfer in Electronic Equipment and of the AIAA Thermophysics Technical Committee. He has been listed in Who’s Who in the South and Southwest (1988–1989) and Who’s Who in the World (1994–1995). Dr. Yeh is also a registered professional engineer in the state of Texas.

Richard C. Chu, who is an IBM Fellow, joined IBM in 1960 after receiving a B.S. in mechanical engineering from National Cheng Kung University, Taiwan, and an M.S. in mechanical engineering from Purdue University. Since joining

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IBM’s Development Laboratory in Poughkeepsie, New York, he has held a variety of technical and managerial assignments. His leadership and creativity in the area of computer cooling technology have earned him numerous awards and increasing responsibilities. He has contributed to the development of critical technologies in methods, apparatus, and systems of temperature control and heat removal for components within large-scale computers. Mr. Chu was an inventor of the Modular Conduction Cooling System and co-inventor of the Ther- mal Conduction Module (TCM) cooling concept that have been applied to most of IBM’s high-performance computers. Earlier, he was responsible for the successful implementation of the Air-Liquid Hybrid Cooling System for the IBM 360/370 computers. His invention achievements are documented in 75 patents and over 140 patent disclosure publications, most of which have been adapted to engineering applications across IBM product lines. Many of his patents have been recognized outside of IBM as industry standards for cooling of electronic equipment. He has published extensively, including a book, book chapters, and over 50 technical papers. His contribution to IBM and the computer industry led to his appointment in 1983 as an IBM Fellow, the company’s highest technical honor. He was elected a member of the National Academy of Engineering in 1987, and a member of the Academia Sinica in 1996. In 1989, Mr. Chu was appointed a founding member of the IBM Academy of Technology, and he was subsequently elected to serve as the Academy’s original vice president, for 1990, and as president, for 1991. He is the recipient of many awards and honors, including thirty-one IBM Invention Achievement Awards, four IBM Outstanding Innovation Awards, and an IBM Corporate Award. He also received the Distinguished Engineering Alumnus Award from Purdue University in 1984, the Distinguished Alumnus Award from National Cheng Kung University in 1986, the American Society of Mechanical Engineers Heat Transfer Memorial Award in 1986, and Purdue University’s Outstanding Mechanical Engineer Award in 1991. Since 1987, he has been listed in Who’s Who in Engineering, Who’s Who in America, and Who’s Who in the World. He was awarded an Honorary Doctor of Sci- ence degree by the American University of the Caribbean in 1992, and he received an Honorary Doctor of Engineering degree from Purdue University in 1996. Mr. Chu is a Fellow of the American Society of Mechanical Engineers (ASME) and a Fellow of the American Association for the Advancement of Science (AAAS) and also a member of the New York Academy of Sciences.

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