Pagina principale
Thermal design of electronic equipment
Thermal design of electronic equipment
Ralph Remsburg
In a field where change and growth is inevitable, new electronic packaging problems continually arise. Smaller, more powerful devices are prone to overheating, causing intermittent system failures, corrupted signals, lower MTBF, and outright system failure. Since convection cooling is the heat transfer path most engineers take to deal with thermal problems, it is appropriate to gain as much understanding about the underlying mechanisms of fluid motion as possible. Thermal Design of Electronic Equipment is the only book that specifically targets the formulas used by electronic packaging and thermal engineers. It presents heat transfer equations dealing with polyalphaolephin (PAO), silicone oils, perfluorocarbons, and silicate esterbased liquids. Instead of relying on theoretical expressions and text explanations, the author presents empirical formulas and practical techniques that allow you to quickly solve nearly any thermal engineering problem in electronic packaging.
Categories:
Anno:
2001
Edizione:
1
Editore:
CRC Press
Lingua:
english
Pagine:
319
ISBN 10:
0849300827
ISBN 13:
9780849300820
Series:
Electronics handbook series
File:
PDF, 20.30 MB
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“Frontmatter” Thermal Design of Electronic Equipment. Ed. Ralph Remsburg Boca Raton: CRC Press LLC, 2001 © 2001 by CRC PRESS LLC THERMAL DESIGN of ELECTRONIC EQUIPMENT © 2001 by CRC PRESS LLC ELECTRONICS HANDBOOK SERIES Series Editor: Jerry C. Whitaker Technical Press Morgan Hill, California PUBLISHED TITLES AC POWER SYSTEMS HANDBOOK, SECOND EDITION Jerry C. Whitaker THE COMMUNICATIONS FACILITY DESIGN HANDBOOK Jerry C. Whitaker THE ELECTRONIC PACKAGING HANDBOOK Glenn R. Blackwell POWER VACUUM TUBES HANDBOOK, SECOND EDITION Jerry C. Whitaker THERMAL DESIGN OF ELECTRONIC EQUIPMENT Ralph Remsburg THE RESOURCE HANDBOOK OF ELECTRONICS Jerry C. Whitaker MICROELECTRONICS Jerry C. Whitaker SEMICONDUCTOR DEVICES AND CIRCUITS Jerry C. Whitaker SIGNAL MEASUREMENT, ANALYSIS, AND TESTING Jerry C. Whitaker FORTHCOMING TITLES ELECTRONIC SYSTEMS MAINTENANCE HANDBOOK Jerry C. Whitaker © 2001 by CRC PRESS LLC THERMAL DESIGN of ELECTRONIC EQUIPMENT Ralph Remsburg Nortel Networks Boca Raton, Florida CRC Press Boca Raton London New York Washington, D.C. © 2001 by CRC PRESS LLC 0082FM.fm Page 4 Wednesday, August 23, 2000 9:50 AM Library of Congress CataloginginPublication Data Remsburg, Ralph. Thermal design of electronic equipment / Ralph Remsburg. p. cm.(Electronics handbook series) Includes bibliographical references and index. ISBN 0849300827 (alk. paper) 1. Electronic apparatus and appliancesThermal properties. 2. Electronic apparatus and appliancesDesign and construction. 3. HeatTransmission. 4. Electronic packaging. I. Title. II. Series. TK7870.25 .R46 2001 621.38104dc21 00057170 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their us; e. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. © 2001 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0849300827 Library of Congress Card Number 00057170 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acidfree paper © 2001 by CRC PRESS LLC 0082FM.fm Page 5 Wednesday, August 23, 2000 9:50 AM Preface The field of electronic packaging continues to grow at an amazing rate. The electronic packaging engineer requires analytical skills, a foundation in mechanical engineering, and access to the latest developments in the electronics field. The emphasis may change from project to project, and from company to company, yet some constants should continue into the foreseeable future. One of these is the emphasis on thermal design. Thermal analysis of electronic equipment is becoming one of the primary aspects of many packaging jobs. An upfront commitment to CFD (Computational Fluid Dynamics) software code, FEA (Finite Element Analysis) software, is the result of realizing that the thermal problems will only get worse. As the size of the electronic circuit is reduced, speed is increased. As the power of these systems increases and the space allotted to them diminishes, heat flux or density (heat per unit area, W/m2) has spiraled. While air cooling is still used extensively, advanced heat transfer techniques using exotic synthetic liquids are becoming more popular, allowing even smaller systems to be designed. This reference book of formulas is the result of sifting through the volumes of data on general heat transfer and extracting the formulas that are needed by today’s electronic packaging engineers. The reader will immediately notice the emphasis placed on fluid dynamics formulas in this book. Since convection cooling is the heat transfer path most engineers take to deal with thermal problems, it is appropriate to gain as much understanding about the underlying mechanisms of fluid motion as possible. The application of advanced thermal management techniques requires a background in fluid dynamics. © 2001 by CRC PRESS LLC 0082FM.fm Page 7 Wednesday, August 23, 2000 9:50 AM Author Ralph Remsburg is currently Senior Thermal Analyst at Nortel Networks, Boca Raton, FL. Previously, he held engineering positions up to the director level. Ford Aerospace, Chrysler Corporation, Delco, Hughes Network Systems, Loral Data Systems, Moog Space Products, Alcon Surgical Labs, and Dell Computer have all benefited from his expertise as a consultant. Remsburg’s name is on over 30 patents, 5 published papers, and a previous book. He attended New York University, received a master’s degree from Columbia University, and is completing a dissertation for a doctorate. © 2001 by CRC PRESS LLC 0082FM.fm Page 9 Wednesday, August 23, 2000 9:50 AM Nomenclature and Symbology SYMBOL DEFINITION Symbol a a A b c CA C C Ċ CD Cf D DAB e e E E f f' Description Velocity of sound; acceleration Thermal diffusivity (a k/c) Area; Ac, crosssectional area Ap, projected area of a body normal to flow As, surface area Ao, outside area Ai, inside area Breadth or width Specific heat cp, specific heat at constant pressure cv , specific heat at constant volume Molar concentration of component A Constant Thermal capacity Rate of hourly heat capacity Ċ c , rate of hourly heat capacity of a colder fluid in a heat exchanger Ċ h rate of hourly heat capacity of a warmer fluid in a heat exchanger Total drag coefficient Skin friction coefficient Cfx, local value of Cf at distance x from leading edge; Cf , average value of Cf Diameter DH, hydraulic diameter Do, outside diameter Di, inside diameter Mass diffusion coefficient Base of natural or Napierian logarithm (2.71828) Internal energy per unit mass Internal energy Emissive power of a radiating body Eb, emissive power of a blackbody E, monochromatic emissive power per micron at wavelength Fanning friction coefficient for flow through a conduit Friction coefficient for flow through pinfins © 2001 by CRC PRESS LLC International Units English Units m/s m2/s m2 ft/s ft2/h ft2 m J/kgK ft Btu/lbm °F kg/mol m3 lb/mol/ft3 Dimensionless J/K Btu/°F W/K Btu/h°F Dimensionless Dimensionless m ft m2/s ft2/h Dimensionless J/kg Btu/lbm J Btu W/m2 Btu/hft2 Dimensionless Dimensionless 0082FM.fm Page 10 Wednesday, August 23, 2000 9:50 AM Symbol F F1 2 1 2 g gc G G h h hl hfg hm H i I I J k K log ln l L Lf ṁ M N p P P Description Force Geometric shape factor for radiation from one blackbody to another Geometric shape and emissivity factor for radiation from one graybody to another Acceleration due to gravity (9.807 m/s2) Dimensional conversion factor Mass velocity or flow rate per unit area (G V ) Irradiation incident on unit surface in unit time Enthalpy per unit mass Combined unitsurface conductance, h hc hr hb, unitsurface conductance of a boiling liquid hc, Local unit convective conductance h c, average unit convective conductance h r , average unit conductance for radiation Head loss Latent heat of condensation or evaporation Local convective mass transfer coefficient Height Angle between sun direction and normal surface Electrical current flow rate Intensity of radiation I, intensity per micron at wavelength Radiosity Thermal conductivity ks, thermal conductivity of a solid kf, thermal conductivity of a fluid Thermal conductance Kk, thermal conductance for conduction heat transfer Kc, thermal convective conductance Kr , thermal conductance for radiation heat transfer Logarithm to base 10 Logarithm to base e General length Characteristic length or length along a heat flow path Latent heat of solidification Mass flow rate Mass General number Static pressure pc, critical pressure pA, partial pressure of component A Wetted perimeter Total pressure © 2001 by CRC PRESS LLC International Units English Units Newton(N) lbf Dimensionless Dimensionless m/s2 1.0 kg m/N s2 kg/m2 s W/m2 J/kg W/m2 K ft/s2 32.2 ft lbm/lbf s2 lbm/h ft2 Btu/h ft2 Btu/lbm Btu/h ft2 °F m J/kg m/s m rad amp W/m2 sr ft Btu/lbm ft/s ft deg amp Btu/h ft2 W/m2 W/m K Btu/h ft2 Btu/h ft °F W/K Btu/h °F Dimensionless Dimensionless m ft or in. m ft or in. J/kg Btu/lbm kg/s lbm/s or lbm/h kg lbm Dimensionless N/m2 lbf /in.2 or lbf/ft2 m N/m2 ft atm 0082FM.fm Page 11 Wednesday, August 23, 2000 9:50 AM Symbol q q̇ G q Q Q̇ r R S SL ST t T u u U U v Description Rate of heat flow qk, rate of heat flow by conduction qr, rate of heat flow by radiation qc, rate of heat flow by convection qb, rate of heat flow by nucleate boiling Rate of heat generation per unit volume Rate of heat generation per unit area Quantity of heat Volumetric rate of fluid flow Radius rH, hydraulic radius ri, inner radius ro, outer radius Electrical resistance Perfect gas constant Shape factor for conduction heat flow Distance between the centerlines of pinfins in adjacent longitudinal rows Distance between the centerlines of pinfins in adjacent transverse rows Time Temperature Tb, temperature of bulk fluid Tf, mean film temperature Ts, surface temperature T∞, temperature of fluid far removed from the heat source Tm, mean bulk temperature of a fluid flowing in a conduit Tsw, temperature at the surface of a wall Tsv, temperature of a saturated vapor Tsl, temperature of a saturated liquid Tfr, freezing temperature Tl, liquid temperature To, total temperature Tas, adiabatic wall temperature Twb, wetbulb temperature Internal energy per unit mass Time average velocity in the x direction u, instantaneous fluctuating x component of velocity u , average velocity Overall unit conductance, overall heat transfer coefficient Freestream velocity Specific volume © 2001 by CRC PRESS LLC International Units English Units W Btu/h W/m3 W/m2 J m3/s m Btu/h ft3 Btu/h ft2 Btu ft3/h ft ohm 8.314 J/K kg mol ohm 1545 ft lbf/lb mol °F Dimensionless m ft m ft s K s or h °F or R J/kg m/s Btu/lbm ft/s W/m2 K m/s m3/kg Btu/h ft2 °F ft/s ft3/lbm 0082FM.fm Page 12 Wednesday, August 23, 2000 9:50 AM Symbol v Description Time average velocity in the y direction , instantaneous fluctuating y component of velocity Volume Rate of work output Distance from leading edge xc, critical distance from the leading edge (beginning of turbulent flow) Coordinate Coordinate Distance from a solid boundary measured in a direction normal to the surface Vertical fin spacing Coordinate Ratio of heat exchanger hourly capacity rate V W˙ x x y y z z Z International Units English Units m/s ft/s m3 W m ft3 Btu ft Dimensionless Dimensionless m ft m ft Dimensionless Dimensionless GREEK LETTERS Symbol (alpha) Description International Units English Units (Delta) Absorptance for radiation , monochromatic absorptance at wavelength Temperature coefficient of volume expansion Temperature coefficient of thermal conductivity Specific heat ratio, cp/cv Body force per unit mass Mass flow rate of condensate per unit breadth m/ D for a vertical tube Boundary layer thickness h, hydrodynamic boundary layer thickness th, thermal boundary layer thickness Difference between values (epsilon) Heat exchanger effectiveness Dimensionless (epsilon) Dimensionless H (epsilon) Emittance for radiation , monochromatic emittance at wavelength , emittance in direction of Thermal eddy diffusivity m2/s ft2/s M (epsilon) Momentum eddy diffusivity m2/s ft2/s (beta) (beta) (gamma) (Gamma) c (Gamma) k (delta) © 2001 by CRC PRESS LLC Dimensionless 1/K 1/R 1/K 1/R Dimensionless N/kg lbf/lbm kg/s m lbm/h ft m ft Dimensionless 0082FM.fm Page 13 Wednesday, August 23, 2000 9:50 AM International Units Symbol Description (zeta) Ratio of thermal to hydrodynamic boundary layer thickness, th/h Fin efficiency Thermal resistance c, thermal resistance to convective heat transfer k, thermal resistance to conductive heat transfer r, thermal resistance to radiative heat transfer jc, thermal resistance from semiconductor junction to semiconductor case ca, thermal resistance from semiconductor case to ambient ja, thermal resistance from semiconductor junction to ambient Angle Wavelength max, wavelength at which monochromatic emissive power Eb is at maximum Latent heat of vaporization Absolute viscosity Kinematic viscosity, Frequency of radiation Mass density; l/v l, density of a liquid , density of a vapor Reflection for radiation StefanBoltzmann constant Surface tension Shearing stress s, shearing stress at surface w, shearing stress at wall of a conduit Transmittance for radiation Angle Quality Geometric parameter Angular velocity f (eta) (theta) (lambda) (lambda) (mu) (nu) τ (nu) (rho) (rho) (sigma) (sigma) (tau) (tau) (phi) (chi) (psi) (omega) English Units Dimensionless Percent (%) K/W °F/Btu rad µm deg Micron J/kg N s/m2 m2/s 1/s kg/m3 Btu/lbm lbm/ft s ft2/s 1/s lbm/ft3 Dimensionless W/m2 K4 Btu/h ft2 R4 N/m lbf/ft lbf/ft2 N/m2 Dimensionless rad deg Percent (%) Dimensionless rad/s rad/s DIMENSIONLESS NUMBERS Symbol Description Ref. hr o ks Bi hL Biot Number ks Bo g ( l )L Bond Number  or 1 2 © 2001 by CRC PRESS LLC ) (Continued) 0082FM.fm Page 14 Wednesday, August 23, 2000 9:50 AM Symbol Ec El Fo Gz Description U Eckert Number  ( T s T ) cp 2 4 g cpz T Elenbass Number  kL t Fourier Modulus 2 or L ṁc Graetz Number p kfL Ref. 2 3 t 2 ro 4 5 Gr L gT Grashof Number 2 6 j Nu 23 Colburn Factor  Pr RePr 7 Ja c p ( T w T sat ) Jakob Number h fg Le Lewis Number D AB M U Mach Number a 3 2 Mo Nu 8 9 k c Mouromtseft Number  hc x Nusselt Number  at point x kf 0.8 0.6 0.4 p 0.4 10 11 Nu hc L  average over surface Nusselt Number kf 11 Nu D hc D  average of diameter Nusselt Number kf 11 Pe Peclet Number Re Pr 12 Pr cp Prandtl Number k 13 Ra Rayleigh Number Gr Pr 14 Re UL Reynolds Number  15 2 Sh h a Boundary Fourier Modulus 2 ks hm L Sherwood Number D AB Sc Schmidt Number  D AB St hc Stanton Number Ucp We U L Weber Number  © 2001 by CRC PRESS LLC 2 3 16 17 or Nu Re Pr 18 0082FM.fm Page 15 Wednesday, August 23, 2000 9:50 AM UNIT CONVERSION FACTORS SI → English Quantity Area (A) Density () Energy (E) Energy per unit mass (e) Force (F) Heat flux generation per unit area (q) Heat generation per unit volume ( q̇ G ) Heat transfer coefficient (hc) Heat transfer rate (q) Length (L) Mass (M) Mass flow rate ( ṁ ) Rate of heat (q) Pressure and stress (p) Specific heat (cp) Surface tension ( ) 1 1 1 1 1 1 1 1 1 m2 10.764 ft2 m2 1550.0 in.2 kg/m3 0.06243 lbm/ft3 kg/m3 1.94032 103 slug/ft3 J 9.4787 104 Btu J 0.73757 lbf ft J 0.23885 cal J 372.44 103 hp h J/kg 4.2995 104 Btu/lbm English → SI 1 1 1 1 1 1 1 1 1 ft2 0.09290 m2 in.2 6.452 104 m2 lbm/ft3 16.0179 kg/m3 slug/ft3 515.38 kg/m3 Btu 1055.06 J lbf ft 1.3558 J cal 4.1868 J hp h 2.685 106 J Btu/lbm 2326.0 J/kg 1 W/m2 0.3171 Btu/(h ft2) 1 lbf 4.448 N 1 lbf 1 slug ft/s2 1 Btu/(h ft2) 3.1525 W/m2 1 W/m3 0.09665 Btu/(h ft3) 1 Btu/(h ft3) 10.343 W/m3 1 W/(m2 K) 0.1761 Btu/(h ft2 °F) 1 Btu/(h ft2 °F) 5.678 W/(m2 K) 1 W 3.41213 Btu/h 1 W 0.239 cal/s 1 m 3.2808 ft 1 m 39.37 in. 1 kg 2.2046 lbm 1 kg 68.521 103 slug 1 kg/s 7936.6 lbm/h 1 kg/s 2.2046 lbm/s 1 W 3.41213 Btu/h 1 W 94.778 106 Btu/s 1 W 0.73757 lbf ft/s 1 W 1.3410 103 hp 1 N/m2 1 Pa = 0.02089 lbf /ft2 1 N/m2 0.14504 103 lbf /in.2 1 N/m2 4.015 103 in H2O 1 N/m2 9.8688 std. atmosphere 1 N/m2 0.10 106 bar 1 J/(kg K) 2.3886 104 Btu/(lbm °F) 1 N/m 0.06852 lbf/ft 1 N/m 1 103 dyn/cm 1 Btu/h 0.2931 W 1 N 0.2248 lbf 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ft 0.30480 m in. 0.02540 m lbm 0.4536 kg slug 14.594 kg lbm/h 126.0 106 kg/s lbm/s 0.4536 kg/s Btu/h 0.2931 W Btu/s 1055.1 W lbf ft/s 1.3558 W hp 745.7 W lbf/ft2 47.88 N/m2 psi 1 lbf /in.2 6894.8 N/m2 in. H2O 249.066 N/m2 std. atm 0.10133 106 N/m2 bar 0.1 106 N/m2 Btu/(lbm °F) 4187.0 J/(kg K) 1 lbf/ft 14.594 N/m 1 dyn/cm 1 103 N/m (Continued) © 2001 by CRC PRESS LLC 0082FM.fm Page 16 Wednesday, August 23, 2000 9:50 AM SI → English English → SI T(K) T(°C) 273.15 T(K) T(°R)/1.8 T(K) [T(°F) 459.67]/1.8 T(°C) [T(°F) 32.0]/1.8 1 K 1°C 1 K 1.8°R 1 K 1.8°F 1 W/(m K) 0.57782 Btu/(h ft °F) T(°R) 1.8T(K) T(°R) T(°F) 459.67 T(°F) 1.8T(°C) 32.0 T(°F) 1.8[T(K) 273.15] 32.0 1°R 1°F 1°R (5/9)K 1°F (5/9)K 1 Btu/(h ft °F) 1.731 W/(m K) 1 m2/s 10.7639 ft2/s 1 m2/s 38750.0 ft2/h 1 K/W 0.52750 °F h/Btu 1 ft2/s 0.0929 m2/s 1 ft2/h 25.81 106 m2/s 1 °F h/Btu 1.896 K/W Quantity Temperature (T) Temperature difference ( T) Thermal conductivity (k) Thermal diffusivity (a) Thermal resistance () Velocity (U) Viscosity, absolute () Viscosity, kinematic () Volume (V) Volumetric flow rate ( q̇ ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 m/s 3.2808 ft/s m/s 196.85 ft/min m/s 11,811 ft/h N s/m2 0.6720 lbm/(ft2 s) N s/m2 2419.1 lbm/(ft2 h) N s/m2 1 103 cP m2/s 10.7639 ft2/s m2/s 38750.0 ft2/h m3 35.3134 ft3 m3 61023.4 in.3 m3 264.17 gal (U.S.) m3/s 35.3134 ft3/s m3/s 1.2713 105 ft3/h m3/s 2118.80 ft3/min m3/s 15850.0 gal (U.S)/min References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. Biot, Jean Baptiste (1774–1862) Eckert, E.R.G. Elenbass, W. Fourier, Baron Jean Baptiste Joseph (1768–1830) Graetz, Leo P. (1856–1941) Grashof, Franz (1826–1893) Colburn, Allan Philip (1904–1955) Lewis, G.W. Mach, Ernst (1838–1916) Mouromtseff, I.E. Nusselt, E. Wilhelm H. (1882–1957) Peclet, Jean Claude Eugene (1793–1857) Prandtl, Ludwig (1875–1953) Rayleigh, Lord (1842–1919) Reynolds, Osborne (1842–1912) Sherwood, Thomas Kilgore (1903–1976) Schmidt, Ernst (1892–1975) Stanton, Sir Thomas Edward (1865–1931) © 2001 by CRC PRESS LLC 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ft/s 0.3048 m/s ft/min 5.080 103 m/s ft/h 8.4667 105 m/s lbm/(ft2 s) 1.488 N s/m2 lbm/(ft2 h) 4.133 104 N s/m2 cP 0.001 N s/m2 ft2/s 0.0920 m2/s ft2/h 25.81 106 m2/s ft3 0.02832 m3 in.3 1.6387 105 m3 gal (U.S.) 3.785 103 m3 ft3/s 2.8317 102 m3/s ft3/h 7.8658 106 m3/s ft3/min 0.47196 103 m3/s gal/min 63.091 106 m3/s 0082FM.fm Page 17 Wednesday, August 23, 2000 9:50 AM Contents Chapter 1 Introduction to Thermal Design of Electronic Equipment 1.1 Introduction to the Modes of Heat Transfer in Electronic Equipment 1.1.1 Convection 1.1.2 Conduction 1.1.3 Radiation 1.1.4 Practical Thermal Resistances 1.2 Theoretical Power Dissipation in Electronic Components 1.2.1 Theoretical Power Dissipation 1.2.2 Heat Generation in Active Devices 1.2.2.1 CMOS Devices 1.2.2.2 Junction FET 1.2.2.3 Power MOSFET 1.2.3 Heat Generated in Passive Devices 1.2.3.1 Interconnects 1.2.3.2 Resistors 1.2.3.3 Capacitors 1.2.3.4 Inductors and Transformers 1.3 Thermal Engineering Software for Personal Computers 1.3.1 Commercial CFD Codes 1.3.2 Flotherm v2.2 References Chapter 2 Formulas for Conductive Heat Transfer 2.1 Conduction in Electronic Equipment: Introduction 2.2 Thermal Conductivity 2.2.1 Thermal Resistances 2.2.2 Conductivity in Solids 2.2.3 Conductivity in Fluids 2.3 Conduction—Steady State 2.3.1 Conduction in Simple Geometries 2.3.1.1 Conduction through a Plane Wall 2.3.1.2 Conduction through Cylinders and Spheres 2.3.1.3 Plane Wall with Heat Generation 2.3.1.4 Cylinders and Spheres with Heat Generation 2.3.1.5 Critical Radius of a Cylinder © 2001 by CRC PRESS LLC 0082FM.fm Page 18 Wednesday, August 23, 2000 9:50 AM 2.3.2 Conduction in Complex Geometries 2.3.2.1 Multidimensional Analytic Method 2.3.2.2 Multidimensional Graphical Method 2.3.2.3 Multidimensional Shape Factor Method 2.3.2.4 Finite Difference Method 2.3.2.5 ResistanceCapacitance Networks 2.4 Conduction—Transient 2.4.1 Lumped Capacitance Method 2.4.2 Application of the Lumped Capacitance Method 2.5 Conduction in Extended Surfaces 2.5.1 Fin Efficiency 2.5.2 Fin Optimization 2.5.3 Fin Surface Efficiency 2.6 Thermal Contact Resistance in Electronic Equipment Interfaces 2.6.1 Simplified Contact Resistance Model 2.6.2 Geometry of Contacting Surfaces 2.6.3 Contact Resistance in a Typical Application 2.7 Discrete Heat Sources and Thermal Spreading References Chapter 3 Fluid Dynamics for Electronic Equipment 3.1 Introduction 3.2 Hydrodynamic Properties of Fluids 3.2.1 Compressibility 3.2.2 Viscosity 3.2.3 Surface Tension 3.3 Fluid Statics 3.3.1 Relationship of Pressure, Density, and Height 3.4 Fluid Dynamics 3.4.1 Streamlines and Flowfields 3.4.2 One, Two, and ThreeDimensional Flowfields 3.5 Incompressible Ideal Fluid Flow 3.5.1 OneDimensional Flow 3.5.1.1 OneDimensional Euler Equation 3.5.1.2 OneDimensional Bernoulli Equation 3.5.1.3 Application of the OneDimensional Equations 3.5.2 TwoDimensional Flow 3.5.2.1 Application of the TwoDimensional Equations 3.6 Incompressible Real Fluid Flow 3.6.1 Laminar Flow 3.6.2 Turbulence and the Reynolds Number © 2001 by CRC PRESS LLC 0082FM.fm Page 19 Wednesday, August 23, 2000 9:50 AM 3.6.3 Boundary Layer Theory 3.6.4 Turbulent Flow 3.7 Loss Coefficients and Dynamic Drag 3.7.1 Expansions 3.7.2 Contractions 3.7.3 Tube Bends 3.7.4 Manifolds 3.7.5 Screens, Grills, and Perforated Plates 3.7.6 Rough Surface Conduits 3.8 Jets 3.9 Fans and Pumps 3.9.1 Fans 3.9.1.1 Fan Operation at Nonstandard Densities 3.9.2 Pumps 3.10 Electronic Chassis Flow References Chapter 4 Convection Heat Transfer in Electronic Equipment 4.1 Introduction 4.2 Fluid Properties 4.2.1 Properties of Air 4.3 Boundary Layer Theory 4.4 Dimensionless Groups 4.5 Forced Convection 4.5.1 Forced Convection Laminar Flow 4.5.1.1 Forced Convection Laminar Flow in Tubes 4.5.2 Forced Convection Turbulent Flow 4.5.2.1 Forced Convection Turbulent Flow in Tubes 4.5.2.2 Forced Convection Flow through Noncircular Tube Geometries 4.5.2.3 Forced Convection Flow through Tubes with Internal Fins 4.5.3 Forced Convection External Flow 4.5.3.1 Laminar Forced Convection along Flat Plates 4.5.3.2 Turbulent Forced Convection along Flat Plates 4.5.3.3 Mixed Boundary Layer Forced Convection along Flat Plates 4.5.3.4 Forced Convection Flow over Cylinders 4.5.3.5 Forced Convection Flow over Spheres 4.5.4 Forced Convection Flow over Complex Bodies 4.5.4.1 Forced Convection Flow along a Populated Circuit Board 4.5.4.2 Forced Convection Flow through PinFin Arrays 4.5.5 Jet Impingement Forced Convection © 2001 by CRC PRESS LLC 0082FM.fm Page 20 Wednesday, August 23, 2000 9:50 AM 4.6 Natural Convection 4.6.1 Natural Convection Flow along Flat Plates 4.6.2 Natural Convection Cooling Using Vertical Fins 4.6.3 Natural Convection along Nonvertical Surfaces 4.6.4 Natural Convection in Sealed Enclosures 4.6.5 Natural Convection in Complex Geometries 4.6.5.1 Natural Convection across Horizontal Cylinders 4.6.5.2 Natural Convection along Vertical Cylinders 4.6.5.3 Natural Convection across Spheres 4.6.5.4 Natural Convection across Cones 4.6.5.5 Natural Convection across Horizontal Corrugated Plates 4.6.5.6 Natural Convection across Arbitrary Shapes 4.6.5.7 Natural Convection through UShaped Channels 4.6.5.8 Natural Convection through PinFin Arrays References Chapter 5 Radiation Heat Transfer in Electronic Equipment 5.1 Introduction 5.1.1 The Electromagnetic Spectrum 5.2 Radiation Equations 5.2.1 StefanBoltzmann Law 5.3 Surface Characteristics 5.3.1 Emittance 5.3.1.1 Emittance Factor 5.3.1.2 Emittance from Extended Surfaces 5.3.2 Absorptance 5.3.3 Reflectance 5.3.3.1 Specular Reflectance 5.3.4 Transmittance 5.4 View Factors 5.4.1 Calculation of Estimated Diffuse View Factors 5.5 Environmental Effects 5.5.1 Solar Radiation 5.5.2 Atmospheric Radiation References Chapter 6 Heat Transfer with Phase Change 6.1 Introduction 6.1.1 Definitions of Phase Change Parameters 6.2 Dimensionless Parameters in Boiling and Condensation 6.3 Modes of Boiling Liquids © 2001 by CRC PRESS LLC 0082FM.fm Page 21 Wednesday, August 23, 2000 9:50 AM 6.3.1 6.3.2 Bubble Phenomenon Pool Boiling 6.3.2.1 Pool Boiling Curve 6.3.2.2 Pool Boiling Correlations 6.3.2.3 Pool Boiling Critical Heat Flux Correlations 6.3.2.4 Pool Boiling Minimum Heat Flux Correlations 6.3.2.5 Pool Boiling Vapor Film Correlations 6.3.3 Flow Boiling 6.3.3.1 External Forced Convection Boiling 6.3.3.2 Internal Forced Convection Boiling 6.4 Evaporation 6.5 Condensation 6.6 Melting and Freezing References Chapter 7 Combined Modes of Heat Transfer for Electronic Equipment 7.1 Introduction 7.2 Conduction in Series and in Parallel 7.3 Conduction and Convection in Series 7.4 Radiation and Convection in Parallel 7.5 Overall Heat Transfer Coefficient Appendix © 2001 by CRC PRESS LLC Hibbeler R. C. “ForceSystem Resultants and Equilibrium” Thermal Design of Electronic Equipment. Ed. Ralph Remsburg Boca Raton: CRC Press LLC, 2001 008201 Page 1 Wednesday, August 23, 2000 9:51 AM 1 Introduction to Thermal Design of Electronic Equipment 1.1 INTRODUCTION TO THE MODES OF HEAT TRANSFER IN ELECTRONIC EQUIPMENT Electronic devices produce heat as a byproduct of normal operation. When electrical current flows through a semiconductor or a passive device, a portion of the power is dissipated as heat energy. Besides the damage that excess heat can cause, it also increases the movement of free electrons in a semiconductor, which can cause an increase in signal noise. The primary focus of this book is to examine various ways to reduce the temperature of a semiconductor, or group of semiconductors. If we do not allow the heat to dissipate, the device junction temperature will exceed the maximum safe operating temperature specified by the manufacturer. When a device exceeds the specified temperature, semiconductor performance, life, and reliability are tremendously reduced, as shown in Figure 1.1. The basic objective, then, is to hold the junction temperature below the maximum temperature specified by the semiconductor manufacturer. Nature transfers heat in three ways, convection, conduction, and radiation. We will explore these in greater detail in subsequent chapters, but a simple definition of each is appropriate at this stage. 1.1.1 CONVECTION Convection is a combination of the bulk transportation and mixing of macroscopic parts of hot and cold fluid elements, heat conduction within the coolant media, and energy storage. Convection can be due to the expansion of the coolant media in contact with the device. This is called free convection, or natural convection. Convection can also be due to other forces, such as a fan or pump forcing the coolant media into motion. The basic relationship of convection from a hot object to a fluid coolant presumes a linear dependence on the temperature rise along the surface of the solid, known as Newtonian cooling. Therefore: qc hc As ( T s T m ) © 2001 by CRC PRESS LLC 008201 Page 2 Wednesday, August 23, 2000 9:51 AM 6.0 Failure Rate per 10 6 h 5.0 4.0 PAL 3.0 2.0 DRAM 1.0 Microprocessor 0.0 20 30 40 50 60 70 80 Junction Temperature, 90 100 110 120 o C FIGURE 1.1 Component failure rates with temperature for Programmable Array Logic (PAL), 256K Dynamic Random Access Memory (DRAM), and Microprocessors. Data from MILHDBK217. where: qc convective heat flow rate from the surface (W) As surface area for heat transfer (m2) Ts surface temperature (°C) Tm coolant media temperature (°C) hc coefficient of convective heat transfer (W/m2) This equation is often rearranged to solve for T, by which: qc T hc As 1.1.2 CONDUCTION Conduction is the transfer of heat from an area of high energy (temperature) to an area of lower relative energy. Conduction occurs by the energy of motion between adjacent molecules and, to varying degrees, by the movement of free electrons and the vibration of the atomic lattice structure. In the conductive mode of heat transfer we have no appreciable displacement of the molecules. In many applications, we use conduction to draw heat away from a device so that convection can cool the conductive surface, such as in an aircooled heat sink. For a onedimensional system, © 2001 by CRC PRESS LLC 008201 Page 3 Wednesday, August 23, 2000 9:51 AM the following relation governs conductive heat transfer: T q k A c L where: q k Ac T L heat flow rate (W) thermal conductivity of the material (W/m K) crosssectional area for heat transfer (m2) temperature differential (°C) length of heat transfer (m) Since heat transfer by conduction is directly proportional to a material’s thermal conductivity, temperature gradient, and crosssectional area, we can find the temperature rise in an application by: qL T kAc 1.1.3 RADIATION Radiation is the only mode of heat transfer that can occur through a vacuum and is dependent on the temperature of the radiating surface. Although researchers do not yet understand all of the physical mechanisms of radiative heat transfer, it appears to be the result of electromagnetic waves and photonic motion. The quantity of heat transferred by radiation between two bodies having temperatures of T1 and T2 is found by q r F 1,2 A ( T 1 T 2 ) 4 4 where: qr amount of heat transferred by radiation (W) emissivity of the radiating surface (highly reflective 0, highly absorptive 1.0) StefanBoltzmann constant (5.67 108 W/m2 K4) F1,2 shape factor between surface area of body 1 and body 2 (1.0) A surface area of radiation (m2) T1 surface temperature of body 1 (K) T2 surface temperature of body 2 (K) Unless the temperature of the device is extremely high, or the difference in temperatures is extreme (such as between the sun and a spacecraft), radiation is usually disregarded as a significant source of heat transfer. To decide the importance of radiation to the overall rate of heat transfer, we can define the radiative heat © 2001 by CRC PRESS LLC 008201 Page 4 Wednesday, August 23, 2000 9:51 AM Fl θsa θcs Die bond Lead Chip θjc Heat spreader Encapsulant FIGURE 1.2 Primary thermal resistances in a chip/heat sink assembly. jc is resistance from the die junction to the device case. cs is resistance from the device case to the heat sink. sa is resistance from the heat sink to the ambient air. (Adapted from Kraus, A. D. and BarCohen, A., Design and Analysis of Heat Sinks, John Wiley & Sons, New York, 1995. With permission.) transfer as a radiative heat transfer coefficient, hr: h r F 1,2 ( T 1 2 1.1.4 2 T 2)(T 1 T 2) PRACTICAL THERMAL RESISTANCES The semiconductor junction temperature depends on the sum of the thermal resistances between the device junction and the ambient environment, which is the ultimate heat sink. Figure 1.2 shows a simplified view of the primary thermal resistances: tot jc cs sa where: tot jc cs sa total thermal resistance (K/W) junction to case thermal resistance (K/W) case to heat sink thermal resistance (K/W) heat sink to ambient thermal resistance (K/W) © 2001 by CRC PRESS LLC 008201 Page 5 Wednesday, August 23, 2000 9:51 AM Thermal resistance between the semiconductor junction and the junction’s external case—This resistance is designated jc and is usually expressed in °C or K/W. This resistance is an internal function of the design and manufacturing methods used by the device manufacturer. Because this resistance occurs within the device, the use of heat sinks or other heatdissipating devices does not affect it. The semiconductor manufacturer decides upon this resistance by weighing such factors as the maximum allowable junction temperature, the cost of the device, and the power of the device. For example, a plastic semiconductor case is often used for a lowpower, inexpensive device. A typical jc for such a device might be 50 K/W. If the device operates in a 35°C environment and dissipates 0.5 W, then the junction temperature Tj is found by: T j Ta jc q 35 C ( 50 K/W ) ( 0.5 W ) 60 C For a higherpowered component, the manufacturer must use a more costly approach to dissipate the power. A typical jc for this type of component might be 2 K/W. Specialized chip assemblies using expensive lead forms, thermally conductive ceramics, and Diamond heat spreaders can further lower this value. Thermal resistance from the case to the heat sink interface surface—This resistance is designated as cs and is expressed in °C or K/W. Case to heat sink thermal energy is transferred primarily by conduction across the contact interface. The field of contact interface thermal resistance is complex and is not well understood. No models are able to predict this value in a variety of cases. Even values arrived at by actual testing may vary by 20%. In any case, this value can be reduced by using thermal greases, pads, and epoxies, and by increasing the pressure at the thermal interface. In some applications, manufacturers mount the semiconductor junction to a copper slug that extends to the surface of the case. This design results in a very low jc. In addition, they design the copper slug to be soldered to a printed circuit board, resulting in an extremely low contact resistance. The thermal resistance from the heat sink contact interface to the ambient environment is designated sa—Like the other resistances, it is also expressed in °C or K/W. This is often the most important resistance of the three as for susceptibility to change by the electronic packaging engineer. The smaller this value, and therefore the resulting total resistance tot, the more power the device can handle without exceeding its maximum junction temperature. For the simplified model, this value depends on the conductive properties of the heat sink, fin efficiency, surface area, and the convective heat transfer coefficient: sa 1 hc As The heat transfer coefficient, hc, introduced earlier, is a complex function and cannot be easily generalized for use. However, many empirical equations result in a reasonable degree of accuracy when generating values of hc. As this formula shows, sa is the reciprocal of the product of the heat transfer coefficient and the sink surface area. © 2001 by CRC PRESS LLC 008201 Page 6 Wednesday, August 23, 2000 9:51 AM Therefore, increasing the surface area, A, of a given heat sink reduces sa. Consequently, increasing the heat transfer coefficient, hc, also reduces the thermal resistance. When we mount a semiconductor on a heat sink, the relationship between junction temperature rise above ambient temperature and power dissipation is given by: T q( jc cs sa ) The focus of the remaining chapters is to explore and expand on these basic resistances to heat transfer, and then predict and minimize them (costeffectively) wherever possible. 1.2 THEORETICAL POWER DISSIPATION IN ELECTRONIC COMPONENTS 1.2.1 THEORETICAL POWER DISSIPATION Electronic devices produce heat as a byproduct of normal operation. When electrical current flows through a semiconductor or a passive device, a portion of the power is dissipated as heat energy. The quantity of power dissipated is found by: Pd VI where: Pd power dissipated (W) V direct current voltage drop across the device (V) I direct current through the device (A) If the voltage or the current varies with respect to time, the power dissipated is given in units of mean power Pdm : t 1 2 P dm  V ( t )I ( t ) dt t t1 where: Pdm mean power dissipated (W) t waveform period (s) I(t) instantaneous current through the device (A) V(t) instantaneous voltage through the device (V) t1 lower limit of conduction for current t2 upper limit of conduction for current © 2001 by CRC PRESS LLC 008201 Page 7 Wednesday, August 23, 2000 9:51 AM 1.2.2 HEAT GENERATION 1.2.2.1 IN ACTIVE DEVICES CMOS Devices The power that is dissipated by bipolar components is fairly constant with respect to frequency. The power dissipation for CMOS devices is a firstorder function of the frequency and a secondorder function of the device geometry. Switching power constitutes about 70 to 90% of the power dissipated by a CMOS. The switching power of a CMOS device can be found by: 2 CV Pd  f 2 where: C input capacitance (F) V peaktopeak voltage (V) f switching frequency (Hz) Shortcircuit power, caused by transistor gates being on during a change of state, makes up 10 to 30% of the power dissipated. To find the power dissipated by these dynamic short circuits, the number of on gates must be known. This value is usually given in units of W/MHz per gate. The power dissipated is found by: Pd Ntot Non q f where: Ntot Non q f 1.2.2.2 total number of gates percentage of gates on (%) power loss (W/Hz per gate) switching frequency (Hz) Junction FET The junction FET has three states of operation: on, off, and linear transition. When the junction FET is switched on, the power dissipation is given as: 2 Pd ON ID R DS ( ON ) where: ID drain current (A) RDS(ON) resistance of drain to source ( ) In the linear and off states the dissipated power is again found by VI. © 2001 by CRC PRESS LLC 008201 Page 8 Wednesday, August 23, 2000 9:51 AM 1.2.2.3 Power MOSFET The power dissipated by a power MOSFET is a combination of five sources of current loss:2,3 a. Pc : conduction losses while the device is on, b. Prd : reverse diode conduction and trr losses, c. PL : power loss due to drainsource leakage current (IDSS) when the device is off, d. PG: power dissipated in the gate structure, and e. PS: switching function losses. Conduction losses, Pc, occurring when the device is switched on, can be found by: 2 Pc I D R DS ( ON ) where: drain current (A) ID RDS(ON) drain to source resistance ( ) Conduction losses when the device is in the linear range are found by VI, as are leakage current losses, PL, and reverse current losses, Prd. Switching transition losses, PS, occur during the transition from the on to off states. These losses can be calculated as the product of the draintosource voltage and the drain current; therefore: t S1 P S f S 0 VDS ( t )I D ( t ) dt t S2 0 VDS ( t )ID ( t ) dt where: fS VDS ID tS1 tS2 switching frequency (Hz) MOSFET draintosource voltage (V) MOSFET drain current (A) first transition time (s) second transition time (s) The MOSFET gate losses are composed of a capacitive load with a series resistance. The loss within the gate is RG PG VGS QG RS RG where: VGS gatetosource voltage (V) QG peak charge in the gate capacitance (coulombs) RG gate resistance ( ) © 2001 by CRC PRESS LLC 008201 Page 9 Wednesday, August 23, 2000 9:51 AM The total power dissipated by the gate structure, PG(TOT), is found by: PG ( TOT ) V GS QG fS 1.2.3 1.2.3.1 HEAT GENERATED IN PASSIVE DEVICES Interconnects The steadystate power dissipated by a wire interconnect is given by Joule’s law: 2 PD I R where: I steadystate current (A) R steadystate resistance ( ) The resistance of an interconnect is L R Ac where: material resistivity per unit length ( /m) (see Table 1.1) L connector length (m) Ac crosssectional area (m2) TABLE 1.1 Resistance of Interconnect Materials Material Alloy 42 Alloy 52 Aluminum Copper Gold Kovar Nickel Silver Resistivity, , /cm 66.5 43.0 2.83 1.72 2.44 48.9 7.80 1.63 Source: King, J. A., Materials Handbook for Hybrid Microelectronics, Artech House, Boston, 1988, p. 353. With permission. © 2001 by CRC PRESS LLC 008201 Page 10 Wednesday, August 23, 2000 9:51 AM Table 1.2 shows the maximum currentcarrying capacity of copper and aluminum wires in amperes:5 TABLE 1.2 Maximum CurrentCarrying Capacity of Copper and Aluminum Wires (in Amperes) Copper MILW5088 Aluminum MILW5088 Underwriters Laboratory National Size, Single Bundled Single Bundled Electrical Wire Wirea Code 60°C AWG Wire Wirea 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 0 00 – – – – 9 11 16 22 32 41 55 73 101 135 181 211 245 283 – – – – 5 7.5 10 13 17 23 33 46 60 80 100 125 150 175 – – – – – – – – – – – 58 86 108 149 177 204 237 – – – – – – – – – – – 36 51 64 82 105 125 146 – – – – – – 6 10 20 30 35 50 70 90 125 150 200 225 0.2 0.4 0.6 1.0 1.6 2.5 4.0 6.0 10.0 16.0 – – – – – – – – American Insurance 500 80°C Association cmilA 0.4 0.6 1.0 1.6 2.5 4.0 6.0 10.0 16.0 26.0 – – – – – – – – – – – – – 3 5 7 15 20 25 35 50 70 90 100 125 150 0.20 0.32 0.51 0.81 1.28 2.04 3.24 5.16 8.22 13.05 20.8 33.0 52.6 83.4 132.8 167.5 212.0 266.0 Rated ambient temperatures: 57.2°C for 105°Crated insulated wire 92.0°C for 135°Crated insulated wire 107°C for 150°Crated insulated wire 157°C for 200°Crated insulated wire a Bundled Wire indicates 15 or more wires in a group. Source: Croop, E. J., in Electronic Packaging and Interconnection Handbook, Harper, C.A., Ed., McGrawHill, New York, 1991. With permission. These values can be rerated at any anticipated ambient temperature by the equation: Tc T I I r Tc T r © 2001 by CRC PRESS LLC 008201 Page 11 Wednesday, August 23, 2000 9:51 AM where: I current rating at ambient temperature (T) Ir current rating in rated ambient temperature (Table 1.2) T ambient temperature (°C) Tr rated ambient temperature (°C) Tc temperature rating of insulated wire or cable (°C) 1.2.3.2 Resistors The steadystate power dissipated by a resistor in given by Joule’s law: 2 PD I R where: I steadystate current (A) R steadystate resistance ( ) The instantaneous power, PD(t), dissipated by a resistor with a timevarying current, I(t), is 2 P D ( t ) I ( t )R where I(t) IM sin( t) and IM peak value of the sinusoidal current (A). The average power dissipation when a sinusoidal steadystate current is applied is 2 PD 0.5I M R 1.2.3.3 Capacitors Although capacitors are generally thought of as nonpowerdissipating, some power is dissipated due to the resistance within the capacitor. The power dissipated by a capacitor under sinusoidal excitation is found by: PD ( t ) 0.5 CV M sin 2 t 2 where: C VM f capacitance (F) peak sinusoidal voltage (V) radian frequency, 2f frequency (Hz) © 2001 by CRC PRESS LLC 008201 Page 12 Wednesday, August 23, 2000 9:51 AM TABLE 1.3 Typical Resistances of Capacitors6–9 Dielectric Material Capacitance (F) RES @ 1 kHz, m 0.1 0.1 0.18 1.0 3.3 2.2 22 33 33 68 19.0 k 16.0 k 10.0 k 2.0 k 0.60 k 1.0 k 0.20 k 0.20 k 0.26 k 0.168 k BX X7R X7R BX Z5U Tantalum Tantalum Tantalum Tantalum Tantalum The equivalent series resistance of a capacitor in an AC circuit can lead to significant power dissipation. The average power in such a circuit is given as: t 1 2 PD  I 2 ( t )R ES dt T t1 where RES equivalent series resistance ( ). Table 1.3 shows the typical resistance of commercial capacitors. 1.2.3.4 Inductors and Transformers Inductors and transformers generally follow the power dissipation of resistors, 2 PD I R L where RL direct current resistance of the inductor or winding ( ). If the highfrequency component of the excitation current is significant, the winding resistance will increase due to the skin depth effect. The power dissipated by the sinusoidal resistance of an inductor is found by: PD ( t ) 0.5LI M sin 2 t 2 where: L inductance (Henry) IM peak sinusoidal current (A) radian frequency (2f ) © 2001 by CRC PRESS LLC 008201 Page 13 Wednesday, August 23, 2000 9:51 AM When a ferromagnetic core is used, the loss consists of two sublosses: hysteresis and eddy current. The rate of combined core power dissipation can be found by: n m Ṗ D ( CORE ) 6.51 f B MAX where: PD(CORE ) power dissipation (W/kg) n, m constants of the core material f switching frequency (Hz) BMAX maximum flux density (Tesla) The power dissipation is then found by: P D Ṗ D ( CORE ) M where M mass of the ferromagnetic core (kg). 1.3 THERMAL ENGINEERING SOFTWARE FOR PERSONAL COMPUTERS The past 10 years have seen a major change in the way we evaluate heat transfer. Whereas mainframe computers were once used to calculate large thermal resistance networks for conduction problems, we now perform FEA (finite element analysis) on desktop personal computers. Ten years ago CFD (computational fluid dynamics) was largely experimental and was almost exclusively used only in research laboratories; it is now also used to provide quick answers on desktop computers. The convective coefficient of heat transfer, the most difficult value to assign in heat transfer, is regularly being estimated within 10%, whereas 30% was formerly the norm. Once we construct and verify a computer model, we can evaluate hundreds of changes in a short time to optimize the model. In the future, as the underlying CFD code becomes more advanced, even the tedious model verification step may be eliminated. As with physical designs, computer models can be a combination of conduction, convection, and radiation modes of heat transfer. Convection problems have the largest variety of permutations, and this has given the CFD engineers the most difficulty: laminar flow changes to turbulent flow, energy dissipation rates change with velocity, at slow velocity natural convection may override the expected forced convection effects, etc. When additional factors such as multiphase flow, compressibility, and fine model details such as semiconductor leads are added, it is easy to see why convective computer modeling is so complex. © 2001 by CRC PRESS LLC 008201 Page 14 Wednesday, August 23, 2000 9:51 AM At the core of these elaborate computer codes are the basic equations of mass, momentum, and energy conservation, shown here in the Cartesian coordinate system for familiarization: Conservation of mass: t  ( u ) x  ( v ) 0 y Conservation of momentum in x:  ( u ) t  ( uu ) x u  ( vu )   y x x u p    y y y Fx v p   y y y Fy Conservation of momentum in y:  ( v ) t  ( uv ) x v  ( vv )   y x x Conservation of energy:  ( h ) t  ( uh ) x k h  ( vh )    y x cp x k h    y cp y q̇ G where: fluid density (kg/m3) t time (s) u, v velocity components in x and y coordinates (m/s) molecular viscosity (N s/m2) p pressure (N/m2) F force per unit volume (Pa/m3) h specific enthalpy (J/kg) k thermal conductivity (W/m K) cp specific heat (J/kg K) q̇ G heat source per unit volume (W/m3) These equations can take many forms and change in different coordinate systems and under different flow conditions. We enter the geometry of a model into a computer CFD program or, more commonly, it is imported in a standard format from a CAD (computeraided drafting) software program. Within the CFD program the required spatial coordinates are chosen to learn the dimensionality of the model, such as , r, and z in the polar coordinate system. By carefully evaluating the problem, a seemingly complex threedimensional problem can sometimes be modeled in two dimensions. An example is the axisymmetric pipe flow model. We require a twodimensional model to calculate © 2001 by CRC PRESS LLC 008201 Page 15 Wednesday, August 23, 2000 9:51 AM the radial, r, and axial, z, variations, in addition to the velocities of v and w. If we require a more realistic and detailed model, adding a circumferential velocity can allow the flow to swirl within the pipe, u, as a function of r and z. Although three momentum equations are used for three velocity components, the flow is still twodimensional because the flowfield variables are a function of just two space coordinates. Once the geometry, coordinate system, and material properties are modeled in a computer, the fluid region is discretized as several smaller domains. A finer or nonuniform grid is often used in areas of greater interest or areas where the flow patterns are so complex that a coarse solution would affect the accuracy of the entire model. We can classify the smaller domains into three broad methods of problem solution: 1. Finite Element Analysis, 2. Finite Difference Analysis, 3. Finite Volume Analysis. The finite element method10,11 uses a weighted residual to obtain the solution to the discrete equations. Some methods use explicit, while others use implicit, formulations with a variety of convergence schemes. As a consequence of the explicit formulation, a solution is found in a timesequencing manner. Time steps are taken to progress toward a final flowfield solution. Usually, finite element methods are easier to use than other methods when adapting irregularshaped elements to complex geometries. The finite difference method12,13 is structured around a Taylor series expansion for each variable adjacent to a grid point. Most codes retain only the first several terms and discard higherorder formulations. The result is a firstorder, secondorder, thirdorder, etc. accuracy. Codes may use explicit, implicit, and semiimplicit methods of domain solution. Usually, we obtain a full solution for a single point before we realize a solution for a subsequent point. Finite difference methods have been used for many years and have a history of optimized solutions. The finite volume analysis method14 is interesting because it attempts to solve the discrete domain solutions by the direct application of the conservation of mass, momentum, and energy equations. The basis of the finite volume method is the fully implicit equation. Solutions are found by iterative methods with a certain flexibility for specific variabilities. Interestingly, different variables are solved by the point to point method while we may solve other variables in a wholefield analysis. We know finite volume methods to be very stable and efficient in their use of computer resources. 1.3.1 COMMERCIAL CFD CODES Turbulence analysis methods15—The typical flow problem encountered in electronic cooling is turbulence. Turbulent flows can be solved by an analysis of the characteristics of the mean (timeaveraged) flow. The most common turbulence models are based on the Boussinesq concept of eddy viscosity. The use of turbulent or eddy viscosity accounts for enhanced mixing (diffusion) due to turbulence. Eddy viscosity is normally magnitudes larger than the effect of molecular viscosity and © 2001 by CRC PRESS LLC 008201 Page 16 Wednesday, August 23, 2000 9:51 AM is a flow property, not a fluid material property. The most commonly used turbulent flow model is the twoequation k ~ model. This model uses two transport equations—one for turbulent kinetic energy, k, and the other for the rate of eddy dissipation, . We apply local calculated values of k and as turbulent viscosity values. When compared with the simpler Prandtl mixing equation, the k ~ model does not require prescribed scales of turbulence length. Although it is a theoretically complex equation, by extensive analysis and comparison with physical models, the k ~ method has been limited to five empirical constants. The k ~ model is being refined and expanded16 for greater applicability in a broad range of fluid problems. Direct Numerical Simulation—A class of CFD that holds great promise is Direct Numerical Simulation (DNS). The hope for DNS is based upon the idea that turbulence, with all its complicated large and smallscale structures, is nothing other than a viscous flow that locally obeys the NavierStokes equations. If a fine enough grid is used, we can calculate all the details of this turbulent flow directly from the NavierStokes equations with no artificial “modeling” of the effects of turbulence. A current limitation of this technique is the enormous amount of computer time required. To use the DNS method to directly solve the NavierStokes equations for a simple problem of flow over a flat plate, Rai and Moin17 had to use 16,975,196 threedimensional grid points and over 400 hours on a CRAY YMP supercomputer. 1.3.2 FLOTHERM V2.2 Several generalpurpose CFD codes are available on the commercial market. These codes have varying degrees of friendliness toward electronic cooling problems but, in general, are very useful. A program by Flomerics claims an 80% share of the CFD market for thermal analysis of electronic packaging. FLOTHERMTM contains a full 3D solver for NavierStokes equations, builtin boundary conditions for common objects such as fans, vents, and filters, and an effective turbulent viscosity solver that accounts for the additional friction and heat transfer due to turbulence. This package is designed specifically for electronics cooling problems. The software is designed to run on personal computers and UNIX platforms. FLOTHERMTM is available from Flomerics, Inc., Southborough, MA. REFERENCES 1. Kraus, A. D. and BarCohen, A., Design and Analysis of Heat Sinks, John Wiley & Sons, New York, 1995. 2. Sergent, J. E. and Krum, A., Thermal Management Handbook for Electronic Assemblies, McGrawHill, New York, 4.7, 1998. 3. CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, E78, 1984. 4. King, J. A., Materials Handbook for Hybrid Microelectronics, Artech House, Boston, 353, 1988. 5. Croop, E. J., Wiring and Cabling for Electronic Packaging, in Electronic Packaging and Interconnection Handbook, Harper, C. A., Ed., McGrawHill, New York, 1991. © 2001 by CRC PRESS LLC 008201 Page 17 Wednesday, August 23, 2000 9:51 AM 6. Hopkins, D. C., Designing Power Hybrid Supplies, Powertechniques Magazine, June, 31–34, 1989. 7. Hopkins, D. C., Jovanovic, M. M., Lee, F. C., and Stephenson, F. W., Offline ZCSQRC ThickFilm Hybrid Circuit, Virginia Power Electronics Center, Sixth Annu. Power Electron. Semin., 71–83, September, 1988. 8 Olean Advanced Products Data Book, SOAP10M295N, AVX Corporation, Myrtle Beach, SC. 9. Kemet Surface Mount Catalog, F3102, 20, September, 1994, Simpsonville, SC. 10. Zienkiewicz, O. C. and Morgan, K., Finite Elements and Approximation, John Wiley & Sons, New York, 1983. 11. Baker, A. J., Computation of Fluid Flow by the Finite Element Method, McGrawHill, New York, 1984. 12. Shih, T. M., Numerical Heat Transfer, Hemisphere, New York, 1984. 13. Anderson, D. A., Tannehill, J. C., and Pletcher, R. H., Computational Fluid Mechanics and Heat Transfer, Hemisphere, New York, 1985. 14. Pantankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980. 15. Launder, B. E. and Spalding, D. B., Lectures on Mathematical Models of Turbulence, Academic Press, New York, 1972. 16. Markatos, N. C., Computer Simulation Techniques for Turbulent Flows, in Encyclopedia of Fluid Mechanics, Vol. 6, Cheremisinoff, N. P., Ed., Gulf Publ., June 1984; J. Appl. Math. Modeling, 10, June 1986. 17. Rai, M. M. and Moin, P., Direct Numerical Simulation of Transition and Turbulence in a Spatially Evolving Boundary Layer, AIAA paper 911607CP, Proc. AIAA 10th Computer Fluid Dynamics Conf., 890–914, 1991. © 2001 by CRC PRESS LLC Hibbeler R. C. “ForceSystem Resultants and Equilibrium” Thermal Design of Electronic Equipment. Ed. Ralph Remsburg Boca Raton: CRC Press LLC, 2001 © 2001 by CRC PRESS LLC 008202 Page 19 Wednesday, August 23, 2000 9:53 AM 2 Formulas for Conductive Heat Transfer 2.1 CONDUCTION IN ELECTRONIC EQUIPMENT: INTRODUCTION Heat transfer by the conduction mode occurs when heat is transferred within a material, or from one material to another. The energy transfer is postulated to occur because of kinetic energy exchange by elastic and inelastic collisions of atoms, and by electron drift. Heat energy is always transferred from a region of higher energy to an area of lower energy. The energy level, or temperature, of a material is related to the vibration level of the molecules within the substance. If the regions are at an equal temperature, no heat transfer occurs. Fourier’s law can be used to predict the rate of heat transfer.1 The law suggests that the rate of heat transfer be proportional to the area of transfer times the temperature gradient dT/dx. dT q k A dx In Fourier’s law, the relation T(x) is the local temperature and x is the distance of heat flow. Although this is an equation of proportionality, the actual rate of heat transfer depends on the thermal conductivity, k, which is a physical property of the heat transfer media. Thermal conductivity is generally expressed in terms of W/m K. Heat transfer can occur by conduction through any material: solid, liquid, or gas. Conduction cannot occur through a vacuum because there is no material to conduct through. Conduction is not usually the predominant method of heat transfer through a gas or liquid. Usually, as we apply heat to a fluid, the heated portion of the fluid expands and sets up density gradients. These density gradients cause motion within the fluid, which leads to convective heat transfer. Convective heat transfer, a macroscopic method of energy transfer, is much more effective than conductive heat transfer. The values used for the thermal conductivities of liquids and solids are generally obtained by experimentation. The thermal conductivity of gases at moderate temperatures closely follows the kinetic theory of gases, and therefore calculated values may be used. © 2001 by CRC PRESS LLC 008202 Page 20 Wednesday, August 23, 2000 9:53 AM 2.2 THERMAL CONDUCTIVITY Fourier’s law presents heat transfer as a proportionality equation that depends on k, the thermal conductivity of the heat transfer media. When we know the steadystate proportionality, the thermal conductivity can be found by q k A dT dx Thermal conductivity is a physical property that suggests how much heat will flow per unit time across a unit area when the temperature gradient is unity, expressed in W/m K. The property of thermal conductivity is important in conduction and convection applications. In some natural convection applications, where we have a confined airspace, heat transfer is actually by conduction, not convection as the designer might assume. The conduction of heat occurs when molecular collisions move the kinetic energy of heat from one molecule to the next. Therefore, thermal conduction can occur only when a temperature differential exists. Usually, metals are good conductors because they have free electrons that are not dedicated to any single nucleus. These free electrons can move through the atomic structure of the metal and collide with other electrons, or with the larger ions and nuclei within the structure. The identical mode of energy transfer also occurs during electrical conduction. This is why most materials that are good thermal conductors are also good electrical conductors. The primary exception to this is diamond. Diamond has a thermal conductivity value approximately 5 times higher than copper, but a dielectric strength 10 times higher than rubber. 2.2.1 THERMAL RESISTANCES Often, the thermal resistances characterize the transmission of heat in the path of heat transfer. Examples of this include thermal pads, dielectric insulators, and adhesive bonding materials. Thermal resistance is most often expressed as temperature rise in units of °C/W or K/W, and is found by: L T cond  k Ac qx where Ac is the crosssectional area available for conduction in units of m2. By comparing the thermal resistances, it sometimes becomes apparent which components in the heat transfer path are contributing most to the heat rise of the power component. Interestingly, we can describe convective heat transfer as a thermal resistance by 1 T conv  hc As qx © 2001 by CRC PRESS LLC 008202 Page 21 Wednesday, August 23, 2000 9:53 AM where As is the surface area in contact with the cooling media. Radiation heat transfer can be described as a thermal resistance by T 1 rad  qx hr AF where AF is the area of radiation based on a geometric factor of shape and emissivity. 2.2.2 CONDUCTIVITY IN SOLIDS Thermal conductivity in a solid material is based upon migration of free electrons and vibrations within the atomic lattice structure. Silver, copper, and aluminum are indicative of materials in this group. These materials have high thermal and electrical conductivity. Figure 2.1 shows how the thermal conductivity of some metals changes with temperature. In nonmetals, the lattice structure vibrations dominate over the movement of free electrons, and thermal conductivity may not be related to electrical conductivity. In materials with highly structured crystalline lattice structures, thermal conductivity can be quite high, while electrical conductivity is quite low. An outstanding example of a material in this group is diamond. Diamond has a thermal conductivity 5 times that of copper, and an electrical breakdown strength of more than 2000 V of direct current per 0.01 mm of length. 2.2.3 CONDUCTIVITY IN FLUIDS Fluids, both liquids and gases, have much greater spacing between molecules than solids and therefore much lower thermal conductivities. The thermal conductivity of a fluid varies with pressure and temperature. Within the pressure range of fluids used in electronic cooling, thermal conductivity variances with pressure can be ignored. Temperature, however, can greatly affect the thermal conductivity of liquids or gases. Within the range of temperatures used in electronic cooling, the thermal conductivity change of a gas is linear with temperature change but is different for each gas. The thermal conductivity change with temperature in liquids is not yet well understood. Figures 2.2 and 2.3 show the thermal conductivity change with temperature for selected gases and liquids, respectively. 2.3 CONDUCTION—STEADY STATE 2.3.1 CONDUCTION IN SIMPLE GEOMETRIES In simple shapes such as a wall or cylinder, the heat flow is onedimensional; that is, we require only a single coordinate to describe the spatial variation of the dependent variables. © 2001 by CRC PRESS LLC 008202 Page 22 Wednesday, August 23, 2000 9:53 AM FIGURE 2.1 Comparison of the variation of thermal conductivity with temperature for typical solid materials used in electronic packaging. 2.3.1.1 Conduction through a Plane Wall In the onedimensional form, T depends only on x. If there is no internal heat generation (qi 0), and we set the plane wall shown in Figure 2.4 to an initial temperature and distance of T(x 0) T1 and a final temperature and distance of © 2001 by CRC PRESS LLC 008202 Page 23 Wednesday, August 23, 2000 9:53 AM FIGURE 2.2 Comparison of the variation of thermal conductivity with temperature for common gases used in electronic cooling applications. T(x L) T2, then: T2 T1 T ( x ) x T1 L © 2001 by CRC PRESS LLC 008202 Page 24 Wednesday, August 23, 2000 9:53 AM FIGURE 2.3 Comparison of the variation of thermal conductivity with temperature for common liquids used in electronic cooling applications. PAO represents polyalphaolefin. Using Fourier’s law, we can find the rate of conductive heat transfer in the onedimensional xdirection dT kA q x kA   ( T 1 T 2 ) dx L © 2001 by CRC PRESS LLC 8 8 8 008202 Page 25 Wednesday, August 23, 2000 9:53 AM L qgen = qG ( A dx ) A dx x Tmax T1 8 T1 FIGURE 2.4 Conduction in a plane wall when the internal heat generation is uniform. In this case the temperature distribution is T1 T2. The heat flux, energy per unit area, is given as q k q x x  ( T 1 T 2 ) A L Rearranging the rate of heat transfer for temperature rise, we have the familiar onedimensional form: qL T k Ac More complex problems of this type may encompass onedimensional heat flow through any number of series and parallel combinations of thermal resistance. Although parallel heat flow is technically a twodimensional problem, we can usually reduce it to a single heat flow direction (see Figure 2.5). The general equation for © 2001 by CRC PRESS LLC 008202 Page 26 Wednesday, August 23, 2000 9:53 AM ,A 8 T Ts ,A TB TC Ts,D L3 k1 k2 k3 1 2 3 T Cold fluid T ,D ,hD x T ,A 8 qx L1 k1A Ts ,A L2 k 2A TB L3 k 3A TC 1 h DA Ts,D 8 1 h AA ,D 8 L2 T ,D 8 8 Hot fluid T ,A ,h1 L1 FIGURE 2.5 Equivalent thermal circuit for heat conduction through a series composite wall. The wall is composed of three sections, with section 2 having the lowest thermal conductivity. heat transfer for these problems, called composite walls, is T ,1 T ,N q x  t Therefore, we can describe a composite wall with three materials (A, B, and C) in series and convective heat transfer along the face of material A and C as T ,1 T ,4 q x 1 L A LB LC 1      h c, 1 A k A A k B A k C A h c, 4 A The overall heat transfer coefficient, U, is sometimes used, which we describe as q U AT © 2001 by CRC PRESS LLC 008202 Page 27 Wednesday, August 23, 2000 9:53 AM L T = T (r ) k = constant qG r r T 0 qk = 0 1 T 1 0 FIGURE 2.6 Radial heat conduction through a cylindrical shell having no internal heat generation. Using the overall heat transfer coefficient, the previous expression for the composite wall of Figure 2.5 becomes 1 1 U   tot A L L L 1 1  A B C  h c, 1 k A k B k C h c, 4 2.3.1.2 Conduction through Cylinders and Spheres In electronic cooling, the most prevalent case of radial heat transfer is the tube containing a flowing coolant. Here, heat flows from the outer surface of the tube to the center of the tube (see Figure 2.6). The rate of heat transfer in the radial direction of the tube is C To Ti dT q k kA  k ( 2 rL ) 1 2 Lk r dr r ln O ri Note that this shows that the distribution of the heat flow is logarithmic, not linear © 2001 by CRC PRESS LLC 008202 Page 28 Wednesday, August 23, 2000 9:53 AM TC rB T¥,D ,hD TB TS ,A rA TS ,D T¥,A,hA rC T¥,A ,h + rD 1 A 3 2 L T¥,A T¥,D,hD Ts,A TB TC Ts,D T¥,A Ts,A 1 ln(r p 2 hA2 rAL p B TB TC T¥,D Ts,D T¥,D / rA )ln(rC / rB )ln(rD / rC ) k1L 2 pkL 2 2 pkL 3 l p hD2 rDL FIGURE 2.7 Depiction of the temperature distribution through a compostie cylindrical wall. The thermal energy is applied at r 0, not at the inner surface, rA. as in the plane wall. The thermal resistance can be expressed as r ln O ri 2 Lk Similar to the method used to calculate combined conduction and convection heat transfer in a composite plane wall, the heat transfer equation for a composite tube (see Figure 2.7) containing three materials and a flowing fluid is T ,1 T ,4 T  q 4 r r r 1 ln  ln  ln  r r r 1 1     h c1 2 r 1 L 2 k A L 2 k B L 2 k C L h c4 2 r 4 L 2 3 4 1 2 3 Using the overall heat transfer coefficient, the previous expression for the composite © 2001 by CRC PRESS LLC 008202 Page 29 Wednesday, August 23, 2000 9:53 AM qk T = T (r ) k = constant qG = 0 r0 r1 T1 T0 FIGURE 2.8 Heat conduction through a hollow sphere having a uniform surface temperature and no internal heat generation. wall tube becomes 1 1 U   tot A r r r r r r r 1 1  1 ln 2 1 ln 3 1 ln 4 1  h c1 k A r 1 k B r 2 k C r 3 r 4 h c4 We can simplify the equation for heat conduction in spherical coordinates to 2 1 d 2 dT 1 d ( rT ) 2  r    0 r dr 2 r dr dr If Ti is the temperature at ri and To is the temperature at ro, then the temperature distribution in the sphere (see Figure 2.8) is ri ro  1  T ( r ) Ti ( To Ti )  r ro ri © 2001 by CRC PRESS LLC 008202 Page 30 Wednesday, August 23, 2000 9:53 AM L T s,A x +L T T (x) ¥ T ,B,hB q . O T (x) T s Ts T s,B T ,A,hA x +L L T ,h T ,h ¥ ¥ ¥ (a) (b) T q . 0 T (x) T s T ,h ¥ (c) FIGURE 2.9 Heat conduction through a plane wall with uniform internal heat generation. (a) Asymmetrical boundary conditions. Surface 2 has better cooling. (b) Symmetrical boundary conditions. (c) Adiabatic surface at midplane. Only surface 2 benefits from convection cooling. The rate of heat transfer through the sphere is then To Ti 2 T q 4 r  ro ri r 4 kro ri and the thermal resistance is found by ro ri 4 kr o r i 2.3.1.3 Plane Wall with Heat Generation In the plane wall studied previously we neglected heat generation, qG, within the wall. If we now calculate for heat generation (see Figure 2.9) and constant thermal © 2001 by CRC PRESS LLC 008202 Page 31 Wednesday, August 23, 2000 9:53 AM conductivity, k, the equation becomes 2 d T ( x)  q̇ G k 2 dx We find the temperature distribution, T(x), by q̇ 2 T 2 T1 q̇ G L x T1 x T ( x ) G x 2k L 2k If the two surface temperatures are equal, T1 T2, the temperature distribution simplifies to a parabolic distribution about the centerline of the plane wall, described as 2 2 q̇ G L x   x T1 T ( x ) 2k L L Since the centerline, which is x L/2, has the maximum temperature, we can find the temperature rise by calculating 2 q̇ G L T 8k Table 2.1 shows the solutions to a variety of conductive plate and wall problems. 2.3.1.4 Cylinders and Spheres with Heat Generation In this section we will examine heat transfer in a radial system such as a cylinder or sphere with internal heat generation. Such cases occur in currentcarrying bus bars, wires, resistors, and a flex circuit rolled into a cylindrical shell. The following equations apply to both cylinders and spheres (see Figure 2.10). The temperature distribution in a cylinder is found by 2 q̇ G r o r 2 T ( r ) 1  T s r o 4k The maximum temperature is at the centerline of the cylinder, r 0; therefore, 2 T max © 2001 by CRC PRESS LLC q̇ G r o T o rk 008202 Page 32 Wednesday, August 23, 2000 9:53 AM TABLE 2.1 Conduction in Plates and Walls2 Description Equations Convectively heated and cooled plate Convectively heated and cooled plate h1 T T 1 T 2 q h1 Bi 1 1 h 2 h1 x T T T 1 k 1 T T2 T1 h1 L h1  1 h k 2 Composite plate Composite plate T T0 T n q n for J˙ Li 1 1    k i h i h 0 1 T T j To  T T n To Plate with temperaturedependent thermal conductivity i 1 j 1 Li 1 x j    1 k k i h i k j h0 n L 1 1  i  h i h 0 i 1 ki i1 Plate with temperaturedependent thermal conductivity3 for k k1 TT−T1 q = T T1 T2 k m L 2 1 k1 2 km T T 2 T 1 x T T T 1   L k1 where k1 k2 km 2 © 2001 by CRC PRESS LLC 008202 Page 33 Wednesday, August 23, 2000 9:53 AM TABLE 2.1 (continued) Conduction in Plates and Walls2 Description Thin rectangular plate on the surface of a semiinfinite solid Equations Thin rectangular plate on the surface of a semiinfinite solid4 kw T T 1 T 2 q  ln 4w b Infinite thin plate with heated circular hole Infinite thin plate with heated circular hole for T T3 at r r1 and r r1  K 0 Br T T T  Br 1 T T 3 T K 0  Infinite thin plate with heated circular hole Infinite thin plate with heated circular hole for q at r r1, and r r1  K 0 Br k T T T  Br Br q 2 1 K 1 1 where: B Bi 1 Bi 2 T1 HT2 T 1H Bi H 1 Bi 2 Finite plate with centered hole Finite plate with centered hole5 w ln r q 2 kT T1 T2 d 2w (Continued) © 2001 by CRC PRESS LLC 008202 Page 34 Wednesday, August 23, 2000 9:53 AM TABLE 2.1 (continued) Conduction in Plates and Walls2 Description Tube centered in a finite plate Equations d Tube centered in a finite plate4 for r 10 2 kT T1 T2 q 4d ln ( r ) c w/d 1.00 1.25 1.50 2.00 2.50 3.00 4.00 Infinite plate with internal heat generation c 0.1658 0.0793 0.0356 0.0075 0.0016 0.0003 1.4 105 0.0 Infinite plate with internal heat generation2 T T1, x0 T T2, xL T T T Po X ( 1 X ) 1 X 2 T T T 2 1 x where X L Infinite plate with convection boundaries and internal heat generation Infinite plate with convection boundaries and internal heat generation3 1  1 1 Po T T T2 Po Po Bi 2     ( 1 X 2 ) Bi 2 2 T T1 T2 1 Bi 2 H 1 ˙ Bi 1 1 Po  0.5 ( 1 X ) Bi2 1 Bi 1 H h2 where H h1 © 2001 by CRC PRESS LLC 008202 Page 35 Wednesday, August 23, 2000 9:53 AM TABLE 2.1 (continued) Conduction in Plates and Walls2 Notes: Bi Biot Number, hL/k q rate of heat flux, W/m2 c value for w/d q linear heat flux, W/m d diameter, m q̇ G volumetric heat flux, W/m3 h heat transfer coefficient, W/m2 K w width k thermal conductivity, W/m K x, y, z Cartesian coordinates L length, m ˙ 2 q̇ G L Po k q rate of heat flow, W X length ratio (x/L) coefficient of thermal expansion (°C1) thickness CL Ts Tmax rB rA dr L Heat generation in differential element is qG L2πrdr FIGURE 2.10 Heat conduction nomenclature for a long circular cylinder with internal heat generation in differential element dr. © 2001 by CRC PRESS LLC 008202 Page 36 Wednesday, August 23, 2000 9:53 AM If we evaluate the temperature distribution at the centerline of the cylinder, we find the nondimensional temperature distribution T (r ) T r 2 s 1  r B T max T s To find the surface temperature of a tube, Ts, having a flowing cold fluid at T , we evaluate with a simplified energy balance equation which yields q̇ G r T s T 2h c The effective heat transfer coefficient for the tube is then 2 2 q̇ G ( r o r i ) h c 2r i ( T i T ) Tables 2.2 and 2.3 show the solutions to a variety of conductive cylinder and sphere problems. 2.3.1.5 Critical Radius of a Cylinder In real problems involving heat dissipation of an insulated cylinder, we must usually account for the effects of convection, whether natural or forced. When the outer radius of the insulation is small, the surface area is also small, and the effect of convection is not too great. As the outer radius of the insulation increases, the surface area also increases. At a critical radius, the effect of convective cooling will outweigh the effect of internal conduction resistance. The rate of heat transfer per unit length of a cylinder is Ti T Ti T Ti T q̇   r tot cond conv ln r 1   o i 2 kL 2 kLr o h c, o where: Ti T∞ ro ri k L hc,o temperature of cylinder, °C temperature of ambient air, °C outer radius of insulation, m inner radius of insulation, m thermal conductivity of insulation, W/m K length of cylinder, m external convective heat transfer coefficient, W/m2 K We can see from this equation that we achieve a maximum heat transfer rate when the total thermal resistance, tot, is at a minimum. If the outer radius of the insulation equals a critical value: k r o r crit h c, o © 2001 by CRC PRESS LLC 008202 Page 37 Wednesday, August 23, 2000 9:53 AM TABLE 2.2 Rods, Tubes, Cylinders, Disks, Pipes, and Wires2 Description Equations Infinite hollow cylinder Infinite hollow cylinder 2 kT To Ti q ro 1 1 ln (  ) Bi Bi r i i o ro ln (  ) r 1 Bi o T T T o T Ti To ro 1 1 ln (  ) Bi Bi r i i i o where h o ro Bio k Composite cylinder Composite cylinder 2 T Tn T1 q n1 i1 for j 1 r i 1  ln  k i r i n i1 1 r i hi 1 j1 T T j T1  T Tn T1 1 1 r 1 r i 1 1  ln    ln  k j r j r hk i r i r i hi j j i1 n1 i1 1 r i 1  ln  ki ri n 1 i hi i 1r where Tj is the temperature in the jth layer Insulated tube Insulated tube 2 kT Ti T f q r 1 o ln   r i Bi o where k ktube and hr o Bio k k Maximum heat loss occurs when ro hInfinite cylinder with temperaturedependent thermal conductivity Infinite cylinder with temperaturedependent thermal conductivity3 with: k ko T T T o k ko at ro k ki at ri 2 km T T i T o q r o ln  ri r o 0.5 T T To 2 k m ln r   T T T  1 2 i o k o ln ro ko 1 ri (Continued) © 2001 by CRC PRESS LLC 008202 Page 38 Wednesday, August 23, 2000 9:53 AM TABLE 2.2 (continued) Rods, Tubes, Cylinders, Disks, Pipes, and Wires2 Description Equations Pipe in semiinfinite solid 4 Pipe in semiinfinite solid 2 k1 T T 2 T 1 q r 2KD 1 2 1  ln   ln 2 D  Bi 2 K r1 Bi 1 where: h1 r 1 Bi1 k1 k K 2 k1 Row of rods in semiinfinite solid h2 d Bi2 k2 d D r2 Row of rods in semiinfinite solid3 For one rod 2 kT T T 2 1 q  d 1 sinh  ln  Bi 1 Dr 1 D 2 D  Bi 2 where: h2 d h1 r 1 d  , D  , Bi2 Bi1 s k k Row of rods in wall3 For each rod Row of rods in wall T2 4 kT T T 2 1 q  h2 d 1  sinh  ln  Bi 1 Dr 1 r1 r1 T1 +h1 where: r1 T1+h1 T1+h1 + + 2d h1 r 1 , Bi1 k + k T2 h2 Circular disk on the surface of a semiinfinite solid ro + T2 z © 2001 by CRC PRESS LLC D  2 D Bi 2 k h2 d , Bi2 k d D s Circular disk on the surface of a semiinfinite solid5 For T T 1 and z → q 4r o k T T 2 T 1 T T T 1 T T2 T1 2 1 2  sin 0.5 0.5 ( R 1 )2 Z 2 ( R 1 )2 Z 2 z r where Z r and R r o o 008202 Page 39 Wednesday, August 23, 2000 9:53 AM TABLE 2.2 (continued) Rods, Tubes, Cylinders, Disks, Pipes, and Wires2 Description Equations Circular disk in an infinite solid Circular disk in an infinite solid3 q 8r o k T T 2 T 1 z ro + T2 k Infinite hollow square rod3 Infinite hollow square rod 2 kT T2 T1 q k k  ln 1.08w h1 r o 2r 2h 2 w o T2 ro w h2 + T1 h1 w Infinite hollow square pipe T2 w T1 Infinite hollow square pipe5 2 kT T1 T2  0.785ln w  d q 2 kT T1 T2  w 0.93ln  0.0502 d w  1.4 d w d 1.4 d Vertical cylinder in a semiinfinite solid Vertical cylinder in a semiinfinite solid3 To 2D  d r k T q 1 T 1 T o D o ln 2D 1  h Bi Bi d ro + T1 where d hd Bid  , k d D ro k (Continued) © 2001 by CRC PRESS LLC 008202 Page 40 Wednesday, August 23, 2000 9:53 AM TABLE 2.2 (continued) Rods, Tubes, Cylinders, Disks, Pipes, and Wires2 Description Equations Two semiinfinite regions of different conductivities connected by a circular disk Two semiinfinite regions of different conductivities connected by a circular disk4 T To z→ z T T1 z→ qz 0 r ro z 0 ro k1 + T T T o T T1 To k2 2k 2 2  sin 1 0.5 0.5 Z 0 (k1 k2) 2 2 (R 1) Z ( R 1 )2 Z 2 T T T o T T1 To 2k 1 2 1 1  sin Z 0.5 0.5 (k1 k2) 2 2 2 2 (R 1) Z (R 1) Z 0 4r o k 1 k 2  T T T q o 1 k1 k2 z r where Z  , R ro ro Heat flow between two rods in an insulated infinite plate5 Heat flow between two rods in an insulated infinite plate 2 kT T1 T2 q w s  ln  r w w + T1 + T2 s 2r © 2001 by CRC PRESS LLC 008202 Page 41 Wednesday, August 23, 2000 9:53 AM TABLE 2.2 (continued) Rods, Tubes, Cylinders, Disks, Pipes, and Wires2 Description Equations Infinite cylinder with convection boundary and internal heat generation To h Infinite cylinder with convection boundary and internal heat generation3 T T Tok 2  0.25  1 R 2 2 Bi q̇ G r o hr where Bi o , k ro r R ro k + . qG Hollow infinite cylinder with convection boundary on outside surface and internal heat generation Hollow infinite cylinder with convection boundary on outside surface and internal heat generation3 with qr 0 and r ri T T T f k 2  0.25  ( 1 R 2i ) 1 R 2 2R 2i lnR 2 Bi q̇ G r o ri Tf k h hr o r r i where R r , Bi k , Ri r o o + • qG ro Hollow infinite cylinder with convectioncooled inside surface and internal heat generation Hollow infinite cylinder with convectioncooled inside surface and internal heat3 generation with qr 0 and r ro T T T f k 2  0.25  ( R 2o 1 ) 1 R 2 2R 2o lnR 2 Bi q̇ G r i ri hr i ro r where R r , Bi k , Ri r i i +h . Tf qG ro (Continued) © 2001 by CRC PRESS LLC 008202 Page 42 Wednesday, August 23, 2000 9:53 AM TABLE 2.2 (continued) Rods, Tubes, Cylinders, Disks, Pipes, and Wires2 Description Equations Electrically heated wire with temperaturedependent thermal and electrical conductivities To ro + Electrically heated wire with temperaturedependent thermal and electrical conductivities1,7 with T To R ro kT  1 t T T T o k To k e 1 e T T T o k eo T T To R 2 R  R 1 B  e  B 8 T 16 2 2 k eo r o E T where B , 2 k To L kt thermal conductivity 2 r R 1 2 ro Notes: Bi Biot Number, hL/k q linear heat flux, W/m d diameter, m q rate of heat flux, W/m2 h heat transfer coefficient, W/m2 K q̇ G volumetric heat flux, W/m3 k thermal conductivity, W/m K r radius, m L length, m s spacing, m q rate of heat flow, W w width, m coefficient of thermal expansion (°C1) A more accurate equation accounts for the variable effect of ro on the heat transfer coefficient hc,o: 1  r o r crit n1  ( 1 n )k where: thermal diffusivity of the convective media, k /cp n 0.5 for laminar forced convection or 0.25 for natural convection k thermal conductivity of insulation, W/m K 2.3.2 CONDUCTION IN COMPLEX GEOMETRIES In the previous section we studied onedimensional heat flow. In this section we will examine heat transfer in multidimensional systems. Multidimensional heat transfer occurs when we transfer the heat from different locations and the temperature may vary in more than one dimension. One example is an active component in a potting compound, an irregularly shaped object, or a corner where we join three chassis walls. Figure 2.11 shows twodimensional conduction. © 2001 by CRC PRESS LLC 008202 Page 43 Wednesday, August 23, 2000 9:53 AM TABLE 2.3 Conduction in Spheres2 Description Equations Spherical shell Spherical shell ro 4 rokT Ti To q  ri ro k ro k  1   r i hi r i ho r o ri ro + T i hi k  1 T T T r o ho r o T Ti To ro k ro k  1   r i r i hi r o ho ri ho To Composite sphere Composite sphere r3 rn rj r2 4 T T1 Tn q n n1 1 1 1 1  r hj+1 r i 1 ki i kn1 r1 i1 2 i 1 r i hi j1 + T1 h1 Tn hn 1 1   1  1  1 1  1  1  k r r i 1 r 2 h k j r j r r 2 h i T T j T1 i i i i1 i i  n n1 T Tn T1 1 1 1 1 r  k r i1 i i i1 2 i 1 r i hi where Tj is the local temperature in the jth layer. Sphere with temperaturedependent thermal conductivity ro ri Sphere with temperaturedependent thermal conductivity3 with T Ti r ri T To r ro k ko T T T o + Ti 4 ro km T T i T o q r o  1 ri To ko ki where km 2 with k ko k ki T T To  k o T To T Ti ro k m r 1  1 1 2 T T i T o 2 r k o o 1 ri (Continued) © 2001 by CRC PRESS LLC 008202 Page 44 Wednesday, August 23, 2000 9:53 AM TABLE 2.3 (continued) Conduction in Spheres2 Description Equations Sphere in a semiinfinite solid T1 4 rokT T1 To q 1 d   1 0.5  r o Bi h1 where ro d Sphere in a semiinfinite solid4 + To Sphere in an infinite medium ro + hr Bi o k Sphere in an infinite medium5 with T T 2 at r → q 4 rokT T1 T2 Ti Two spheres separated by a large difference in an infinite medium T1 T2 + Two spheres separated by a large difference in an infinite medium4 4 rk T T 1 T 2 q  2r + r 2 1 s s for s Spherical shell with specified inside surface heat flux and internal heat generation ro ri + qi 2r, error 1% Spherical shell with specified inside surface heat flux and internal heat generation with T To r ro qr qi r ri . qG r where R  , ri Solid sphere with internal heat generation in an infinite medium k1 ko 5r T T Tok Ro R q̇ G r i 2 ( R R o )    R 2o R 2 RR o RR o qi ri 6qi To ro + s . qG h © 2001 by CRC PRESS LLC ro Ro ri Solid sphere with internal heat generation in an infinite medium6 k ko 0 r ro with k k1 r ro q̇ G q̇ Go 0 r ro q̇ G 0 r ro h contact coefficient at r ro T T r→ T T T ko 2k o 2 2  1  1 R   2 k1 Bi 6 q̇ Go r o 0R1 008202 Page 45 Wednesday, August 23, 2000 9:53 AM TABLE 2.3 (continued) Conduction in Spheres2 Description Equations T T T k 1  2 3R q̇ Go r o where hr o Bi  , ko r ro r R ro Notes: A area, m q Bi Biot number, hL/k q rate of heat flux, W/m2 d diameter, m q̇ G volumetric heat flux, W/m3 h heat transfer coefficient, W/m2 K r radius, m k thermal conductivity, W/m K s spacing, m L length, m rate of heat flow, W coefficient of thermal expansion (°C1) y qy" Lines of constant T1 temperature (isotherms) T2 < q" = iq"x + jqy" T1 qx" Heat flow lines x Isotherm FIGURE 2.11 Twodimensional conduction showing the lines of heat flow. There is no internal heat generation. Heat conduction in multidimensional systems can be calculated by analytic, analogic, graphical, and numerical methods. Fourier, in 1822, made the first major contribution to the analysis of multidimensional heat transfer.8 Fourier’s method of separating variables led to the requirement of transform methods such as the Laplace transform, to express the arbitrary Fourier series expansion. These methods can yield accurate results but are quite timeconsuming. Graphical methods (see Figure 2.12) include the flux plot method and the Schmidt9 method. Both methods involve drawing skill and considerable time but © 2001 by CRC PRESS LLC 008202 Page 46 Wednesday, August 23, 2000 9:53 AM T2 F ∆q1, ∆q2 Heat flow lines E B A T1 T1 > T2 C ∆q1, ∆q2 D ∆q15 Isotherms ∆q15 (a) (b) ∆ι ∆ι q" ∆T (c) FIGURE 2.12 Construction of a network of curvilinear squares for an adiabatic corner section with no internal heat generation. (a) Scale model. (b) Heact flux plot. (c) Typical curvilinear square. can yield accurate results. These techniques are generally used only for very simple geometries with simple boundary conditions. Nevertheless, these solutions can be used to provide exact and relatively quick answers to complex geometries that we can simplify. These equations can also be used to find partial solutions to simple areas of very complex geometries. Analytic techniques provide a solution at every point in time and space within the prescribed boundaries of the problem. Finite difference methods provide a solution only at a finite number of points (see Figure 2.13) within the problem and are an approximation of the analytic solution. Using a finite number of points simplifies the calculation to repetitive arithmetic instead of the complex calculations involved with the analytic solutions. Although many texts devote considerable space to numerical and finite difference methods, computers are now used to solve problems of this complexity. For this reason, we will concentrate only on a broad view of the analytic techniques. © 2001 by CRC PRESS LLC 008202 Page 47 Wednesday, August 23, 2000 9:53 AM FIGURE 2.13 A simple finite difference mesh for a rectangular plate with steadystate conduction. 2.3.2.1 Multidimensional Analytic Method In a twodimensional system without internal heat generation and with uniform thermal conductivity, the general conduction equation has been found as 2 2 T T 2 2 0 x y The total rate of heat transfer is a vector. The vector is dependent upon the rate of heat flow in x, which is qx, and the rate of heat flow in y, which is qy. The total rate of heat transfer is then perpendicular to an isotherm within the boundaries of the geometry. Therefore, if we solve for the temperature distribution, the heat flow can be found easily. Examine a rectangular plate that is insulated at two opposite sides (see Figure 2.14). Since the problem is linear, T XY, X X(x), and Y Y(y). The solution to the temperature distribution is T ( x, y ) y sinh  L x T m  sin b L sinh  L The solution to the temperature distribution is shown graphically in Figure 2.15. When we specify more complex boundary conditions, the series can become infinite. © 2001 by CRC PRESS LLC 008202 Page 48 Wednesday, August 23, 2000 9:53 AM FIGURE 2.14 A rectangular adiabatic plate with steadystate sinusoidal temperature distribution on the upper edge. FIGURE 2.15 A depiction of the resulting isotherms and heat flow lines for the adiabatic plate shown in Figure 2.14. © 2001 by CRC PRESS LLC 008202 Page 49 Wednesday, August 23, 2000 9:53 AM FIGURE 2.16 Twodimensional heat conduction through a square channel of length L. There is no heat generation and the heat flows to the outer surface of the channel. (a) Symmetry planes. (b) Heat flux plot. (c) Typical curvilinear square. See Ozisik10 for a more detailed explanation of these conditions. Schneider11 provides a more detailed analysis of threedimensional heat conduction. 2.3.2.2 Multidimensional Graphical Method We can use the graphical method to find a good approximation of the heat flow within a complex twodimensional object when the problem is isothermal and we insulate the boundaries. In this method the designer draws a set of lines that represent constant temperature in one direction and constant heat flux lines perpendicular to the temperature lines. Therefore, heat cannot flow across constant heat flux lines, and some constant heat profile flows between any two heat flux lines. To find the temperature distribution, we use a twodimensional scale drawing of the object. Lines are drawn through trial and error until we form a network of intersecting lines with rightangle junctions, as shown in Figure 2.16. Flux lines are perpendicular to the object boundaries except at the corners. Flux line that lead to or from a corner bisect the angle between the surfaces that form the corner. The graphic solution, like the Laplace transform analytical solution, is unique to each geometry. Therefore, any curvilinear network, whatever the size of the mesh, that satisfies the specified boundary conditions represents a correct solution. We know that the rate of heat flow remains constant across any square of a heat flow lane of the graphical solution from the boundary at T1 to the boundary of T2. The temperature differential across the heat flow lane is then given by T2 T1 T N © 2001 by CRC PRESS LLC 008202 Page 50 Wednesday, August 23, 2000 9:53 AM where N is the number of temperature increments between two boundaries at T1 and T2. The total rate of heat flow from the prescribed boundary at T2 to the boundary at T1 is equal to the sum of the heat flow through all of the heat flow lanes. We can then write the total rate of heat transfer as nM q n1 M M q n  k ( T2 T1 )  k T tot N N where: qn rate of heat flow through the nth lane M number of heat flow lanes Ttot total temperature difference between surfaces transferring heat In the graphical method, even a crude sketch may yield a good approximation of the solution. 2.3.2.3 Multidimensional Shape Factor Method For a twodimensional system, the rate of heat transfer per unit depth from surface 2 to surface 1, q2,1, is related to the temperature differential T, the thermal conductivity of the medium k, and the ratio of the number of heat flow lanes to the number of temperature increments, M/N. This can be expressed as M q k  T tot N If each of the two surfaces is isothermal and the other surfaces are adiabatic, and if we let the ratio M/N become a geometrical shape factor, S, we use a simple formula for heat flow between two surfaces: q kS T where: q k S T heat flow, W thermal conductivity, W/m K shape factor, m temperature difference, °C Multidimensional analysis with shape factors can only be used when both objects have no heat generation and when both have isothermal surface temperatures. Table 2.4 lists values for some typical shape factor configurations. © 2001 by CRC PRESS LLC 008202 Page 51 Wednesday, August 23, 2000 9:53 AM TABLE 2.4 Shape Factors for SteadyState Conduction Configuration Restrictions Conduction Shape Factor Plane wall A area L wall thickness A L Concentric cylinders L r2 r1 inner cylinder radius r2 outer cylinder radius Concentric spheres r1 inner sphere radius r2 outer sphere radius if r 2 → Eccentric cylinders Concentric square cylinders 2 L r ln r 2 1 4 1 1  r1 r2 4 r1 L r2 e axial centerline offset r2 radius of outer cylinder r1 radius of small cylinder 2 L 2 2 2 1 r 2 r 1 e  cosh 2r r L a if a/b 1.4 2 L 0.93ln ab 0.0502 if a/b 1.4, where a side of large square, b side of small square 2 1 2 L 0.785ln ab (Continued) © 2001 by CRC PRESS LLC 008202 Page 52 Wednesday, August 23, 2000 9:53 AM TABLE 2.4 (continued) Shape Factors for SteadyState Conduction Configuration Circular cylinder in a square cylinder, concentric Restrictions a 2r a side of square Conduction Shape Factor 2 L ln 0.54 a r Buried sphere h distance below surface h r1 if h → Buried cylinder L 4 r1 r1 1 2h 4 r1 r1 2 L  1 2h cosh h distance below surface if h 3r1 1 2 L  ln 2h r 1 h if  → , s → 0 ri Buried rectangular box L h,a,b h h distance below surface 2.756L ln 1  a a box width b box height Edge of adjoining walls L wall thickness W length of attachment W L/5 © 2001 by CRC PRESS LLC  r 0.59 0.54W h b 0.078 008202 Page 53 Wednesday, August 23, 2000 9:53 AM TABLE 2.4 (continued) Shape Factors for SteadyState Conduction Configuration Restrictions Corner of three adjoining walls W Disk on semiinfinite medium r radius of disk Vertical cylinder in semiinfinite medium L Conduction between two parallel cylinders L/5 0.15L 2.3.2.4 4r D L 2 L  ln 4L D D1, D2 z axial centerline spacing D1 diameter of cylinder 1 D2 diameter of cylinder 2 Buried thin horizontal disk Conduction Shape Factor D diameter of disk z distance below surface 2 L 2 2 2 cosh D 1 D 2  2D 1 D 2 1 4z 4.45D D 1 5.67z Finite Difference Method Finite difference equations are constructed of nodal networks. These networks are composed of discrete points placed on the surface of an object or, in the case of a threedimensional analysis, throughout an object. Each point is connected to at least one other point by a line. Each point is numbered and called a node. The network of lines that connect the nodes is called a grid or a mesh. For a twodimensional system, the x and y location of each node is indicated by m and n indices, respectively. © 2001 by CRC PRESS LLC 008202 Page 54 Wednesday, August 23, 2000 9:53 AM Since the indicated temperature at each node is an average temperature of the area around the node, the number of nodes affect the accuracy of the solution. These equations are normally performed on computers, and this chapter serves to present only a brief overview of the underlying equations and methods used. Complete solutions w