Thermal design of electronic equipment [1 ed.] 0849300827, 9780849300820

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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

Library of Congress Cataloging-in-Publication Data Remsburg, Ralph. Thermal design of electronic equipment / Ralph Remsburg. p. cm.--(Electronics handbook series) Includes bibliographical references and index. ISBN 0-8493-0082-7 (alk. paper) 1. Electronic apparatus and appliances--Thermal properties. 2. Electronic apparatus and appliances--Design and construction. 3. Heat--Transmission. 4. Electronic packaging. I. Title. II. Series. TK7870.25 .R46 2001 621.38104--dc21

00-057170

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 use. 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 0-8493-0082-7 Library of Congress Card Number 00-057170 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

© 2001 by CRC PRESS LLC

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 up-front 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

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

Nomenclature and Symbology SYMBOL DEFINITION Symbol a a A

b c

CA C C C˙

CD Cf

D

DAB e e E E

f f'

Description Velocity of sound; acceleration Thermal diffusivity (a  k/c) Area; Ac, cross-sectional 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 C˙ c , rate of hourly heat capacity of a colder fluid in a heat exchanger C˙ 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 pin-fins

© 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

Symbol F F1 2

ᑠ1 2 g gc G G h h

hl hfg hm H i I I J k

K

log ln l L Lf m˙ 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 unit-surface conductance, h  hc  hr hb, unit-surface 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

Symbol q

q˙ G q Q Q˙ r

R ᑬ

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 pin-fins in adjacent longitudinal rows Distance between the centerlines of pin-fins 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, wet-bulb 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 Free-stream 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

Symbol v

V W˙ x

x y y z z Z

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

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)

k (beta)  (gamma) (Gamma) c (Gamma)  (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

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, ␮Ⲑ␳ Frequency of radiation Mass density; l/v l, density of a liquid , density of a vapor Reflection for radiation Stefan-Boltzmann 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 N/m2 lbf/ft2

Dimensionless rad deg Percent (%) Dimensionless rad/s rad/s

DIMENSIONLESS NUMBERS Symbol Bi

Description hL Biot Number  -----ks

Ref. or

hr --------o ks

1

2

Bo

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g (  l    )L Bond Number  -----------------------------

) (Continued)

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 m˙ c p Graetz Number  --------kfL

Ref. 2 3

t -----2 ro

4 5

3 2

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

Mo Nu

8 9 0.8 0.6 0.4 p 0.4

 k c Mouromtseft Number  ----------------------- hc x Nusselt Number  -------- at point x kf

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

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2 

U L Weber Number  -------------

3 16 17

or

Nu ------------Re Pr

18

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 ( q˙G ) Heat transfer coefficient (hc) Heat transfer rate (q) Length (L) Mass (M) Mass flow rate ( m˙ ) 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

1 N  0.2248 lbf

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 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)

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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 ( q˙ )

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)

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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

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

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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 Resistance-Capacitance 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 Three-Dimensional Flowfields 3.5 Incompressible Ideal Fluid Flow 3.5.1 One-Dimensional Flow 3.5.1.1 One-Dimensional Euler Equation 3.5.1.2 One-Dimensional Bernoulli Equation 3.5.1.3 Application of the One-Dimensional Equations 3.5.2 Two-Dimensional Flow 3.5.2.1 Application of the Two-Dimensional 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

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 Pin-Fin Arrays 4.5.5 Jet Impingement Forced Convection © 2001 by CRC PRESS LLC

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 U-Shaped Channels 4.6.5.8 Natural Convection through Pin-Fin 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 Stefan-Boltzmann 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

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

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Hibbeler R. C. “Force-System Resultants and Equilibrium” Thermal Design of Electronic Equipment. Ed. Ralph Remsburg Boca Raton: CRC Press LLC, 2001

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 by-product 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 )

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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 MIL-HDBK-217.

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  h---------c 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 air-cooled heat sink. For a one-dimensional system, © 2001 by CRC PRESS LLC

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) cross-sectional 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 cross-sectional area, we can find the temperature rise in an application by: qL  T  -------k Ac

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 4

4

q r   F 1,2 A ( T 1  T 2 ) where: qr  amount of heat transferred by radiation (W)   emissivity of the radiating surface (highly reflective  0, highly absorptive  1.0)   Stefan-Boltzmann 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

Fl

θsa

θcs Die bond Lead

Chip

Heat spreader

θjc

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 Bar-Cohen, A., Design and Analysis of Heat Sinks, John Wiley & Sons, New York, 1995. With permission.)

transfer as a radiative heat transfer coefficient, hr: 2

2

h r   F 1,2 ( T 1 T 2 ) ( T 1 T 2 )

1.1.4

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)

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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 heat-dissipating 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  T a jc q  35C ( 50 K/W ) ( 0.5 W )  60C For a higher-powered 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: 1 sa  ---------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.

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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 (cost-effectively) wherever possible.

1.2 THEORETICAL POWER DISSIPATION IN ELECTRONIC COMPONENTS 1.2.1

THEORETICAL POWER DISSIPATION

Electronic devices produce heat as a by-product 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  --- 冮 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

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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 first-order function of the frequency and a second-order 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  peak-to-peak voltage (V) f  switching frequency (Hz) Short-circuit 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.

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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 drain-source 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: ID  drain current (A) 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 drain-to-source voltage and the drain current; therefore: t

t

S2  S1  P S  f S  冮0 VDS ( t )I D ( t ) dt 冮0 VDS ( t )ID ( t ) dt  

where: fS VDS ID tS1 tS2

    

switching frequency (Hz) MOSFET drain-to-source 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  gate-to-source voltage (V) QG  peak charge in the gate capacitance (coulombs) RG  gate resistance ( ) © 2001 by CRC PRESS LLC

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 steady-state power dissipated by a wire interconnect is given by Joule’s law: 2

PD  I R where: I steady-state current (A) R steady-state 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 cross-sectional 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.

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Table 1.2 shows the maximum current-carrying capacity of copper and aluminum wires in amperes:5 TABLE 1.2 Maximum Current-Carrying Capacity of Copper and Aluminum Wires (in Amperes) Copper MIL-W-5088

Aluminum MIL-W-5088

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°C-rated insulated wire 92.0°C for 135°C-rated insulated wire 107°C for 150°C-rated insulated wire 157°C for 200°C-rated 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., McGraw-Hill, 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

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 steady-state power dissipated by a resistor in given by Joule’s law: 2

PD  I R where: I  steady-state current (A) R  steady-state resistance ( ) The instantaneous power, PD(t), dissipated by a resistor with a time-varying 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 steady-state current is applied is 2

PD  0.5I M R 1.2.3.3

Capacitors

Although capacitors are generally thought of as non-power-dissipating, some power is dissipated due to the resistance within the capacitor. The power dissipated by a capacitor under sinusoidal excitation is found by: 2

PD ( t )  0.5  CV M sin 2  t where: C VM  f

   

capacitance (F) peak sinusoidal voltage (V) radian frequency, 2f frequency (Hz)

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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  --- 冮 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 high-frequency 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: 2

PD ( t )  0.5LI M  sin 2  t where: L  inductance (Henry) IM  peak sinusoidal current (A)   radian frequency (2f )

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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 P˙ 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  P˙ 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.

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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:

   ------ ----- (  u ) ----- (  v )  0 t  x y Conservation of momentum in x:

    u  u  p ---- (  u ) ----- (  uu ) ----- (  vu )  -----  ------ -----  ------  ------ Fx t x y  x   x  y   y  y Conservation of momentum in y:

    v  v  p ---- (  v ) ----- (  uv ) ----- (  vv )  -----  ----- -----  -----  ------ Fy t x y x  x y  y y Conservation of energy:

    k h  k h ---- (  h ) ----- (  uh ) ----- (  vh )  -----  ----- ------ -----  ----- ------ q˙ G t x y x  cp x y  cp y where:

 t u, p F h k cp

 fluid density (kg/m3)  time (s) v  velocity components in x and y coordinates (m/s)  molecular viscosity (N s/m2)  pressure (N/m2)  force per unit volume (Pa/m3)  specific enthalpy (J/kg)  thermal conductivity (W/m K)  specific heat (J/kg K) q˙ 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 (computer-aided 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 two-dimensional model to calculate © 2001 by CRC PRESS LLC

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 two-dimensional 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 time-sequencing 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 irregular-shaped 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 higher-order formulations. The result is a first-order, second-order, third-order, etc. accuracy. Codes may use explicit, implicit, and semi-implicit 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 whole-field 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 (time-averaged) 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

is a flow property, not a fluid material property. The most commonly used turbulent flow model is the two-equation 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 small-scale structures, is nothing other than a viscous flow that locally obeys the Navier-Stokes equations. If a fine enough grid is used, we can calculate all the details of this turbulent flow directly from the Navier-Stokes 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 three-dimensional grid points and over 400 hours on a CRAY Y-MP supercomputer.

1.3.2

FLOTHERM V2.2

Several general-purpose 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 3-D solver for Navier-Stokes equations, built-in 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 Bar-Cohen, 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, McGraw-Hill, New York, 4.7, 1998. 3. CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, E-78, 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., McGraw-Hill, New York, 1991.

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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., Off-line ZCSQRC Thick-Film Hybrid Circuit, Virginia Power Electronics Center, Sixth Annu. Power Electron. Semin., 71–83, September, 1988. 8 Olean Advanced Products Data Book, S-OAP10M295-N, AVX Corporation, Myrtle Beach, SC. 9. Kemet Surface Mount Catalog, F-3102, 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 91-1607-CP, Proc. AIAA 10th Computer Fluid Dynamics Conf., 890–914, 1991.

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Hibbeler R. C. “Force-System Resultants and Equilibrium” Thermal Design of Electronic Equipment. Ed. Ralph Remsburg Boca Raton: CRC Press LLC, 2001

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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.

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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 steady-state 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 cross-sectional 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

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where As is the surface area in contact with the cooling media. Radiation heat transfer can be described as a thermal resistance by 1 T  rad  --------  ----------hr AF qx 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 one-dimensional; that is, we require only a single coordinate to describe the spatial variation of the dependent variables.

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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 one-dimensional 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

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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

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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 x-direction kA dT q x   kA -------  ------ ( T 1  T 2 ) L dx

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8 8 8

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 one-dimensional heat flow through any number of series and parallel combinations of thermal resistance. Although parallel heat flow is technically a two-dimensional problem, we can usually reduce it to a single heat flow direction (see Figure 2.5). The general equation for

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8

T

,A

Ts ,A

TB TC Ts,D L3

k1

k2

k3

1

2

3

T

8

L2

T

8

qx

,A

L1 k1A Ts ,A

L2 k 2A TB

L3 k 3A TC

1 h DA Ts,D

8

1 h AA

,D

Cold fluid T ,D ,hD

x

T

8

8

Hot fluid T ,A ,h1

L1

,D

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  ---------------------------冱 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

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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

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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 冱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

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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 ( rT ) 1 d 2 dT ----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

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-L T s, A

x +L

T

T(x)

¥

T ,B,hB

q .

O

T(x) Ts

Ts

T s, B T ,A,hA

x +L

-L

T ,h

T ,h

¥

¥

¥

(a)

(b)

T

q .

0

T(x) Ts

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

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conductivity, k, the equation becomes 2

d T ( x) -   q˙ G k ---------------2 dx We find the temperature distribution, T(x), by q˙ G L q˙ 2 T 2  T1 -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 q˙ 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

q˙ 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 current-carrying 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

q˙ 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

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q˙ G r o  T o  ---------rk

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

冱

i 1

Li 1  1  --- ----  ---- k i h i h 0

for J˙ 1

T T j  To -------------------- T T n  To 

Plate with temperature-dependent thermal conductivity

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

冱

j 1

i1

冱

Plate with temperature-dependent 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

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TABLE 2.1 (continued) Conduction in Plates and Walls2 Description Thin rectangular plate on the surface of a semi-infinite solid

Equations Thin rectangular plate on the surface of a semi-infinite solid4 kw  T T 1  T 2 q  --------------------------------4w ln  ----- b

Infinite thin plate with heated circular hole

Infinite thin plate with heated circular hole for T  T3 at r  r1 and r r1 Br K 0  ---- T T  T  ------------------------  ----------------Br  T T 3  T K 0  -------1- 

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)

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TABLE 2.1 (continued) Conduction in Plates and Walls2 Description Tube centered in a finite plate

Infinite plate with internal heat generation

Equations d Tube centered in a finite plate4 for r  ----10 2 kT T1  T2 q  -------------------------------4d ------ )  c ln (

r w/d c 1.00 0.1658 1.25 0.0793 1.50 0.0356 2.00 0.0075 2.50 0.0016 3.00 0.0003 4.00 1.4  105 0.0 Infinite plate with internal heat generation2 T  T1, x0 T  T 2, 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

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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

q˙ 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 q˙ 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.

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If we evaluate the temperature distribution at the centerline of the cylinder, we find the nondimensional temperature distribution T (r )  T s r 2 ------------------------  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 q˙ G r T s  T  -------2h c The effective heat transfer coefficient for the tube is then 2

2

q˙ 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 q˙  -------------------  ----------------------------  --------------------------------------------------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

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 T n  T 1 q  ----------------------------------------------------------------------------n1

冱

i1

1   r i  1  --- ln ---------------   k i  r i 

n

冱

i1

1 -------r i hi

for j 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 --------------------------------------------------------------------------------------------------------------------------

冱

i1

n1

冱

i1

1  r i  1 ---- ln  ---------------  ki  ri 

n

冱

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 temperature-dependent 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  r----o- ko

1

 ri

(Continued)

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TABLE 2.2 (continued) Rods, Tubes, Cylinders, Disks, Pipes, and Wires2 Description

Equations Pipe in semi-infinite solid 4

Pipe in semi-infinite 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 semi-infinite solid

h2 d Bi2  -------k2 d D  ---r2

Row of rods in semi-infinite solid3 For one rod 2 kT T  T

2 1 q  -------------------------------------------------------------------------------------

 d   1 D  sinh 2  D  --------------  ln   ------------ Bi 1   Dr 1 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  D  1 - sinh 2  D  --------  --------  ln   ------------ Bi 2   Bi 1  Dr 1

r1

r1 T1 +h1

where:

r1 T1+h1

T1+h1 +

+

2d

h1 r 1 -, Bi1  --------k

+

k h2

T2

Circular disk on the surface of a semi-infinite solid

ro +

T2

z

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k

h2 d -, Bi2  ------k

d D  --s

Circular disk on the surface of a semi-infinite solid5 For T  T 1 and z → q  4r o k  T T 2  T 1

T T  T ---------------------1-  T T2  T1    2 2 1  - ---- 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

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 k  T T 2  T 1 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 ---- 1.4 d

d

Vertical cylinder in a semi-infinite solid

Vertical cylinder in a semi-infinite solid3 

To

Bi



2D  -  ------d  r k  T q   -----------------------------------------1 T 1 T o D  o  ln 2D  1  --------

h





Bi d



ro +

T1

where d

hd Bid  ------ , k

d D  ---ro

k

(Continued)

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TABLE 2.2 (continued) Rods, Tubes, Cylinders, Disks, Pipes, and Wires2 Description

Equations

Two semi-infinite regions of different conductivities connected by a circular disk

Two semi-infinite 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 0.5 0.5

(k1  k2)   2 2 2 2  (R  1)  Z  (R  1)  Z 

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  ------------------------------- s  w -------  ln  ------

r w

w + T1

+ T2 s

2r

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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  q˙ 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 q˙ G r o ri k

Tf

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 convection-cooled 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 q˙ G r i ri

hr i ro r ---where R  --r- , Bi  -----k , Ri  r i i

+h . Tf qG ro

(Continued)

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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 temperature-dependent 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

q˙ 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 one-dimensional 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 two-dimensional conduction.

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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 T 1  T n q  --------------------------------------------------------n n1

hj+1

1  1 冱 ---1-  ---1-  ----------- 冱 ---------r 

r i  1 ki i

kn-1

r1

i1

2 i  1 r i hi

j1

+ T1 h1

冱

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 冱 ---1-  ---1-  ----------- 冱 ---------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 temperature-dependent thermal conductivity

ro

ri

Sphere with temperature-dependent 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)

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TABLE 2.3 (continued) Conduction in Spheres2 Description

Equations

Sphere in a semi-infinite solid

T1

4 rokT T1  To q  ------------------------------------1 d -  ----- 1  0.5  --- r o Bi

h1

ro

d

where

+ To

Sphere in an infinite medium ro

+

Sphere in a semi-infinite solid4

hr Bi  --------o k

Sphere in an infinite medium5 T  T 2 at r → with q  4 rokT T1  T2

Ti

Two spheres separated by a large difference in an infinite medium T2

T1

+

2r

+

Two spheres separated by a large difference in an infinite medium4 4 rk  T T 1  T 2 q  ----------------------------------r 2  1  -s  

for s 2r, error ⯝ 1%

s

Spherical shell with specified inside surface heat flux and internal heat generation ro

ri

+ qi

Spherical shell with specified inside surface heat flux and internal heat generation with T  To r  ro qr  qi r  ri

T T  Tok Ro  R q˙ G r i 2 ( R  R o ) -----------------------  --------- -------------------------  R 2o  R 2  ---------------RR o RR o qi ri 6qi

.

qG

r where R  --- , ri

To

Solid sphere with internal heat generation in an infinite medium k1

ro

+ ko

s 5r

.

qG h

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ro Ro  ---ri

Solid sphere with internal heat generation in an infinite medium6 k  ko 0  r  ro with k  k1 r ro q˙ G  q˙ Go 0  r  ro q˙ 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 Bi k1  6 q˙ Go r o

0R1

TABLE 2.3 (continued) Conduction in Spheres2 Description

Equations

T T  T k 1 -----------------------  ------2 3R q˙ 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

 rate of heat flow, W

d  diameter, m

q˙ 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

 coefficient of thermal expansion (°C1)

y qy"

Lines of constant T1 temperature (isotherms)

T2


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.

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FIGURE 2.13 A simple finite difference mesh for a rectangular plate with steady-state conduction.

2.3.2.1

Multidimensional Analytic Method

In a two-dimensional 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.

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FIGURE 2.14 A rectangular adiabatic plate with steady-state 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.

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FIGURE 2.16 Two-dimensional 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 three-dimensional 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 two-dimensional 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 two-dimensional scale drawing of the object. Lines are drawn through trial and error until we form a network of intersecting lines with right-angle 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

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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 

冱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 two-dimensional 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.

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TABLE 2.4 Shape Factors for Steady-State 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  a--b-

(Continued)

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TABLE 2.4 (continued) Shape Factors for Steady-State 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 r1

4 r1

-----------------r1 1  ----2h

4 r1

2 L --------------------- 1 2h

cosh

if h 3r1

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

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 ------ r  1

h  distance below surface

 0.59

0.54W

 h---  b

 0.078

TABLE 2.4 (continued) Shape Factors for Steady-State Conduction Configuration

Restrictions

Corner of three adjoining walls

W L/5

Disk on semi-infinite medium

r  radius of disk

Vertical cylinder in semi-infinite medium

L D

Conduction between two parallel cylinders

2.3.2.4

0.15L

4r

2 L ------------------- ln  4L D

L 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

 1  4z

 D 1  D 2  -------------------------------------- 2D 1 D 2  

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 three-dimensional 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 two-dimensional system, the x and y location of each node is indicated by m and n indices, respectively.

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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 will require not only nodal points within the object, but should also include initial and boundary conditions. The temperature distribution for each node can be written as a form of the conservation equation. These equations are then solved simultaneously to determine the temperature at each node. The appropriate form of the conservation equation for nodes connected in a two-dimensional array is called a finite difference equation. By manipulating these equations in the x and y directions and stipulating that the distance between any four nodes is a square, x  y, we arrive at T m, n  1  T m, n  1  T m  1, n  T m  1, n  4T m, n  0 This is an approximate algebraic form of the finite difference equation, stating that the sum of four nodal temperatures surrounding the node of interest is four times the temperature of the central node. An energy balance on a corner node, as shown in Figure 2.17, with terms added for convection, can be written as hc  x hc  x 1 T m  1, n  T m, n  1  --- ( T m  1, n  T m, n  1 )  ------------ T   3  ------------ T m, n  0  k 2 k 

FIGURE 2.17 Finite difference mesh approximation for the internal corner of a plate. Both surfaces of the internal corner have an identical heat convection coefficient.

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It is often more appropriate to describe the equation in terms of conduction thermal resistance, . For example, the heat transfer by conduction from nodal point (m1, n) to nodal point (m,n) can be written as T m  1, n  T m , n T m  1, n  T m , n -  ---------------------------------q (m  1, n) → ( m, n )  ---------------------------------x  1  --------------ky

Table 2.5 shows the finite difference approximations for steady-state conduction in a square mesh. 2.3.2.5

Resistance-Capacitance Networks

The resistance-capacitance network method is often a useful alternative to the finite difference methods. We know that thermal resistances can be added in series as

 tot   1   2   3  … or can be combined for a parallel path total resistance as 1 1 1 1 -------  -----  -----  -----  …  tot 1 2 3 For the common problem of combinations of series and parallel heat flow, we see that the thermal resistance for the material in parallel is

B C  2  ------------------B  C and the rate of heat flow is found by

T n3 q k  -------------------冱 n n1

For a composite material, N represents the number of layers in series, n is the thermal resistance of the nth layer, and T is the overall temperature difference across the exterior walls. Figure 2.18 shows a node, m, surrounded by a volume, Vm, having a thermal capacitance of Cm  mcmVm. The nodes that surround node m are called n. Thermal conduction from the outer volumes of Vn into Vm is expressed as thermal resistance mn. Table 2.6 shows the relationship of finite control volumes, Vm, and internal thermal resistance, , in different coordinate systems. Table 2.7 depicts boundary condition equations for use in resistance capacitance networks.

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TABLE 2.5 Finite Difference Approximations for Steady-State Conduction in a Square Mesh,  x  y Configuration

Finite Difference Equation

Interior Node

T 0  0.25 ( T 1  T 2  T 3  T 4 )

Plane surface with convection

1 T 0  ---------------- [ T 1  0.5 ( T 2  T 3 )  BiT ] 2  Bi

Plane wall with an unknown heat flux

qs  x T 0  0.5T 1  0.25 ( T 2  T 3 )  ----------2k

Exterior corner with convection

1 T 0  ---------------- [ 0.5 ( T 1  T 2 )  BiT ] 1  Bi

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TABLE 2.5 (continued) Finite Difference Approximations for Steady-State Conduction in a Square Mesh,  x  y Configuration

Finite Difference Equation

Interior corner with convection

1 T 0  ---------------- [ T 2  T 3  0.5 ( T 1  T 4 )  BiT ] 3  Bi

Interior node near a curved nonisothermal surface

Ta T1 T2 Tb 1 -  -------------  ----------------------  --------------------T 0  ---------- ------------a ( 1  a ) 1  b b ( 1  b) 1 1 1a ---  --a

b

2.4 CONDUCTION—TRANSIENT The solution for some heat conduction problems varies with time. Even problems involving steady-state conditions require time for the system to reach a steady state. We call such problems unsteady, or transient, conduction. Such cases may occur when the boundary conditions change, as when an object is immersed in a different temperature bath or with variable internal heat generation. There are two basic types of transient conduction problems, those that have boundary conditions that change once—balanced systems—and those that have continually changing boundary conditions—unbalanced systems. An example of the first type is a resistor in a system that is powered up. The temperature of the resistor will increase until there is an energy balance between the internal heat generation and the cooling effects of convection, radiation, and conduction. An example of an unbalanced system is the same resistor with a sine wave input voltage. The temperature of the device will continue to change with the changing input voltage conditions.

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FIGURE 2.18 A finite control volume for a resistance capacitance network. This control volume, Vm, surrounds the node, m. Thermal resistances, mn, extend from the control volume to other nodes.

TABLE 2.6 Internal Nodal Resistance Equations for Cartesian, Cylindrical, and Spherical Coordinate Systems Coordinate System Coordinates used

Cartesian x, y, z

Cylindrical r, , z

Spherical r, , 

Volume element

xyz

r m  r   z

r m sin r

m

x ---------------- y  zk

r -------------------------------------r r   ------  zk  m 2

r ----------------------------------------------------- r 2  r  -----sin   k 2  m

r ----------------------------------------  r  zk  r m  -----2

r  m

m

x ---------------- y  zk

2

r ----------------------------------------------------- r 2 ----- 2  sin   k

n

x ---------------- y  zk

r m  --------------- r  zk

 sin  ------------------- r  k

n

x ---------------- y  zk

r m  --------------- r  zk

 sin  ------------------- r  k

l

x ---------------- y  zk

z -----------------------r m  rk

 ---------------------------------------------  -  r  k sin    ------2 

l

x ---------------- y  zk

z ------------------------r m  rk

 ------------------------------------------------  r  k sin    ------- 2

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TABLE 2.7 Control Volumes and Resistances for Various Two-Dimensional Boundary Conditions Configuration

Control Volume and Resistances

V  (x)2

Interior node

1  01   02   03   04  --k

2

Plane surface with convection

 01

(x)  V  -------------2 2 1  02   03  -- --- ; k k 1  0  -----------hc  x

Plane surface with unknown surface heat flux

2

 01

(x)  V  -------------2 2 1  02   03  -- --- ; k k Q˙ s  q s  x

Exterior corner with convection

2

(x)  V  -------------4

2  01   02  --k 1  0  ----------hc  x

Interior corner with convection

2

 01   04

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3(x)  V  ----------------4 1 2  02   03  -- --- ; k k 1  0  -----------hc  x

2.4.1

LUMPED CAPACITANCE METHOD

Several methods are commonly used to find the solution to transient conduction problems. One approach used quite often with good results is the lumped capacitance method. Consider a microprocessor in a powered computer system. The microprocessor operates at a steady-state temperature of Ti. We turn off the system and the microprocessor begins to cool and approach T . If we turn off the computer at t  0, the temperature of the device will decrease until it reaches T , at t 0. To use the lumped capacitance method we assume that the temperature of the device is uniform. Although this assumption is not always accurate, the level of accuracy can be determined. Consider the thick plane wall shown in Figure 2.19. One surface is at Ts,1. The opposite surface is in a coolant media having a temperature of T . Therefore, the opposite surface is at some temperature between Ts,1 and T , which we will call Ts,2. We can describe the steady-state surface energy balance by the equation kA --------s ( T s, 1 T s, 2 )  h A s ( T s, 2  T ) L where k is the thermal conductivity of the solid object. We then rearrange the equation to find the temperature characteristics:

T s, 1  T s, 2 ---------------------------  T s, 2  T

L   ------- k A s -------------1   -------h A  s

We then see that the equation is equal to the ratio of the conductive thermal resistance cond to the convective thermal resistance conv , and

 cond hL ----------  --------c  Bi  conv k where: Lc  V/As,m Bi  Biot number (dimensionless) The Biot number is the basis for determining the validity of the lumped capacitance model of negligible internal temperature gradients. The Biot number is a measure of the temperature drop in the solid object relative to the temperature differential between the object’s surface and the coolant media (see Figure 2.20). If Bi  1.0, then the lumped capacitance model is valid, and we assume the internal temperature distribution within the solid object is uniform. For a simple object such © 2001 by CRC PRESS LLC

FIGURE 2.19 Nomenclature for the analytical solution of a transient temperature plate. The plate is initially at a uniform temperature but is then subjected at time zero to a change in environmental temperature through a unit surface conductance, hc.

FIGURE 2.20 The distribution of transient temperatures through a symmetrically convectioncooled plane wall. The Biot number varies from Bi  1 to Bi 1, while the temperature remains steady at T  T(x,t).

as a plate, cylinder, or sphere, when Bi  0.1, the error in the assumption of uniform temperature is less than 5%. Because we are disregarding Fourier’s conduction law, which requires an internal thermal resistance, we must use an energy balance equation to find the transient temperature response of the object. We will balance the surface heat loss of the © 2001 by CRC PRESS LLC

object to the rate of change of the object’s internal energy. This can be written dT  h A s ( T  T )   Vc ------dt where:  As  T  T    V  c  dT  h

average heat transfer coefficient, W/m2 K surface area of object, m2 average temperature of object, K ambient temperature, K density of object, kg volume of object, m3 specific heat of object, J/kg K temperature change, K, during time period dt, s

After separating the variables and integrating, we can find the temperature of an object at a point in time, t: h As

  ---------- t T T   Vc T ---------  ------------------ -  e T i Ti  T

We also note that the equation shows that the temperature differential between an object and its surroundings decays at an exponential rate to zero as t approaches infinity. We call the term Vc/ h As the thermal time constant, t, which can also be expressed as 1  i   --------- (  Vc )   t C t h As where:

t  resistance to convection heat transfer Ct  lumped thermal capacitance of the object (cpm) Since the terms on the right-hand side of the equation represent capacitance and resistance, an increase in either will cause the system to respond more slowly to boundary condition changes. To find the total heat transfer over the time period of transient conduction we can solve the equation t

 --  Q  (  Vc )  T i  1  e t  

where Q is the total energy transfer (J), and q is the rate of heat transfer (W). © 2001 by CRC PRESS LLC

2.4.2

APPLICATION

OF THE

LUMPED CAPACITANCE METHOD

For a simple analysis of transient conduction, we can use the previous formulas to find the linear temperature rise with respect to time: q  T  --------- Vc where T  temperature/time differential, K/s. We can also manipulate the equation to solve for the temperature difference if the object is constructed of two materials. Since the lumped capacitance method does not allow for anything but infinite internal thermal conductivity, we use the lumped thermal capacitance, Ct, of each material. Therefore: q  T  -------------------------------------C t1  C t2  C t3 Note that this equation does not represent an energy balance because it does not account for heat dissipation by conduction to other components, convection to the air, or radiation. By using the thermal time constant t and combining previous equations, we find a useful formula for calculating the transient temperature rise with respect to time when we know the steady-state temperature rise. We can use the equation to find the maximum temperature of a component that operates at some nominal power but occasionally experiences a maximum power mode: t 1

T H

--------     T ss  1  e t   

where:

TH  full-power cycle temperature rise, K Tss  steady-state normal power temperature rise, K t  power cycle time duration, s When we turn off a device, the equation to find the time it will take to reach the ambient temperature is much the same as the equation to find when the device will reach its maximum temperature: t

  ----  T   Ti  e t  

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2.5 CONDUCTION IN EXTENDED SURFACES One of the most widely used methods to solve problems in heat transfer is the extended surface. The extended fins of heat sinks are available in a variety of sizes, shapes, and forms. Usually constructed of aluminum, extended fin arrays are popular in low-cost extrusions and in higher-performance and more costly bonded configurations. It is important when evaluating fin designs to decide whether a simple fin shape will suffice or whether a more expensive fin cross section is needed. In a subsequent chapter we will explore convection, but in this chapter we will investigate the heat conductance through the most popular fin cross sections. There are three basic fin shapes: longitudinal, spine, and annular. These three fin configurations are shown in Figure 2.21. The longitudinal fin, sometimes called a straight or plate fin, has one dimension, L, that is in the direction of flow and is usually greater than the other dimensions. The spine fin may be circular or any other shape. If the fin is circular, the thickness is the diameter, D. If the fin is another

FIGURE 2.21 Types of fins, terminology, and nomenclature. (a) Longitudinal or plate fin. (b) Spine fin. (c) Annular or radial fin.

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shape, the length, L, and thickness, , are usually similar in value. The annular fin, also called the radial fin, follows a curved surface. The height, b, of this fin is the difference between the outer radius, ro, and the inner radius, ri. In a fin of finite height, optimizing the fin geometry is necessary so that we conduct the maximum heat from the wall to the tip of the fin. Beyond this, we should maximize the conductance of the fin throughout the height of the fin. We can accomplish this using different cross sections. If the fin consists of a highly conductive material, we will enhance the temperature gradient along the fin and the heat characteristics of the wall. We accomplish maximum cooling of the wall if the entire fin is at the same temperature as the wall. We must weigh the shape, size, and material of a fin against the cost of the array to meet this goal.

2.5.1

FIN EFFICIENCY

We call the effectiveness of a fin the fin efficiency, f. We define fin efficiency as the ratio of the actual amount of heat transferred by the fin to the heat that we may have transferred if the entire fin was at the wall temperature. We can describe the fin efficiency for a circular spine fin, normally called a pin-fin, as: 2

tanh 4L h c  kD  f  -----------------------------------------2 4L h c  kD For a longitudinal fin of rectangular cross section, the theoretical formula becomes: 2

tanh hPL  k A  f  ------------------------------------------s2 hPL  k A s Most practical extruded rectangular fins are long and wide and have a thin cross P 2 - ⬵ --- . This simplification maintains the surface area from which the section, where ---As  heat is lost. The fin efficiency for this simplification becomes: 2 c c

b tanh 2h -------------k  f  --------------------2

2h c b c ------------k

where: b  bc  hc  k  D  

height of fin, m (b  /2), m average heat conductance, W/m2 K thermal conductivity of fin material, W/m K fin diameter, m fin thickness, m

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P  wetted perimeter of fin, m As  surface area of fin, m2 The error from this approximation will be less than 8% when: h c   ------ 2k 

0.5

 0.5

If we let m equal a fin performance parameter, we can substitute different equations for m to find the fin efficiency of different fin configurations. For a rectangular fin, the equation simplifies to: tanh mL  f  ---------------------c mL c where:

m 

2h c -------k

Views and values of f for different fin configurations are shown in Table 2.8. Since Bessel functions increase the difficulty of some equations for fin efficiency, graphs are sometimes used. Figure 2.22 shows the efficiency of longitudinal fins, while Figure 2.23 shows the efficiency of annular fins. To calculate the total efficiency, tot, of a finned array, combine the wall surface at 100% efficiency with the efficiency of the surface area of the fins, f: A tot  tot  ( A tot  A b ) A f  f where: Atot  total heat transfer area (m2) Ab  base area of the fins (m2) Af  heat transfer area of the fins (m2) For a simple plane flat surface wall in contact with air, the thermal resistance, 1 - . If fins are added to this plane surface, the fin surface area will , is given as --------h A increase, but we must conduct the heat a greater distance through the fins. Occasionally, because of this extra distance, the addition of fins may not increase the h As heat transfer rate. Fins are usually required and are beneficial when ----------  1.0. Pk Because of this, fin efficiency reaches a maximum when L  0. Therefore, calculating a maximum fin efficiency is not possible based on height. Fins are usually optimized based on the amount of material in the fin array. For most large-run c

s

c

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TABLE 2.8 Efficiency Factors for Fins of Various Shapes Fin Configuration Longitudinal fin of rectangular profile

f 1  f  ------- tanh mb mb

yt Longitudinal fin of parabolic profile

I  4--- mb 1 2  33   f  ------- ------------------------mb I  --4- mb 1  3  3



1  x 0.5 y  t  -------------  b  Longitudinal fin of triangular profile

1 I 1 ( 2mb )  f  ------- -------------------mb I 0 ( 2mb )

1x y  t  -------------  b  Longitudinal fin of parabolic profile

2  f  ------------------------------------------------˙ 2 4 ( mb )  1  1

1x 2 y  t  -------------  b  (Continued)

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TABLE 2.8 (continued) Efficiency Factors for Fins of Various Shapes Fin Configuration Annular fin of rectangular profile

f 2r 1

-------- K ( mr )I ( mr )  I ( mr )K ( mr ) m 1 1 1 2 1 1 1 2 -2 --------------------------------------------------------------------------------------- f  -----------2 r 2  r 1 K 0 ( mr 1 )I 1 ( mr 2 )  I 0 ( mr 1 )K 1 ( mr 2 )

yt Annular fin of hyperbolic profile



r 



r 



r 1

I 2  3  2--3- mr 1 I 2  3  2--3- mr 2 ----2-  I 2  3  2--3- mr 2 ----2- I 2  3  2--3- mr 1 r 1 r 1    f   ----------------- ---------------------------------------------------------------------------------------------------------------------------------r2  r1 r r     I 1  3  2--3- mr 1 I 2  3  2--3- mr 2 ----2-  I 2  3  2--3- mr 2 ----2- I 1  3  2--3- mr 1 2r 1 -------m



r 1

r y  t  ----1  r Spine fin of circular profile

1  f  -------------------- tanh ( 2 )mb ( 2 )mb

yt Spine fin of parabolic profile

1  x 0.5 y  t  -------------  b  © 2001 by CRC PRESS LLC

I 1 --43- ( 2 )mb 2  f  -------------------------- ------------------------------ 4--- ( 2 )mb I 4--- ( 2 )mb 3  0 3

TABLE 2.8 (continued) Efficiency Factors for Fins of Various Shapes Fin Configuration Spine fin of triangular profile

f I 2 [ 2 ( 2 )mb ] 4  f  ----------------------- -------------------------------2 ( 2 )mb I 1 [ 2 ( 2 )mb ]

1x y  t  -------------  b  Spine fin of parabolic profile

2  f  --------------------------------------------------2 8 --- ( ( mb )  1 )  1 9

1x 2 y  t  -------------  b  Where : m 

2h c -------k

extrusions, the amount of material a greater bearing on cost than fin cross-sectional complexity. Usually, placing fins on the side of a heat exchanger where the heat transfer coefficient is lower is more desirable. Thin, closely spaced fins are generally more efficient than other patterns, and the fins should be constructed of a highly thermally conductive material. We prefer fins that are an integral part of the wall to fins that are bonded to a wall because of the penalty of contact resistance at the thermal interface.

2.5.2

FIN OPTIMIZATION

For longitudinal fins of rectangular profile and for longitudinal fins of triangular profile, we can solve the efficiency equations to yield an optimum fin. There is a specific value of fin height, b, and fin thickness, , that results in maximum heat dissipation for a given area. For the longitudinal fin of rectangular profile, the heat flow through the base is found by q 0  k  Lm  T 0 tanh mb © 2001 by CRC PRESS LLC

FIGURE 2.22 Efficiency of straight longitudinal fins of rectangular (Ap  Lct), triangular (Ap  Lt/2), and parabolic (Ap  Lt/3) profiles. (Adapted from Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley & Sons, New York, 1990. With permission.)

FIGURE 2.23 Efficiency of annular fins of rectangular (Ap  Lct) profile. (Adapted from Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley & Sons, New York, 1990. With permission.)

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where: q0  k    L  m  T0  b 

heat transfer from fin, W thermal conductivity of fin, W/m K fin thickness, m fin length, m fin performance factor, m1 temperature difference between fin base and ambient environment, K fin height from base surface, m

If we rearrange the equation in terms of the profile area, Ap, we see that 0.5

2h 0.5 0.5  1.5 q 0  L  T 0 ( 2h c k )  tanh A p  --------c   k  The measure of fin width that results in the maximum heat dissipation is found when the derivative of this equation equals zero. Kern and Kraus12 performed this differentiation to find R, a dimensionless fin-optimizing parameter, where 0.5

2h  1.5  1.419

R  A p  --------c  k Since the value of R, is known, we can solve for the fin thickness 1 2 ---

2h c A p 3   0.791  -------------k  Now we can solve for the fin height:

1 ---

A kA 3 b  ------p  1.262  ---------p   2h c 

When we insert the solution of the fin width into the equation for heat transfer, we find 2 1 q  1.26 ( h c A p k ) ---  T 0 3

Since we know the optimum condition of mb is the value 1.419, we can rearrange the formula to find the optimum profile area for a specific heat and temperature for a rectangular fin: 0.50  q  3.0 - ---------A p  --------2 h c k   T 0 © 2001 by CRC PRESS LLC

For a longitudinal fin of triangular profile, Kern and Kraus12 have differentiated the equation for the rate of heat transfer using Bessel functions and a fin-optimizing parameter. The equation for the triangular profile fin thickness is 2 2h   1.328 A p  --------c k

1/3

and the triangular profile fin height is 2 Ap 1/3  A p k b  -------  1.506  --------- 2h c If we substitute the solution for the optimum fin thickness into the equation for the rate of heat transfer, we obtain 2

1/3

q  1.422 ( h c A p k )  T 0 Since we know the optimum condition of mb is the value 1.422, we can rearrange the formula to find the optimum profile area for a specific heat and temperature: 0.347  q  3.0 - ---------A p  -----------2 h c k   T 0 Examination of the equations reveals that, for identical conditions, the triangular profile fin requires only about 69% as much material as the rectangular fin. Also, since the fin volume increases as the cube root of heat flow, two identical fins are equal to a single fin eight times as large.

2.5.3

FIN SURFACE EFFICIENCY

Besides the fin efficiency, , there is also a term for overall fin surface efficiency, o. The fin efficiency characterizes the conductive efficiency of a single fin, while the overall fin surface efficiency characterizes an array of fins and the attachment surface. If the area of the fin array is separated into a finned area and a base area, we see that Atot  Af  Ab, and we can describe the overall surface efficiency as q tot q tot -  ---------------------- 0  -------q max h c A tot  T b where: qtot  total rate of heat transfer, W qmax  maximum possible rate of heat transfer, As  Tb, W © 2001 by CRC PRESS LLC

hc  convective heat transfer coefficient, W/m2 K Atot  total surface area of fins and base, m2 Tb  temperature difference between fin base and ambient media, K Using the separate values of the fin area, Af, and the fin base area, Ab, the overall fin surface efficiency can be expressed by Af  0  1  -------- ( 1   f ) A tot

2.6 THERMAL CONTACT RESISTANCE IN ELECTRONIC EQUIPMENT INTERFACES Most electronic applications have at least one interface where heat flow must cross between two surfaces. At each interface there is a measurable temperature difference across the joint. This can occur dramatically within two rough surfaces under light joining pressure, or slightly in the contact between a device soldered to a heat sink, but it still exists. Each of these contacts contributes to the overall thermal contact resistance. Together, these additional resistances can cause excessive component temperatures. Altoz13 estimates that when we join seemingly identical cross-sectionalarea components, only 5% of the apparent surface areas actually make intimate physical contact. We cannot ignore thermal contact resistances, but they are very difficult to quantify. No accurate models exist that are applicable across the range of electronic packaging applications.

2.6.1

SIMPLIFIED CONTACT RESISTANCE MODEL

Consider that heat flows from a microprocessor into a heat sink, both of which are in intimate contact. We assume that heat flow occurs in the axial direction, x, only. Heat flows through the microprocessor according to Fourier’s law. In a real interface, the surfaces are not perfectly smooth. As heat flows from the microprocessor into the heat sink, the heat transfer area seems constant—we call this the apparent contact area, Aa. We call the temperature difference caused by the thermal resistance at the interface the thermal contact resistance, Tc. We call the interface coefficient hi. The apparent contact area is composed of the cross-sectional area where we make actual hard contact, Ac, and a void area where contact does not occur, Av. The apparent area of contact, Aa, then, is the sum of both the actual contact area Ac and the void area Av . A magnified view of the thermal interface is shown in Figure 2.24. Heat flow across the thermal interface is quite complex. Conduction in the interface is parallel and three-dimensional as the heat appears to “squeeze” through the points of contact. Heat flow also occurs by radiation and by conduction through the interface material, which might be air. We can show that convection in the void fluid is insignificant if the Rayleigh number is 1700, which is almost always the case.

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FIGURE 2.24 Heat flow across contacting surfaces: (a) without interstitial material, (b) with interstitial material. The interstitial material greatly increases the contact area available for heat transfer.

It would appear that the thermal path through the hard points of contact is much more significant than the thermal conductance through the void material. However, researchers have shown air to have a large impact on the overall thermal conductance when compared with a vacuum. Since it has been estimated that the actual solid material point-to-point contact area may be only 5% of the apparent area, it follows that 95% of the heat path is through the void material. Because finding the exact values of Ac and Av is quite difficult, empirical equations are commonly used.

2.6.2

GEOMETRY

OF

CONTACTING SURFACES

The actual surface of the contact area may contain peaks and valleys superimposed on larger waves. We call the superimposed irregularities, roughness. The actual depth of the interface can range from about 2 m for a very smooth polished surface to about 25 m for a very rough surface. We call the peaks in the roughness asperities while we call the recesses between the asperities, valleys. Current models use the ratio of a line drawn through the peaks to the RMS roughness, which is determined by the mechanical finishing operation. Kraus and Bar-Cohen14 give typical values of this ratio in Table 2.9 in the next section. Figure 2.25 depicts the vertex angle, 2, the radius of curvature, , and the slope, tan, of an asperity. We find the vertex angle by measuring the included angle of the majority of the asperity’s slope. The lay of a surface is the primary orientation of the surface pattern that results from the production method employed. The transverse roughness is determined for a section normal to the lay, and the longitudinal roughness is in the direction of the lay. The heat transfer across an interface of two materials in contact is a very complex phenomenon and is a function of many parameters. The following appear to be of

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FIGURE 2.25 Nomenclature for an asperity. The slope of the asperity is the tangent of the angle . The radius of curvature of the asperity is called . Half of the vertex angle is the angle .

the greatest importance: 1. 2. 3. 4.

the pressure at the interface, the hardness of the contacts, the size of the contact asperities, the geometry of the contacting surfaces with regard to roughness and waviness, 5. the average gap thickness of the void space, 6. the thermal conductivity of the fluid in the void spaces, and 7. the thermal conductivity of the contact materials.

2.6.3

CONTACT RESISTANCE

IN A

TYPICAL APPLICATION

Cooper et al.15 formulated a reasonably useful relationship for the heat transfer interface coefficient, hi, as P k  -----a  H

0.985

h i  1.45 ------------------ tan   where: k k

1 2 - and k1 and k2 are the thermal conductivities of material 1 and  2 ---------------k1  k2 material 2 (W/m K) Pa  contact pressure at the thermal interface (N/m2) H  hardness of the softer material (N/m2  108)    21   22 and 1 and 2 are the RMS roughness of material 1 and material 2 (m) tan   21   22 and 1 and 2 are the absolute asperity angles of material 1 and material 2

k

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This correlation shows a good relationship to test data within the following range of applicability: Pa

0.35  103  ---H  0.01 13.80  k  133.2 (W/m K) 1.0    85.0 m 0.08  tan   0.160

Shlykov16 developed a correlation under the assumption that both the point of contact conductance and the void material layer conductance are based on the two heat flow paths being independent, but that the sum effect is a parallel combination. The relationship for the heat transfer gap interface coefficient is h gi  k g  Y where: kg  thermal conductivity of gap fluid (W/m K) 1 3 10 10 2 4 Y  ------  ------  ----2 4  ----3  ----2  --- ln ( 1  x )  x 3 x x x x 7 ( 1  2 ) x  -------------------------2l 2l  0.4416 for air To simplify the correlation for the asperity’s peak line, Y, Kraus and Bar-Cohen14  provide Table 2.9 for use when Y  --- . Since an increase in asperity contact reduces the contact resistance, increasing the pressure is often a good way in which to reduce T. In some applications it may be possible to insert a thin foil of a soft, thermally conductive metal in the interface. Table 2.10 shows the results of some researchers’ experiments.

TABLE 2.9 Typical Values of the Ratio Y/ as a Function of the Surface-Finishing Operation

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Finishing Operation

Y

Grinding Hyperlap Sandpaper Superfinish Lap with loose abrasive

4.5 6.5 7.0 7.0 10.0

TABLE 2.10 Thermal Contact Resistances of Various Interfaces

  103 (m2 K/W)

Interface Silicon chip to lapped aluminum in air (27–500 kN/m2) Aluminum to aluminum with indium foil (⬃100 kN/m2) Aluminum to aluminum with lead coating Aluminum to aluminum with Dow Corning 340 grease (⬃100 kN/m2) Silicon chip to aluminum with 0.02-mm-thick epoxy Ceramic to ceramic in air Ceramic to metal in air Graphite to metal in air Stainless steel to stainless steel in air Aluminum to aluminum in air Aluminum to aluminum in silicone oil Stainless steel to aluminum in air Copper to copper in air Iron to aluminum in air Brass to brass with 15-m-thick tin solder coating

3.0 to 6.0 ⬃0.70 0.10 to 1.0 ⬃0.70

Ref. 17 18,19 20 14,15

2.0 to 9.0 0.50 to 3.0 1.5 to 8.5 3.0 to 6.0 1.7 to 3.7 27.5 5.25 3.0 to 4.5 10.0 to 25.0 4.0 to 40.0 0.025 to 0.14

21 7 7 7 7 22 18 7 7 7 23

2.7 DISCRETE HEAT SOURCES AND THERMAL SPREADING In many problems the cross-sectional area available for heat conduction is not constant. Consider the case of a heat source mounted to a large heat sink plate, as shown in Figure 2.26. In these cases we use the concept of heat spreading. As the cross-sectional area available for heat transfer increases, the thermal resistance decreases. It does not always follow, though, that an increase in cross-sectional area will amount to a corresponding decrease in thermal resistance. There is a penalty in allowing the heat to spread to the larger area that we call the thermal spreading resistance, sp. Yovanovich and Antonetti24 found that the spreading resistance can be expressed as: 3

5

7

1  1.410   0.344   0.043   0.034   sp  ----------------------------------------------------------------------------------------------------------4ka where:

  ratio of heat transfer area 1 to heat transfer area 2 k  thermal conductivity of material of heat transfer area 2 (W/m K) a  square root of the area of heat transfer area 1 (m2)

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FIGURE 2.26 Heat conduction of an energy source on a plane substrate.

FIGURE 2.27 The thermal resistance for a circular heat source on a two-layer substrate. (FromYovanovich, M. M. and Antonetti, V. M., Advances in Thermal Modeling of Electronic Components and Systems, Hemisphere, Washington, D. C., 1988. With permission.)

The limitation on the thermal spreading correlation is that the thickness of heat transfer area 2 must be 3.0 to 5.0 times greater than the square root of the area of heat transfer area 1. If this is not true, Figure 2.27, which applies to a circular heat source, may be useful. Mikic25 has compiled several relationships for thermal spreading. To use these relationships, Tc is added to the average surface temperature that is found by spreading the heat over the entire surface area. In each equation we define the © 2001 by CRC PRESS LLC

FIGURE 2.28 Heat conduction for a circular energy source on an infinite conducting substrate.

FIGURE 2.29 Heat conduction for a circular heat source on a finite conducting substrate.

variables in the corresponding figures. For a circular heat source on an infinite substrate (see Figure 2.28) the correlation is q  T c  -----------------2 ak For a circular heat source on a finite substrate (see Figure 2.29): a q  T c  ------------------  1  --- b 2 ak © 2001 by CRC PRESS LLC

1.5

FIGURE 2.30 Heat conduction for a long strip energy source on a finite conducting substrate.

FIGURE 2.31 Heat conduction for a short strip heat source on a finite conducting substrate.

For a long heat strip on a finite substrate (see Figure 2.30): q 1  T c  ----------ln ----------------- a

Lk sin ( ------) 2b For a rectangular heat source on a rectangular finite substrate (Figure 2.31), the overall temperature rise, Tc, is the sum of three temperature rises, Tc1, Tc2, and

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Tc3, where: m a



m d

 T c1

 T c2

-) q c 冱 sin ( ----------b ---- -------------------------------2 2 2 k db m  1 m

 T c3



-) q b 冱 sin ( ----------b ---- -------------------------------2 2 2 k ac m  1 m

q 2 - ----- ----------2 2 k ad

冱 m1



n d

m a

b

c

- ) sin ( ------------ ) sin ( ---------c b 冱 --------------------------------------------------2 2 0.5 n1



n m mn  --------   -------

For the case of a circular heat source on a finite circular substrate with a finite depth (Figure 2.32): 2 J   a--- L 1 m  b 4q b   T c  ---------- --- 冱 tanh   m --- ---------------------b  3m J 20  m

ak a m  0

FIGURE 2.32 Heat conduction for a circular energy source on a finite conducting substrate having a finite thickness.

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REFERENCES 1. Fourier, J. B. J., The Analytical Theory of Heat, transl. by Alexander Freeman, Cambridge University Press, London, 1878. 2. VanSant, J. H., Conduction Heat Transfer Solutions, Natl. Tech. Info. Service, UCRL52863, Rev. 1, 1983, in Handbook of Applied Thermal Design, Guyer, E. C., Ed., McGraw-Hill, New York, 1984. 3. Jakob, M., Heat Transfer, Vol. 1, John Wiley & Sons, New York, 1949. 4. Kutateladze, S. S. and Borishanksii, V. M., A Concise Encyclopedia of Heat Transfer, Pergamon Press, New York, 1966. 5. Hahne, E. and Grigull, U., Formfaktor und Formwiderstand der stationären mehrdimensionalen Wärmeleitung, Int. J. Heat Mass Transfer, 18, 751–767, 1975. 6. Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, 2nd ed., Oxford University Press, London, 1959. 7. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, John Wiley & Sons, New York, 1960. 8. Fourier, J. B. J., The Physical Theory of Heat, Dover Publishers, New York, 1955. (Originally published in 1822.) 9. Schmidt, E., On the Application of the Calculus of Finite Differences to Technical Heating and Cooling Problems, in Festschrift zum Siebzigsten Geburstag August Foeppls, Springer, Berlin, 1924, p. 179. 10. Ozisik, M. N., Boundary Value Problems of Heat Conduction, International Textbook, Scranton, PA, 1968. 11. Schneider, P. J., Conduction Heat Transfer, Addison-Wesley, Cambridge, MA, 1955. 12. Kern, D. Q. and Kraus, A. D., Extended Surface Heat Transfer, McGraw-Hill, New York, 1972. 13. Altoz, F. E., Thermal Management, in Electronic Packaging and Interconnection Handbook, Harper, C. A., Ed., McGraw-Hill, New York, 1991. 14. Krause, A. D. and Bar-Cohen, A., Thermal Analysis and Control of Electronic Equipment, McGraw-Hill, New York, 1983, p. 202. 15. Cooper, M. G., Mikic, B. B., and Yovanovich, M. M., Thermal Contact Resistance, Int. J. Heat Mass Transfer, 12, 279–300, 1969. 16. Shlykov, Y. L., Calculating Thermal Contact Resistance of Machined Metal Surfaces, Teploenergetika, 12(10), 79–83, 1965. 17. Eid, J. C. and Antonetti, V. W., Small Scale Thermal Contact Resistance of Aluminum Against Silicon, in Heat Transfer—1986, Vol. 2, Tien, C. L., Carey, V. P., and Ferrel, J. K., Eds., Hemisphere, New York, 1986. 18. Fried, E., Thermal Conduction Contribution to Heat Transfer at Contacts, in Thermal Conductivity, Vol. 2, Tye, R. P., Ed., Academic Press, London, 1969. 19. Snaith, B., O’Callaghan, P. W., and Probert, S. D., Interstitial Materials for Controlling Thermal Conductances Across Pressed Metallic Contacts, Appl. Energy, 16, 175, 1984. 20. Yovanovich, M. M., Theory and Application of Constriction and Spreading Resistance Concepts for Microelectronic Thermal Management, Int. Symp. Cooling Technol. Electronic Equipment, Honolulu, 1987. 21. Peterson, G. P. and Fletcher, L. S., Thermal Contact Resistance of Silicon Chip Bonding Materials, Int. Symp. Cooling Technol. Electronic Equipment, Honolulu, 1987, p. 438. 22. Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley & Sons, New York, 1990.

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23. Yovanovich, M. M. and Tuarze, M., Experimental Evidence of Thermal Resistance at Soldered Joints, AIAA J. Spacecraft Rockets, 6, 1013, 1969. 24. Yovanovich, M. M. and Antonetti, V. M., Application of Thermal Contact Resistance Theory to Electronic Packages, Advances in Thermal Modeling of Electronic Components and Systems, Bar-Cohen, A. and Kraus, A. D., Eds., Hemisphere, New York, 1988, p. 79. 25. Mikic, B. B., Course notes, for M. I. T. Special Summer Session on Thermal Control of Modern Electronic Components, 1978 and 1979.

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Remsburg R. “ Fluid Dynamics for Electronic Equipment” Thermal Design of Electronic Equipment. Ed. Ralph Remsburg Boca Raton: CRC Press LLC, 2001

3

Fluid Dynamics for Electronic Equipment

3.1 INTRODUCTION To fully understand heat transfer by natural or forced convection, having a basic understanding of fluid dynamics is necessary. More advanced cooling techniques that involve fluid dynamics include liquid immersion, multiphase cooling, and jet impingement. The general term fluid conveys any substance that does not have a form. Liquids and gases are both fluids. Further, to differentiate liquids from gases, liquids have a surface, whereas a gas will expand to fill a volume. Liquids are generally considered nearly incompressible, whereas gases are readily compressible. The vast majority of coolant flows are of the incompressible type. However, the equations may require correction for the properties of a compressible fluid due to the altitude. This text emphasizes incompressible flow.

3.2 HYDRODYNAMIC PROPERTIES OF FLUIDS 3.2.1

COMPRESSIBILITY

Altitude affects the density of the air due to compressibility. Along with the change in density, important changes are found in how much heat the air can absorb, turbulence, heat transfer coefficients, and many other parameters. The application of pressure can compress all materials, with the result being the storage of elastic energy. Assuming perfect energy conversion, the materials expand to their original volumes when they release the pressure. Fluids do not have the rigidity of solid materials, so the change in volume defines the modulus. The change in volume is called the Bulk Modulus. With fluids, especially gases, the curve plotted by the bulk modulus becomes steeper with increasing pressure. This shows that as we compress the fluid, it becomes more difficult to compress it further. The bulk modulus of a fluid is given by: P  B   V --------  V

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where: B V p V

   

bulk modulus (N/m2) original volume (m3) change in pressure (N/m2) change in volume (m3)

Values of the bulk modulus are positive. In the previous equation, the negative sign compensates for the fact that the volume is compressing. The bulk modulus of most liquids is so small that it is generally of concern only in hydraulics. Water’s bulk modulus is 0.2  1010 N/m2, whereas the bulk modulus of air is 1.01  105 N/m2. Compression and expansion of gases follow the laws of thermodynamics. If the temperature remains constant (isothermal) during the expansion or compression process, the operation follows Boyle’s law p ---  Constant 

or

p ---  Constant p

If the process is theoretically frictionless, and no heat is exchanged, the operation follows the isentropic relationship p -----  Constant k

or

p -----k  Constant p

where: p  pressure (N/m2)   g  specific weight (N/m3) k  ratio of specific heats For perfect gases, cp  cv  ᑬ, which, in combination with cp /cv  k, is found to yield the relationship

ᑬk c p  ------------k1 where ᑬ is the universal gas constant, 286.8 J/kg K.

3.2.2

VISCOSITY

The viscosity of a fluid is one of its most important material properties. Viscosity is a measure of a fluid’s internal friction. A fluid’s viscosity affects how much energy must be used to move a fluid through a cooling duct. The fluid viscosity also affects the transition from laminar to turbulent flow, the heat transfer coefficient, the overall cooling efficiency of a system, and many other parameters. Molecular cohesion forces cause viscous drag in liquids and molecule collisions in gases. The viscous drag is proportional to the speed of the fluid and varies linearly © 2001 by CRC PRESS LLC

FIGURE 3.1 Effect of viscosity, , on the velocity profile of fluid between two parallel plates.

in the laminar flow range. In the turbulent regime, the drag increases more quickly, approximately equal to the square of the velocity. Viscosity is the shearing force that exists between the fluid layers and is expressed as a frictional stress per unit of contact area, . Temperature affects viscosity differently for gases than for liquids. The viscosity of a gas increases with temperature. In liquids, an increase in temperature leads to a decrease in viscosity. Two terms describe the thickness of a fluid: absolute viscosity, , and kinematic viscosity, u. By examining the two plates shown in Figure 3.1, we can better understand the effect of absolute viscosity. The lower plate is stationary while the upper plate is moving at a steady velocity of u, caused by a force F. According to boundary layer theory, the fluid sticking to the stationary plate has a velocity of zero. The velocity of the fluid sticking to the upper plate is the same as the plate, u. We can represent the absolute viscosity of the fluid by:

  ---------uy where:

   u  y 

absolute viscosity (N s/m2) fluid shear stress (N/m2) velocity of the upper plate (m/s) fluid film thickness (m)

Kinematic viscosity, , is sometimes used in calculations instead of absolute viscosity. Kinematic viscosity equals absolute viscosity divided by the mass density:



 --- where:

 kinematic viscosity (m2/s)  absolute viscosity (N s/m2)   mass density (kg/m3) © 2001 by CRC PRESS LLC

FIGURE 3.2 Laminar fluid velocity profile along a flat plate.

Figure 3.2 shows the velocity profile of a fluid along a surface. Note that as the distance from the stationary wall increases, so does the fluid velocity. Internal shear stresses occur within the fluid due to the differing velocities of the layers within the fluid. In solids, the shear stress is proportional to the strain (relative displacement). However, a fluid will flow with very little pressure. A constant application of pressure will result in an infinite strain. Consequently, stresses in fluids are based on the rate of strain rather than total strain, as in solids. Although the linear Newtonian relationship is only an approximation, it is surprisingly good for a wide class of fluids. Most electronic coolants such as water, air, and other gases are essentially Newtonian; nevertheless, consideration of fluids that are non-Newtonian can be important in fluid mechanics, although generally not so well understood or appreciated.

3.2.3

SURFACE TENSION

All molecules and atoms exhibit cohesive forces that are electrical in nature. In solids, the forces are very strong, while in gases the forces are much weaker. In a liquid, the forces are not strong enough to give rigidity but are strong enough to form a surface. Within the liquid, a molecule is acted upon equally by all other molecules. When a molecule reaches the liquid surface, through random motion, there is more cohesive force acting on it from the other molecules in the liquid than from outside the surface. The unbalanced cohesive forces are a form of potential energy and can be measured. A larger surface would represent more unbalanced cohesive forces and therefore more potential energy. We define the coefficient of surface tension, , as the potential energy per unit of surface area. If we can pull the liquid surface away from the mass of liquid by a force, F, the surface tension can be expressed, in units of N/m, by the equation F  -----------2 D

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TABLE 3.1 Surface Tension of Various Liquids to Air Liquid Benzene Ethyl alcohol Freon-12 Jet fuel (JP-4) Mercury Perfluorocarbon (FC-72) Silicate ester (Flo-Cool 180) Water Water

Temperature, °C

Surface Tension, , N/m

20 20 27 16 20 25 25 20 100

0.029 0.022 0.008 0.029 0.489 0.012 0.025 0.073 0.059

where is the surface tension in N/m, and D is the diameter of the ring in m. A typical value for water in contact with air is 0.073 N/m. Because these values are so small, the dyne (1.0 N) per centimeter is often the unit used in tabular data. In heat transfer, the surface tension becomes important for determining the wettability of a surface by a liquid. This property affects the thermal hysteresis of a liquid when in contact with a surface above the liquid’s boiling point. In practical applications, low surface tension affects capillarity and can cause fluid leaks. Fluids with low surface tension can “wick” through very small openings. While water has a coefficient of surface tension of about 0.073 N/m, an electronic coolant such as perfluorocarbon has a value of about 0.012 N/m and is known as a fluid that is difficult to seal. Table 3.1 shows a comparison of the surface tensions of various liquids.

3.3 FLUID STATICS Fluid statics is the relationship of the forces of pressure, density, and height of a fluid to the fluid’s surroundings, when the fluid is at equilibrium. Acceleration and velocity can fall within the field of fluid statics while there is no relative motion of the fluid particles under study. A quick way to decide whether a problem involves fluid statics or fluid dynamics is the presence of viscosity as a variable in any equation. Since viscosity is a measure of fluid shear, the presence of a viscosity variable suggests that there is fluid shear, and hence, relative motion between fluid particles. The lack of a viscosity variable may indicate dynamic flow of an ideal fluid or a problem in fluid statics.

3.3.1

RELATIONSHIP

OF

PRESSURE, DENSITY,

AND

HEIGHT

The equation for the relationship of pressure, density, and height is derived by examining the static equilibrium of the differential fluid element shown in Figure 3.3. We show the z-axis as parallel to the force of gravity, vertical. Since there is no pressure differential in the horizontal plane, pressure can vary only with z, vertical height, in a static fluid. If we assume that the fluid has constant density, we find a

© 2001 by CRC PRESS LLC

FIGURE 3.3 Differential fluid element dx dz in static equilibrium.

simple relationship for the increase of pressure with height p1  p2 z 2  z 1  H  -----------------g

or

p 1  p 2   ( z 2  z 1 )   gH

We often call the pressure difference ( p1  p2) a head, H, of fluid of specific weight, g. This leads to the expression of pressure as a head of millimeters of mercury (mm Hg), meters of water (m H2O), etc. We can also rearrange the equation to find the relationship with flowing fluids as p p -----1-  z 1  -----2-  Constant g g For variable density gases, such as the atmosphere, the polytropic equation p1 ------------n  Constant (g) can be used to find the relationship between pressure, density, temperature, and height. Pressure, like temperature, can be described in relative or absolute terms. In the SI system, we can describe temperature in K or in °C. We know that 0°C is offset 273.15°C above 0K, which is absolute zero. In the SI system, pressure is often offset 101,330 N/m2 above a vacuum of 0.0 N/m2, to describe atmospheric pressure at sea level. When we measure pressure using a vacuum as the reference point, we term it absolute pressure. When the pressure is measured using atmospheric pressure as a starting point, we term it gage pressure.

© 2001 by CRC PRESS LLC

3.4 FLUID DYNAMICS Fluid dynamics is the study of fluids in motion. Fluid motion, whether naturally induced or artificially produced, causes heat to be transferred by convection. Convective heat transfer is arguably the most useful mode of heat transfer for electronic equipment. Fluid dynamics derives equations to describe fluid motion in terms of displacements, velocities, and accelerations without regard to the forces that cause the motion.

3.4.1

STREAMLINES

AND

FLOWFIELDS

We can describe the motion of fluid particles from two different points of view. In the Eulerian view, the focus is on particular points in the space filled by the fluid. We give a description of the state of fluid motion at each point as a function of time. The values and variations of velocity, density, and other fluid parameters are determined at various spatial points. The Langrangian view labels each fluid particle, and then traces the path, density, velocity, etc. of each fluid particle over time. This analysis results in a trajectory path for each fluid particle, which we call a path line. Although engineers have written complex software to follow the Langrangian view, general engineering calculations for fluids used in heat transfer are generally better served by the Eulerian equations. These equations are simple and will allow the determination of the average flow properties in the area of interest. Lines can be drawn in a steady flow of fluid, parallel to the direction of flow. If we draw these lines so that the tangent at any point is in the direction of the velocity vector at that point, we can call these curves streamlines. Figure 3.4 shows a streamline flowfield.

FIGURE 3.4 Typical streamlines in fluid flow.

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FIGURE 3.5 A typical streamtube in one-dimensional fluid flow.

When streamlines are drawn on a fluid flow through a closed curve such as shown in Figure 3.5, the lines form a boundary that the fluid particles cannot cross. Crosswise flow is prohibited because the velocity is always tangent to the boundary. Therefore, the space between the individual streamlines becomes a tubular passage called a streamtube. The streamtube concept broadens the application of the principles of fluid flow. For example, streamtubes will allow the same laws to describe such different problems as flow in an enclosed passage and flow around an immersed object.

3.4.2

ONE-, TWO-,

AND

THREE-DIMENSIONAL FLOWFIELDS

Variations in fluid variables characterize one-dimensional flow along, but not across, streamlines. A one-dimensional streamtube describes a cross-sectional fluid area where all fluid properties are uniform. In practice, flow throughout a pipe is considered one-dimensional, so that we can apply average properties at each finite cross section. Flow along each individual streamline is strictly one-dimensional. The equations of one-dimensional flow can easily approximate many two- and three-dimensional nonturbulent real flows. Flowfield is a term used to describe the ideas of two- and three-dimensional arrays. Two-dimensional flows occur over a flat plane, while three-dimensional flows occur in a three-dimensional volume. Figure 3.6 shows a two-dimensional flow over a cylinder. We can apply two-dimensional equations to most three-dimensional problems when we make corrections to account for the slight three-dimensional effects present in real fluid flow. Fully three-dimensional flowfields are quite complex. These flows are difficult to visualize and difficult to predict, and often contain secondary flowfields, as shown in Figure 3.7. The generalization of the kinematics and dynamics of true threedimensional flow is beyond this reference. Flows of this type are almost exclusively the domain of specialized computer programs. With the additional complexity of

© 2001 by CRC PRESS LLC

FIGURE 3.6 Two-dimentional streamlines over a cylinder.

FIGURE 3.7 Simplified three-dimensional flow over a cylinder, showing primary flow, V1, secondary flow, V2, and tertiary flow, V3.

heat transfer and turbulence, this field has become a testing ground for the fastest and most powerful supercomputers.

3.5 INCOMPRESSIBLE IDEAL FLUID FLOW In this section we examine the equations of a hypothetical ideal fluid. We assume that ideal fluids have no viscosity. Viscosity in real fluid problems complicates the problem considerably by introducing the effects of boundary surface roughness, frictional effects, eddies of turbulence, and energy dissipation. Since there is no friction in the flow of an ideal fluid, the fluid motion conforms to Newton’s second law. These equations do not account for the complications caused by compressibility. In electronic cooling problems, there is almost never a reason to consider the compressibility of gases or liquids caused by flow. Even if the gas is compressible, these effects do not affect the accuracy of a dynamic solution until the compression approaches 10%.

© 2001 by CRC PRESS LLC

3.5.1

ONE-DIMENSIONAL FLOW

As we have learned previously, variations in fluid variables characterize one-dimensional flow along, but not across, streamlines. The one-dimensional streamtube describes a cross-sectional fluid area where all fluid properties are uniform. Flow along each individual streamline is strictly one-dimensional. The equations of onedimensional flow can easily approximate many two- and three-dimensional nonturbulent real flows. 3.5.1.1

One-Dimensional Euler Equation

If we examine the one-dimensional cylindrical fluid element streamline shown in Figure 3.8, we see that two forces tend to accelerate the element: a. Pressure forces on the ends of the element, pdA  (p  dp)dA  dpdA, and -----  gdAdz. b. Weight component in the direction of flow, pgdsdA  dz  ds

Acceleration follows the radius r of the streamline, or 2

V a r   -----r The ideal one-dimensional Euler equation for an incompressible fluid of uniform density is 2 P  U d  ------  ------  z  0  g 2g  

FIGURE 3.8 One-dimensional flow along a streamline. The motion of single particles can be described using velocity and acceleration.

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3.5.1.2

One-Dimensional Bernoulli Equation

If the flow is incompressible and the density is uniform, we can integrate the onedimensional Euler equation between any two arbitrary points. We can define a relationship among pressure, p, magnitude of velocity, U, and height, z, for all points on the streamline with the equation 2

p U ------  ------  z  H  Constant  g 2g We call this relationship Bernoulli’s equation. The Bernoulli constant, H, is known as the total head. We most often write the Bernoulli equation in terms of pressure, rather than head, which results in 2

2

p 1  0.5  U 1   gz 1  p 2  0.5  U 2   gz 2 where: p  static pressure 0.5U 2  dynamic pressure gz  potential pressure The total pressure, also called the stagnation pressure, p, is defined as 2

p s  p o  0.5  U o Strictly speaking, Bernoulli’s equation applies only to an infinitesimal streamtube element on a single streamline. The forces normal to the streamlines are ( p1  p2)ds and the component of weight, which is g h ds cos, where cos  (z2  z1)/h. In a theoretical one-dimensional flow, crosswise acceleration is zero, which means that these forces are in equilibrium, which will yield p p -----1-  z 1  -----2-  z 2 g g 3.5.1.3

Application of the One-Dimensional Equations

The Bernoulli equation suggests that as velocity increases, the sum of pressure and potential head ( p/  z) will decrease. Since the variance of z is small in most real engineering applications, the relationship becomes the statement: as velocity increases, pressure decreases. The Torcelli theorem, which is a special case of the Bernoulli equation, shows that the velocity of a jet of an ideal fluid under a static head varies with the square root of the head and can be written as U 

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2gz

Ideally, with no friction, the velocity, U, is equal to that of a falling body from a height of z. In real applications we must consider the effects of fluid shear, wall friction, three-dimensional effects, and surface tension. The combination of Torcelli’s theorem and the Bernoulli equation can be used to solve a wide variety of problems in gravity-fed fluid dynamics.

3.5.2

TWO-DIMENSIONAL FLOW

The solutions to two-dimensional flowfield problems are generally more complex than the one-dimensional problems we covered previously. The formal analysis of two-dimensional problems usually requires partial differential equations. Because of the availability of relatively low-cost software, when a fully two-dimensional problem is encountered, computers are generally the preferred calculation method. Because of the growing use of computers to solve such problems, we will not develop the two-dimensional flowfield equations. 3.5.2.1

Application of the Two-Dimensional Equations

Although quite complex, for irrotational flow of an ideal incompressible fluid we can apply the Bernoulli equation over the entire two-dimensional flowfield. In this type of flowfield, all of the velocity components are related and can be determined by the definition of the streamline and of the differential equation of continuity. Another semiquantitative approach is applicable to common problems. For any element of a streamline having a radius of curvature r, the normal acceleration component, a, is directed toward the center of the radius curvature. Therefore, a  U 2/r. When the flowfield has a single center of curvature, then Ur  Constant. Of course, we cannot integrate the equation over the entire flowfield because the equation applies only to individual streamlines or groups of streamlines with the same radius of curvature. Figure 3.9 shows the effect of the flowfield streamline curvature in a convergentdivergent conduit. The tubes show the pressure at different points in the flow. Streamlines AA along the conduit walls are the most sharply curved. The streamline in the center of the conduit has no curvature. The streamlines between these two extremes have an intermediate curvature. According to the previous equation, because the streamlines close to the wall in section 2 have the most curvature, they also have the highest acceleration. Because of the streamline curvature, we must use two-dimensional solutions to solve these types of problems. If we apply the continuity equation to the conduit of Figure 3.9, we see that Q  A1 U 1  A2 U 2  A3 U 3 and the actual mean velocity of the flow in section 2 of Figure 3.9 is found by A2

1 U 2  ------ 冮 d A A2

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FIGURE 3.9 Two-dimensional conduit flow through a convergent-divergent section.

The limitation of this approach is that the velocity profile is nonuniform at section 2. Therefore, the terms p, z, and also vary. To compensate for this, we can write the Bernoulli equation in terms of total power instead of unit energy. Following the method of previous equations, we can rewrite the Bernoulli equation as: 2 p U Q  -----1-  z 1  ------1  2g  g

冮

3

A2

p

 ------  z d A  冮 ------ d A 2g g

A2

In some flowfields sharp bends in the streamlines are noticeable, such as shown in Figure 3.10. We call the point at the sharp bend a stagnation point. Stagnation points are important in heat transfer because of their effect on the localized heat transfer coefficient. Since the flow dynamics are interrelated, being able to identify possible stagnation points in the flowfield is important. Stagnation points occur at abrupt direction changes on a streamline, when the streamline contacts a solid boundary. The velocity at a stagnation point is zero. In a gravity-fed system, the vertical distance between a stagnation point and the energy level is a measure of fluid pressure at the stagnation point. Since the points in a flowfield are interrelated, the pressure at the stagnation point will allow us to calculate the velocities at other points. For simple geometries, obtaining the complete kinematics of a flowfield by analysis is possible. Consider the pin-fin shown in Figure 3.11. The radial and tangential components of the velocity at any point in the flowfield can be expressed by the equations 2  R

r  U  1  -----2  cos  r  

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and

2  R

t   U  1  -----2  sin  r  

FIGURE 3.10 Stagnation points in two-dimensional flowfields. The stagnation points occur at stationary boundaries, where the local velocity falls to zero.

FIGURE 3.11 Two-dimensional flowfield over a pin-fin.

where:

r

t R r U

    

radial velocity (m/s) tangential velocity (m/s) cylinder radius (m) radius to flow point (m) rectilinear velocity (m/s)

In heat transfer, the velocity along the surface is the primary interest; therefore, R  r, so r  0, and t  2Usin. At the forward and rearward stagnation points © 2001 by CRC PRESS LLC

A and B, respectively,   0 and   ; therefore, t  0. When we apply the Bernoulli equation to the flow, we obtain 2 2 p (  2U sin  ) U p -----o  ------  z o   ---  z  ------------------------------  2g 2g 

In real fluids, we must account for turbulence and the inertial effects of viscosity. Because of these limitations, this solution is accurate only for the front surface of a cylinder at a Reynolds number (Re  (UD)/ ) less than about 10.0.

3.6 INCOMPRESSIBLE REAL FLUID FLOW Up to this point we have studied ideal fluids, that is, fluids that have no viscosity and are therefore frictionless. The Euler and Bernoulli equations are strictly applicable only to ideal fluids. In a real fluid, viscosity causes a resistance to fluid motion by producing shear forces within the fluid and along all surfaces that have relative motion to the fluid. Viscosity is also responsible for the two states of fluid flow called laminar flow and turbulent flow. These dynamic fluid conditions can cause the actual flow measurements to differ appreciably from ideal flow. Researchers have modified the Euler equation to include the shear stress caused by fluid viscosity. When we include these effects, the Euler equation becomes a set of nonlinear, second-order partial differential equations called the Navier-Stokes equations. Unfortunately, since real fluid flow is so complex and varies with time, the Navier-Stokes equations can be used only in the most simple geometries.

3.6.1

LAMINAR FLOW

In this section we will analyze the fluid and heat transfer dynamics of fully developed laminar flow through an isoflux round tube, as shown in Figure 3.12. The velocity

FIGURE 3.12 Force balance on a cylindrical fluid element.

© 2001 by CRC PRESS LLC

distribution within the tube is given by 2

2

r  r dp u ( r )  -----------------s -----4 dx The maximum velocity at the center of the tube cross section is 2

r dp u max  ------s- -----4 dx The fully developed velocity distribution is then found as a parabolic wavefront (see Figure 3.13) in the dimensionless form of

r 2 ----------  1   ----  r s max We obtain the pressure loss inherent in forcing the fluid through a tube of L length from a pressure balance on a fluid element within the tube by 2

 p r s  2 r s s L where:

p  pressure drop per unit length L (p  (dp/dx)L) (N/m2) s  wall shear stress (s  du/drⱍr  rs) (N/m2) We more commonly relate the pressure drop to the Darcy1,2 friction factor f according to 2

L U  p  f ---- ---------D 2g c

FIGURE 3.13 Developing velocity distribution in a laminar flow circular tube.

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Another friction factor is the Fanning friction coefficient, which is four times the Darcy* friction factor. In this text we will use the Darcy friction factor exclusively. We usually know the mass flow in a tube, but it can also be found, when the p is known, by the formula 4

p rs m˙   ---------------8L Likewise, the average velocity in the tube can also be found when the p or the mass flow rate is known: 2

 pr s m˙ U  ------------2   ------------- 8L  r s Since we know that the flowfront is parabolic (U  0.5Umax), we can rearrange the formula to find the p based on the Reynolds number: 2

2

64L U 64 L  U  p  -------------- ------  --------- ---- ---------2 Re 2 D D 2g c U D and from that we can see that rearrangement of the formula yields the relationship of the laminar flow Reynolds number to the Darcy friction factor 64 f  --------Re D In a laminar flow circular conduit, the flowfront becomes stable and fully developed after a number of diameter lengths. According to Chen3, for 1  Re  2000, the number of lengths is given by 0.072 L ----  0.061Re  ---------------------------0.04 Re  1 D Table 3.2 presents other depictions of the velocity profile in a pipe. The laminar profile is a simple parabola, while the blunter turbulent profile can be expressed only as an approximation. Figure 3.14 shows the profiles for fully developed flow.

* Besides the Darcy friction factor, there is also a unit called a Darcy, which is the volume of liquid of unit viscosity passing through a unit area of a porous medium in unit time when subjected to a pressure gradient of one standard atmosphere per unit distance.

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TABLE 3.2 Velocity Profiles in Pipes Centerline Velocity

Description and Range of Applicability 1. Laminar flow profile4 a. 0  Re  2000 b. 0  r  R 2. Viscous sublayer for turbulent flow in smooth pipes5 a. Re  4000 u*  -  5.0 b. ------

u* y u*  -  10.0 c. --------  ------



Flow Profile

Average Velocity

r 2 u ------  1.0   --- R Uc

Uc ------  2.0 U

u* y u -----  ------v u*

—

3. Power law for turbulent core flow6 a. Re  4000 u* y -  70 or y  , b. ------v whichever produces the greater limit; y  R 4. Prandtl’s profile for turbulent core flow a. Re  4000 u* y b. -------  70 or y  , v whichever produces the greater limit; y  R

5. Nikuradse’s profile for turbulent core flow in smooth pipes a. Re  4000 u* y -  70 b. ------v u*  c. ------  1.0 6. Nikuradse’s profile for turbulent core flow in rough pipes5 a. Re  4000 b. y  

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Uc n  1 ) ( 2n  1 ) ------  (---------------------------------------2 U 2n

1 ---

u y n ------  --- U c  R

0.5 1 where: ---  f for f  0.10 n

Uc  u ----------------   Alog  --y-  B 10   u* R A 5.755 5.626

B 0.05 0.86

Ref 5 6

u *y u -----c  Alog 10  --------  B

u* A 5.75 5.60

B 5.50 4.90

Prandtl’s profile can not be used to determine Uc ( n  1 ) ( 2n  1 ) ------  ---------------------------------------2 U 2n

U f 0.5 a * R ------c   --- A log 10  --------- B  8  v  U

Ref. 5 7

u -----  5.75 log 10  --y  B u*  u *  u *   5.5  5.75log  ------- , --------  5.0 10  



  *   70 B  9.5 to 8.5 , u------

 u*    8.5, ------  70

Uc f 0.5 y ------   --- 5.75log 10  --  B  8 U

TABLE 3.2 (continued) Velocity Profiles in Pipes Notes: D  pipe diameter (m)

2

dp U f  friction factor defined such that ------   0.5 -------------dx D R  pipe radius (m)

f

 radius from pipe centerline (m)

r

U  average velocity over the cross section (m/s) Uc  maximum flow velocity along centerline (m/s)  local flow velocity (m/s) f 0.5  0.5 u  friction velocity   ---  U  ---  8   y  distance from pipe wall along radii to centerline (m) u

*

 

 equivalent sand roughness (m) 2 U f  fluid shear stress  --------- --2 4

FIGURE 3.14 Profile of turbulent flow in a smooth-walled pipe: u  local time-averaged velocity; U  average velocity over cross section; R  pipe radius. Prandtl profile has been normalized to power law result at centerline. (Adapted from Blevins, R. D., Applied Fluid Dynamics Handbook, Van Nostrand Reinhold, New York, 1984. With permission.)

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We call the energy required to move the fluid through the tube the pumping power, Pp, which is found by

 pm˙ P p  ----------- p where: Pp  pumping power (W) p  pump efficiency (%) Table 3.3 shows the parameters for laminar flow for a variety of cross sections. For laminar flow the value of K is not significantly affected by surface roughness in the conduit. Table 3.4 shows the solutions for some problems in confined twodimensional flows. The flows are of two types: (1) flows through fixed-wall passages and (2) flows between a moving and a fixed wall. Fixed walls require a pressure gradient to move the fluid, while moving wall flow can exist in uniform pressure if the wall spacing is uniform.

3.6.2

TURBULENCE

AND THE

REYNOLDS NUMBER

Referring to Figure 3.15, at location A we see fluid flowing parallel to the surface of a plate at a constant speed, U. We call this laminar flow. If there are obstacles, or if there is a large degree of surface roughness along the plate, viscous action quickly

FIGURE 3.15 Developing turbulence on a flat plate. (Adapted from Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley & Sons, New York, 1960. With permission.)

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TABLE 3.3 Properties of Laminar Flow Cross Sections8

Cross Section Circle

Semicircle

Laminar Flow Friction Coefficient, f Re Resistance Coefficient, K

Area, A, and Hydraulic Diameter, DH

Circle9

Circle 2

D A ---------4 DH  D

f Re  64 38 K  1.20  -----Re

Semicircle

Semicircle f Re  63.07 K  1.463

2

R A  ---------2

2 R D H  -------------- 2 Circular Sector

Circular Sector 2

R  A  --------2 2R  D H  ------------2

Circular Segment

10 30 45 60 90 120 160 180

Circular Segment 2

R A  ----- (   sin  ) 2 2R (   sin  ) D H  ---------------------------------   2sin --2

Annulus



Annulus A  (a2  b2) DH  2(a  b)

 0 20 60 80 120 180 240 300 b/a 0.0 0.05 0.10 0.50 0.75 1.00

Circular Sector10 f Re K 50.02 53.24 55.13 56.68 59.04 60.80 62.44 63.07

2.410 1.855 1.657 1.580 1.517 1.488 1.468 1.463

Circular Segment11 f Re K 62.22 62.24 62.39 62.51 62.76 63.07 63.36 63.66

1.740 1.734 1.686 1.650 1.571 1.463 1.385 1.341

Annulus4,12 f Re 64.0 86.27 89.37 95.25 95.87 96.0

K 1.25 0.830 0.784 0.688 0.678 0.674 2

64 ( a  b ) f Re  -------------------------------------2 22 2 (a  b ) a  b  ---------------------a log e --b

(Continued) © 2001 by CRC PRESS LLC

TABLE 3.3 (continued) Properties of Laminar Flow Cross Sections8

Cross Section

Laminar Flow Friction Coefficient, f Re Resistance Coefficient, K

Area, A, and Hydraulic Diameter, DH

Eccentric Annulus

Eccentric Annulus4

Eccentric Annulus A  (a2  b2) DH  2(a  b)

f (c  0) ----------------------f (c  0)

c ----------ab 0.01 0.10 0.30 0.50 0.70 0.90

Sector of Annulus

b/a 0.1

0.3

0.5

0.7

0.9

0.9952 0.9909 0.9875 0.9861 0.9855 0.9852

0.9597 0.9256 0.9004 0.8877 0.8829 0.8819

0.9018 0.8246 0.7587 0.7421 0.7315 0.7276

0.8405 0.7219 0.6345 0.5987 0.5827 0.5769

0.7942 0.6391 0.5253 0.4797 0.4594 0.4522

Sector of Annulus13 Values of f Re 

Sector of Annulus b ---

DH

Ellipse

a  2 2 10 A  --- ( a  b ) 2 0.1 57.96 2 2 2(a  b )  ---------------------------------------------------- 0.3 68.28 (a  b)  2(a  b) 0.5 69.48 0.7 62.24 0.9 59.92

30 58.36 60.96 57.88 58.84 76.28

60 58.76 57.88 59.52 68.14 84.72

90 59.22 59.28 64.12 74.48 88.12

180 62.46 67.45 75.06 83.36 92.00

Ellipse4

Ellipse14 A  ab

2

2

2

8D ( a  b ) f Re  -------------------------------2 2 a b

2

4ab 64  16c  D H  --------------  -----------------------a  b  64  3c 4  ab c  -------------ab 0.1  a/b  10

Right Triangle

Right Triangle A 

1 --- ab 2

2ab D H  ------------------------------------------------2 2 0.5 a  b  (a  b )

  tan1 (b/a)

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 10 30 45 60 70 90

Right Triangle15,16 f Re 49.96 52.14 52.62 52.14 51.32 48.00

K 2.40 1.95 1.88 1.95 2.10 2.97

TABLE 3.3 (Continued) Properties of Laminar Flow Cross Sections8

Cross Section Isosceles Triangle

Isosceles Triangle17 f Re K

Isosceles Triangle 2

a A  ----- sin  2 a sin  D H  --------------------1  sin --2-

Square

Rectangle

Laminar Flow Friction Coefficient, f Re Resistance Coefficient, K

Area, A, and Hydraulic Diameter, DH

 10 30 45 60 90 120 150

49.90 52.26 53.06 53.33 52.61 50.98 49.90

Square f Re  56.91 K  1.433

Square A  a2 DH  a

Rectangle A  ab DH

2ab  -------------ab

2.409 1.966 1.853 1.818 1.909 2.165 2.543

Rectangle12,18 f Re

a/b 1.0 2.0 5.0

56.91 62.19 76.28

K 1.433 1.281 0.931

for b/a  1.0 64 f Re  -----------------------------------b 11 b 2 ---  ------ --- ( 2  --- ) a 24 a 3

Rhombus

Rhombus 2 A  a sin   ab D H  a sin   b

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 90 60 45 30 10 0

Rhombus17 f Re 56.91 55.32 53.52 51.21 48.29 48.00

K 1.433 1.673 1.850 2.120 2.693 2.971 (Continued)

TABLE 3.3 (continued) Properties of Laminar Flow Cross Sections8

Cross Section Trapezoid

Area, A, and Hydraulic Diameter, DH Trapezoid b A  --- ( a  c ) 2

Trapezoid17   30 f Re

b/a 4.0 1.0 0.25

2b ( a  c ) D H  ------------------------acd 1 2 2 a  b  --- ( c  a ) ) 4

Laminar Flow Friction Coefficient, f Re Resistance Coefficient, K

51.13 52.98 65.14

0.5

Trapezoid17   45 f Re

b/a 4.0 1.0 0.25

53.29 55.31 71.75

b/a

55.66 56.60 72.21

K 1.716 1.618 1.084

Trapezoid17   85 f Re

b/a 4.0 1.0 0.25

66.96 56.94 73.19

K 1.343 1.552 1.069

Polygon with n Sides

Polygon with n Sides19,20

1 2 A  --- a n cot  4

n Sides

f Re

3 4 5 6 7 8 9 10 20 

53.33 56.91 58.95 60.22 61.24 61.65 62.08 62.40 63.52 64.00

D H  a cot  a a R 1  --------------, R 2  -------------2 sin  2 tan 

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K 1.876 1.744 1.142

Trapezoid17   60 f Re

4.0 1.0 0.25

Polygon with n Sides

K 2.141 1.969 1.288

TABLE 3.3 (continued) Properties of Laminar Flow Cross Sections8

Cross Section Parallel Plates

Rectangular Duct with Round Edges

Laminar Flow Friction Coefficient, f Re Resistance Coefficient, K

Area, A, and Hydraulic Diameter, DH

Parallel Plates9 f Re  96.0 38 K  0.64  -----Re

Parallel Plates A  ab DH  2b a  b

Rectangular Duct with Round Edges 2 A  2ab  (4  )b 2

4ab  2 ( 4  )b D H  ----------------------------------------------a  ( 2  4 )b

Tube with Internal Fins

Tube with Internal Fins 2

d A  ---------  nth 4

Rectangular Duct with Round Edges21 2b/a f Re 0.15 83.76 0.20 80.52 0.25 77.64 0.40 71.04 0.50 68.12 0.667 65.28 1.00 64.0 Tube with Internal Fins22–24 D 1.18 f Re  64.3  ----  d

2

d  4nth D H  ---------------------------- d  2nh n  number of fins

Arbitrary Cross Section

Arbitrary Cross Section 4A D H  ------P

Arbitrary Cross Section 48.0  f Re  96.0 0.5  K  2.5

(Continued)

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TABLE 3.3 (continued) Properties of Laminar Flow Cross Sections8 Notes: A

 area of cross section (m2)

DH  hydraulic diameter (m) K  inlet resistance coefficient for laminar flow k

 f Re  friction coefficient for laminar flow 2

fU dp  friction factor defined so that the spanwise pressure gradient is ------   0.5 ---------DH dx Re  Reynolds number, based on average velocity

f



 angle (degrees)

dampens the internal disturbances. As the distance, x, from the leading edge increases, the fluid begins to develop internal disturbances, shown at location B. At this distance, the inertial effects of the fluid overcome the dampening effects and small disturbances are formed within the fluid. As x increases further, the disturbances grow until they destroy the uniform motion of the flow, shown at location C. We call the fluid motion after location C, fully turbulent. In the fully turbulent flow regime, macroscopic parcels of the fluid move across the parallel streamlines and vigorously transport momentum and thermal energy. Note that in this region of large-eddy, chaotic motion, particles of fluid can actually move in the opposite direction of the bulk fluid flow. The shift from laminar to turbulent flow is not a well-defined point but a zone in which the chaotic motion overwhelms the laminar flow. The transition to turbulent flow along a flat surface is based on the relationship among the fluid properties, the flow distance x, the roughness of the surface, and the flow velocity. We call the parameter that indicates the turbulence of a flowing fluid the Reynolds number, Re. It is one of the most important dimensionless groups in fluid dynamics and heat transfer. This number commemorates Osborne Reynolds.28 The Reynolds number describes the ratio of inertial force to viscous force within the fluid stream and is found by:

U  x Ux Re  ------------ --------

where: Re  U x

     

Reynolds number, dimensionless fluid density (kg/m3) free-stream velocity (m/s) length from leading edge (m) absolute viscosity (N s/m2) kinematic viscosity (m2/s)

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TABLE 3.4 Confined Laminar Flow25–27 Geometry Two-Dimensional Channel

Flow Parameters Two-Dimensional Channel y dp u  -------- ( h  y ) -----2 dx h dp   --- -----2 dx 3

h b dp Q  ---------- -----12 dx x p  p 1   p p1  p2 --L dp ------  constant dx Two-Dimensional Shear Flow

Two-Dimensional Shear Flow y u  --- U h U    ---- , on upper surface h Uhb Q  ----------2 p  constant

Two-Dimensional Flow through a Converging or Diverging Slot

2

2

2

h2 ( h1  h ) -  p p1  p2 p  p 1  --------------------------2 2 2 h ( h1  h2 ) 2 2

h 1 h 2 b  p p1  p2 Q  ---------------------------------6 L ( h1  h2 ) L  p p1  p2 b -  p 1 Lb T  -------------------------h 1  ----1h2

h1  h2 --------------------  1.0. Valid for either h  h 1 2 L (as shown) or h2  h1.

(Continued) © 2001 by CRC PRESS LLC

TABLE 3.4 (Continued) Confined Laminar Flow25–27 Geometry Two-Dimensional Slipper Bearing

Flow Parameters Two-Dimensional Slipper Bearing 6 U x(L  x) p  p 0  --------------------------------------------------------------------------x 2 1 ˙ 2h 2  1  h 2 - ------------------  --- ( h 1  h 2 )  ----------------- h1  h2   h1  h2 L 2 h1 2 ( h1  h2 ) 6 L Ub T  -------------------------2 log e -----  -------------------------h1  h2 h2 ( h1  h2 )

h1 h 1 ( h 1  2h 2 ) -------------------------------- log e ----h2 h1  h2

 0.5 ( 5h 1  h 2 ) x P  L -------------------------------------------------------------------------------h1 -  2h 2  2h 1 ( h 1  h 2 )log e ---h 2

h T ( h2  h1 ) ULb -  ------------------ log e  -----1 F  ------------------------- h 2 2L h1  h2 h 1 h 2 Ub Q  -----------------h1  h2 X P  hydrodynamic center Two-Dimensional Squeeze Film Bearing Load

Two-Dimensional Squeeze Film Bearing Load bx dh u  ------ -----h dt dh Q  2 xb -----dt 2 6 dh  L 2 p  ------3- ------  -----  x  4 dt h   L2 3

冮

b L dh - -----pb dx  -----------3 h dt L  2 3 1 bL 1 t  -------------  -----2  -----2 , time to close gap h1 to h2. 2W  h h W 

2

Axisymmetric Radial Flow Between Two Stationary Disks

1

Axisymmetric Radial Flow between Two Stationary Disks 2 3Q  y y  u  ----------  ---  -----2

hr  h h 

h dp   --- -----2 dr 3

h  p p1  p2 Q  ----------------------------R˙2 6 log e  ---- R1 6 Q r p  p 1  -----------3- log e ----R1

h Fluid is injected along axis and flows out radially

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TABLE 3.4 (continued) Confined Laminar Flow25–27 Geometry Flow between a Rotating Disk and a Stationary Disk

Flow Parameters Flow between a Rotating Disk and a Stationary Disk 4 2 y  r  u  -------------  hy  -----2 12  h 2

6 Q 3  2 r 2 p  p 1  ------------- ( r  R 1 )  -----------3- log e ----20 R1

h ry 

 ---------h 3

2

2

2

h  ( R 2  R 1 ) Q  -------------------------------------------R˙2 40 log e  ---- R1 r   ----------h Axial Flow in Annulus between Two Concentric Cylinders T3-4h

Axial Flow in Annulus between Two Concentric Cylinders 2 2 R2  R1 r  p p1  p2 1 2 2 -log e  ----- ------------------u  ------- R 1  r  -----------------R2   R 4 L 1 log ( ----- ) e

R1

2 2 R 2  R 1  p p1  p2

2 2 2 2 - -------------------Q  ------- ( R 2  R 1 ) R 2  R 1  -----------------R 8 L log ( -----2 ) e

R1

p  p 1   p p1  p2 --x-, dp ------  constant t L dx Circumferential Flow between Outer Stationary Cylinder and Inner Rotating Cylinder

Circumferential Flow between Outer Stationary Cylinder and Inner Rotating Cylinder 2

2

R1 R2   1 r -2 ---  -----2

 -----------------2 R 2  R 1  r R 2 2

2R 1  -2 , on inner cylinder   -----------------2 R1  R2 2

2

2 LR 1 R 2   R R Q  ------------------------1 -----2 2 2 0.5  1  2  log e R1 R1  R2 R 2 

p  constant Also valid for R1  R2; i.e., outer cylinder rotates. (Continued)

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TABLE 3.4 (continued) Confined Laminar Flow25–27 Axisymmetric Squeeze Film Bearing Load

Axisymmetric Squeeze Film Bearing Load r dh u  --- -----h dt 2 dh Q  r -----dt

3 dh 2 2 p  ------3- ------ ( R  r )  p 0 h dt R

W 

冮  p p  p ( 2 r ) dr  0

0

4

3 R dh ----------------- -----3 2h dt

Time for constant load W to close gap h1 to h2: 4

1 3 R 1 t  ------------------  -----2  -----2 4W  h 2 h 1

Notes: b  width (m)

u  axial velocity (m/s)

F  force per unit width (N/m)

 transverse velocity (m/s)

h  gap width (m)

W  force (N)

L  length (m)

x  axial distance (m)

p  static pressure (N/m2)

y  transverse distance (m) 3

Q  volumetric flow rate (m /s)

z  vertical velocity (m/s)

r  radius (m)

  shear stress on surface (N/m2)

R  overall radius (m)

 absolute viscosity (N s/m2)

T  thrust (N)

  rate of rotation (rad/s)   density (kg/m3)

On a flat plate, the transition from laminar to turbulent flow begins at a Reynolds number of about 200,000 for a very rough surface. The transition ranges up to about 1,000,000 for an extremely smooth surface. The value is normally considered to be 300,000. The Reynolds number for fluid flow within an enclosed channel such as a round pipe or hose is based on the diameter, D, not the flow length x:

U D Re  ------------- We consider the Reynolds number that suggests the transition from laminar to turbulent flow within a round pipe to be from about 2300 to 4000. For flow © 2001 by CRC PRESS LLC

FIGURE 3.16 Flow through a noncircular duct may be calculated by using the principle of equivalent hydraulic diameters.

between two parallel plates, turbulence begins at ⬃1000; flow in an open channel becomes turbulent at ⬃500, and for flow around an immersed sphere or cylinder in cross flow, the critical Reynolds number is 10.0. These velocity profiles are not at all similar to the flow dynamics of a round tube, but we can still find the Reynolds number by using a hydraulic diameter, DH. We sometimes call this characteristic dimension the equivalent diameter (see Figure 3.16). We find the hydraulic diameter by the equation: 4A 2LW D H  ---------c  ---------------LW P where: DH  hydraulic diameter (m) A  area of flow (m2) P  wetted perimeter (m) After overcoming the entrance effect, a fluid moving slowly through a pipe will always remain at the same laminar Reynolds number throughout its length, if the velocity remains below a critical velocity. The critical velocity is the speed that a fluid will start the transition to turbulence, if other parameters remain constant. Flow along a flat plate will always become turbulent after the critical length, x, is reached.

3.6.3

BOUNDARY LAYER THEORY

In real fluid flows, friction and drag losses are confined to the thin fluid layer surrounding a stationary object. The dynamics of this layer most directly affect the rate of heat transfer. © 2001 by CRC PRESS LLC

In 1904, Prandtl clarified the interaction of a dynamic fluid and a stationary surface by introducing the theory of boundary layers.29 Prandtl’s concept allowed the application of inviscid solutions to viscous problems. If the boundary layer is thin compared with the flow channel, and no flow separation occurs, friction and drag are confined to this layer. Therefore, the flow outside the boundary layer can be considered ideal. The boundary layer phenomenon is found in both natural and forced-convection modes of heat transfer. We will define the thickness of the boundary layer as the distance from the stationary surface to a distance where the moving fluid is less than 99% of the free-stream velocity. To understand further the ideas of the velocity boundary layer, we will examine forced air flow over a flat plate. As stated previously, the fluid particles directly in contact with an object are stationary and have a velocity of zero. The stagnant layer impedes the velocity of the fluid particle layer directly above the layer of zero velocity. In turn, each underlying layer of fluid particles impedes the velocity of the layer directly above it. At some distance above the surface, the velocity impediment is negligible. We call this distance the velocity boundary layer thickness, . The thickness of this layer is primarily dependent upon the shear stress within the fluid and therefore its turbulence, as described by the Reynolds number. The velocity boundary layer increases in thickness at greater distances from the initiation point, in the direction of fluid motion. We find the thickness of the velocity boundary layer, , at a distance of x, by the equation: 5.0x   -----------Re x Since the shear stress has an influence on the surface friction, we can find the skin friction coefficient at a distance from the initiation point by applying the equation: Cf , x  0.6641Re x

 0.5

and we can find the average friction coefficient over the flat surface by:  0.5

C f,x  1.3282Re x

Similar to the velocity boundary layer, , there is a thermal boundary layer, th, which forms when there is a difference in temperature between the free stream and an object. To learn other significant parameters of the hydrodynamic flow, we can also find a displacement boundary layer, designated d. The displacement boundary layer represents the decrease in the volume rate of flow, per unit width, due to the presence of the boundary layer. The boundary layer displaces the free-stream lines of flow by the distance d. We relate the displacement boundary layer to the Reynolds number by the equation  0.5

 d  1.7208xRe x

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Viscous motion characterizes laminar flow of a fluid. Parcels of fluid follow well-defined parallel flowpaths called streamlines. Molecular diffusion transfers heat and momentum across streamlines and cross flow is negligible. Since fluid motion at the wall surface is zero, heat transfer through this layer is by conduction. To simplify the problem we will assume that the flow is laminar and parallel to the heat transfer surface. The flow is steady and incompressible, and the fluid properties are constant. With these assumptions, we can reduce the boundary layer equations to more simple forms by the assumption of constant fluid properties and constant hydrodynamic boundary layer conditions. By solving for the hydrodynamic variables, we can then solve for the energy and species equations that depend on the variables u and . Blasius30 found a solution for the hydrodynamic components by defining the velocity components as a dimensionless stream function such that the continuity equation is automatically satisfied and no longer needed. Blasius introduced a similarity solution by using dependent and independent variables to reduce the partial differential equations of the momentum equation to ordinary differential equations. If we define the boundary layer as the distance where u reaches 99% of the free stream value, we find y U  x  5.0 at  -- ------------ x

U

and therefore, at 99% of the free-stream velocity, we relate the boundary layer thickness to the Reynolds number, (Ux)/ , by 5x   -----------Re x This agrees with what we know of the boundary layer:   0 at the beginning of a plate (x  0) and increases with increasing length along the flowpath, and the boundary layer thickness will decrease as the velocity increases. The shear force of the fluid friction at the wall can be found from the velocity gradient at y  0, so that at any value of length, x, the velocity gradient at the heat transfer surface is

u -----y

y0

U  0.332 ------- Re x x

Substituting this resulting velocity gradient into the equation for shear, the fluid shear at the wall per unit area s is seen as U  s  0.332 ------- Re x x Related to a Reynolds number using length as the characteristic dimension, we see that the fluid shear at the leading edge of the wall surface is very large and decreases © 2001 by CRC PRESS LLC

with increasing distance from the leading edge. If we divide both sides of the equation 2 by the velocity pressure of the free stream,  U  /2, we find the dimensionless local drag coefficient, also called the skin friction coefficient, Cfx; therefore, 0.664 C fx  ------------Re x The average value of the local friction coefficient can be found be integrating between the leading edge at x  0 and the local length at x  L: C f  1.328 Re L For laminar flow over the flat plate, we see that at a given length the average friction coefficient is twice that of the local friction coefficient. Table 3.5 shows the characteristics of boundary layer flow over a flat plate. The properties are dependent on the Reynolds number and the plate roughness. Table 3.6 shows the properties of two-dimensional laminar boundary layers.

TABLE 3.5 Boundary Layer Flow over a Flat Plate31 Turbulent, Re > 106 Property

Laminar, Re < 10

5

Smooth Plate

Fully Rough Plate –

Displacement Thickness, d

1.7208 x ------------------0.5 Re

0.018 x ---------------1/7 Re

Momentum Thickness, m

0.6641 x ------------------0.5 Re

0.015 x ---------------1/7 Re

0.5 x --------------------------------------------------------------2.57 [ 2.635  0.618log e ( --x ) ]

Skin Friction Coefficient, c

0.6641 x ------------------0.5 Re

0.026 -----------1/7 Re

1 --------------------------------------------------------------2.46 [ 3.476  0.707log e ( --x ) ]

Drag Coefficient, CD

1.328 x ---------------0.5 Re

0.0303 --------------1/7 Re

1 --------------------------------------------------------------2.57 [ 2.635  0.618log e ( --x ) ]

Notes: CD  drag coefficient 2

FD  total drag on one side of plate between 0 and x, F D  0.5  U xbC D b

 breadth of plate (m) 2

c  skin friction coefficient, where shear stress  s  0.5  U c  U  free stream velocity (m/s)

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TABLE 3.5 Boundary Layer Flow over a Flat Plate31 (Continued) x

 distance from origin of boundary layer (m)

d  boundary layer streamline displacement distance (m) m  decrease in momentum flux due to the boundary layer 

 surface roughness height (m)

Laminar formulas are exact. Turbulent solution errors are 5%.

TABLE 3.6 Properties of Two-Dimensional Laminar Boundary Layers Property Skin Friction Coefficient, c

Similarity Solution 2 v -----------------------  ------- 0.5 ( 2   )  Ux

0.5

2

d f ---------2 d

0

Flat Plate (␤ ⴝ 0)

0.5 0.6641  -------  Ux

—

v 0.5 1.3282  -------  Ux

Displacement Thickness, d

v 0.5 * 0.5 ( 2   ) x  -------  d  Ux

v 0.5 1.7208 x  -------  Ux

Momentum Thickness, m

v 0.5 * 0.5 ( 2   ) x  -------  m  Ux

v 0.5 0.6641 x  -------  Ux

Drag Coefficient, CD

Shear Thickness, sh

v 0.5  d 2 f 0.5 ( 2   ) x  -------  ------- Ux  -2 d

 1.0   0

Notes:

s C  -----------˙ 2U   --------2   FD CD  -----------˙ 2U   -------- 2  U  free stream velocity (m/s) x

 axial distance from point of zero boundary layer thickness (m)

d  boundary layer streamline displacement distance (m) d 0.5 * -Re  d  -------------------------0.5 (2  ) x m  decrease in momentum flux due to the boundary layer m 0.5 * -Re  m  -------------------------0.5 (2  ) x sh  frictional shear resistance of the body surface

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v 3.0116 x  -------  Ux

0.5

FIGURE 3.17 A view of the transition to turbulence along a flat plate. (Adapted from White, F. M., Viscous Fluid Flow, McGraw-Hill, New York, 1974. With permission.)

3.6.4

TURBULENT FLOW

As the flow over a plate exceeds a critical Reynolds number, the flow becomes unstable, as shown in Figure 3.17. The streamlines are mixed within a short distance after the transition to turbulence. Turbulence is characterized by rapidly fluctuating eddies that transport fluid parcels in random and irregular patterns. Groups of particles collide with each other and cross flow occurs on a large scale. Because of this fluid particle motion, the thickness of the boundary layer is much smaller than in laminar flow. The thickness of the boundary layer increases much more rapidly in turbulent flow than in laminar flow and is related to the Reynolds number by  0.20

  0.37xRe x

Within this range, the derivation of a unifying relationship between fluid flow and the Nusselt number is difficult. Turbulence promotes the continual transportation of mass and momentum across a plane normal to the y-axis. Figure 3.18 shows the velocity profile for turbulent flow along several surfaces.

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FIGURE 3.18 Turbulent boundary layer profiles for a flat plate, flows with pressure gradient, and flow in a smooth tube. (From Mills, A. F., Heat and Mass Transfer, Irwin, Chicago, 1995. With permission.)

3.7 LOSS COEFFICIENTS AND DYNAMIC DRAG So-called minor losses occur downstream from an anomaly that causes an extraction of the useful energy within the flow. This large-scale turbulence in the flow is superimposed upon, and mixed with, the normal turbulence pattern in the flow. To calculate the effect of the minor loss turbulence, decoupling the minor loss turbulence from the normal turbulence is necessary. We may then state the minor loss as a loss of head, HL, and restrict it to a section of flow downstream from the anomaly. In this manner we can simply add several minor losses to the overall head loss caused by wall friction. Experimenters have found that head loss is related to the square of the fluid velocity and a loss coefficient, KL, which varies with the geometry of the anomaly; therefore, 2

U H L  K L -----2g The loss coefficient, KL, follows the same trend as the friction factor, f. That is, the value increases with increasing roughness and decreasing Reynolds number. These effects stabilize as the Reynolds number increases, so that the values are nearly constant across a range of high Reynolds numbers. In measurements made by researchers, the loss coefficient, K, represents the sum of three losses: 1. the pressure loss due to the anomaly, 2. the pressure loss in the upstream flow section in excess of the normal pressure loss, and 3. the pressure loss in the downstream flow section in excess of the normal pressure loss.

3.7.1

EXPANSIONS

When a constrained flow reaches an abrupt expansion, such as a small pipe flow entering a larger pipe as shown in Figure 3.19, the fluid velocity rapidly decreases. The deceleration causes large-scale turbulence. Researchers have found that simultaneously applying the Bernoulli equation with the continuity and momentum principles can relate the head loss to the flow variables. In this relationship, the loss coefficient has a value of 1.0; therefore, 2

(U1  U2) H L  K L --------------------------2g When the abrupt expansion is very large, for example, when a small pipe enters a large reservoir, U2 may be taken as zero, so that the head loss is equal to the velocity head in the pipe.

© 2001 by CRC PRESS LLC

FIGURE 3.19 Fluid flow through an abrupt expansion at a high Reynolds number.

FIGURE 3.20 Geometry of a gradual expansion.

The following equation can calculate the value of KL for an abrupt expansion more closely: A1 2 K   1  ------  A 2 When the expansion occurs gradually, as shown in Figure 3.20, the loss coefficient is related to the shape of the expansion. Gibson32 explored the relationship of conical expansions to loss coefficients and found the loss coefficient to be related primarily to the angle of the expansion, but it was also affected by the ratio of the pipe areas. Alternately, the following equation can be used to find the loss coefficient for a conical expansion when the included angle of the expansion is 45°: 2  D 1  K   1.0  ------2 2.6 sin --2 D 2 

© 2001 by CRC PRESS LLC

FIGURE 3.21 Fluid flow through an abrupt contraction at a high Reynolds number.

When the included angle of the expansion is 45°, the loss coefficient is found by: 2  D 2 K   1.0  ------2 D 1 

2

Table 3.7 shows the solutions to pressure loss in a variety of expansion and contraction configurations. The data were taken from experiments using round pipes. The values can be used for turbulent flow through noncircular conduits by using DH. Table 3.8 shows the discharge coefficients for several nozzles. The actual flow rate through one of these geometries is almost never equal to the theoretical flow rate. The discharge coefficient, C, corrects for the actual flow rate. A flow rate through a geometry where C  1.0 conforms exactly to the theoretical flow rate.

3.7.2

CONTRACTIONS

The so-called vena contracta characterizes an abrupt contraction, such as shown in Figure 3.21. Because of the complex flow effect caused by the vena contracta at location Ac, experimental results conflict on the exact loss coefficient. When the area ratio of A2/A1 is zero, the loss coefficient is 0.50. When the area ratio is 1.0, the loss coefficient is zero. The loss coefficient varies in a fairly linear relationship between these two extremes. A formula that has very close results to the existing data is 2

D 2  K  0.5  1.0  ------2 D 1  An extreme case is the so-called re-entrant contraction, shown in Table 3.7c. This type of entrance, when the pipe wall is very thin, can result in a loss coefficient of 1.0. Harris33 concluded that if the pipe wall thickness is 0.50D, the loss coefficient drops to 0.50. Idel’chik34 showed that the loss coefficient is based not only on the ratio of the wall thickness to the diameter, but also on the ratio of the length of the protrusion to the diameter. © 2001 by CRC PRESS LLC

TABLE 3.7 Pressure Loss in Abrupt Contractions and Expansions Description and ⌬ p Pressure Change

Loss Coefficient, K

Entrance Flush with Wall at Right Angle

Entrance Flush with Wall at Right Angle K  0.5 where L D

p fL ------------------2  1 K -----D 0.5  U Entrance Flush with Wall at Arbitrary Angle

Entrance Flush with Wall at Arbitrary Angle36 2

K  0.5 0.3 cos  0.2cos 

fL p ------------------2  1 K -----D 0.5  U Protruding Entrance

Protruding Entrance33,34 b/D a/D 0 0.005 0.01 0.02 0.03 0.05 0.05

0.005 0.01 0.10 0.50 0.63 0.57 0.54 0.51 0.50 0.50 0.50

0.68 0.62 0.57 0.52 0.51 0.50 0.50

0.86 0.79 0.71 0.60 0.54 0.50 0.50

1.0 0.93 0.86 0.72 0.61 0.5 0.5

0.5 1.0 0.90–0.93 0.82–0.86 0.66–0.72 0.50–0.60 0.43–0.50 0.44–0.50

fL p ------------------2  1 K -----D 0.5  U

(Continued)

© 2001 by CRC PRESS LLC

TABLE 3.7 (continued) Pressure Loss in Abrupt Contractions and Expansions Description and ⌬ p Pressure Change

Loss Coefficient, K Rounded Entrance34 R/D K

Rounded Entrance

0 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.12 0.16 0.20 0.60

fL p ------------------2  1 K -----D 0.5  U

Rounded Entrance in Wall

1.00 0.87 0.74 0.61 0.51 0.40 0.32 0.20 0.10 0.06 0.03 0.01

Rounded Entrance in Wall34,37 R/D K 0 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.12 0.16 0.16

fL p ------------------2  1 K -----D 0.5  U

Conical Entrance34 

Conical Entrance a/D

p fL ------------------2  1 K -----D 0.5  U

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0.44–0.5 0.35–0.43 0.28–0.36 0.22–0.31 0.17–0.26 0.13–0.22 0.10–0.20 0.07–0.15 0.03–0.09 0.0–0.06 0.0–0.03

0.025 0.050 0.100 0.250 0.600 1.000

0

10

20

40

1.0 1.0 1.0 1.0 1.0 1.0

0.96 0.93 0.80 0.68 0.46 0.32

0.93 0.86 0.67 0.45 0.27 0.20

0.86 0.75 0.48 0.22 0.14 0.11

60

100

140

0.80 0.67 0.41 0.17 0.13 0.10

0.69 0.58 0.41 0.22 0.21 0.18

0.59 0.53 0.44 0.34 0.33 0.30

TABLE 3.7 (continued) Pressure Loss in Abrupt Contractions and Expansions Description and ⌬ p Pressure Change

Loss Coefficient, K

Conical Entrance in Wall

Conical Entrance in Wall34  a/D

0

10

20

40

60

100

140

180

0.025 0.050 0.075 0.100 0.150 0.600

0.5 0.5 0.5 0.5 0.5 0.5

0.47 0.45 0.42 0.39 0.37 0.27

0.45 0.41 0.35 0.32 0.27 0.18

0.41 0.33 0.26 0.22 0.16 0.11

0.40 0.30 0.23 0.18 0.15 0.12

0.42 0.35 0.30 0.27 0.25 0.23

0.45 0.42 0.40 0.38 0.37 0.36

0.5 0.5 0.5 0.5 0.5 0.5

p fL ------------------2  1 K -----D 0.5  U Rounded Entrance in Wall with Neighboring Wall

Rounded Entrance in Wall with Neighboring Wall34 R/D a/D

0.10 0.125 0.150 0.200 0.300 0.500 0.800

0.20 0.30 0.50

– 0.80 0.45 0.19 – 0.50 0.32 0.17 0.65 0.36 0.25 0.10

.09 .07 .05

0.05 0.05 0.04 0.04 0.03 0.03

p fL ------------------2  1 K -----D 0.5  U Entrance with Neighboring Wall

Entrance with Neighboring Wall34 (a) For Straight Entrances: K with wall  K without wall  A 2 (b) For Flanged Edges: K with wall  K without wall   ------2  A 1 0.3

0.4

0.5

0.6

0.7

0.8 1.00

a/D

0.2



1.60 0.65 0.37 0.25 0.15 0.07 0.04

0.0

p fL ------------------2  1 K -----D 0.5  U (Continued)

© 2001 by CRC PRESS LLC

TABLE 3.7 (continued) Pressure Loss in Abrupt Contractions and Expansions Description and ⌬ p Pressure Change

Loss Coefficient, K

Entrance with Inlet Screen

Entrance with Inlet Screen For Straight Entrances: K with screen  K without screen K screen For Flanged Edges: A 2 K with screen  K without screen K screen  ------2  A 1

p fL ------------------2  1 K -----D 0.5  U

See Table 3.10 for Kscreen values

Abrupt Contraction34,38

Abrupt Contraction

A 2 K  0.5  1  ----A 1  Abrupt Contraction K  0.5781 0.3954 0.5  4.5385 14.24 1.5 19.22 2 8.540 2.50 A2 2 f 2 L2 f 1 L1 p ------------------2  1   ------ K ---------- ---------A   D D2 1 1 0.5  U 2

A where:  ------2 A1

A1/A2 1.2 1.5 2.0 3.0 5.0 10.0

© 2001 by CRC PRESS LLC

A1 U ------2  -----A2 U1

Gradual Contraction34 For conical contraction ᐉ/D2

Gradual Contraction

f 2 L2 f 1 L1 A 2 p ------------------2  1   ------2 K ---------- ---------- A 1 D2 D1 0.5  U

and

0 0.08 0.17 0.25 0.33 0.40 0.45

0.05 0.10 0.15 0.60 0.06 0.12 0.23 0.31 0.38 0.45

0.04 0.09 0.17 0.27 0.35 0.41

0.03 0.07 0.14 0.23 0.31 0.39

0.03 0.06 0.06 0.08 0.18 0.27

TABLE 3.7 (continued) Pressure Loss in Abrupt Contractions and Expansions Description and ⌬ p Pressure Change

Loss Coefficient, K

d/D

Sharp-Edged Orifice39,40 0.20 0.25 0.30 0.35 0.40 0.45 0.50

K

65.0 39.0 27.0 19.0 14.0 10.0 7.7

d/D

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

K

5.8 4.2

Sharp-Edged Orifice

3.1

2.3

1.5

0.97 0.55 0.26

p fL ------------------2  K -----D 0.5  U

Flow into Multi-Channel Core

Flow into Multi-Channel Core2,5 A2 K  0.5  1  ------  A 1 A2 

冱 A 2i i

 sum of flow areas

p f 2 L2 f 1 L1 A 2 2 ------------------2  1   ----- ---------- K --------- A 1 D1 D2 0.5  U 2

Exit From Straight Pipe

Exit From Straight Pipe K  1.0 (K is independent of ᐉ)

fL p ------------------2  ------  1 K D 0.5  U (Continued)

© 2001 by CRC PRESS LLC

TABLE 3.7 (continued) Pressure Loss in Abrupt Contractions and Expansions Description and ⌬ p Pressure Change

Loss Coefficient, K

Exit from Straight Pipe with Screen

Exit from Straight Pipe with Screen K  1.0 see Table 3.10 for Kscreen values

p fL ------------------2  K screen ------  1 K D 0.5  U Abrupt Expansion

Abrupt Expansion A 1 2 K   1  ----A 2 

A 2 p fL ------------------2   ------2  1 K -----  D A 0.5  U 1 1 Gradual Expansion41 2ᐉ/D1

Gradual Expansion

f L2 A 2 p ------------------2   ------2  1 K --------D2  A 1 0.5  U 1

A2/A1

0.1 0.2

0.3

0.5

1.0

2.0

3.0

1.2 1.4 1.6 2.0 2.5 3.0 4.0

0.06 0.10 0.17 0.25 0.35 0.45 0.60

– 0.08 0.12 0.23 0.32 0.45 0.60

– 0.07 0.10 0.20 0.35 0.45 0.60

– 0.06 0.08 0.15 0.25 0.37 0.55

– – 0.06 0.08 0.10 0.22 0.42

– – – – – – 0.06 – 0.08 0.06 0.15 0.10 0.40 0.30

Expansion from Multichannel Core

– 0.09 0.13 0.25 0.35 0.45 0.60

5.0

Expansion from Multichannel Core A1 2 - K   1  ----A 2  A1 

A2 2 fL p ------------------2   ------  1 K -----D A1 0.5  U 1

© 2001 by CRC PRESS LLC

冱 A 1i i

 sum of the flow area on the left-hand side.

TABLE 3.7 (continued) Pressure Loss in Abrupt Contractions and Expansions Notes: A  cross-sectional area (m2)

R  radius (m)

D  diameter or hydraulic diameter (m)

U  average flow velocity (m/s)

f  friction factor

  angle

K  loss coefficient

  fluid density (kg/m3) ᐉ  flow length of conical section (m)

L  flow length (m) 2

p  static pressure (N/m ) All solutions are for L  D Results are for turbulent flow, Re  104

FIGURE 3.22 Geometry used to describe a gradual contraction.

Table 3.7e shows a rounded or bell-mouthed entrance. Hamilton35 studied these intakes and determined that if the radius of the rounding is 0.14D, we eliminate the vena contracta and the loss coefficient drops to about 0.10. The effect of a gradual contraction varies, depending on the degree of streamlining. A well-designed contraction has a loss coefficient of about 0.040. An optimized contraction may reduce the loss coefficient to about 0.020. An optimized contraction does not necessarily mean a long transition, because as the length of the contraction section grows, so does the loss attributed to wall friction. Figure 3.22 shows the nomenclature used to describe a gradual contraction. The following equation can be used to find the loss coefficient for a conical contraction when the included angle of the contraction is 45°: 2  D 2  K  0.5  1.0  ------2 1.6 sin --2 D  1

© 2001 by CRC PRESS LLC

TABLE 3.8 (continued) Nozzle Discharge Coefficients Description

Discharge Coefficient, C Conical Entrance from Plenum34 ␪

Conical Entrance from Plenum ᐉ/D

0

10

20

40

60

100

140

0.025 0.050 0.10 0.25 0.60 1.00

0.71 0.71 0.71 0.71 0.71 0.71

0.71 0.72 0.75 0.77 0.83 0.87

0.72 0.73 0.77 0.83 0.89 0.91

0.73 0.76 0.82 0.91 0.94 0.95

0.75 0.77 0.84 0.92 0.94 0.95

0.77 0.80 0.84 0.91 0.91 0.92

0.79 0.81 0.83 0.86 0.87 0.88

Conical Entrance in Wall1 ␪

Conical Entrance in Wall ᐉ/D 0.025 0.050 0.075 0.10 0.15 0.82 Rounded Entrance from Plenum

0 0.82 0.82 0.82 0.82 0.82 0.89

Rounded Entrance R/D C 0.0 0.71 0.01 0.73 0.02 0.76 0.03 0.79 0.04 0.81 0.05 0.85

20 0.83 0.84 0.86 0.87 0.89 0.95

40 0.84 0.87 0.89 0.91 0.93 0.94

60 0.85 0.88 0.90 0.92 0.93 0.90

100 0.84 0.86 0.88 0.89 0.89 0.86

140 0.83 0.84 0.85 0.85 0.85 0.82

from Plenum1 R/D C 0.06 0.87 0.08 0.91 0.12 0.95 0.16 0.97 0.20 0.99 0.60 1.00

Rounded Entrance in Wall34,37 C

Rounded Entrance in Wall R/D 0.0 0.01 0.02 0.03 0.04 0.05

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10 0.82 0.83 0.84 0.85 0.85 0.92

C34 0.82 0.84 0.86 0.87 0.89 0.91

C37 0.83 0.86 0.88 0.91 0.92 0.94

R/D 0.06 0.08 0.12 0.16 0.16

C34 0.91 0.93 0.96 0.97 0.99

C37 0.95 0.97 0.99 1.00 1.00

180 0.82 0.82 0.82 0.82 0.82 0.60

TABLE 3.8 (continued) Nozzle Discharge Coefficients Description

Discharge Coefficient, C Gradual Contraction34 ᐉ/D2

Gradual Contraction A1/A2 0.0

0.05

0.10

0.15

0.60

1.2 1.5 2.0 3.0 5.0 10.0

0.91 0.91 0.88 0.86 0.85 0.83

0.94 0.93 0.90 0.88 0.86 0.84

0.95 0.94 0.92 0.89 0.87 0.85

0.95 0.95 0.96 0.92 0.92 0.89

0.89 0.88 0.87 0.85 0.84 0.83

Angled Exit Nozzle42 ␪

Angled Exit Nozzle d/D

5

15

30

40

90

0.5 0.6 0.8 0.9

0.91 – – 0.91

0.89 0.88 0.88 0.86

0.79 – – 0.80

0.75 0.72 0.76 –

0.60 0.64 0.69 0.76

d/D

Ring Exit Nozzle43 C d/D

0.0–0.5 0.5 0.6 0.7 0.8

0.61 0.63 0.65 0.68 0.71

Ring Exit Nozzle

Curved Exit Nozzle

Re-Entrant Tube in Reservoir

0.85 0.90 0.95 1.00

C 0.73 0.77 0.87 0.975

Curved Exit Nozzle44 d/D C 0.279 0.805 0.335 0.806 0.391 0.807 0.446 0.811 0.501 0.817 0.557 0.825 Re-Entrant Tube in Reservoir44

ᐉ/D 0 0.5 2.0–3.0 3.0

C 0.61 0.53 0.72 0.71

(Continued)

© 2001 by CRC PRESS LLC

TABLE 3.8 (continued) Nozzle Discharge Coefficients Description Short Exit Tube from Reservoir

Discharge Coefficient, C Short Exit Tube from Reservoir ᐉ/D C 0 2.0–3.0 3.0

0.61 0.61–0.82 0.82

Notes: A  cross-sectional flow area (m2) l  conical section length (m) actual mass flow rate C  discharge coefficient, -------------------------------------------------------------theoretical mass flow rate D, d  diameter or hydraulic diameter (m) U  average flow velocity (m/s)

  angle (deg.)

When the included angle of the expansion is 45°, the loss coefficient is found by: 2  D 1  ----K   1.0  2 sin --2 D 2 

Table 3.7 shows the solutions to pressure loss in a variety of expansion and contraction configurations. The data were taken from experiments using round pipes. The values can be used for turbulent flow through noncircular conduits by using DH. Table 3.8 shows the discharge coefficients for several nozzles. The actual flow rate through one of these geometries is almost never equal to the theoretical flow rate. The discharge coefficient, C, corrects for the actual flow rate. A flow rate through a geometry where C  1.0 conforms exactly to the theoretical flow rate.

3.7.3

TUBE BENDS

Loss coefficients in smooth tube bends have been studied extensively by Ito.45 These losses are affected by complex three-dimensional flowfields consisting of simultaneous wall separation, wall friction, and twin-eddy secondary flows. The force required to impel a fluid through a curved pipe is greater than that required for an © 2001 by CRC PRESS LLC

FIGURE 3.23 Geometry used to describe a large-radius tube bend.

identical length of straight pipe. For tight-radiused, compact bends, the pressure drop is localized. In these cases, the pressure drop in the pipe section will be the sum of the losses due to pipe spans upstream and downstream of the bend, in addition to the pressure loss through the bend. This loss can be represented by the equation 1 2 1 2 L 1  L 2  p  ---  U K ---  U f  -----------------2 2 D  where L1 and L2 are the lengths of the bend inlet and bend outlet tangents, as shown in Figure 3.23. For smooth, turbulent flow bends, when r/D  2.0, and the factor Re(D/r)2  360, the loss coefficient can be found by an equation due to Ito r K  0.0175  f c  ---D For smooth, turbulent flow bends, when r/D  2.0, and the factor Re(D/r)2  360, the loss coefficient can be found by K  0.00431  Re where:



 1.0 if r  50D or

D 45°  1.0 5.13  -- r

© 2001 by CRC PRESS LLC

1.47

 0.17

r  -- D

0.84

D 90°  0.95 4.42  -- r

90°  1.0 when fc



r ---D

1.96

when

r ---D

 9.85

 9.85

0.336 ----------------------------0.2 Re ( ---Dr- )

0.5

Sharp bends when r/D  2.0, and miter bends where r/D is zero, as shown in Figure 3.24 and Figure 3.25, respectively, result in a loss coefficient of about 1.10 for a 90° bend. In sharp bends, the pressure loss at the inside wall of the bend substantially exceeds the pressure rise at the outside wall of the bend. For bends of about r/D  2.0, these pressures are nearly equal. Idel’chik,34 Miller,41 and Kirchbach54 devoted considerable study to turbulent losses in these bends. The data of these

FIGURE 3.24 Fluid flow through a sharp-radius tube bend at a high Reynolds number.

FIGURE 3.25 Geometry used to describe a sharp-radius tube bend.

© 2001 by CRC PRESS LLC

researchers are applicable when Re  4000 for miter bends, and when Re  50,000 for sharp bends, and are shown in Table 3.9. When the bend occurs in a rectangular section of a smooth or a sharp-edged bend, the aspect ratio, , of the section must be accounted for. For rectangular sections the loss coefficient is first found for the above geometries for round tubes and then multiplied by the aspect ratio correction factors attributed to Idel’chik,34 Ito,45 and Madison and Parker,46 and Ward-Smith.47 That is, K  K . Table 3.9 shows the pressure loss for a variety of curved pipes and bends.

3.7.4

MANIFOLDS

When coolant flows to several branches, we call this a dividing manifold. This class of manifold extends to cases where the manifold, or tube, is regularly perforated, and when the branches are the flow channels between circuit boards, similar to that shown in Figure 3.26. We can estimate the p between points 1 and 2 from the total pressure change across the branch junction and the friction loss in the interval between branches. Ito and Imai55 studied these branches and found the loss coefficient across the branch junction as 2

U3 K 1,2  1.55  0.22  ------  0.03  U 1 when 0  U3/U1  0.22, and 2

U K 1,2  0.65  ------3  0.22  0.03 U1  when 0.2  U3/U1  1.0. The loss coefficient going into the branch is found by 0.5

0.5

r r K 1,3  0.99  0.23  ----  0.82 0.29  ----  D  D

r U 0.30 ---- ------3 D U1

r 0.5 r U 2 1.02  0.64  ---- 0.76 ----  ------3  D D  U 1 The pressure drop across points 1 and 2 reduces to 1 2 1 1 2 L 2 2  p  ---  U 2 K 1,2  ---  ( U s  U 1 ) ---  U 1 f ---2 2 2 D The pressure change is the sum of the frictional losses and a pressure rise due to the deceleration of the manifold flow as we channel the coolant to the branches. We can generalize the equation to find the pressure drop in nonhorizontal manifolds © 2001 by CRC PRESS LLC

TABLE 3.9 (continued) Pressure Loss in Curved Pipes and Bends Description and pressure loss, p Smooth Bend in Circular Pipe

Loss Coefficient, K Smooth Bend in Circular Pipe48–50 Re  2000

Re  4000 Moderate Bends, R/D 1.8 f ( L1 L2 ) p ------------------2  K -------------------------D 0.5  U

D 2 Re  ---- R 0 to 360

K R 0.0175  f c  ---D

360

0.00431  Re

 0.17

R

-) ( --D

0.84

where: 



45

D 1.0 5.13  ----  R

90

 D 1.96 R   9.85  0.95 4.42  ---- for  --  D- R     R-  9.85  1.0 for  --D  D 1.0 5.06  ----  R

180 Sharp Bends, R/D 2 for Re 5  105

© 2001 by CRC PRESS LLC

1.47

4.52

K



R/D

20

30

45

75

90

180

0.50 0.75 1.00 1.50 2.00

0.053 0.038 0.035 0.040 0.045

0.120 0.070 0.058 0.060 0.065

0.270 0.140 0.100 0.090 0.089

0.80 0.31 0.20 0.15 0.14

1.10 0.40 0.25 0.18 0.16

– 0.70 0.28 0.21 0.19

TABLE 3.9 (continued) Pressure Loss in Curved Pipes and Bends Description and pressure loss, p

Loss Coefficient, K for Re  5  105 K

Re

 (K

5 0.17

Re 5  10

5

5  10 )  -------------------  Re 

Coil of Circular Pipe51–53

Coil of Circular Pipe

D For Re  2100 1 12  ----  d

0.5

D 0.5 N  Re  ----  d

f -----c f

0 to 11.6

1.0 11 ------------------------------------------------2.2 0.45 ---------- ) 1  1  ( 11.6 N

11.6 to 2000

0.11N0.5

2000 D For Re  2100 1 12  ----  d

0.5

0.336 f c  ----------------------------0.2 0.5 Re ( ---Dd- )

f cL p ------------------2  -------D 0.5  U

0.01  d/D  0.15 Spiral of Circular Pipe

Spiral of Circular Pipe53



 ( n1  n2 )  For 7.3  2s/D  15.5, f  ------------------------------ ----- ) Re ( 2s D

2s 0.5 Re  -----  D

fL P ------------------2  -----D 0.5  U

500 to 2000 2  103 to 4 103 4  103 to 150  103





2.5 0.7 2.1 0.75 0.03 0.9

0.5





2.0 1.0 1.5

0.6 0.5 0.2

(Continued)

© 2001 by CRC PRESS LLC

TABLE 3.9 (continued) Pressure Loss in Curved Pipes and Bends Description and pressure loss, p

Loss Coefficient, K Miter Bend in Circular Pipe 34,41,54 For Re  2  105 5 0.2 2  10 K  K Re  2  105  -------------------  Re  For Re  2  105 K  K 0.025 60 0.50 0.055 70 0.70 0.10 80 0.90 0.20 90 1.10 0.35 120 1.50

Miter Bend in Circular Pipe

 10 20 30 40 50

f ( L1 L2 ) p ------------------2  K -------------------------D 0.5  U

Compound Miter Bend in Circular Pipe34,41 For Re  2  105 5 0.2 2  10 K  K Re  2  105  -------------------  Re 

Compound Miter Bend in Circular Pipe

< < T A B L E f ( L1 L2 ) P ------------------2  K --------------------------D 0.5  U

For Re  2  105 K

 45 60

90

180 Smooth Bend in Rectangular Duct

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n2n3 0.11 – 0.15 – 0.70 0.75 0.45 0.40 0.35 0.35 0.30 0.30 0.35 0.20 0.40 0.25 0.45 0.25 4.0 –

n4 – – 0.75 0.40 0.30 0.25 0.20 0.20 0.20 –

Smooth Bend in Rectangular Duct34,46,47,50 K  Ksmooth bend in circular pipe a/b 0.25 0.50 1.0 2.0 43.0

f ( L1 L2 ) p ------------------2  K -------------------------D 0.5  U

R/D 2.95 2.95 0.50 1.0 1.5 2.0 3.0 4.0 5.0 0.5

0.5  R/b  1.5 1.30 0.95 0.95 0.85 0.90

R/b  1.5 1.50 1.00 0.90 0.75 0.85

TABLE 3.9 (continued) Pressure Loss in Curved Pipes and Bends Description and pressure loss, p Miter Bend in Rectangular Duct

Loss Coefficient, K Miter Bend in Rectangular Duct K  Kmiter bend in circular pipe

f ( L1 L2 ) p ------------------2  K -------------------------D 0.5  U Notes: a  height of rectangular duct cross section

p  pressure drop between point 1 and point 2

b  breadth of rectangular duct cross section

R

d  diameter of coil, centerline to centerline

Re  Reynolds number

D  pipe diameter or hydraulic diameter

s

 distance between coils, centerline to centerline

f  friction factor for straight pipe

U

 average flow velocity

fc  friction factor for curved pipe

,  dimensionless coefficients

K  loss coefficient



L  length of pipe

Error range of K  10 to 20%

FIGURE 3.26 A uniform dividing manifold.

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 bend radius

 angle in degrees

between the i and the i 1 branches as 1 2 1 1 2 f i  1 Li  1 2 2  p i  ( i  1 )  ---  U i K i  ---  ( U i  U i  1 ) ---  U i  1 -----------------------  g ( z i  1  z i ) Di  1 2 2 2 A marching technique solves these equations, starting at the closed end of the manifold and working backward and assuming a flow velocity through the last branch. By starting the calculations at different branches, the full range of pressure and velocity across the manifold can be determined. When flow combines into a manifold, the manifold static pressure must fall with distance from the closed end because of frictional losses and because the flow will tend to accelerate as the branch flows contribute to the flow stream. For a manifold of uniform diameter with equally spaced branches that flow at the same rate, Abramovich56 found the static pressure variation along the manifold is 2 3  U  x2 fL x  p  p 0  ---------L-  -----2 ------- -----3 3D L  2 L

when the flow is turbulent, and 2

f L L x 2 U - ----p  p 0  ---------L-  1 -------2D  L 2 2  when the flow is laminar. The variable x is the distance from the closed end, where p0  0; UL is the velocity in the manifold at x  L; and fL is the friction factor at x  L. The differential pressure between the closed end of the manifold and the manifold outlet was found by Austin and Seader57 as 2

U fL  p 0  L  ---------L-  1 ------- 3D 2 for turbulent flow, and 2

f L L U  p 0  L  ---------L-  1 -------2D  2 for laminar flow.

3.7.5

SCREENS, GRILLS,

AND

PERFORATED PLATES

Many electronic equipment cabinets have various fire safety and EMI shield obstructions in the flow path. The screen or grill acts as a porous structure. As the fluid coolant flows through the screen, a static pressure drop occurs across the screen which results in a net drag force on the screen and head loss of the fluid pressure. © 2001 by CRC PRESS LLC

FIGURE 3.27 Common porous structures used in electronic packaging: (a) woven screen, (b) rod screen, (c) square-edged perforated plate, (d) square-edged grill.

We show common nomenclature for porous structures in Figure 3.27. We can define the porosity, , of the screen as the ratio of open area to the total area. N holes A hole   ----------------------A total For a woven, round wire screen, the porosity can be found by 2

2

(L  D) D -   1.0  ----  ( 1.0  ND ) 2   --------------------2  L L where: L  center-to-center spacing of wires (m) D  wire diameter (m) N  the number of wires per unit length, 1.0/L The velocity through the holes can be found as a function of the approach velocity and the porosity by U screen 1 --------------  -- U The loss coefficient through a round wire screen or plate with rounded hole edges was studied by Pinker and Herbert58 and can be found by 2

(1   ) K  -----------------------2  © 2001 by CRC PRESS LLC

where is a function of the Reynolds number when 0.05    0.85. This function is shown in Table 3.10. If the flow is turbulent and the porosity is greater than 85%, the loss coefficient for a rounded perforation screen was found by Annund59 as K  0.95 ( 1   ) 0.20 ( 1   )

2

Table 3.10 shows the pressure loss equations for a variety of porous structures in a flowstream.

3.7.6

ROUGH SURFACE CONDUITS

In many applications the coolant fluid is forced over a populated circuit board. Sometimes the circuit boards are aligned so that a small channel separates each board. If the components on the circuit board are small, or if the flow is laminar, the problem is very similar to flow in a noncircular duct. The protrusion of components into the fluid stream does not have a significant impact in laminar flow if the height of the protrusion does not exceed the height of the boundary layer. In turbulent flow, however, a surface only as rough as a sand casting can affect the pressure drop. The increase in friction is due to form drag on the surface roughness asperities. Local vortex shedding causes a higher pressure on the front of the asperity than on the rear elements. In 1944, Moody63 generated the Moody chart, shown in Figure 3.28, from the pioneering work of Nikuradse in 1933.64 Fluid engineers have widely used the chart to estimate the friction factor in a pipe. The chart uses the roughness parameter /D to correlate the friction factor to the Reynolds number in a pipe with a rough surface, where  is the mean diameter of the sand grains used to roughen the surface. In place of the Moody chart, Zigrang and Sylvester65 recommend an explicit formula to find the friction factor  2.0 ˙   13  /r 5.02  /r f    2.0log 10 -------  ---------- log 10  ------- ---------   7.4 Re D  7.4 Re D 

which has as its asymptote in the fully rough regime Nikuradse’s formula r f  1.74 2.0log 10  --  

 2.0

We find the dimensionless sand grain roughness for a pipe by the relation

U f +   ----------  ---  8

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0.5

TABLE 3.10 (continued) Pressure Loss Through Screens, Grills, and Perforated Plates Description Screen with Round Fibers

Loss Coefficient, K Screen with Round Fibers58–60 2 ( 1   ) for 0.05    0.8 K  -----------------------2  where: Re



Inclined Screen

20

30

1.3 1.1

40

60

80

100

200

For turbulent flow through a very open mesh,   0.85 and Re 500 K  0.95 (1  ) 0.2 (1  )2 round fiber screen K  0.67 (1  ) 1.3 (1  )2 diamond-pattern wire net K  0.70 (1  ) 4.9 (1  )2 knotted polyethylene net K  0.72 (1  ) 2.1 (1  )2 knotless polyethylene net Inclined Screen61 K  K  0 where: K  0 is the uninclined screen loss coefficient, and  is a function of  For 0  θ  45°,   cos2  0.0 10 20 30 40 50 60 70 80 85



1.0 0.97 0.88 0.75 0.59 0.45 0.30 0.23 0.15 0.09

Square Wire Screen or Perforated Square Wire Screen or Perforated Plate with Plate with Square Edges  K  0  0.50 0.05 1000 0.55 0.10 250 0.60 0.15 85 0.65 0.20 52 0.70 0.25 30 0.75 0.30 17 0.80 0.35 11 0.85 0.40 7.7 0.90 0.45 5.5 1.00 Grill or Perforated Plate with Square Edges

400

0.95 0.83 0.75 0.70 0.60 0.52

Square Edges34, 62 K 3.8 2.8 2.0 1.5 1.1 0.78 0.53 0.35 0.08 0

Grill or Perforated Plate with Square Edges For D  ᐉ  50D 2

0.5 ( 1   ) ( 1   ) K  -------------------------------------------------------2  For ᐉ  50D ᐉ 2 0.5 ( 1   ) ( 1   ) f  ----  D K  -----------------------------------------------------------------------------2  © 2001 by CRC PRESS LLC

(Continued)

TABLE 3.10 (continued) Pressure Loss Through Screens, Grills, and Perforated Plates Description

Loss Coefficient, K

Grill or Perforated Plate with Rounded Entrance and Tapered Exit

Grill or Perforated Plate with Rounded Entrance and Tapered Exit For D  ᐉ  50D

 K

0.3 0.4 2.0 1.0

0.5 0.6

0.6 0.4

0.7 0.2

For ᐉ  50D fᐉ Kᐉ50D --------2  D

Grill or Perforated Plate with Rounded Entrance and Exit

Multiple Screens

Grill or Perforated Plate with Rounded Entrance and Exit fD

K entrance K exit -----ᐉK  --------------------------------------------------2  where: Kentrance  loss coefficient of contraction Kexit  loss coefficient of expansion

Multiple Screens For SL  3D, use Tube Array Correlation N

For SL  3D, K 

冱K  0 i

Notes: D  fiber diameter for screens, hydraulic diameter for perforated plates (m) f  friction factor

ᐉ  thickness of grill or plate (m)  p p1  p2 K  loss coefficient  ------------------2 0.5  U p  static pressure (N/m2) U  approach velocity (m/s)

  porosity   angle (deg) Turbulent flow, Re  500, error 20% Larger errors for   0.10 or   0.85.

© 2001 by CRC PRESS LLC

and the same dimensionless sand grain roughness parameter for a surface is

U  C f 0.5 +   ----------  ---------x 2

The criteria for the flow regimes follow: 0    5.0; hydrodynamically smooth 5.0    60.0; transitional roughness 60.0  ; fully rough For flow along a fully rough plate, Mills and Huang66 recommend the following relationships for the local Cfx: x 2.46 C fx   3.476  0.707 ln --   where 150  x/  1.5  107, and for the fully rough plate average Cfx: L 2.57 C f   2.635  0.618 ln ---   where 150  L/  1.5  107. Coleman et al.67 correlated some values for  by attaching small balls onto a flat surface. Table 3.11 and Figure 3.29 show Coleman’s results. Dalle Donne and Meyer68 experimented with transverse ribs on a surface and found the following relationship to , when 2.0  S/H  6.3:

  H

S  0.73 3.4  3.7  ----  H

and when 6.3  S/H  20.0:

  H

S 3.4  0.42  ----  H

0.46

Often, flow over a populated circuit board is turbulent, because the components at the leading edge act as a trip that hastens the transition to turbulence. The protrusion that causes the transition to turbulence must meet the following two requirements, recommended by Schlichting5 and by White,4 and shown in Figure 3.30: 820  -----------U

and

 0.30  d

where

  height of the protrusion (m)  0.5 d  displacement boundary-layer thickness for laminar flow (m)  1.7208x Re x v  kinematic viscosity (m2/s)

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© 2001 by CRC PRESS LLC

FIGURE 3.28 Moody Chart.63,64

TABLE 3.11 Equivalent Surface Roughness for the Surfaces Tested by Coleman Ball (D ⴛ 10ⴚ3) 4.10 4.10 4.10 4.10 4.10 2.10 2.10

m m m m m m m

Center-to-Center Spacing (S ⴛ 10ⴚ3) 10.0 20.0 10.0 6.0 4.1 10.0 5.0

Height of Protrusion (H ⴛ 10ⴚ3)

m m m m m m m

4.10 4.10 4.10 4.10 4.10 2.10 2.10

m m m m m m m

Equivalent ⑀ (⑀ ⴛ 10ⴚ3) 0.492 m 1.68 m 9.96 m 10.6 m 1.55 m 0.903 m 5.19 m

FIGURE 3.29 Equivalent roughness, , of flat surfaces.

FIGURE 3.30 Two forms of turbulence initiators: (a) component trip, (b) distributed roughness.

© 2001 by CRC PRESS LLC

FIGURE 3.31 Drag of a two-dimensional fence in a turbulent boundary layer. (Adapted from Raju, K. G. and Singh, V., J. Ind. Aerodyn., 1, 301, 1975-76. With permission.)

These requirements apply only to free surfaces. If the populated board is part of a wall that causes channel flow, these requirements do not apply. There, the boundary layers on each wall merge, and the flow becomes turbulent at a Reynolds number of about 2300. The drag is influenced by the viscous sublayer. The theoretical viscous sublayer yu * depth is found by 0  ------- -  10.0. The thickness of the layer for a smooth plate is 100v ----------given as 0  y  U . The approximate value is 0  y  0.005 . A small protuberance increases the skin friction and the drag on the surface. If the height of the body, h, exceeds the thickness of the viscous sublayer but is not much greater than the boundary layer thickness, then the drag of the protuberance is a function of the ratio h/, as shown in Figure 3.31. Table 3.12 shows turbulent drag coefficients for protuberances. Table 3.13 shows drag coefficients for two-dimensional bluff bodies. Table 3.14 shows drag coefficients for three-dimensional bodies.

3.8 JETS Submerged jets are some of the most intensively studied fluid phenomena. Several books have been written that exclusively cover jets. There are two broad types of jets, submerged jets and open jets, also called two-phase jets. An example of a submerged jet is an underwater hose flowing in a pool; an open jet would be a firefighter’s hose. Technically, a jet is described as a source of momentum and energy in a fluid reservoir. In this section our discussion will be limited to submerged jets. Table 3.15 presents equations to determine the dimensions of a submerged jet.

© 2001 by CRC PRESS LLC

TABLE 3.12 Drag of Protruding Bodies Protuberance

Drag Coefficient, CD

Plate Section

1.4

Square Section

1.2

Equilateral Triangle Section

1.0

Right Triangle Section

Gap Section69

Semicircle Section

→ 1.3 ← 1.0

0.01h e 0.1h 0.25 8h e 20h

0.8

(Continued)

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152

Thermal Design of Electronic Equipment

TABLE 3.12 (continued) Drag of Protruding Bodies Protuberance

Drag Coefficient, CD

Bump Section

h 2 15  --- for 0  h/e  0.16  e

Sheet Metal Joint Section

Cuboid

Rectangular Solid

Cylinder

Hemisphere

→ 0.4 ← 0.2

Normal to face, CD  1.0570 to 1.6571 Diagonal to face, CD  0.8070 to 1.6571 A  b2

Normal to face, CD  1.371 to 1.570 Diagonal to face, CD  1.0571 to 1.2570 A  b2 1  h/b  4

CD  0.75 0.5  h/b  5

0.4  CD  0.6 for 103  Re  2  104 CD  0.1 for Re 2  105 A  D2/8

Fluid Dynamics for Electronic Equipment

153

TABLE 3.12 (continued) Drag of Protruding Bodies Notes: CD  drag coefficient where heigh h

boundary layer thickness  FD  total drag, FD  0.5 U2 ACD A  projected area, bh CD error is 20%

FIGURE 3.32 Streamline pattern for a round laminar jet.

Andrade101 found that a plane laminar jet, formed by a long narrow slot, was very sensitive to disturbances at a Reynolds number greater than 30. McNaughton and Sinclair102 found that axisymmetric (round) jets are laminar up to ReD  1000 and become fully turbulent at ReD  3000, when D is the diameter of the jet nozzle and U is the nozzle velocity. Solutions for a plane laminar jet and a round jet are shown in Table 3.16 using the nomenclature of Figure 3.32. Several characteristics of these jets can be surmised by Table 3.16: 1. The centerline flow velocity decreases inversely with axial distance, and the lateral width increases with axial distance. 2. The volume of fluid moving with the jet (entrainment) increases with axial distance. 3. The Reynolds number of the jet increases with axial distance for a plane jet and remains constant for a round jet. Table 3.17 lists the characteristics of turbulent jets. Figures 3.33 and 3.34 depict turbulent jet geometries. Turbulent jets are the sum of three flow regions: an initial region consisting of the core flow and the surrounding shear layer; the transition region; and the fully developed turbulent region.

TABLE 3.13 Drag of Two-Dimensional Bluff Sections8 Geometry Circular Cylinder

Drag Coefficient, CD Circular Cylinder72 Re CD

102 1.4

103 1.0

104 1.1

105 1.2

106 0.4

107 0.8

8 For Re  1.0, --------------------------------- Re log e  7.4 Re  Stranded Cylinder

Stranded Cylinder73 Type Stranded Steel Cable Jacked Steel Cable Braided Synthetic Cable Plaited Synthetic Cable Link Chain

Circular Cylinder with Thin Fin

0.33 1.2

0.67 1.15

1.0 1.1

CD 2.2 2.5 2.7 2.5 2.1

CL 0.30 0.25 0.0 0.25 0.40

 30 35 40 50 60

Cylinder Near a Wall76 for Re  2  104 E/D 0 0.25 0.5 1.0 1.5 2.0 4.0 6.0

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0.0 1.25

1.5 1.07

2.0 1.02

Cylinder with Five Thin Fins75 for Re  104

 20 10 0 10 20

Cylinder Near a Wall

104  Re  105 1.3 1.1 1.1 0.99 0.85

Circular Cylinder with Thin Fin74 for Re  105 and T  D L /D CD

Cylinder with Five Thin Fins

103  Re  104 1.5 1.5 1.1 1.2 –

CD 0.8 1.1 1.2 1.3 1.2 1.2 1.2 1.2

CL 0.6 0.25 0.15 0.05 0.02 0 0 0

CD 1.5 1.1 1.4 2.0 2.4

CL 0.25 0.0 0.2 0.4 0.35

2.5 1.0

TABLE 3.13 (continued) Drag of Two-Dimensional Bluff Sections8 Geometry Two Cylinders Side By Side

Drag Coefficient, CD Two Cylinders Side By Side77 for 104  Re  105 E/D 0 0.25 0.5 1.0 1.5 2.0 4.0 6.0

CD per cylinder 1.6 1.0 0.9 1.1 1.3 1.2 1.2 1.2

CL per cylinder 0.8 0.6 0.4 0.2 0.1 0.05 0.0 0.0

Cylinder Downstream of Another Cylinder Downstream of Another Cylinder77 for 104  Re  105 Cylinder CD on downstream cylinder T/D

Rectangle

L/D

0

0.5

1.0

2.0

1.0 1.5 2.0 2.5 3.0 4.0

0.4 0.2 0.0 0.2 0.2 0.3

0.65 0.50 0.45 0.45 0.40 0.40

1.1 1.0 0.7 0.7 0.65 0.65

1.1 1.0 1.0 1.0 1.0 1.0

Rectangle70,78 for Re  105 L/D 0.1 0.2 0.4 0.5 0.65 0.8

Rectangle with Thin Splitter Plate

CD 1.9 2.1 2.35 2.5 2.9 2.3

L/D 1.0 1.2 1.5 2.0 2.5 3.0 6.0

CD 2.2 2.1 1.8 1.6 1.4 1.3 0.89

Rectangle with Thin Splitter Plate79,80 for Re  5  104 CD L/D 0.1 0.2 0.4 0.6 0.8 1.0 1.5 2.0

T/D 0 1.9 2.1 2.35 1.8 2.3 2.0 1.8 1.6

5 1.40 1.40 1.39 1.38 1.36 1.33 1.30 –

10 1.38 1.43 1.46 1.48 1.47 1.45 1.40 1.33 (Continued)

© 2001 by CRC PRESS LLC

TABLE 3.13 (continued) Drag of Two-Dimensional Bluff Sections8 Geometry Rectangle in a Channel

Drag Coefficient, CD Rectangle in a Channel79 for Re 103 n H C D   1  ----  D for 0  D/H  0.25 L/D n

0.1 0.25 2.3 2.2

0.50 2.1

1.0 1.2

2.0 0.4

Rectangle with Rounded Corners Rectangle with Rounded Corners70,76,81 for Re  105

Inclined Square

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L/D

R/D

CD

0.5

0 0.021 0.083 0.250

2.5 2.2 1.9 1.6

1.0

0 0.021 0.167 0.333

2.2 2.0 1.2 1.0

2.0

0 0.042 0.167 0.50

1.6 1.4 0.7 0.4

6.0

0 0.5

0.89 0.29

Inclined Square82 for Re  4.7  104  CD  CD 0 2.2 25 2.1 5 2.1 30 2.2 10 1.8 35 2.3 15 1.3 40 2.4 20 1.9 45 2.4

TABLE 3.13 (continued) Drag of Two-Dimensional Bluff Sections8 Geometry Diamond with Rounded Corners

Rounded Nose Section

Drag Coefficient, CD Diamond with Rounded Corners81 for Re  105 Lo /Do

R/Do

CD

0.5

0.021 0.083 0.167

1.8   1.7  Fore and aft corners not 1.7  rounded

1.0

0.015 15 0.118 1.5 0.235 1.5

2.0

0.040 0.167 0.335

Rounded Nose Section70 L/D 0.5 1.0 2.0 4.0 6.0

Thin Plate Normal to Flow

1.1   1.1  Lateral corners not rounded  1.1 

CD 1.16 0.90 0.70 0.68 0.64

Thin Plate Normal to Flow78 for Re  105 CD  1.9

Thin Flat Plate Inclined to Flow

Thin Flat Plate Inclined to Flow83–85 where T  0.1D  2 tan  for   8  1 C N   ------------------------------for 90   12  0.283  0.222  -----------sin  CL  CN cos CD  CN sin There is a discontinuity in the range 8°    12° with CN  0.8 (Continued)

© 2001 by CRC PRESS LLC

TABLE 3.13 (continued) Drag of Two-Dimensional Bluff Sections8 Geometry Two Thin Plates Side by Side

Drag Coefficient, CD Two Thin Plates Side by Side86 for Re  4  103 CD (each plate)

E/D 0.5 1.0 2.0 3.0 5.0 10.0 15.0 Two Thin Plates in Tandem

1.42 to 2.20 1.52 to 2.13 1.9 to 2.10 2.0 1.96 1.9 1.9

Two Thin Plates in Tandem86 for Re  4  103 E/D

CD1

CD2

2 3 4 6 10 20 30 

1.80 1.70 1.65 1.65 1.90 1.90 1.90 1.90

0.10 0.67 0.76 0.95 1.00 1.15 1.33 1.90

Thin Plate Extending from a Wall Thin Plate Extending from a Wall79,80 for Re  5  104 CD  1.4

Thin Plate Extending Part Way Across a Channel

Thin Plate Extending Part Way Across a Channel79 for Re 103 and 0  D/H  0.25 1.4 C D  ---------------------2.85 1  D  ----   H

Ellipse

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Ellipse81,87 for Re  105 D/L

CD

0.125 0.25 0.50 1.0 2.0

0.22 0.30 0.60 1.0 1.6

TABLE 3.13 (continued) Drag of Two-Dimensional Bluff Sections8 Geometry

Drag Coefficient, CD

Isosceles Triangle with Point into Isosceles Triangle with Point into Flow Flow87 for Re  104

 30 60 90 120

CD 1.1 1.4 1.6 1.75

Isosceles Triangle with Base into Isosceles Triangle with Base into Flow Flow87 for Re  104

 30 60 90 120 Two Isosceles Triangles, Side by Side with Points into Flow

Two Isosceles Triangles, Side by Side with Points into Flow76 for Re  2  104 E/D 0 0.25 0.50 1.0 2.0 3.0

Rounded Isosceles Triangle

Cup Section

CD 1.9 2.1 2.15 2.05

CD 1.7 1.2 or 0.4 1.2 or 1.8 1.1 or 1.9 1.75 1.75

CL 0.5 0.3 or 0.4  (multiple  0.2 or 0.3  values due to jet switch) 0.15 or 0.25 0.1 0.08

Rounded Isosceles Triangle81 for Re  105

R/D

CD →

←

0.0 0.021 0.083 0.25

1.4 1.2 1.3 1.1

2.1 2.0 1.9 1.3

Cup Section87 for Re  2  104 → CD  2.3 ← CD  1.1

(Continued)

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TABLE 3.13 (continued) Drag of Two-Dimensional Bluff Sections8 Geometry

Drag Coefficient, CD

Hexagon

Hexagon75 → CD  1.0 ↑ CD  0.7

I-Shape

I-Shape CD L/D Flow Direction 0.5 → 2.05 ↑ 0.9

L-Shape

L-Shape88

Flow Direction 0.5 → 2.0 ↑ – ↑ – ← 1.9 1.8 ↓ 1.7 ↑

Channel Section

Channel Section88 Flow Direction → ← ↑

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1.0 1.6 1.9

CD 1.8 2.05 0.6

CD

CL

L/D

L/D

1.0 2.0 2.0 2.0 1.8 2.5 1.8

0.5 1.0 0.1 0.3 – 0.0 – 0.3 0.95 2.1 0.7 0.0 1.7 2.1

TABLE 3.13 (continued) Drag of Two-Dimensional Bluff Sections8 Geometry Built-Up Section

Drag Coefficient, CD Built-Up Section88 Flow Direction → ↑

CD 1.4 2.2

Notes: CD  drag coefficient CL  lift coefficient D  characteristic width (m) Drag  force exerted parallel to the flow Re  Reynolds number

  angle (deg) For turbulent flow, Re 104, uncertainty 5%

FIGURE 3.33 Submerged turbulent jet. (From Blevins, R. D., Applied Fluid Dynamics Handbook, Van Nostrand Reinhold, New York, 1984. With permission.)

© 2001 by CRC PRESS LLC

TABLE 3.14 Drag of Three-Dimensional Bodies8 Geometry and Characteristic Area

Drag Coefficient, CD Sphere72

Sphere Re CD

102 1.0

103 0.41

104 0.39

105 0.52

106 5  106 0.12 0.18

24 For Re  1.0, C D  ---------------------------------3 - Re ) Re ( 1  ----16 2

D A  ---------4

Hemisphere

Hemisphere70 → 0.42 ←1.17

2

D A  ---------4

Thin Hemispherical Cup

Thin Hemispherical Cup70 → 0.38 ← 1.42

2

D A  ---------4

Ellipsoid of Revolution

Ellipsoid of Revolution70 L 5 For 1  ----  10.0, Re  2  10 D D L D 0.5 C D  0.44 ----   0.016 ----   0.016 ----  L  D  L 

2

D A  ---------4

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TABLE 3.14 (continued) Drag of Three-Dimensional Bodies8 Geometry and Characteristic Area

Drag Coefficient, CD Sphere in a Circular Duct89,90

Sphere in a Circular Duct

D  0.92, Re  2  10 5 For 0  ----Do D 4.5 C D  1  1.45  ------ C D  D o

Do -------   D

2

D A  ---------4

Thin Circular Disk

Thin Circular Disk91 Re CD

1 25

2 15

5 6

10 3.6

102 1.5

103 1.1

104 1.1

105 1.15

2

D A  ---------4

Circular Rod in Parallel Flow

2

D A  ---------4

Thin Annular Disk with Hole

 2 2 A  ---- ( D o  D i ) 4

Circular Rod in Parallel Flow92 L /D 0.0 0.5 1.0 1.5 2.0 3.0 4.0 5.0

CD 1.15 1.10 0.93 0.85 0.83 0.85 0.85 0.85

Thin Annular Disk with Hole70 Di/Do 0.0 0.2 0.4 0.6 0.7 0.75 0.8 1.0

CD 1.20 1.22 1.25 1.30 1.35 1.60 1.80 2.00 (Continued)

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TABLE 3.14 (continued) Drag of Three-Dimensional Bodies8 Geometry and Characteristic Area Two Thin Disks in Tandem

Drag Coefficient, CD Two Thin Disks in Tandem91 For Re  105 D1/D2

2

D A  ---------4

Thin Disk in Tandem with Cylindrical Rod

G/D 0.0 0.125 0.25 0.50 0.75 1.00 1.50 2.00

0.25 1.15 1.13 1.10 0.96 0.87 0.80 0.97 –

0.50 1.15 1.05 0.80 0.80 0.75 0.68 0.73 –

0.65 1.15 0.80 0.88 0.42 0.45 0.55 0.73 –

0.80 – 0.55 0.30 0.22 0.30 0.42 0.65 0.85

1.0 1.15 1.16 1.16 1.15 1.10 1.05 0.85 0.93

1.2 1.70 1.72 1.73 1.75 1.70 1.67 1.58 1.38

Thin Disk in Tandem with Cylindrical Rod93 For Re  5  105 D1/D2

2

D A  ---------4

G/D 0.05 0.10 0.25 0.50 0.75 1.00 1.25 1.50

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0.25 0.73 0.70 0.60 0.43 0.35 0.33 0.40 0.50

0.50 0.72 0.53 0.38 0.25 0.23 0.25 0.27 0.32

0.625 0.67 0.48 0.16 0.04 0.14 0.23 0.30 0.37

0.75 0.55 0.23 0.03 0.04 0.11 0.18 0.30 0.43

1.0 0.75 0.73 0.73 0.71 0.70 0.68 0.65 0.63

Cylindrical Rod Perpendicular to Flow94 For Re  8.8  104

Cylindrical Rod Perpendicular to Flow

A  LD

0.125 0.74 0.72 0.71 0.55 0.45 0.53 0.55 0.57

L/D 1.0 1.98 2.96 5.0 10 20 40 

CD 0.64 0.68 0.74 0.74 0.82 0.91 0.98 1.20

TABLE 3.14 (continued) Drag of Three-Dimensional Bodies8 Geometry and Characteristic Area Thin Hollow Perforated Circular Cylinder with End Caps

Drag Coefficient, CD Thin Hollow Perforated Circular Cylinder with End Caps95 For Re  8.8  104 L/D 2.07 4.02 5.45 6.5 7.92 

CD 0.80 0.89 0.91 0.95 0.95 1.0

Holes are 0.01 D in diameter and cover 40% of surface A  LD Long Inclined Circular Rod

Long Inclined Circular Rod73,96 Normal Force 2 C N  1.2 sin  Tangent Force 2 C T  0.083 cos   0.035 cos 

Normal, A  LD Tangent, A   DL Cone70

Cone

2

D A  ---------4

Cube

 10 20 30 40 60 75 90 180

CD 0.30 0.40 0.55 0.65 0.80 1.05 1.15 1.40 Cube70 → 1.05 ↑ 0.80

A  D2

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(continued)

TABLE 3.14 (continued) Drag of Three-Dimensional Bodies8 Geometry and Characteristic Area

Drag Coefficient, CD Cube Above Surface97 For Re 3  105

Cube Above Surface Boundary layer thickness,   0.09D

A  D2 Thin Rectangular Plate Perpendicular to Flow

H/D

CD

0.0 0.25 0.50 1.0 1.5 2.0

1.18 1.10 1.15 1.13 1.10 1.08

Thin Rectangular Plate Perpendicular to Flow94 For Re  105 L/D

CD

1.0 2.0 4.0 8.0 10.0 12.0 17.8 

A  LD Inclined Thin Rectangular Plate

CL 1.1 1.5 0.70 0.30 0.20 0.0

1.05 1.10 1.12 1.20 1.22 1.22 1.33 1.90

Inclined Thin Rectangular Plate CN /CN (  90°) L/D



A  LD

0 10 20 30 40 50 60 70 80 90

1/6

1/3

1.0

1.5

2.0

3.0

0.0 0.0 0.0 0.0 0.0 0.0 0.15 0.22 0.38 0.37 0.45 0.50 0.40 0.53 0.83 0.80 0.90 0.76 0.67 0.87 1.22 0.80 0.73 0.76 0.92 1.20 1.40 0.85 0.78 0.85 1.04 1.13 1.05 0.90 0.83 0.90 1.05 1.05 1.03 0.94 0.90 0.94 1.04 1.04 1.02 0.97 0.95 0.97 1.03 1.02 1.01 0.98 0.97 0.98 1.0 1.0 1.0 1.0 1.0 1.0

CN (  90°) from this rectangular plate perpendicular to flow L/D

1/6

1/3

1.0

1.5

2.0

3.0

CN (  90°)

1.16

1.11

1.05

1.07

1.10

1.11

CL  CN cos CD  CN sin © 2001 by CRC PRESS LLC

TABLE 3.14 (continued) Drag of Three-Dimensional Bodies8 Geometry and Characteristic Area

Drag Coefficient, CD Square Rod Parallel to Flow92 For Re  1.7  105

Square Rod Parallel to Flow

L/D

CD

0.0 1.25 0.5 1.25 1.0 1.15 1.5 0.97 2.0 0.87 2.5 0.90 3.0 0.93 4.0 0.95 5.0 0.95 Square Rod with Rounded Corners CD Re

A  D2

Square Rod with Rounded Corners

R/D

5.5  105

0.0 0.025 0.50 0.10 0.20 0.25

A  D2

Various Plate Sections

8.2  106

0.75 0.60 0.55 0.32 0.17 0.17

0.75 0.35 0.25 0.15 0.15 0.15

Various Plate Sections70 CD  1.2 where solidity is 40 to 70% of square plate of same overall dimensions

A  Solid Area Porous Parabolic Dish

Porous Parabolic Dish70 For Re  2  106 Porosity

0

→CD ←CD

1.42 1.33 0.95 0.92

0.1

0.2

0.3

0.4

0.5

1.20 0.90

1.05 0.86

0.95 0.83

0.82 0.80

2

D A  ---------4

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(Continued)

TABLE 3.14 (continued) Drag of Three-Dimensional Bodies8 Notes: A

 reference area

CD

 drag coefficient

CL

 lift coefficient

D

 reference width

Drag  force exerted parallel to the flow Re

 Reynolds number



 angle (deg)

For turbulent flow, Re 104, uncertainty 5%

FIGURES 3.34 Radial turbulent jet. (From Blevins, R. D., Applied Fluid Dynamics Handbook, Van Nostrand Reinhold, New York, 1984. With permission.)

Figure 3.35 compares the geometries of plane wall, radial wall, impinging, and cylindrical wall jets. Table 3.18 presents the characteristics of these turbulent jets. Exact solutions have been developed for laminar jets with swirl by Loityanskii106 and quoted by Rose.107 To determine the axial velocity: 2

- 1   ------------- 4  1 2 1  u  --------------------------------2 ---  --------- --------------------------------3 ----2 2 2 2 2 2 x  x   -----------  1 1   ----------   3 

2

4

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2

4

2

TABLE 3.15 Development of Jets, Wakes, Shear Layers, and Plumes99,100 Laminar Flow Flow Plane Jet Round Jet Plane Wake Round Wake Plane Shear Layer Plane Plume Round Plume Notes:

Turbulent Flow

Width, b

Velocity, um

Species Tm

Width, b

Velocity, um

Species Tm

x 2/3 x x 0.5 x 0.5 x 0.5 x 0.4 x 0.5

x1/3 x1.0 x0.5 x1.0 1.0 x 0.2 1.0

x1/3 x1.0 x0.5 x1.0 1.0 x0.6 x1.0

x x x 0.5 x1/3 x x x

x0.5 x1.0 x0.5 x2/3 1.0 1.0 x1/3

x0.5 x1.0 x0.5 x2/3 1.0 x1.0 x5/3

b  half width of jet, i.e., transverse width required for axial flow velocity to 0.5 centerline velocity (m)

Tm  centerline concentration of temperature relative to reservoir um

 centerline velocity (m/s)

x

 axial distance from origin of jet parallel to flow (m)

To determine the radial velocity:

v 

  2 2 2 2 h 3     2  1   ---------- 2  1   ---------------   4 4   1 1 ------------------------------------------- ----2  - ---  v  --------------------------------------2 2 3 2 2 2 x   x    1   ---------- ---------- 1  4   4     

To find the circumferential velocity:

 1    --------------------------------2 ----2 2 2 h x 1   --------- 4

The static pressure for a laminar jet with swirl is 1 2 2 1 p  p   ---  ---------------------------------3 ----4 2 2 3 x   1   ------------ 4

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TABLE 3.16 Submerged Laminar Jets5 Jet Characteristics

Plane Jet

Round Jet

Stream Function, 

  1.651 ( M o vx ) tanh  where M 13 y   0.2752  ------2-o -------23 x v

13

2

   vx --------------------------2 1  0.25  where 0.5 3 0.5 M o r - -   ---------- --------v x 16 



Mo 

 u2 dy

 constant



M o  2  0 u rdr  constant 2

2 13

Centerline Velocity, um

M 13 2 0.4543  -------o ( 1  tanh  )  vx 

3 M 1 ------- -------o --------------------------------8  vx ( 1  0.25  2 ) 2

M o v  0.5 2 [ 2  (1  tanh  )  tanh  ] 0.5503  -------- x2 

3 0.5 M   0.25  0.25  ---- -------o -------------------------------  x ( 1  0.25  2 ) 2

Inline Velocity, u

Transverse Velocity,

3 M ------- -------o 8  vx

M 0.4543  -------o  vx 

2

13

Width, b

v 3.203  -------  M o

Volumetric Flow Rate, Q

3.3019 ( M o vx )

x

23

2

3

2

v 5.269  -------  M o

0.5

x

8vx

13

0.5

Reynolds Number, um b Re  -------v

M o x 1  3 1.455  -------- v 

Mo 0.6289 --------v

Maximum Nozzle Re for Laminar Flow

U o ( 2b o ) -------------------  30 v

U o ( 2r o ) -------------------  1000 v

Notes: b

 half width of jet, i.e., transverse width required for axial flow velocity to 0.5 centerline velocity (m)

Mo  axial momentum flux of jet, momentum flux per unit length of slot or plane jet r

 radius from centerline (m)

x

 coordinate from origin of jet parallel to flow (m)

y

 transverse coordinate (m)



 nondimensional coordinate

  stream function constant

The dimensionless parameters are 3W 0.5    ------------ 16

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r   ---------0.5 x

3T   ----------------1.5 16 

TABLE 3.17 Submerged Turbulent Jets103–105 Jet Characteristics

Plane Jet

Round Jet

Radial Jet

12bo

10ro

4.0(Robo)0.5

r 12.0 ----o u o x

( Ro bo ) 3.5 --------------------------------0.5 0.5 r ( Ro  r )

Length of Initial Region, x1 Centerline Velocity, um

0.5

b 3.4  -----o u o  x

Velocity Profile, u/um e Width, b

y 2  57  --  x

e

r 2  94  --  x

0.5

e

y 2  57  --  r

0.11x

0.086x

0.11r

Volumetric Flow Rate, Q

x 0.5 0.44  ----- Q o  b o

x 0.16 ---- Q o ro

–

Entrainment Velocity, ue

0.053um

0.031um r 8.2 ----o E o x

Kinetic Energy, E 2

U o   E o  --------2    Nozzle T above reservoir

b 0.5 2.6  -----o E o  x

b 0.5 3.0  -----o  T o  x

r 10.0 ----o  T o x

– –

–

Notes: b

 half width of jet, i.e., transverse distance for axial velocity, u, to fall to 0.5 centerline velocity, um (m)

b0  half width of jet at nozzle (m) bT  transverse distance for species concentration to fall to 0.5 centerline value (m) E

 kinetic energy of fluid within jet

E0  kinetic energy of fluid exiting nozzle Q

 volumetric flow rate (m3/s)

Q0  volumetric flow rate through nozzle (m3/s) r

 radial distance from centerline of nozzle (m)

r0

 radius of nozzle (m)

R0  radial distance from central axis to orifice of radial jet (m)

T  concentration of temperature from reservoir level Tm  concentration of temperature at jet centerline T0  concentration of temperature at nozzle above reservoir level u

 axial velocity (m/s)

um  axial velocity at jet centerline (m/s) U0  nozzle flow velocity (m/s) x

 axial distance from nozzle exit (m)

y

 transverse distance to central plane (m)

Uncertainty 10%

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FIGURE 3.35 Four submerged turbulent wall jets. (From Blevins, R. D., Applied Fluid Dynamics Handbook, Van Nostrand Reinhold, New York, 1984. With permission.)

The flow rate is Q  8  x  2  When the flow in the round jet with swirl is turbulent, the equations of Table 3.19 are used. These equations are generally reliable for weak or moderate swirl, 0  S  0.60, and when the maximum axial velocity exceeds the maximum circumferential velocity, um m. The maximum axial velocity and the minimum static pressure fall with increasing distance from the nozzle, but maximum circumferential velocity is located at roughly r/x  0.12.

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TABLE 3.18 Submerged Wall and Impinging Jets104 Jet Characteristics Initial Length, x1

Plane Wall Jet 6.4bo

10ro 0.5

Maximum Velocity, um

Radial Wall Jet

b 3.5  -----o U o  x

Impinging Radial Jet 4.0(Robo)

0.5

Cylindrical Wall Jet

0.5

–

r 2.1 ----o U o r

b o r o - Uo 3.5  -------- r 

2

 b----o-  ri  b -------------  ----o2 ri

Uo

-----------------------------------------------------------------˙ 2 0.5 0.073  ---rx   0.0034  r---x   i i Width, b

0.068x

0.078r

0.087r

0.080x

Boundary Layer Thickness, 

0.014x

0.016r

0.020r

0.012x

Velocity Profile, u/um

e

Volumetric Flow Rate, Q

x 0.248 ----- Q o bo

r 0.201 -------------------Q 0.5 o ( bo r o )

–

–

Entrainment Velocity, ue

0.035um

0.081um

–

–

y  2  0.693  -----------  b 

e

r  2  0.693  -----------  b 

e

y  2  0.693  -----------  b 

e

y  2  0.693  -----------  b 

Notes: b

 transverse distance for axial velocity to fall to 0.5 centerline velocity (m)

b0  width of jet at nozzle (m) Q  volumetric flow rate (m3/s) Q0  volumetric flow rate through nozzle (m3/s) r

 radial distance from centerline of nozzle (m)

r0  radius of nozzle (m) u

 axial component of velocity (m/s)

um  maximum inline component of velocity (m/s) U0  flow velocity through nozzle (m/s) ve  entrainment velocity, i.e., transverse velocity with which flow enters jet (m/s) x

 inline distance (m)

For turbulent flow only. Uncertainty 20–30%

3.9 FANS AND PUMPS Fans and pumps are used to convert mechanical energy into fluid energy and are usually classified by the manner in which they transmit the mechanical energy to the fluid. We generally classify pumps as kinetic pumps or positive displacement pumps, and we categorize fans as axial flow or centrifugal.

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TABLE 3.19 Round Submerged Jets with Swirl104,107–109 Jet Characteristic

Equation r c 1 ----o U 0 x

Centerline Velocity, um Maximum Circumferential Velocity, m Centerline Static Pressure, p∞  pm

2

r c2  ----o mo  x r 4 c3  ----o (p  pmo)  x

Width, b

c4x

Parameters S

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

c1

12

9

6

4

2.8

2.0

1.4

0.9

Considerable uncertainty in c2 4  c2  11 suggested104 c3 as a function of S has not been established104 p  p∞ for S  0 0.0 0.2 0.084 0.13

S c4

0.4 0.18

0.6 0.23

 c 4  tan ( 4.8  14S ) --------180 Volumetric Flow Rate, Q

x c 5 ---- Q 0 ro

0.0 0.16

S c5

0.2 0.24

0.8 0.29

1.0 0.34

0  S  1.0

0.4 0.32

0.6 0.40

0.8 0.48

1.0 0.56

0.8 16

1.0 13

c5  0.16  0.4S, 0  S  1.0 Inline Velocity Profile, u/um

e

r   1  --  x

2

0.0 92

S 1

0.2 52

92  1  ----------------1  6S Circumferential Velocity Profile, /m

r r 2 r 3 a --  b  --  c  --    x x x

Static Pressure Profile,

e

r   2  --  x

2

p  p ------------------p  pm

S 0.0

0.4 27

0  S  1.0

0.066

0.134 0.234

a 0.0 7.7 10.7 b 0.0 71.5 20.0 c 0.0 542 326 S 2

0.0 

0.2 64

150  2  ----------------- , 1  8S

0.6 18

0.4 32

0.6 22

0  S  0.6

Notes: b

 transverse distance for axial velocity to fall to one-half centerline value um (m)

p

 static pressure (N/m2)

pm  static pressure at centerline (N/m2) p∞  reservoir static pressure (N/m2) Qo  volume flow rate through nozzle (m3/s) r

 radial distance from centerline of nozzle (m)

ro  radius of nozzle (m) S

 dimensionless swirl parameter

Uo  nozzle exit flow velocity (m/s)

© 2001 by CRC PRESS LLC

0.416

18.1 15.1 98.8 67.2 138 75.4

0.60 22.8 155 275

TABLE 3.19 (continued) Round Submerged Jets with Swirl104,107–109 m  maximum circumferential velocity at a given value of x (m/s) m0  maximum circumferential velocity at the nozzle (m/s) x

 inline distance from nozzle exit (m)



 boundary layer thickness (m)

For turbulent flow only Uncertainty 20%

Positive displacement pumps are generally used to greatly increase the fluid pressure. These types of pumps are most often found in hydraulic power and control systems. Kinetic energy pumps, also called dynamic pumps, are low-pressure pumps used to circulate fluids and for fluid handling and can be found in electronic cooling systems. Centrifugal pumps are the most common forms of kinetic pumps. Axial flow fans, also called propeller fans, develop static pressure by changing the air flow velocity. Axial flow fans are normally used to develop increased velocity at low static pressures. Centrifugal fans develop static energy by increasing the kinetic energy of the fluid and imposing a centrifugal force on the rotating air column. There are many types of custom fans and pumps, but our discussion will center primarily on axial flow fans and to a lesser extent on kinetic energy pumps.

3.9.1

FANS

Three parameters are important in the selection of fans: air power, pressure developed, and the operating efficiency. These three parameters vary with the volume flowing and, therefore, are usually presented graphically. Moving or stationary air confined in a duct will exert a pressure perpendicular to the duct wall. The fan must supply this pressure, which we call the static pressure. We usually measure static pressure in millimeters or inches of water. To achieve consistent units, we can use meters or feet of water. The total pressure always decreases along the direction of flow. However, as we have found in our study of expansions and contractions, the static pressure can change with diameter changes. The loss in pressure due to friction should not be confused with the change in pressure due to diameter changes. The total pressure is the sum of the static and velocity pressures. In an ideal fluid (no friction losses), the total pressure would be constant along the length of the duct. Typical fan characteristic curves for axial fans and forward- and backwardcurved centrifugal fans are shown in Figure 3.36. The dips in the total pressure plots are characteristic of each type of fan. The highest efficiency is found to the right of the dip and should be chosen as the operating point.

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FIGURE 3.36a Typical f an curv es sho wing pressure, po wer, andficienc ef y: axial flow or propeller f an.

FIGURE 3.36b Typical f an curv es sho wing pressure, po wer, andficienc ef y: centrifug al fan with forw ard-curv ed blades.

© 2001 by CRC PRESS LLC

FIGURE 3.36c Typical fan curves showing pressure, power, and efficiency: centrifugal fan with backward-curved blades.

When choosing between fan types, the following points should be considered: Axial Flow, Figure 3.36a • • • •

Higher outlet velocities are possible than with centrifugal fans. Overloading is much less likely due to the flat power curve. Minimum sound occurs at maximum efficiencies. Lowest cost.

Centrifugal (Forward-Curving Blades), Figure 3.36b • Maximum efficiency occurs near the point of maximum static pressure. • Power rises rapidly with increases in the delivery rate. • Motor overloading is possible if we have not calculated the duct losses carefully. • Minimum sound occurs at maximum pressures. Centrifugal (Backward-Curving Blades), Figure 3.36c • The fan may operate over a greater range without encountering unstable air. • Overloading is less likely. • Efficiency is often higher than forward-curved.

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FIGURE 3.37 A typical fan curve.

• The fan is noisier than the forward-curved centrifugal. • Minimum sound occurs at the highest efficiencies. • Very sensitive to obstacles near the outlet. For these reasons, • axial flow fans are normally chosen for low static pressure electronic chassis, where a somewhat higher noise level is more acceptable than higher cost; • forward-curved blade centrifugal fans are chosen for applications requiring higher static pressure and low noise; and • backward-curved blade centrifugal fans are not very common due to installation limitations and a higher noise level. Most manufacturers provide fan-rating tables similar to the charts in Figure 3.37 for their fans. Such a table usually provides static pressure vs. volumetric flow rate but usually does not include efficiency. If the application requires it, fans can be placed in parallel or in series. For fans in parallel, the air flow volume will be the sum of the individual fans, but only at free delivery when static pressure is zero. At the highest static pressure the fan can deliver, the volumetric flow is the same as a single fan, as shown in Figure 3.38. For fans in series, the static pressure will be the sum of the individual fans but, again, only at the point on the curve where air flow volume is zero. In both series and parallel operation, particularly when more than two fans are used, some areas of the combined performance curve may result in unstable and unpredictable performance. © 2001 by CRC PRESS LLC

FIGURE 3.38 Single fan performance vs. the performance curves of two fans in parallel and two fans in series.

3.9.1.1

Fan Operation at Nonstandard Densities

Most rating and performance charts are given for a standard sea-level air density of 1.201 kg/m3. Small variations in density due to normal temperature and humidity fluctuations need not be considered. However, if we expect the system to operate at elevated temperatures or reduced atmospheric pressures, corrections are necessary. Fans are constant-volume devices. That is, a fan, under changing conditions of air density, will deliver the same volume of air. Thus, if we increase the temperature or altitude (density decreases), the air volume moved will remain constant, but the mass of the air moved, and therefore the effective cooling, will decrease. The mass flow is proportional to the density change. Table 3.20 shows some applicable fan laws that can be used to find the proper correction factors for fans used in nonstandard air density, or speeds.

3.9.2

PUMPS

In positive displacement rotary pumps, the action of the pump forces (rather than induces) the liquid to flow in the required direction. This positive displacement action causes the pump to consistently deliver a similar quantity of liquid with each revolution. Fluid is moved by elements (which may be gears, lobes, vanes, or screws) rotating within an enclosed housing. As the elements rotate, they push the liquid along from the intake of the pump to the point of discharge. Centrifugal pumps move fluids by inducing flow. They use a rotating impeller to move liquid from the intake of the pump to the point of discharge by centrifugal force. Approximately 90% of all pumps in use, other than hydraulics, are the

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TABLE 3.20 Basic Fan Laws Variable

Nonstandard Speed

Nonstandard Density

Air Flow, Q (m3/s)

N Q 2  Q 1  ------2  N 1

Constant

Pressure, p (N/m2)

N 2 p 2  p 1  ------2  N 1

p 2  p 1  -----2  1

Power, q (W)

N q 2  q 1  ------2  N 1

Noise, qN (W)

3

N q N 2  q N 1  50 log  ------2 10  N 1

q 2  q 1  -----2  1 q N 2  q N 1  20 log 10  -----2  1

Notes: N  fan frequency of rotation (RPM)

 air density (kg/m3)

centrifugal type because they are the least expensive type of pump. They are commonly used to transfer large volumes of liquid at relatively low pressures where accuracy is not required. Centrifugal pumps handle liquids of low viscosity, deliver a relatively steady discharge flow, and are generally accurate to within 20%. Centrifugal and axial pumps provide smooth continuous flow but, like fans, they reduce their flow output as the flow resistance increases. When the resistance of the external system becomes infinitely large (for example, a closed valve blocks the outlet line), the pump will produce no flow and, thus, its volumetric efficiency becomes zero. These pumps are well suited for low-pressure, high-volume flow applications, such as electronic cooling. The operation of a centrifugal pump is simple. The fluid enters at the center of the impeller and is picked up by the rotating impeller. As the fluid rotates with the impeller, the centrifugal force causes the fluid to move radially outward. This causes the fluid to flow through the outlet discharge port of the housing. The fact that there is no positive internal seal against leakage is the reason that the centrifugal pump is not forced to produce flow when there is no demand. When demand for the fluid occurs (for example, the opening of a valve), the pressure delivers the fluid to the source of the demand.

3.10 ELECTRONIC CHASSIS FLOW The flow of coolant in a typical electronic chassis involves numerous obstacles. Separating a chassis into arbitrary zones of pressure drop is often helpful. In an air-cooled system we often pull the air into the chassis through louvers. When the air is in the chassis, it must normally negotiate several turns through electronic components. The airflow over the components causes an impingement effect on the

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upstream side and a wake or shrouding effect on the downstream side of the components. In many applications we pull the airflow through parallel spaces between circuit boards. Eventually the air passes through a fan and then out into the ambient environment. To achieve forced convection in an electronic chassis, effecting a differential pressure at the inlet and the outlet of the chassis is necessary. If the differential pressure at these two points is greater than the chassis flow impedance between the two points, coolant will flow through the chassis. The intersection of the fan or pump curve determines the velocity and volume of the flow and the chassis impedance curve. The coolant flow results in a static pressure and a velocity pressure. The system pressure determines static pressure or, in the case of a fan-cooled chassis, the atmosphere. Velocity pressure is the force that causes the fluid movement within the chassis. As the velocity pressure increases, the coolant velocity increases. From our study of the friction factor, we also realize that the pressure losses increase to the square of the velocity, so that increasing the velocity of the flow can result in a very large fan or pump requirement. In a common electronic chassis, the expression of pressure loss is most often expressed in terms of velocity head, Hv, instead of the standard pressure units of N/m2. The velocity head is related to the velocity by the equation U 

2gH

where: U  velocity (m/s) g  gravitational acceleration (9.807 m/s2) H  height (m) Pressure drop can then by calculated as velocity head loss as mH2O by multiplying the velocity head and the loss coefficient, K. The formula for static pressure loss for a velocity head of unity is found by  U 2  ----------------- H v   2g H O 2 -  --------------- air  We can express the velocity head loss as a function of standard 20.0°C water and standard atmospheric air by inserting the proper variables into the equation. To obtain mH2O we find    2 U U   -------------------------------------------------------------------------------  U 2 H   2g H2 O   ( 2 ) ( 9.807m/s 2 ) ( 998.2kg/m3 )   -------------  -----------------  127.6  ------------------------------------------------------------------------ 2  1.202kg/m air     

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FIGURE 3.39 A system will operate at only one point. The system operating point is the intersection of the chassis impedance curve and the fan pressure curve.

The static pressure can be expressed in terms of the velocity head at a specific point in the flow. If we divide a problem into zones of pressure drop that are each dominated by a specific geometry, then the loss caused by that geometry can be expressed as the ratio of the velocity head at the entrance to that zone. For example, the pressure loss occurring when flow is separated and parallel across either side of a circuit board can be expressed as the ratio of velocity head at the point the flow separated. When the chassis impedance curve is combined with the fan characteristic curve, we can define the operating point, as shown in Figure 3.39. This point will indicate the volumetric air flow through the chassis and, therefore, the air velocity and cooling effectiveness.110,111

REFERENCES 1. Darcy, H., Recherches expérimentales reatives aux mouvements de l’eau dans tuyauz, Mem. Prés. Acad. Sci. Inst. France., 15, 141, 1858. 2. Wyckoff, R. D., Botset, H. G., Muskat, M., and Reed, D. W., Rev. Sci. Instrum., 4, 395, 1933. 3. Chen, R.-Y., Flow in the Entrance Region at Low Reynolds Numbers, J. Fluids Eng., 95, 153, 1973. 4. White, F. M., Viscous Fluid Flow, McGraw-Hill, New York, 1974. 5. Schlicting, H., Boundary Layer Theory, 6th ed., McGraw-Hill, New York, 1968. 6. Hinze, J. O., Turbulence, 2nd ed., McGraw-Hill, New York, 1975. 7. Clauser, F. H., The Turbulent Boundary Layer, in Advances in Applied Mechanics, Vol. IV, Dryden, H. L. and Kármán, Th. von., Eds., Academic Press, New York, 1956, 51.

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8. Blevins, R. D., Applied Fluid Dynamics Handbook, Van Nostrand Reinhold, New York, 1984. 9. Chen, R.-Y, Flow in the Entrance Region at Low Reynolds Numbers, J. Fluids Eng., 95, 153, 1973. 10. Haji-Sheikh, A., University of Texas at Arlington, 1970, as quoted by Shah, R. K. and London, A. L., Laminar Flow Forced Convection in Ducts, Advances in Heat Transfer, Suppl. 1, Academic Press, New York, 1978, 265. 11. Sparrow, E. M. and Haji-Sheikh, A., Flow and Heat Transfer in Ducts of Arbitrary Shape with Arbitrary Thermal Boundary Conditions, J. Heat Transfer, 88, 351, 1966. 12. Shah, R. K., A Correlation for Laminar Hydrodynamic Entry Length Solutions for Circular and Noncircular Ducts, J. Fluids Eng., 100, 177, 1978. 13. Sparrow, E. M., Chen, T. S., and Jonsson,V. K., Laminar Flow and Pressure Drop in Internally Finned Annular Ducts, Intl. J. Heat Mass Transfer, 7, 583, 1964. 14. Rektorys, K., Ed., Survey of Applicable Mathematics, MIT Press, Cambridge, MA, 1969, 140. 15. Shah, R. K. and London, A. L., Laminar Flow Forced Convection in Ducts, Advances in Heat Transfer, Suppl. 1, Academic Press, New York, 1978, 237. 16. Iqbal, M., Kharty, A. K., and Aggarwala, B. D., On the Second Fundamental Problem of Combined Free and Forced Convection Through Vertical Non-Circular Ducts, Appl. Sci. Res., 26, 183, 1972. 17. Shah, R. K., Laminar Flow Friction and Forced Convection Heat Transfer in Ducts of Arbitrary Geometry, Intl. J. Heat Transfer, 18, 849, 1975. 18. Jones, O. C., An Improvement in the Calculation of Turbulent Friction in Rectangular Ducts, J. Fluids Eng., 98, 173, 1976. 19. Cheng, K. C., Laminar Flow and Heat Transfer Characteristics in Regular Polygon Ducts, Proc. 3rd Intl. Heat Transfer Conf., Vol. 1, AIChE, New York, 1966, 64. 20. Shih, F. S., Laminar Flow in Axisymmetric Conditions by a Rational Approach, Can. J. Chem. Eng., 45, 285, 1967. 21. Zarling, J. P., Application of the Schwarz-Neumann Technique to Fully Developed Laminar Heat Transfer in Non-Circular Ducts, J. Heat Transfer, 99, 332, 1977. 22. Nandakmar, K. and Masliyah, J. H., Fully Developed Viscous Flow in Internally Finned Tubes, Chem. Eng. J., 10, 113, 1975. 23. Scott, M. J. and Webb, R. L., Analytic Prediction of the Friction Factor for Turbulent Flow in Internally Finned Channels, J. Heat Transfer, 103, 423, 1981. 24. Soliman, H. M. and Feingold, A., Analysis of Fully Developed Laminar Flow in Longitudinally Finned Tubes, J. Chem. Eng., 14, 119, 1977. 25. Wilcock, D. F., Understanding the Fundamental Behavior of Fluid-Film Bearings, in Fundamentals of the Design of Fluid Film Bearings, Rohde, S. M. et al., Eds., American Society of Mechanical Engineers, New York, 1979. 26. Fuller, D. D., Theory and Practice of Lubrication for Engineers, John Wiley & Sons, New York, 1956. 27. Zuk, J., Fundamentals of Fluid Sealing, NASA Technical Note NASA TN-D8151, Lewis Research Center, Cleveland, OH, 1976. 28. Reynolds, O., An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall be Direct or Sinuous, and of the Law of Resistance in Parallel Channels, Philos. Trans. R. Soc. London, 174, 935, 1883. 29. Prandtl, L., Uber Flüssigkeittsbewegung mit kleiner Reibung, 1904 (reprinted in Vier Abhandlungen zur Hydrodynamik, Göttingen, 1927. 30. Blasius, H. Z. Math. Phys., 56, 1, 1908. (English translation in NACA TM-1256.) 31. Blevins, R. D., Applied Fluid Dynamics, Van Nostrand Reinhold, New York, 1984, 305.

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32. Gibson, A. H., Hydraulics and Its Applications, 4th ed., D. Van Nostrand, New York, 1930, 93. 33. Harris, C. W., The Influence of Pipe Thickness on Reentrant Intake Losses, University of Washington Eng. Expt. Stn. Bull., 48, Seattle, 1928. 34. Idel’chik, I. E., Handbook of Hydraulic Resistance, U. S. Department of Commerce, Nat1. Tech. Inf. Service, Report AEC-TR-6630, 1960. (Translated from Russian data.) 35. Hamilton, J. B., The Suppression of Intake Losses by Various Degrees of Rounding, University of Washington, Eng. Exp. Stn. Bull., 51, Seattle, 1929. 36. Weisbach, G., Lehrbuch der Ingenieur und Maschinenmechanik, 2nd ed., 1950. 37. Harris, C. H., Elimination of Hydraulic Eddy Current Loss at Intake, University of Washington, Eng. Exp. Stn., Bull. 54, Seattle, 1930. 38. Kays, W. M., Loss Coefficients for Abrupt Changes in Flow Cross-Section, Trans. ASME, 72, 1067, 1950. 39. Irwin, R. W. and Motycka, J., Friction Factors for Corrugated Plastic Drainage Pipe, ASCE J. Hydraulics Div., 105, 29, 1979. 40. Jain, A. K., Accurate Explicit Equations for Friction Factor, ASCE J. Hydraul. Div., 102, 674, 1976. 41. Miller, D. S., Internal Flow Systems, BHRA Fluids Engineering, Cranfield, U.K., 1978. 42. Grey, R. E. and Wilsted, H. D., Performance of Conical Jet Nozzles in Terms of Flow and Velocity Coefficients, NACA Tech. Note No. 1757, Lewis Flight Propulsion Laboratory, Cleveland, OH, 1948. 43. Freeman, J. R., Experiments Relating to Hydraulics of Fire Streams, Trans. ASCE, 21, 303, 1888. 44. Rouse, H. et al., Experimental Investigations of Fire Monitors on Nozzles, Trans. ACHE, 117, 1147, 1952. 45. Ito, H., Pressure Losses in Smooth Pipe Bends, Trans. ASME (Ser. D), 82, 1, 1960. 46. Madison, R. D. and Parker, J. R., Pressure Losses in Rectangular Elbows, Trans. ASME, 58, 167, 1936. 47. Ward-Smith, A. J., Pressure Losses in Ducted Flows, Butterworths, London, 1971. 48. Kittredge, C. P. and Rowley, D. S., Resistance Coefficients for Laminar and Turbulent Flow Through One-Half Inch Valves and Fittings, Trans. ASME, 79, 1759, 1957. 49. Bruins, P. F. et al., Friction of Fluids in Solder Type Fittings, Trans. AIChE, 36, 721, 1940. 50. Itoh, H., Pressure Losses in Smooth Pipe Bends, J. Basic Eng., 82, 131, 1960. 51. White, C. M., Streamline Flow Through Curved Tubes, Proc. R. Soc. London, 123A, 645, 1929. 52. Baylis, J. A., Experiments on Laminar Flow in Curved Channels of Square Sections, J. Fluid Mech., 48, 417, 1971. 53. Srinivasan, P. S., Nandapurkar, S. S., and Holland, F. A., Friction Factors for Coils, Trans. Inst. Chem. Eng., 48, T156, 1970. 54. Kirchbach, H., Loss of Energy in Miter Bends, Trans. Munich Hydraul. Inst., Bull. No. 3, 1920, 43. 55. Ito, H. and Imai, K., Energy Losses at 90° Pipe Junctions, ASChE J. Hydraul. Div., 99, 747, 1973. 56. Abramovich, G. N., Fluid Motion in Curved Ducts, Trans. Central Aero Hydrodyn. Inst., U.S.S.R., Moscow, 1953. (Translated by NACA.) 57. Austin, L. R. and Seader, J. D., Fully Developed Viscous Flow in Coiled Circular Pipes, AIChE J., 19, 85, 1973. 58. Pinker, R. A. and Herbert, M. V., Pressure Loss Associated with Compressible Flow Through Square Mesh Wire Gauges, J. Mech. Sci., 9, 11, 1967.

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59. Annund, W. J. D., The Resistance to Airflow of Wire Gauges, J. R. Aeronaut. Soc., 57, 1953. 60. Murota, T., An Experimental Study of the Drag Coefficient of Screens for Building Use, in Wind Effects on Structures, Ishizaki, H. and Chiu, N. L., Eds., University Press of Hawaii, Honolulu, 1976, 105. 61. Carrothers, P. J. G. and Baines, W. D., Forces on Screens Inclined to a Fluid Flow, J. Fluids Eng., 97, 116, 1965. 62. SAE, Aerospace Applied Thermodynamics Manual, Society of Automotive Engineers, New York, 1969, 25. 63. Moody, L. F., Friction Factors for Pipe Flow, Trans. ASME, 66, 1944. 64. Nikuradse, J., Laws of Flow in Rough Pipes, NACA TN 1292, 1950. (English translation of VDI-Forschungsheft, 361, 1933). 65. Zigrang, D. J. and Sylvester, N. D., Explicit Approximations to the Solution of Colebrook’s Friction Factor Equation, AIChE J., 28, 514, 1982. 66. Mills, A. F. and Huang, Xu, On the Skin Friction Coefficient for a Fully Rough Flat Plate, J. Fluids Eng., 105, 364, 1983. 67. Coleman, H. W., Hodge, B. K., and Taylor, R. P., A Reevaluation of Schlichting’s Surface Roughness Experiment, J. Fluids Eng.,106, 60, 1984. 68. Dalle Donne, M. and Meyer, L., Turbulent Convective Heat Transfer from Rough Surfaces with Two-Dimensional Rectangular Ribs, Int. J. Heat Mass Transfer, 20, 583, 1977. 69. Owens, R. and Palo, P., Wind Induced Steady Loads on Moored Ships, Naval Civil Engineering Laboratory, Tech. Mem. TM-44-81-7, Port Hueneme, CA, June, 1981. 70. Hoerner, S. F., Fluid Dynamic Drag, published by the author, New Jersey, 1965. 71. Akins, R. E. et al., Mean Force and Momentum Coefficients for Buildings in Turbulent Boundary Layers, J. Ind. Aeronaut., 2, 195,1977. 72. Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, U.K., 1967, 246. 73. Nordell, W. J. and Meggitt, D. J., Undersea Suspended Cable Structures, ASCE J. Struct. Div., 107, 1025, 1981. 74. Geiger, G. E. and Collucio, S. J., The Heat Transfer and Drag Behavior of a Heated Circular Cylinder with Integral Heat-Conducting Downstream Splitter Plate in Crossflow, J. Heat Trans., 96, 95, 1974. 75. Mahrenholtz, O. and Bardowicks, H., Aeroelastic Problems at Masts and Chimneys, J. Ind. Aerodyn., 4, 261, 1979. 76. Roshko, A. et al., Flow Forces on a Cylinder Near a Wall or Near Another Cylinder, Second Natl. Conf. Wind Eng. Res., Colorado State University, Ft. Collins, CO, 1975. 77. Zdravkovich, M. M. and Pridden, D. L., Interference Between Two Circular Cylinders; Series of Unexpected Discontinuities, J. Ind. Aerodyn., 2, 255,1970. 78. Courchesne, J. and Lanville, A., A Comparison of Correction Methods Used in the Evaluation of Drag Coefficient Measurements for Two-Dimensional Rectangular Cylinders, J. Fluids Eng., 101, 506, 1979. 79. Raju, K. G. and Singh, V., Blockage Effects on Drag of Sharp-Edged Bodies, J. Ind. Aerodyn., 1, 301, 1975–76. 80. Bearman, P. W. and Trueman, S. M., An Investigation of the Flow Around Rectangular Cylinders, Aeronaut. Q., 23, 229, 1972. 81. Delaney, N. K. and Sorenson, N. E., Low Speed Drag of Cylinders of Various Shapes, NACA Tech. Note 3038, Ames Aeronautical Laboratory, Moffet Field, CA, 1953.

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82. Obasaju, E. D., On the Effects of End Plates on the Mean Forces on Square Sectioned Cylinders, J. Ind. Aerodyn., 4, 179, 1979. 83. Fage, A. and Johansen, F. C., On the Flow of Air Behind an Inclined Flat Plate of Infinite Span, Proc. R. Soc. London, 116A, 170, 1927. 84. Schlichting, H. and Truckenbrodt, E., Aerodynamics of the Airplane, McGraw-Hill, New York, 1979. 85. Loftin, L. K. and Smith, H. K., Aerodynamic Characteristics of 15 NACA Airfoil Sections at Seven Reynolds Numbers from 0.7  106 to 9.0  106, NACA Tech. Note 1945, Langley Memorial Aeronautical Laboratory, Langley Field, VA, 1949. 86. Ball, D. J. and Cox, N. J., Hydrodynamic Drag Forces on Groups of Flat Plates, ASCE J. Waterways, 104, 163, 1978. 87. Lindsey, W. F., Drag of Cylinders of Simple Shapes, NACA Tech. Rep. TR-619, Langley Memorial Aeronautical Laboratory, Langley Field, VA, 1937. 88. Scruton, C. and Newberry, C. W., On the Estimation of Wind Loads for Buildings and Structural Design, Inst. Civil Eng., 25, 97, 1963. 89. Achenbach, E., The Effects of Surface Roughness and Tunnel Blockage on Flow Past Spheres, J. Fluid Mech., 65, 113, 1974. 90. Awbi, H. B. and Tan, S. H., Effect of Wind Tunnel Walls on the Drag of a Sphere, J. Fluids Eng., 103, 461, 1981. 91. Morel, T. and Bohn, G., Flow over Two Circular Disks in Tandem, J. Fluids Eng., 102, 104, 1980. 92. Nakaguchi, H., Recent Japanese Research on Three-Dimensional Bluff-Body Flows Relevant to Road-Vehicle Aerodynamics, in Aerodynamic Drag Mechanisms of Bluff Bodies and Road Vehicles, Sovran, G. et al., Eds., Plenum Press, New York, 1978, 227. 93. Roshko, A. and Koenig, K., Interaction Effects on the Drag of Bluff Bodies in Tandem, in Aerodynamic Drag Mechanisms of Bluff Bodies and Road Vehicles, Sovran, G. et al., Eds., Plenum Press, New York, 1978, 253. 94. Goldstein, S., Ed., Modern Developments in Fluid Dynamics, Oxford University Press, London, 1938. 95. Alridge, T. R. et al., The Drag Coefficient of Finite Aspect Ratio Perforated Circular Cylinders, J. Ind. Aerodyn., 3, 251, 1978. 96. Springston, G. B., Generalized Hydrodynamic Loading Functions for Bare and Faired Cables in Two-Dimensional Steady-State Cable Configurations, Naval Research and Development Report 2424, Washington, D.C., 1967. 97. Bearman, P. W., Bluff Body Flows Applicable to Vehicle Aerodynamics, Aerodynamics of Transportation, American Society of Mechanical Engineers, New York, 1979. 98. Pawlowski, F. W., Wind Resistance of Automobiles, SAE J., 27, 5, July-Dec., 1930. 99. Chen, C. J. and Rodi, W., Vertical Turbulent Buoyant Jets: A Review of Experimental Data, Pergamon Press, New York, 1980. 100. Fujii, T., Theory of Steady Laminar Natural Convection Above a Horizontal Line Heat Source and a Point Heat Source, Int. J. Heat Mass Transfer, 6, 597, 1963. 101. Andrade, E.N.C., The Velocity Distribution in a Liquid-into-Liquid Jet. II. The Plane Jet, Proc. Phys. Soc., 51, 784, 1939. 102. McNaughton, K. J. and Sinclair, C. G., Submerged Jets in Short Cylindrical Flow Vessels, J. Fluid Mech., 25, 367, 1966. 103. Abramovich, G. N., The Theory of Turbulent Jets, MIT Press, Cambridge, MA, 1963, 99. 104. Rajaratnam, N., Turbulent Jets, Developments in Water Science, Vol. 5, Elsevier, Amsterdam, 1976, pp. 21, 48, 59. 105. Hinze, J. O., Turbulence, 2nd ed., McGraw-Hill, New York, 1975, 536.

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106. Loityanskii, L. G., The Propagation of a Twisted Jet in an Unbounded Space Filled with the Same Fluid, Prikl. Mat. Mekh., 17, 3, 1953. 107. Rose, W. G., A Swirling Round Turbulent Jet, 1-Mean-Flow Measurements, J. Appl. Mech., 29, 615, 1962. 108. Chiger, N. A. and Chervinsky, A., Experimental Investigation of Swirling Vortex Motion in Jets, J. Appl. Mech., 34, 443, 1967. 109. Kerr, N. M. and Fraser, D., Swirl 1, Effect of Axisymmetrical Turbulent Jets, J. Inst. Fuel, 38, 519, 1965. 110. Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley & Sons, New York, 1960. 111. Mills, A. F., Heat and Mass Transfer, Irwin, Chicago, 1995.

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Remsburg R. “Force-System Resultants and Equilibrium” Thermal Design of Electronic Equipment. Ed. Ralph Remsburg Boca Raton: CRC Press LLC, 2001

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4

Convection Heat Transfer in Electronic Equipment

4.1 INTRODUCTION Of the three modes of heat transfer (conduction, convection, and radiation), the convection mode has the most varied applications. Convection is the result of two energy transfer mechanisms: fluid motion and molecular motion. The molecular motion at the heat transfer interface is due to conduction through the stagnant thermal boundary layer. Heat transfer through this layer is based upon Fourier’s law,1 T  qL/k Ac. In convective heat transfer, the engineer is faced with estimating the heat transfer coefficient, hc, for a surface. Usually, this coefficient comes from texts of empirical formulas which are based on actual experiments and observations. We cannot calculate the heat transfer coefficient exactly because we can only analytically solve the differential equations governing convection for the simplest flows and geometries. The empirical equations for the Nusselt number2 are often tedious and complex.

4.2 FLUID PROPERTIES Specific heat—Every material has a thermal capacity. In the SI system, we measure thermal capacity as the heat required to make 1.0 kg of material 1.0°C warmer. Since this capacity is proportional to a material’s mass, we call this the specific heat. We use the specific heat of water as the reference standard of one calorie per gram per degree Celsius. Since a calorie is 4.184 kJ, the specific heat of water at 20°C can be expressed as 4.184 kJ/kg K. The lower the specific heat, the easier it is for the material to absorb heat energy. This property is significant in calculating how readily the fluid can absorb heat from an electronic component. Thermal expansion—The thermal expansion of a fluid is especially important in determining heat transfer under conditions of natural convection. The temperature differential between the electronic component and the ambient environment causes the fluid to expand and become less dense. Heat transfer has increased because of the temperature-induced motion of the fluid. Just as the structure of a liquid allows easier compression than a solid material, it also allows greater thermal expansion. The coefficient of thermal expansion is the increase in volume per degree change

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in temperature. We note the increase in volume as:

 V  V  T where:

V V  T

   

change in volume (m3) original volume (m3) coefficient of volumetric expansion (1/K) change in temperature (K)

Unlike solids and liquids, the cohesive forces affecting the widely spaced molecules of a gas are very small. Therefore, we relate the thermal expansion of a gas to the initial pressure of the gas. We must always note the coefficient of volumetric expansion for a gas at constant pressure. Density—The weight of an object is proportional to the object’s mass. Density () is the object’s mass per unit volume. One cubic centimeter of water at 4°C has a mass of one gram. Therefore, the density of water at 4°C is 1000 kg/m3. Density changes with temperature, especially for gases. Most fluids expand on a generally linear slope with temperature; therefore, density will decrease. Water exhibits a peculiar behavior near its freezing temperature. Water reaches its greatest density at 4°C, not 0°C. Density for a liquid is usually found in tables. We use the general gas law to define the density of a gas: p   --------ᑬT where:

  p  ᑬ T 

density (kg/m3) absolute pressure (kg/m2) real gas constant (286.8 J/kg K) absolute temperature (K)

Appendix I shows some of the characteristics of common gases.

4.2.1

PROPERTIES

OF

AIR

Viscosity is largely a function of temperature in the range of cooling electronics, that is, when p -----  0.2 pc © 2001 by CRC PRESS LLC

and

T -----  1.0 Tc

where: T  Tc  p  pc 

absolute temperature (K) absolute critical temperature (K) pressure (N/m2) critical pressure (N/m2)

Chapman and Enskog3 developed a theoretical expression for the viscosity of a pure dilute gas that is accurate to about 2%: 0.5

0.5

7 M w T  26.69  10 -----------------2

v

where:

 absolute viscosity (N s/m2) Mw  molecular weight

 collision diameter (Å) T  v  1.147  -----  T 

0.145

T   -----  0.5 T 

2.0

T  effective force potential temperature (K) Values for constants for air are found in Appendix I.

4.3 BOUNDARY LAYER THEORY The boundary layer phenomenon is found in both natural and forced-convection modes of heat transfer. The fluid turbulence affects the thickness of the boundary layer and therefore the rate of heat transfer. Figure 4.1 depicts a heated stationary surface at temperature Ts, surrounded by a cooler, moving fluid at a bulk temperature of T and free-stream velocity of U . Note that the fluid velocity decreases closer to the stationary surface. Since the fluid at the interface is also stationary, Fourier’s conduction equation determines the heat transfer through this region. In small fluidfilled spaces, the thermal boundary layer can be thick enough to prevent appreciable natural convection. In these cases, heat transfer by conduction is dominant. We will define the thickness of the boundary layer as the distance from the stationary surface to a distance where the moving fluid is less than 99% of the free-stream velocity. To further understand the ideas of the velocity and the thermal boundary layers, and how these affect the heat transfer coefficient, we will study forced air flow over a heated flat plate. As stated previously, the fluid particles directly in contact with an object are stationary and have a velocity of zero. This stagnant layer impedes the velocity of the fluid particle layer directly above the layer of zero velocity. In turn, each underlying layer of fluid particles impedes the velocity of the layer directly © 2001 by CRC PRESS LLC

FIGURE 4.1 Transition to turbulence on a flat plate. This view shows the laminar sublayer and the buffer layer that separates the laminar and turbulent layers.

FIGURE 4.2 Development of the velocity boundary layer along a flat plate. The thickness of the boundary layer, , can be described as 5 x / Re x .

above it. As we have previously discovered, the distance from the stationary surface to the point where the velocity impediment is negligible is called the velocity boundary layer thickness, . Figure 4.2 shows the development of the velocity boundary layer. The velocity boundary layer increases in thickness at greater distances from the initiation point, in the direction of fluid motion. From our examination of fluid dynamics, we know that we can find the thickness of the velocity boundary layer, , at a distance of x, by the equation 5.0x   -----------Re x Also, from our studies of fluid dynamics, we know that the fluid shear stress has an influence on the surface friction and that we can find the skin friction coefficient at a distance from the initiation point by applying the equation:

0.5

C f , x  0.6641Re x

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FIGURE 4.3 Development of the thermal boundary layer along an isothermal flat plate. The thickness of the thermal boundary layer, t, can be described as 0.975  /Pr1/3.

and we can find the average friction coefficient over the flat surface by

0.5

C f , x  1.3282 Re x

Similar to the velocity boundary layer, , there is a thermal boundary layer, th, which forms when there is a difference in temperature between the free stream and an object. Fluid particles directly in contact with a hot object reach the temperature of the object. Measuring the fluid temperature at greater distances from the object, the temperature decreases to the free-stream temperature. We call the fluid region where the fluid temperature gradient occurs the thermal boundary layer. Figure 4.3 shows the development of the thermal boundary layer. We describe the thermal boundary layer as the distance at which the fluid reaches 99% of the free-stream fluid temperature. Just like the velocity boundary layer, the thermal boundary layer thickness increases with distance from the initiation point. We relate the thickness of the thermal boundary layer, th, to the Prandtl number and the velocity boundary layer by:

   th  .975  ----------13 Pr  In our examination of fluid dynamics, we learned about the displacement boundary layer, designated d, which represents the decrease in the volume rate of flow per unit width due to the presence of the boundary layer. The displacement boundary layer displaces the free-stream lines of flow by the distance d and is related to the Reynolds number by the equation

0.5

 d  1.7208x Re x

The heat transfer coefficient, hc, is the basis for quantifying convective heat transfer. Consequently, calculating hc with any degree of accuracy for all but the simplest geometries is impossible. Nevertheless, many empirical equations are sufficiently accurate to give the design engineer an understanding of the heat transfer problem. We can describe the heat transfer coefficient as: q h c  ------------A s T © 2001 by CRC PRESS LLC

Using the knowledge that the heat transfer coefficient is related to the thermal boundary layer, we can find that the local hc at a specific plate length is approximately: 3k h c  --------2  th If we use a more rigorous analysis, we obtain the local Nusselt number equation for laminar flow over a flat plate: hc x -  0.332Pr 1  3 Re 1x  2 Nu x  ------k which is valid for fluids with a Prandtl number greater than 0.1. The average Nusselt number for laminar flow over a flat plate is found as: 12

Nu L  0.6641 Re L Pr

13

The Reynolds analogy can then equate the friction coefficient to the heat transfer coefficient for the flat plate in laminar flow: C Nu x  2  3 0.322  -------------------fx - Pr  -----------12  2  Re x Pr Re x Here, we call the ratio Nux /Rex Pr the Stanton number, Stx.

4.4 DIMENSIONLESS GROUPS Besides the Reynolds number, the Prandtl number, and the Nusselt number discussed previously, we can use many other dimensionless numbers to describe the physical phenomenon in heat transfer. Researchers have named the dimensionless groups after some outstanding researchers in this field. Dimensional analysis requires a consistent set of units to equate the variables. Table 4.1 shows some important variables used in this text and their dimensions. We can use the Buckingham  theorem to learn the number of independent dimensionless groups required to find a relationship that describes a physical parameter. This rule states that the number of required independent dimensionless groups that can be formed by combining the physical variables related to a problem is equal to the total number of these physical quantities, minus the number of primary dimensions required to express the dimensional formulas of the physical quantities. We find that in forced convection heat transfer problems, we obtain seven physical quantities and four primary dimensions from the Buckingham  theorem, as shown in Table 4.2. The convective heat transfer coefficient, hc, is the variable that we eventually want to evaluate. Solving the simultaneous equations, we obtain the resulting first © 2001 by CRC PRESS LLC

TABLE 4.1 Heat Transfer Physical Quantities Nomenclature and Dimensions Quantity Absolute viscosity Acceleration Coefficient of thermal expansion Density Enthalpy Force Heat Heat transfer coefficient Internal energy Kinematic viscosity Length Mass Mass flow rate Pressure Shear per unit area Specific heat Surface tension Temperature Thermal conductivity Thermal diffusivity Thermal resistance Time Velocity Work

Nomenclature

Dimensions

a,g   h F Q h e v ( ) L, x M m˙ P  c

T k   t u, U W

M/Lt L/t2 I/T M/L3 L2/t2 ML/t2 ML2/t2 M/t3T L2/t2 L2/t L M M/t M/t2L M/Lt2 L2/t2T M/t2 T ML/t3T L2/t Tt3/ML2 t L/t ML2/t2

dimensionless group: hc D -  Nu  1  -------k This is the Nusselt number that was discussed earlier. Simultaneous solving of the equations determine 2: U D  2  --------------  Re which forms the Reynolds number, with the critical characteristic dimension being D. The resulting third dimensionless group is c p -  Pr  3  -------k which forms the Prandtl number. © 2001 by CRC PRESS LLC

TABLE 4.2 Variables for Buckingham  Theorem for Forced Convection Variable

Nomenclature

Dimension

D k U  cp hc

[L] [MT/t3T] [L/t] [M/L3] [M/Lt] [L2/t2T] [M/t2T]

Characteristic Physical Dimension Thermal Conductivity Velocity Density Absolute Viscosity Specific Heat (constant pressure) Coefficient of Heat Transfer

TABLE 4.3 Variables for Buckingham  Theorem for Natural Convection Variable

Nomenclature

Characteristic Physical Dimension Coefficient of Thermal Expansion Characteristic Velocity Temperature Difference (T T ) Kinematic Viscosity Acceleration Due to Gravity Thermal Diffusivity

L  U T v g 

Dimension [L] [1/T] [L/t] [T] [L2/t] [L/t2] [L2/t]

Although the heat transfer coefficient depends on six variables, with the aid of dimensional analysis we have combined the seven original variables into three dimensional groups: Nu, Re, and Pr. We can then write the functional relationship as Nu D  f ( Re D, Pr ) which is a form we will see in many empirical forms. We find that in natural convection, like forced convection, we also have seven physical quantities and four primary dimensions, but they are different, as shown in Table 4.3. From this we can expect that we can describe the Nusselt number in three (seven physical four dimensional) dimensionless groups. Using the previous Buckingham  method, we obtain the first dimensionless group U L  1  ----------v which is the Reynolds number, v  2  --

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FIGURE 4.4 Correlation of various experimental data for cylinder in a cross flow using dimensional analysis. (From Kreith, F. and Bohn, M. S., Principles of Heat Transfer, 4th ed., Harper & Row, New York, 1986. With permission.)

which is the Prandtl number, and 3 2

L  g(T T )  3  ----------------------------------------2 which we call the Grashof number. The Grashof number represents the ratio of buoyant forces to viscous forces within the fluid. In natural convection we know that no external element is forcing the fluid, and that temperature differences supply the motion. In these cases the Reynolds number, 1, is not an independent variable. Therefore, we eliminate the dependence of the Nusselt number on the Reynolds number, and we replace it by an empirical formulation. Nu   ( Gr )  ( Pr ) Dimensional analysis allows the researcher to apply the results of one experimental test to a variety of other problems. After dimensional analysis, shown in Figure 4.4, we see that test results for water, oils, and air flow over different diameter cylinders can all be correlated by using the Reynolds number and the Nusselt number. Biot number—Bi  hcro/k is used in problems concerning simultaneous conduction and convection modes of heat transfer, usually relating to transient heat flow. The Biot number is the ratio of conductive thermal resistance, Rk, to a convective resistance, Rc; therefore, r0  k Rk Bi  -----  -----------Rc 1  hc © 2001 by CRC PRESS LLC

The physical limits of the ratio are Bi → 0

when

r R k  ----o → 0 k

or

1 R c  ---- → hc

Bi →

when

1 R c  ---- → 0 hc

or

ro R k  ---- → k

which states that the Biot number approaches zero when the conductivity of the solid (or the convective resistance) is so large that the solid body is essentially isothermal. Therefore, the temperature change is mostly in the fluid at the solid/fluid interface. The Biot number approaches infinity when the thermal resistance within the solid is so great that the temperature change within the fluid is very small. Colburn factor4 for heat transfer—j  (Nu/RePr)Pr2/3 suggests the relationship between heat transfer and fluid friction. We use the Colburn factor for heat transfer quite often when comparing the efficiencies of extended surface designs for a given application. We use the Colburn factor for mass transfer, jm  (Sh/ReSc)Sc2/3, less often. Eckert Number5—Ec  U /cp(Ts T ) is used primarily in high-speed flow problems. In high-speed flow the boundary layer may support very large temperature gradients. Since the physical properties of the fluid depend on the temperature of the fluid, setting the fluid properties is difficult. The Eckert number shows that we can still use the constant-property heat transfer equation if we evaluate all of the fluid properties at a reference temperature. This number is proportional to the ratio of the temperature rise of a fluid in an adiabatic compression to the temperature difference between the wall and the fluid at the edge of the boundary layer. The Eckert number can also be expressed in terms of the Mach number, M, for a perfect gas, as follows: 2

(  1 )M e T e Ec  -------------------------------Tw Te Elenbaas number6—El  z42gTcp / kL is a gap-based Rayleigh number. We use this number to find the Nusselt number of natural convection flow between vertical plates. This problem occurs in the design of heat sinks and flow between cards in an electronic system. Kraus and Bar-Cohen7 proposed the Elenbaas number as a tribute to W. Elenbaas. The modified Elenbaas number, El, can be expressed for problems where the temperature is unknown but the heat flux, q, is known: 5 2

z  gBc p q El  --------------------------2 k L 2

Fourier modulus—Fo  tr /L2 or tr / r o is seen quite often in transient heat conduction problems. This number is the ratio of the rate of heat transfer by conduction to the rate of energy storage in the system. The variable  represents thermal © 2001 by CRC PRESS LLC

diffusivity and is equal to k/c. If we accept that the nondimensional form of the conduction equation is 2

2

2 L    q˙ G L r -------2-  ----------  ------r- -----kT  t r  r  2

then we call the reciprocal of the dimensionless group, L r /tr , the Fourier modulus, or the Fourier number, Fo:

tr Fo  ------2Lr Grashof number8—Gr  L32gT/ 2 is used frequently in problems of natural convection and represents the ratio of buoyant forces to viscous forces. The Grashof number is the third dimensionless group of the conservation of energy equation for natural convection. 3 2

L  g(T T )  3  ----------------------------------------2 We often multiply the Grashof number by the Prandtl number to arrive at the Rayleigh number. The ratio of the Grashof number and the Reynolds number shows whether natural or forced convective forces are dominant. Lewis number9—Le  /DAB  Sc/Pr is the relation of the Prandtl and the Schmidt numbers. The Lewis number is important in situations where heat transfer and mass transfer occur simultaneously. For example, a Lewis number of 1.0 would suggest that psychrometric and thermodynamic wet-bulb temperatures are equal. Nusselt number—Nu  hcx/kf is the key to finding the heat transfer coefficient, which is usually the unknown variable in most heat transfer problems. In dimensionless parameters, we obtain: k (T T s) T * ---------h c  -----f -----------------------L ( T s T )  y*

y*  0

k T   -----f --------L  y*

y*  0

Upon close inspection of these parameters, we can see it suggests that a form of the heat transfer coefficient is the Nusselt number, Nu, which we define: hc L  T * Nu  -------- ⬅ ---------kf  y*

y*  0

We know that the Nusselt number for a prescribed geometry depends only on x*, ReL, and the Prandtl number Pr, or: Nu  f 1 ( x*, Re L, Pr )

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Once we know the functional relationship for a particular fluid, we can use it to obtain the Nu for other fluids and for any value of U and L. Also, we can use the local value of Nu to find the local value of hc, the average value of hc, and the average value of NuL. Since integrating over the heat transfer surface obtains the average heat transfer coefficient, it is independent of x*. Therefore, the average Nusselt number becomes a function of only the Reynolds number and the Prandtl number: hc L -  f 2 ( Re L , Pr ) Nu L  ------kf Peclet number—Pe  UL/  Re Pr plays the same role in the transfer of thermal energy as the Reynolds number does for transfer of momentum. The Peclet number, which is the product of the Reynolds number and the Prandtl number, is often used in problems involving creeping external laminar flow, where Re ⬃1.0. Such problems are mainly encountered in analysis of porous media flow. The Pe can be thought of as the ratio of transport by convection to the transport by thermal diffusion. Prandtl number—Pr  cp /k is the ratio of momentum to energy in a fluid. This dimensionless group predicts the relation between velocity and temperature distributions. We can also describe the Prandtl number as the ratio of momentum diffusivity to thermal diffusivity. In many plots of experimental or theoretical data points, researchers combine the Nusselt number and the Prandtl number in the y-axis and use the Reynolds number in the x-axis. Rayleigh number—Ra  GrPr is the product of the Grashof and Prandtl numbers. In our discussion of dimensionless parameters we saw that 3 2 L   g  T   c p  Nu    ------------------------- ---------   ( Gr )  ( Pr ) 2    k 

Quite often the experimental data will collapse the Nusselt number to the product of a constant and the Rayleigh number and the Prandtl numbers: Nu   ( Gr )  ( Pr )  C ( RaPr ) The Rayleigh number can suggest the critical value at which the flow of fluid will become unstable and turbulent in a natural convection system. In systems of closely spaced walls, the Rayleigh number suggests whether heat transfer between adjacent walls is predominantly convective or conductive. Reynolds number—Re  LU  / indicates the degree of turbulence in a moving fluid system. The Reynolds number describes the ratio of inertial force to viscous force within the fluid stream. When the ratio exceeds a certain value, which is different for different geometries, the fluid no longer moves in discreet streamlines. At higher values of the Reynolds number than laminar flow there is a region of mixed boundary layer flow, and then full turbulence. Schmidt number10—Sc  v/DAB describes the fluid velocity and concentration distributions. This number provides a measure of the relative effectiveness of momentum and mass transport by diffusion in the boundary layer. The Schmidt

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number relates the Lewis and Prandtl numbers and often replaces the Prandtl number in empirical equations involving the Sherwood number for mass transfer. If we let the kinematic viscosity v equal the momentum diffusivity /, then we see that the ratio of v/ is equal to the Prandtl number: c p v / ---  ---------------  ------- k  c p k  Pr Therefore, if v   then Pr  1.0, and the momentum and energy equations are identical. It is apparent then that the Prandtl number, the ratio of fluid properties, describes the relation between velocity and temperature distribution. If the kinematic viscosity, called the momentum diffusivity, v, equals the binary diffusion coefficient DAB, then the mass and momentum conversion equations are identical. We then refer to the ratio of v/DAB as the Schmidt number, Sc; therefore: v Sc  --------D AB Sherwood number11—Sh  hmx/DAB is similar to the Nusselt number for heat transfer but is used to describe mass transfer. We call the dimensionless form of the mass transfer coefficient the Sherwood number. This nondimensional number has a definition of

C* hm L Sh  --------  ---------A y* D AB

y*  0

The local value of the Sherwood number for a prescribed geometry depends only on x*, ReL, and the Schmidt number, Sc, or Sh  f 1 ( x*, Re L, Sc ) and the average value of the Sherwood number depends on only ReL and Sc, or hm L Sh L  --------  f 2 ( Re L ,Sc ) D AB Stanton number—St  hc/U cp or Nu/RePr is the number used to evaluate the heat transfer coefficient to the fluid friction coefficient. By using the Stanton number, we can rate a variety of heat transfer surfaces according to the rate of heat transfer achieved for each unit of fluid pressure loss. The wall heat flux can be nondimensionalized equal to the Stanton number and we define the Stanton number as qw St  --------------------------------- e u e ( i w i aw )

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We can obtain a relationship between the local Nusselt number Nux and the corresponding skin friction coefficient Cfx by dividing the local Nusselt number by the product of the local Reynolds number and the Prandtl number to the 1/3 power, or C Nu x  2  3 0.322  -------------  -------fx - Pr  -----------1/2  Re x Pr 2 Re x This equation states that the Stanton number (Nux /Rex Pr) times the Prandtl number to the 2/3 power is equal to 1/2 the value of the friction coefficient.

4.5 FORCED CONVECTION The field of convection pertains to heat transfer aided by a moving fluid. Within the field of convective heat transfer we have two basic types of flow, external and internal. Within each of these flow regimes there are natural convection and forced convection. Density gradients in the fluid cause fluid movement in natural convection. Fluid movement in forced convection is usually by some artificial means such as a pump or fan. Since the fluid is forced over a heat transfer surface, the velocities are usually much higher than natural convection flows. The forced fluid decreases the thickness and therefore the thermal resistance across the boundary layer, increasing the heat transfer coefficient from the surface. Forced air convection heat transfer rates are often a magnitude higher than natural convection, and forced liquids can raise the heat transfer rate by another magnitude. Because the complex three-dimensional fluid dynamics of convection are not yet solvable, most of the formulas presented in this chapter are correlations of experimental data. Besides the actual heat transfer, pressure drop, p, becomes very important in forced convection. Engineers have designed cooling fans to move air from a low pressure region to an area of higher pressure. If the pressure of the area of higher pressure is too high, the fan may not operate as expected. This commonly occurs if the fluid outlet of an electronic chassis does not have sufficient area or if components, bends, and heat sinks obstruct the flow path. Effect of the Reynolds number—The Reynolds number can be used to find the effect that turbulence will have on both pressure drop and the Nusselt number. In a long tube of constant dimensions, the Reynolds number suggests a laminar flow condition up to Re  2000. At about 2100, the flow is generally considered to be converting to turbulence. At a Reynolds number of 10,000, flow is fully turbulent. Depending on the surface roughness of the tube, the Reynolds number that shows turbulence can be quite different from these averages. The characteristic length for flow in tubes is the hydraulic diameter, DH, where we find the Reynolds number by U DH  UD  -----------HRe DH  --------------

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FIGURE 4.5 Variation of Nusselt number with Reynolds number for flow in a long straight tube. (From Kreith, F. and Bohn, M. S., Principles of Heat Transfer, 4th ed., Harper & Row, New York, 1986. With permission.)

In laminar flow, the thermal boundary layer is stratified and heat transfer occurs only by heat conduction within the fluid. At the initiation of turbulent flow, the Nusselt number increases rapidly. During turbulent flow, eddy currents disrupt the layers of the thermal boundary layer which cause heat transfer by convection. After full turbulence is reached the Nusselt number increases, but at a slower rate. The relationship of the Reynolds number and the Nusselt number is shown in Figure 4.5. Even in fully turbulent flow, heat transfer at the fluid wall occurs by conduction. In the fluid layer directly against the wall, the viscous forces dampen turbulent fluid disturbances that dominate in the wall region. Most of the heat transfer in forced convection occurs at the border between the viscous sublayer region and the turbulent region. Because heat transfer in the turbulent region is already very good, the only way to increase heat transfer is to reduce the thickness of the thermal boundary layer. If this is to occur, we must increase the Reynolds number. A side effect of turbulence is the large energy losses that occur. We see these energy losses as increased pressure drop through the tube. We have previously seen that in convective flow along a flat plate, the Nusselt number is related to Re0.8. Since, as we have seen, the velocity and thermal boundary layers are based on the same principles, this relationship is also true for tube flows.

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Effect of the Prandtl number—As we have found previously, the Prandtl number is the ratio of kinematic viscosity of a fluid to the thermal diffusivity of the fluid, and we can describe this ratio as c p  Pr  ---  -------k  We sometimes call the kinematic viscosity term  or (cp ) the molecular diffusivity of momentum because it is a measure of the rate of momentum transfer between the molecules in the fluid. The thermal diffusivity term (k/cp) is sometimes called the molecular diffusivity of heat because it is a measure of the ratio of heat transmission and energy storage capacities of the fluid molecules. If the fluid in a tube has a Prandtl number of 1.0, then the velocity and thermal profiles are identical to that of identical conditions along a flat plate. If the Prandtl number is less than 1.0, the temperature gradient of the surface layers is less steep than the velocity gradient. For fluids with a Prandtl number greater than 1.0, the thermal gradient is steeper than the velocity profile gradient. In the temperature range of electronics, the Prandtl number of gases, which is generally from about 0.6 to 0.9, shows that the velocity gradient is steeper than the temperature gradient. For the same temperature range, the Prandtl number of liquids starts at about 0.9. Oils may have a high Prandtl number of 5000 due to their high viscosity and low thermal conductivity. Effect of the entrance—When working with internal flow problems, the shape of the entrance can affect flow and heat transfer. If the tube is short, the correlations used for fully developed flow will be inaccurate because the effects of the entrance dominate. Consider the case of a tube with fluid entering at a constant velocity. Wall drag brings the fluid layer immediately beside the tube wall to a rest. For a given distance from the entrance, the fluid forms a laminar boundary layer along the wall. If the Reynolds number is high, the fluid is turbulent and the boundary layer will become turbulent. The boundary layer will increase in thickness as x increases. The boundary layer in a tube will increase until it merges at the geometric center of the tube’s diameter. At this point, the fluid flow is called a fully developed flow. A fully developed flow will remain essentially unchanged if the tube geometry and fluid velocity remain constant. Similar to the hydrodynamic boundary layer that forms on a tube wall, a thermal boundary layer also forms. The actual shape of the fully developed flow’s velocity or thermal profile depends on the Reynolds number. The shape of the profile in the entrance region depends on the type of entrance. If the entrance to a round tube is square-edged, the profile resembles that of flow along a flat plate. In this case, the local heat transfer coefficient is greatest near the entrance and decreases along the flow path. When the flow dynamics have established the steady-state velocity and thermal profiles, we see a fully developed flow. In internal flow, turbulence begins at a Reynolds number of about 2200. If the Reynolds number is below this, flow is laminar and the entrance effects may extend as much as 100 hydraulic diameters along the length of the flow path. In a round tube with fluid having a laminar flow profile, the length for entrance effects on the

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velocity profile can be found by  x fully developed  0.05 Re D -  -----------------------laminar D In a round tube with fluid having a laminar flow profile, the length for entrance effects on the thermal profile can be found by  x fully developed  0.05 Re D Pr -  -----------------------laminar, thermal D In a tube having a turbulent profile, the mixing that occurs in the boundary layer quickly overcomes the entrance effects. When the Reynolds number is greater than 2200, the entrance effects are independent of the Prandtl number and dissipate within 10 to 20 tube diameters along the flow path. In a later section we will calculate the fully developed Nusselt number for a round tube with a uniform wall temperature as 3.657. However, this value is not adequate for calculating the heat transfer coefficient when the tube length is within the zone where entrance effects dominate. For a round smooth tube, Edwards et al.12 developed a correlation for the laminar Nusselt number that includes the effects of the entrance: 0.065 ( D/L )RePr Nu  3.66  -----------------------------------------------------------23 1  0.04 [ ( D/L )RePr ] where Re  2300. For laminar flow between parallel plates at a constant axial wall temperature, the fully developed Nusselt number in laminar flow is 7.541. Edwards et al.12 also correlated the entrance effects for this configuration as 0.030 ( D h /L )RePr Nu  7.54  ----------------------------------------------------------------23 1  0.016 [ ( D h /L )RePr ] where Re  2800 and DH  hydraulic diameter. When the Reynolds number suggests turbulence, the entrance effect is more difficult to obtain. For a rough approximation of the effect of the entrance shape for turbulent gases at a Prandtl number of about 1.0, Mills13 found that the resulting Nusselt number is a multiple of the fully developed Nusselt number. Table 4.4 and Figure 4.6 show the result of Mills’ work. Effect of property variations—The thermophysical properties of a fluid can affect the value of the Nusselt number and heat transfer coefficient. As fluid moves along a surface, the fluid temperature approaches that of the surface. Consequently, the fluid properties will vary from the start point to the end. Because of the cohesive structure of a liquid, only the viscosity changes appreciably and is important. For gases, thermal

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TABLE 4.4 Effect of the Entrance on the Ratio of the Average Nusselt Number to the Fully Developed Turbulent Nusselt Number (For Pr ⬃ 1.0) Tube Length (L/D) Entrance Shape Long calming section Open end, 90° edge 90° Elbow Tee (confluence) 90° Round bend 180° Return bend

2

4

6

8

10

20

40

80

160

320

1.49 2.36 2.15 1.77 1.63 1.54

1.34 1.95 1.86 1.56 1.44 1.37

1.26 1.73 1.68 1.44 1.34 1.28

1.21 1.60 1.57 1.36 1.28 1.23

1.17 1.54 1.49 1.31 1.24 1.19

1.10 1.32 1.32 1.19 1.16 1.12

1.06 1.18 1.18 1.10 1.10 1.08

1.03 1.09 1.09 1.06 1.05 1.04

1.01 1.05 1.05 1.03 1.03 1.02

1.01 1.02 1.02 1.01 1.01 1.01

FIGURE 4.6 Entrance region heat transfer for turbulent flow of gases with various entrance configurations; hc∞ is the fully developed heat transfer coefficient. (From Mills, A. F., J. Mech. Eng. Sci., 4, 63, 1962. With permission.)

conductivity (k), viscosity ( ), and density () are temperature related. For gases,   T 1,  T 0.7, and k  T 0.7. Therefore, for fluids, the Reynolds number because it is based on density and viscosity; the Prandtl number because it is related to thermal conductivity; and, in natural convection, the Grashof number because it is density and viscosity related are all affected by the fluid evaluation temperature. In liquids, the absolute viscosity decreases with increasing temperature, while gases act in reverse. When we increase the temperature of a liquid, the layer near the heat transfer surface becomes less viscous than the bulk liquid. Therefore, the velocity of the higher-temperature liquid is greater than that of an unheated liquid near the wall but is less in the center. These changes affect the shape of the parabolic velocity profile.

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In problems of fluid dynamics, which are isothermal, we always evaluate the fluid properties at the bulk fluid temperature. In cases of heat transfer, theoretically, the fluid property variation should appear as a variable within an equation. In evaluating friction loss, the viscosity variation should appear as s / b, where s  fluid viscosity at the surface temperature and b  fluid viscosity at the bulk temperature. Usually, we evaluate the fluid at the average film temperature. We define this as the temperature halfway between the wall surface temperature and the fluid bulk, or ambient temperature. For internal flows, the property ratio or the temperature ratio approach is often used in correlations. For internal liquid flow, we use a viscosity ratio or Prandtl number ratio, since viscosity is the important variable. For internal gaseous flows, we use the temperature ratio since density, viscosity, and thermal conductivity are all functions of the temperature. When parallel plates are distance z apart and uniformly heated, Swearingen and McEligot14 accounted for gaseous property variations using the correlation (q s ) ( 0.5z )  0.75 - Gz b Nu  Nu  0.024  ------------------------- ( kT ) entrance  0.3

where: Nu qs z G

   

Nusselt number, evaluated at constant properties heat flux (W/m2) plate spacing (m) flow rate per unit area, V (kg/m2 s)

Another complexity of internal flows is while the fluid temperature will vary according to the length traveled, the temperature of an isoflux wall will also vary. Fortunately, correlations for isoflux conditions usually account for this effect and, if not, we use the mean temperature conditions within an acceptable margin of error. If we know the fluid bulk temperature, the power applied to the heat transfer surface, and the mass flow rate, we can establish the fluid temperature rise measured at the exit point by using the relationship q  T  --------m˙ c p where m˙  mass flow rate (kg/s).

4.5.1

FORCED CONVECTION LAMINAR FLOW

In this section we will explore the fluid and heat transfer dynamics of laminar flow in tubes and noncircular ducts. Flow is generally considered laminar up to Reynolds number values of 500,000 for flat plates to 100 for pin-fins, and up to 2200 for internal flows. Above these values the flow can become turbulent quickly or can

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FIGURE 4.7 Development of the thermal boundary layer profile in a heated circular tube. This view shows that the thermal boundary layer is fully developed at a distance of xfd,t.

remain in a transition region. Surface effects can lower the transition to turbulence of a flat plate from a Reynolds number of 500,000 down to a value of 50,000. Cylinders begin a transition to turbulence at about 100, and we measure the degree of turbulence by how much of the surface boundary layer is in turbulence. Fortunately, researchers are developing correlations that can span the laminar, transition, and turbulent regions of flow with an acceptable degree of accuracy. 4.5.1.1

Forced Convection Laminar Flow in Tubes

Since we have previously studied laminar flow dynamics, we can now evaluate cases of isoflux and isothermal heat transfer in laminar flow. We assume that the temperature gradient does not affect the laminar velocity profile. Referring to Figure 4.7, we see the thermal boundary layer development for a circular tube. Except for the entrance region, where the heat flux is uniform, the temperature difference between the surface and the bulk fluid remains constant along the length of the duct. Knowing that the flow profile in a laminar tube is parabolic, we can write the average heat transfer coefficient as k (  T /  r )r  r qc h c  ---------------------------------s  --------------------------A(T s T b) Ts Tb Then we evaluate the radial temperature gradient at r  rs, and we arrive at the heat transfer coefficient for laminar isoflux flow: 48k 24k h c  ----------  ---------11D 11r s

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FIGURE 4.8 Variations of axial temperature in a long circular tube. (a) Condition of constant surface temperature, also called isothermal. (b) Condition of constant heat flux, also called isoflux.

which also describes the Nusselt number for laminar isoflux flow in a circular tube: hc D -  4.364 Nu D  -------k The problem of heat transfer in an isothermal tube is somewhat more complex because the temperature difference between the bulk fluid temperature and the wall temperature varies along the tube. Figure 4.8 shows a comparison of isoflux and isothermal temperature profiles. Because the profiles are not constant, we require an iterative solution process. We find the rate of heat transfer by forced convection from the tube by q c  m˙ c p [ ( T s T b, in ) ( T s T b, out ) ]  m˙ c p (  T in  T out ) We find that the Nusselt number is a constant for isothermal conditions: hc D -  3.657 Nu D  -------k

4.5.2

FORCED CONVECTION TURBULENT FLOW

In tubes, when the Reynolds number reaches about 2200, the flow is becoming turbulent. The region from 2200 to about 10,000 is very difficult to analyze because the turbulence often has an intermittent character. That is, at any stationary point in the flow, turbulence may or may not be present at that instant in time. As the Reynolds

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number rises from 2200 to 10,000, the probability that the flow at the stationary point will be turbulent increases until the flow becomes fully turbulent. A complex eddying motion that causes fluctuations of the velocity components, pressure, and temperature characterizes turbulent flow. In gaseous compressible flow, the eddies also influence the gas density. Because of the unpredictability of flow in this transition range, designing a system that does not operate in this region is best. When a system operates in the transition range, engineering calculations for the heat transfer coefficient may have more than a 50% margin of error. Since we cannot accurately predict the exact conditions at a point in turbulent flow, engineers use correlations that are based on empirical data and simplified models of the flow. The Prandtl mixing length theory15 is one successful model of fluid turbulence. In this model the heat transfer process is analogous to the kinetic theory of gases. Eddies replace molecules as carriers of momentum and thermal energy, and a mixing length replaces the mean free path. 4.5.2.1

Forced Convection Turbulent Flow in Tubes

As we noted in the previous section, flow in tubes is considered fully turbulent when the Reynolds number reaches 10,000. Therefore, the correlations in this section do not apply to Reynolds number flows of less than 10,000. Also, since researchers cannot yet solve the exact mathematical analysis of turbulent flow, the correlations are based on empirical data and simplified models. Figure 4.9 shows the sublayers of the turbulent boundary layer in a circular tube with fully developed flow and the velocity profile under the same conditions. Experimental results of friction in tubes are presented in the Moody16 diagram, which accounts for the surface roughness inside the tube. We relate the Fanning friction coefficient to the Reynolds number in turbulent flow by the correlation

0.25

f  0.316 Re D for Reynolds numbers less than 2  104, and

0.20

f  0.184 Re D

for Reynolds number above 2  104. Colburn studied solutions to the local Nusselt number for fully developed turbulent flow in circular tubes. The Chilton and Colburn17 analogy relates the friction factor to the Nusselt number: Nu D 0.667 Cf 0.667 f - Pr  -------------------  ---  St Pr Re D Pr 8 2 Substituting the previous correlation for the friction coefficient, we arrive at the local Nusselt number for isothermal flow, 0.8

Nu D  0.023 Re D Pr © 2001 by CRC PRESS LLC

0.333

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FIGURE 4.9 Fully developed turbulent flow in a smooth circular tube. (a) The three layers of flow of the turbulent boundary layer are shown. (b) Profile of the turbulent hydrodynamic velocity in a circular tube.

When heat transfer occurs, that is, the tube and the fluid are at different temperatures, we prefer the Dittus and Boelter18 equation: 0.8

Nu D  0.023 Re D Pr

n

where: n n 0.7 ReD L/D

 0.4 when Ts  Tm  0.3 when Tm  Ts Pr 160  10,000  10.0

If the T is large enough to cause property variations, the Sieder and Tate19 correlation is preferred, due to the correction for the variations in viscosity: 0.8

Nu D  0.027 Re D Pr

0.333

 ------b  s

0.14

where:

b  absolute viscosity at fluid bulk temperature (N s/m2) s  absolute viscosity at heat transfer surface temperature (N s/m2) Gnielinski20 proposed an expression for the Nusselt number that we can use for Reynolds number flow in the transition region, but which is applicable only for smooth tubes since the heat transfer coefficient increases with increasing surface roughness: ( f /8 ) ( Re D 1000 )Pr Nu D  --------------------------------------------------------------------0.5 0.667 1  12.7 ( f /8 ) ( Pr

1) where: f  (0.79 ln ReD 1.64) 2 0.5  Pr  2000 2300  ReD  5  106 Average Nusselt numbers are of more use to the engineer, but including the effects of the entrance region complicates these correlations. Usually, if L/D  60 then the previous correlations will have an error of 15% or less. Figure 4.10 shows the relationship of various correlations for the turbulent flow of water. 4.5.2.2

Forced Convection Flow through Noncircular Tube Geometries

We have a varied choice of correlations for the Nusselt number in particular tubes and ducts. The important geometry in these shapes is the shape of the duct—square © 2001 by CRC PRESS LLC

FIGURE 4.10 Comparison of various Nusselt number correlations for turbulent water flow in a circular tube using experimental data points (26.7°C, Pr  6.0). (From Kreith, F. and Bohn, M. S., Principles of Heat Transfer, 4th ed., Harper & Row, New York, 1986. With permission.)

or circular—and the ratio of the long side to the short side. As in other flows, the correlations can be separated into laminar correlations and turbulent flow correlations. Although the heat transfer coefficient is much higher in turbulent flow, the need to avoid excessive pumping losses and noise often mandates laminar flow. Table 4.5 summarizes the results compiled by Shah and London21 and Kays and Crawford22 for the mean Nusselt numbers for duct shapes. Researchers23,24 proposed a correlation for heat transfer for turbulent forced convection flow in triangular and rectangular ducts in air in the form:

0.7 L 0.7  T s 0.7 0.8 0.4 T ----Nu DH  0.021 Re DH Pr  -----s 1   -------  T b  D H  T b

4.5.2.3

Forced Convection Flow through Tubes with Internal Fins

We know that the boundary layer along the internal surface of a tube inhibits heat transfer. We have previously examined the effect of rough internal surfaces, which result in a higher Nusselt number because of the disruption of the boundary layer. Researchers have also studied the effect of internal fins that travel the length of a tube. Carnavos25 studied the heat transfer performance of air in turbulent flow in tubes having different © 2001 by CRC PRESS LLC

Table 4.5 Nusselt Numbers for Various Duct Geometries Geometry

L/W

NuH1

NuH2

NuT

f

Round

N/A

4.364

4.364

3.657

64/Re

Hexagon

N/A

4.002

3.862

3.340

30/Re

Ellipse

1.11:1

5.009

4.350

3.660

75/Re

Square

1:1

3.608

3.091

2.976

57/Re

Rectangle

1.43:1

3.730

3.054

3.080

59/Re

Rectangle

2:1

4.123

3.017

3.391

62/Re

Rectangle

3:1

4.800

2.974

4.000

69/Re

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TABLE 4.5 (continued) Nusselt Numbers for Various Duct Geometries Geometry

L/W

NuH1

NuH2

f

NuT

Rectangle

4:1

5.331

2.930

4.439

73/Re

Rectangle

8:1

6.490

2.904

5.597

82/Re

Rectangle

:1

8.235

8.235

7.541

96/Re

Triangle

1:1:1

3.111

1.892

2.470

53/Re

Triangle (rounded corners)

1:1:1

4.205

3.780

2.470

64/Re

Notes: ReD  2200. NuH1  average Nusselt number for uniform heat flux in the flow direction and uniform wall temperature at any cross section. NuH2  average Nusselt number for uniform heat flux, both axially and circumferentially. NuT  average Nusselt number for uniform wall temperature.

internal spiral and longitudinal fins. Figure 4.11 depicts the tubes that Carnavos studied. Carnavos determined that the average Nusselt number could be correlated within 6% by the empirical formula 0.1

A 0.8 A fa -  ----n Nu DH  0.023Re DH  ---- A fc  A a

© 2001 by CRC PRESS LLC

0.5

3

(sec  )Pr

0.4

FIGURE 4.11 A digitally enhanced axial view of the internally finned tubes used by T. C. Carnavos. (Adapted from Carnavos, T. C., Heat Transfer Eng., 1, 43, 1979.

where: 104  Afa  Afc  An  Aa   

Re  105 actual free-flow cross-sectional area (m2) open core flow area inside fins (m2) nominal heat transfer area based on tube I.D. without fins (m2) actual heat transfer area (m2) helix angle for spiral fins (deg.)

For the tube patterns shown, a correlation for the Fanning friction coefficient was found to agree with the experimental data to within 7%: 0.046  A fa 0.5 0.5 - ------- cos  f  -----------0.20  A  Re DH fn where Afn  nominal flow area based on tube I.D. without fins (m2). Kreith and Bohn26 estimate that the capacity of a heat exchanger can be increased by 12 to 66% if we replace the existing smooth round tubes with internally finned tubes, and the pumping power remains constant.

4.5.3

FORCED CONVECTION EXTERNAL FLOW

Many researchers have examined the analysis of external flow along an external surface. When a fluid is forced along an external surface, a velocity and thermal boundary layer form which control the heat transfer and fluid drag components of the boundary layer equations. It is worth the effort to develop an understanding of the basic analysis used to solve this theoretical problem in both the laminar and turbulent flow regions for flat surfaces. We can then apply the knowledge gained here to more complex

© 2001 by CRC PRESS LLC

5

Radiation Heat Transfer in Electronic Equipment

5.1 INTRODUCTION Radiation cooling of electronic components and boxes is not usually a concern to thermal engineers. The three factors that affect cooling by radiation are the temperature difference between an object and its surroundings, the surface characteristics of the object and its surroundings, and the view that the object has of its surroundings. Typically, the temperature difference is the delta between the device case and an outer chassis, or an outer chassis and the walls of a room. Since radiation heat transfer is based on this temperature difference, when the delta between a component and its surroundings becomes high enough for radiation cooling to matter, the device is most likely already above its maximum junction temperature. At room temperature, the total emissive power of a perfect emitter is about 450 W/m2. This is less than 10% of the heat that convection in air can transfer. The characteristics of the surface that covers the device or the chassis in question is another important variable. Materials used in electronics are normally opaque to radiation. Therefore, the surface characteristics are important to a depth of only about 2.5  106 m for metals and about 0.5  103 m for nonmetals. In the range of temperatures used for electronics, the color of the surface does not affect the radiation emittance. The third variable is the view factor. This is the fraction of the radiation that leaves one surface and is intercepted by another surface. This can be as high as 1.0 for the case of a sphere inside a larger sphere, or quite low, such as two plates at an angle approaching 180°. Algebraic equations can calculate the view factor, and this is usually done with the aid of a computer. Although radiation is not usually a concern when trying to cool an electronic package, the package may absorb radiation heat by being near a high-temperature source. Such a case may occur in the engine compartment of a car, where electronic modules are exposed to the radiative heat of hot engine components and exhaust manifolds. Although the color of an object is not important in radiative cooling, color is important when the object may absorb heat energy from a wide-band radiation source. This case may occur when we expose an electronic package to the sun.

5.1.1

THE ELECTROMAGNETIC SPECTRUM

Radiation heat transfer occurs by electromagnetic waves traveling at the speed of light, about 3  108 m/s. These waves may travel through a vacuum or through a gas. Some gases absorb radiation and this must be considered in exact calculations.

© 2001 by CRC PRESS LLC

Examples of gases that affect the radiation transfer are water vapor, carbon dioxide, and air containing a large quantity of particulate matter. Standard air does not affect the results enough to be considered in the power range and temperature range of electronic cooling. The electromagnetic spectrum extends from a fraction of a cycle per second (Hz) having a wavelength of 3  104 m, to waves having frequencies greater than 3  1019 Hz and wavelengths shorter than 1011 m. At the lower-frequency end of the spectrum are radio waves. As the frequency increases, the bands of infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays are encountered. Heat waves are primarily found between radio waves and visible light in the infrared spectrum, but they extend into both regions. Figure 5.1 shows the electromagnetic spectrum and the portion of the spectrum that concerns radiation heat transfer. As the temperature of an object increases the radiated heat energy increases, with more of this energy being emitted in the visible light range. The human eye does not respond to electromagnetic waves having a wavelength greater than 0.7 m. This wavelength corresponds to a temperature of about 800 K or about 530°C. Therefore, if the temperature of an object is below 800 K, we cannot see the heat.

5.2 RADIATION EQUATIONS Radiation heat transfer involves emittance, absorptance, reflectance, and transmittance at the surface of an object. In most engineering problems, transmittance is not a factor. The reflected energy is very small and is accounted for in the emittance and absorptance of an object in the radiation exchange. We call a theoretical surface that absorbs all incident radiation and that also emits all radiation a blackbody. We can find the rate of energy emitted by a blackbody by the equation 4

qr   A1 T 1 where: qr    A1  T1 

rate of heat flow (W) Stefan-Boltzmann constant, 5.66961  108 W/m2 K4 surface area (m2) absolute temperature of object 1 (K)

This equation shows that heat is radiated proportional to the fourth power of the absolute temperature. If the blackbody is in an enclosure, and we know the temperature of the enclosure, the net rate of heat transfer is found by 4

4

qr   A1 ( T 1  T 2 ) where T2 is the absolute temperature of the enclosure. © 2001 by CRC PRESS LLC

© 2001 by CRC PRESS LLC

FIGURE 5.1 (a) The electromagnetic spectrum. (b) The thermal radiation portion of the electromagnetic spectrum, generally considered to be 0.1 to 100 m.

If we are working with real surfaces, which are not perfect emitters of energy and are called graybodies, the equation for the net rate of heat transfer is changed to 4

4

qr   A1 1 ( T 1  T 2 ) where 1 is the emittance of the graybody. Emittance is the ratio of the emission from a graybody compared to a blackbody at the same temperature. If both the object and the enclosure are graybodies, the net rate of radiation transfer is found by 4

4

q r   A 1  1 ᑠ 1,2 ( T 1  T 2 ) where ᑠ 1,2 is a dimensionless modulus, called the transfer factor, that modifies the equation for perfect radiators to account for the emittances and relative geometries of graybody 1 to graybody 2. Sometimes it is convenient to express the radiation heat loss as a radiation heat transfer coefficient, hr . We find this coefficient by manipulating the radiation heat transfer equation: 4

4

 1 ( T 1  T 2 ) qr h r  --------------------------------  --------------------------------T1  T2 A1 ( T 1  T 2 )

5.2.1

STEFAN-BOLTZMANN LAW

We call the rate that radiation is emitted per unit area at all possible wavelengths, in every possible direction, the total hemispherical emissive power, E, which has units of W/m2. Therefore, we can show this in the form

E 

0 E  (  ) d

The spectral distribution of energy emitted by a blackbody has been well documented. In 1900, Planck1 derived the equation as 2

2hc o I , b ( , T )  ----------------------------------------hc o 5 -)  1]  [ exp ( --------kT where: h k co T

   

6.6256  1034 J s/molecule (Planck’s constant) 1.3805  1023 J/K molecule (Boltzmann’s constant) 2.9979  108 m/s (speed of light in a vacuum) absolute temperature (K)

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The blackbody is a theoretical diffuse emitter. That is, the surface emits radiation equally, no matter the direction. The total output from a blackbody, based on Planck’s theories is Eb  T 4 where we see that the term  is the Stefan-Boltzmann constant, which depends on C1 and C2, and has the value 5.66961  108 W/m2 K4. J. Stefan discovered the formula for radiation heat transfer in 1879. L. Boltzmann derived the relationship theoretically in 1884. The Stefan-Boltzmann constant allows the designer to calculate how much radiation is emitted in all directions and over all wavelengths due to the temperature of the blackbody.

5.3 SURFACE CHARACTERISTICS A hot object emits heat as electromagnetic energy waves. How much energy is radiated is related to an emittance factor, . The emittance factor concerns only the surface characteristics of an object. For example, if we paint an object, the material we construct the object from does not affect its emittance factor. If we frost a clear glass window, the glass can absorb heat and the temperature of the glass will increase. This is because we have altered the surface characteristics. The emittance factor of a surface is rated by the percentage of energy it can radiate when compared with a perfect radiator, which we call a blackbody. A blackbody is an object that emits 100% of its energy; therefore,   1.0. Since this is a theoretical idea, all real objects have an emittance less than 1.0. When the waves strike another object, the object partially absorbs and partially reflects them. In a somewhat transparent object, the waves may be transmitted through the object. This is shown in Figure 5.2. The radiation energy balance can then be written G  G,reflec G,absorb G,trans and therefore, 1     where: G  

    

irradiation (W/m2) wavelength (m) reflectance absorptance transmittance

The specific amount of energy distributed to these variables is determined by the temperature difference, the surface characteristics, the object’s geometry, the

© 2001 by CRC PRESS LLC

FIGURE 5.2 A semitransparent body displays the three characteristics of radiation heat transfer when energy, G, is applied: absorption, G, abs; reflection, G, ref; and transmission, G, tr.

material, and the electromagnetic wavelength. If the values of reflectance, absorptance, and transmittance are averaged over the entire spectrum, the energy balance becomes 1

Since we do not normally have transparent surfaces in electronic cooling, we can neglect transmittance, , which results in 1   Of course, this shows that if we know one value the other is easily found. A perfect blackbody not only emits 100% of its energy but also absorbs 100% of the energy it receives. We call the ability of an object to absorb heat energy absorptance, which we represent by . The efficiency of a real object to absorb heat energy is always less than 100%. Similar to emittance, the amount of energy the object absorbs is based on the surface characteristics and the temperature. Usually, dull surfaces have good emittance and good absorptance. When we include visible light energy, dark surfaces have good emittance and absorptance. Shiny surfaces are usually poor emitters and absorbers of radiation. In visible light the same

© 2001 by CRC PRESS LLC

is true of light-colored surfaces. As an example, a polished metal such as aluminum has an emittance of about 0.06. This increases to about 0.09 when the aluminum has a commercial finish and increases to 0.10 when iridited. If the aluminum has a clear, dull, anodized finish, the emittance increases to about 0.80. Again, the color of the surface does not have an appreciable effect in the temperatures that are common in electronics cooling. This is because the wavelength of infrared heat, about 7m, is well below that of visible light. Therefore, if the color is not visible it does not have an effect. In the temperature range of concern, black paint will have the same characteristics as white paint. Because of the effect of color in visible light such as the sun, we will classify solar absorptance as s.

5.3.1

EMITTANCE

The actual spectral radiation emitted and the direction of the radiation emitted by a real surface differs from the Planck distribution. The actual emittance can also have different values when measured in different directions and at different wavelengths. Figure 5.3 shows the emission from a differential element into a theoretical hemisphere. We define the total hemispherical emittance as the radiation emittance over all possible directions and wavelengths: E(T )  ( T )  -------------Eb(T )

FIGURE 5.3 Radiation from a differential area dA into a surrounding hemisphere having a center at the differential area dA.

© 2001 by CRC PRESS LLC

FIGURE 5.4 Spectral dependence of the spectral normal emissivity ,n of selected materials. (Adapted from Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley & Sons, 1995. With permission.)

Researchers can use these formulas to categorize the emittance of a variety of surface finishes and materials. Figure 5.4 shows the spectral emittance dependence on wavelength for several materials.2 If we want to find the total emissive radiation power at any temperature, and we know the value of  (T), we can use the integration of Planck’s distribution, Eb  T 4 along with the equation for total hemispherical emittance E(T )  ( T )  -------------Eb(T ) If we want to find the spectral emissive radiation power at any wavelength and temperature, and we know the value of (, T), we use the formula for spectral emissive power C1 B , b ( , T )   I , b ( , T )  -----------------------------------C  ˙T 5   e 2  1 © 2001 by CRC PRESS LLC

with the formula for spectral hemispherical emittance E  ( , T )   ( , T )  ----------------------E  ,b ( , T ) Table 5.1 contains the emittance and absorptance values for a variety of surfaces.

TABLE 5.1 Total Emittance and Solar Absorptance at ~27oC (~300 K) Surface Aluminum, anodized hard Aluminum, anodized soft Aluminum film, evaporated Aluminum foil, bright side Aluminum, oxidized Aluminum, polished Aluminum, 6061 commercial finish Aluminum oxide Brass, polished Brass, oxidized Carbon graphite Chrome, black deposited on metal Chromium, blued Chromium, plating Chromium, polished Copper, polished Copper, oxidized black Copper, oxidized chemical conversion Copper, electroplated Glass Glass, second surface mirror Gold, polished Gold foil, bright side Iron, cast oxidized Iron, new galvanized Iron, old galvanized Iron oxide Iron, polished Magnesium Molybdenum Nickel, black deposited on metal Nickel, electroplated

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Solar Absorptance, 0.23 0.55 0.09

0.37

0.96 0.96 0.78

0.91 0.87 0.47 0.13 0.16

0.43 0.90 0.22

␣s

Emittance, 0.80 0.76 0.03 0.07 0.11 0.04 0.04 0.33 0.10 0.61 0.88 0.15 0.18 0.10 0.08 0.04 0.16 0.13 0.03 0.90 0.81 0.03 0.07 0.63 0.23 0.28 0.96 0.06 0.07 0.03 0.15 0.03

⑀

␣s Ⲑ ⑀ 0.288 0.724 3.0

9.25

1.09 6.40 4.33

5.69

15.7 0.161 5.33

14.3 6.0 7.33 (Continued)

TABLE 5.1 (continued) Total Emittance and Solar Absorptance at ~27oC (~300 K) Surface Paint, aluminized Paint, black lacquer Paint, red Paint, yellow Paint, black epoxy Paint, black silicone Paint, white acrylic Paint, white zinc oxide Paint, white epoxy Palladium Platinum Rhodium Rubber Silver, polished Snow, fresh Steel, stainless polished Stainless steel, dull Tantalum Teflon Water Zinc, galvanized Zinc, polished

5.3.1.1

Solar Absorptance,

␣s

0.96

0.95 0.94 0.26 0.25 0.41 0.33 0.28 0.90 0.07 0.13 0.50 0.59 0.12 0.98

Emittance, 0.65 0.96 0.96 0.95 0.87 0.90 0.90 0.95 0.85 0.03 0.03 0.02 0.90 0.01 0.82 0.17 0.21 0.02 0.85 0.90 0.25 0.02

⑀

␣s Ⲑ ⑀ 1.0

1.09 1.04 0.289 0.294 13.7 11.0 14.0 1.0 7.0 0.159 2.38 29.5 0.141

Emittance Factor

The Stefan-Boltzmann law strictly applies to only blackbodies. Usually, we have an interchange of thermal energy between two or more bodies. In real engineering problems we commonly use a transfer factor, ᑠ1,2, to represent the interactive emittances and views of the bodies. Calculating this factor involves a complex series of integrals. Table 5.2 contains the equations needed to find a number of transfer factors. 5.3.1.2

Emittance from Extended Surfaces

Extended surfaces such as plate fin arrays are often used to increase the surface area of a product. The additional surface area may offer a substantial improvement in the temperature of the part. Fins also offer a benefit in radiative cooling. The channels between the fins act as cavities that act on radiant energy and increase the emittance of a part. The deeper the channel compared to the channel width, the more the channel acts as a deep cavity, increasing the emittance. Bilitzky8 obtained values of the increased emittance, Ê, for longitudinal fins of rectangular profile when q  4 4 ÊA( T 1  T 2 ). His results are shown in Figures 5.5 through 5.9. Harper9 suggests © 2001 by CRC PRESS LLC

TABLE 5.2 Transfer Factors for Various Geometries

ᑠ

Surface Geometry Infinite parallel plates

1

---------------------1 1 ---- + ----  1   1

2

Concentric spheres, surface A1 surrounded by surface A2.

1 ----------------------------------------A 1 1  ---- ------1  --- 1 A 2   2  1

Concentric cylinders, surface A1 surrounded by surface A2.

1 ----------------------------------------A 1 1  ---- ------1  --- 1 A 2   2  1

Surface A1 surrounded by very large surface A2

General case for two surfaces

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1

1  2

3.0

b/z = 10.0

Effective channel emittance, E

2.5

2.0 b/z = 5.0

b/z = 2.0

1.5

b/z = 1.0

1.0

0.5

L/b = 1.0 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Channel surface emissivity,

0.8

0.9

1.0



FIGURE 5.5 Channel radiative emittance for L/b  1.0. (Adapted from References 8 and 19.)

2.0 b/z = 10.0 1.8

1.6

Effective channel emittance, E

b/z = 2.0 1.4 b/z = 5.0 1.2

1.0 b/z = 1.0 0.8

0.6

0.4

0.2

L/b

= 2.0

0.0 0.1

0.2

0.3

0.4

0.5

0.6

Channel surface emissivity,

0.7

0.8

0.9

1.0



FIGURE 5.6 Channel radiative emittance for L/b  2.0. (Adapted from References 8 and 19.)

© 2001 by CRC PRESS LLC

1.6

1.4

Effective channel emittance, E

b/z = 10.0

1.2

b/z = 2.0 b/z = 5.0

1.0 b/z = 1.0

0.8

0.6

0.4

0.2

L/b = 5.0 0.0 0.1

0.2

0.3

0.4

0.5

0.6

Channel surface emissivity,

0.7

0.8

0.9

1.0



FIGURE 5.7 Channel radiative emittance for L/b  5.0. (Adapted from References 8 and 19.)

1.4

1.2

Effective channel emittance, E

b/z = 10.0

1.0

b/z = 5.0

0.8 b/z = 2.0 b/z = 1.0

0.6

0.4

0.2

L/b = 10.0 0.0 0.1

0.2

0.3

0.4

0.5

0.6

Channel surface emissivity,

0.7

0.8

0.9

1.0



FIGURE 5.8 Channel radiative emittance for L/b  10.0. (Adapted from References 8 and 19.)

© 2001 by CRC PRESS LLC

1.2

b/z = 10.0

Effective channel emittance, E

1.0

b/z = 5.0

0.8

b/z = 2.0

0.6 b/z = 1.0

0.4

0.2

L/b

= 100

0.0 0.1

0.2

0.3

0.4

0.5

0.6

Channel Surface emissivity,

0.7

0.8

0.9

1.0



FIGURE 5.9 Channel radiative emittance for L/b  100.0. (Adapted from References 8 and 19.)

that the effective emittance of a plate fin array is found by 1   flat  eff   flat  -----------------2 

where:

eff  effective emittance flat  emittance of the flat surface

5.3.2

ABSORPTANCE

When radiation is intercepted by an object the radiation is partially reflected, partially absorbed, and sometimes partially transmitted. If we define the spectral irradiation an object receives at wavelength  as G, then in units of W/m2 the energy balance has the form G  G,reflec G,absorb G,trans Usually, the object is opaque and we can neglect G,trans. Reflected radiant energy does not affect the object. The irradiation that is absorbed will raise the energy level of the object, causing an increase in temperature. Absorptance, like directional vectors and wavelength, affects emittance. The formulas for working with absorptance are similar to those previously discussed for emittance. The spectral directional absorptance, ,θ (, , ), of a surface is the ratio © 2001 by CRC PRESS LLC

of the intensity of the radiation absorbed at the individual wavelength, , and in the direction of  and  to the intensity of the radiation absorbed by a blackbody at the identical wavelength. Since the effect of temperature on absorptance is exceedingly small in real applications, we neglect the term T. Therefore, the formula for spectral directional absorptance is I  ,i,abs ( , ,  )  , (  ,  ,  )  ----------------------------------I  ,i ( , ,  ) where:

, (, , )  spectral directional absorptance (dimensionless)   wavelength (m)   zenith angle (rad)   azimuth angle (rad) I  spectral intensity (W/m2) As we learned about emittance, absorption in real objects may occur at different intensities depending on the wavelength and directional vector of the radiation. To use the absorptance for real engineering problems, we define a term representing the directional average of the spectral irradiation, which we call the spectral hemispherical absorptance, (): G , abs (  )  (  )  ---------------------G (  )

5.3.3

REFLECTANCE

The definition and the type of reflectance may take one of several forms. This is because reflectance is a bidirectional phenomenon. That is, the reflectance depends not only on the direction of the irradiation, but also the direction of the reflected radiation.10 Figure 5.10 shows the difference between diffuse and specular reflection.

FIGURE 5.10 Two types of reflected radiation from a plane surface: diffuse reflection and specular reflection.

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Because reflectance is such a small part of heat transfer in electronics cooling, we will present only the most basic form, which is an integrated average over a hemisphere. 5.3.3.1

Specular Reflectance

The spectral directional reflectance, , (,,), is the fraction of the spectral incident radiation in the direction  and , reflected by the surface. Therefore, I , i, reflec ( , ,  )   , (  , , )  --------------------------------------I  ,i ( , ,  ) The spectral hemispherical reflectance, designated (), is the fraction of the spectral irradiation reflected: G , reflec (  )   (  )  ------------------------G (  ) The total hemispherical reflectance, , is then G reflec   ------------G In most occurrences of reflectance,  is a combination of diffuse and specular. Occasionally, the reflectance is specular, resembling a mirror-like surface.

5.3.4

TRANSMITTANCE

Although we display the formula for transmittance here, note that this is not one value but may change with depth and with the wavelength. For example, glass and water are transparent at the wavelengths of visible light but become opaque at longer wavelengths. The hemispherical transmittance of a material can be found by G , trans (  )

  ----------------------G (  )

and

G trans

 ----------G

5.4 VIEW FACTORS The view factor, F1,2, is the portion of the radiation leaving surface A1 that surface A2 intercepts. If both surfaces are blackbodies, the energy transfer can be from surface A1 to surface A2, F1,2, or from surface A2 to surface A1, F2,1. The net rate of radiation between © 2001 by CRC PRESS LLC

FIGURE 5.11 The area of radiation importance of some complex structures can be found by measuring the projected area of the irregular surface.

FIGURE 5.12 Geometry and nomenclature used for deriving the shape factor formula of two surfaces.

two blackbodies can be found by determining the radiation from either of the surfaces to the other surface. After determining the radiation we replace its emissive power with the difference between the emissive powers of the two surfaces. Since the result does not depend on which emitting surface is chosen, the surface that is easiest to calculate is usually selected. If the surface is irregular, such as that shown in Figure 5.11, the projected area is used, that is, the area that would result if a string were drawn across the surface. Figure 5.12 shows the geometry for derivation of the view factor. To learn how much radiation is interchanged, we use differential surface dA1 and dA2. To find the radiation surface dA2 intercepts, we evaluate cos  1 cos  2 dA 2 dq 1,2  E b1 dA 1  -----------------------------------2   L © 2001 by CRC PRESS LLC

where: Eb1  intensity of radiation from surface dA1 cos1  area of surface dA1 seen from surface dA2 cos2  area of surface dA2 seen from surface dA1 L  distance between surfaces The net rate of radiation heat transfer between surface dA1 and dA2 is then cos  1 cos  2 dA 1 dA 2 dq 1, 2  ( E b1  E b2 )  --------------------------------------------2   L To find the radiation transferred between the surfaces, we integrate the term in the right parentheses over both surfaces; therefore, dq 1, 2  ( E b1  E b2 ) 

cos  1 cos  2 dA 1 dA 2   --------------------------------------------2 

A1 A2

L

We can write this double integral for radiation from dA1 to dA2 as A1F1,2. Researchers have derived many complex equations for determining the double integral and the view factor. Table 5.3 shows view factor equations and illustrations of the geometries. Howell11 gathered the most useful view factors into a single volume. In cases where the surfaces are significantly different from those in the tables, it is best to use a computer program that can calculate the view factors of multiple complex surfaces, such as TRASYS®.12*

5.4.1

CALCULATION

OF

ESTIMATED DIFFUSE VIEW FACTORS

Since the calculation of the double integrals is so complex, other methods estimate the view factor and may yield acceptable accuracy. These estimated calculation methods do not apply to specular reflection, only to diffuse view factors. The contour integral method is based on the application of Stokes’ theorem to reduce surface integrals to contour integrals. By superpositioning and reciprocity the view factor may be expressed as a function of simple shapes view factors that are already known by a method called Diffuse View Factor Algebra. If we have an infinitely long triangle enclosure whose surfaces are flat or convex, we may express the following system of six equations with six unknown variables: N

 F 1, 2

 1.0

i  1,2,3

j1

A i F i, j  A j  F j,i * TRASYS is a registered trademark of Lockheed Martin.

© 2001 by CRC PRESS LLC

i, j  1,2,3

TABLE 5.3 View Factors for Two- and Three-Dimensional Geometries Ref. Infinite parallel surfaces of same width

Infinite parallel surfaces of different widths on the same centerlines

F 1, 2  F 2, 1 

2

1 (h  w )  (h  w)

1 F 1, 2  -----b 2 --a-

14

2

2



Infinite 90° surfaces of different widths with a common edge

13

b c 4 --- --a a

c b ---  --- 4 a a

h h  2 1 -  1  --F 1, 2  ---  1 --w  w  2

10

(Continued)

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TABLE 5.3 (continued) View Factors for Two- and Three-Dimensional Geometries Ref. Infinite surface to a row of cylinders

a 2 0.5 F 1, 2  1   1   ---   b  

13

2 0.5

 1   a---  b a 1   --------------------

--- tan  b   a 2    --b- 

For n rows of in-line cylinders

Infinite surfaces of same width with one common edge, at an angle

Infinite triangle

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F 1, n

rows

 1  ( 1  F 1, 2 )

n

 F 1, 2  F 2, 1  1  sin --2

10

A1 A2  A3 F 1, 2  --------------------------------2 A1

15

TABLE 5.3 (continued) View Factors for Two- and Three-Dimensional Geometries Ref. Rectangle at an angle to an infinite surface

1  cos  F 1, 2  ---------------------2

15

Right triangle to congruent triangle, common short edge

1 1 F 1, 2  ---  1  ------- 3 2

11

Infinite concentric cylinders for concentric spheres r1  A1 and r2  A2

F 1, 2  1 F 2, 2 F 2, 1

Exterior of infinite cylinder to interior of concentric semicylinder

10

r1  1  ---r2 r1  --r2

1 F 1, 2  --2-

15

(Continued) © 2001 by CRC PRESS LLC

TABLE 5.3 (continued) View Factors for Two- and Three-Dimensional Geometries Ref. Interior surface of infinite semicylinder to itself when a concentric cylinder is obstructing

Interior of infinite semicylinder 1 to interior of semicylinder 2 when concentric parallel cylinder 3 is present

Infinite parallel cylinders, same diameter

2 0.5 2 F 1, 1  1  ----  1   r----1  r----1 sin 1 r----1    r 2  r2 r2

2 r 1 2 0.5  r 1  r -1 1 r F 1, 2  ----  1   -- ---- sin  ----1   --  r 2  r 2   r 2  r 2

s 2 1   1 F 1, 2  F 2, 1  ----   1 ----2r      1  s 1 

sin  -----------------   1 ----2r s   ----1

 2r 

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15

15

10

TABLE 5.3 (continued) View Factors for Two- and Three-Dimensional Geometries Ref. Inside surface of a right circular cylinder to itself

Base of a cylinder to inside surface of cylinder

Radius in cylinder base to inside surface of cylinder

h h 2 0.5   F 1, 1   1 ----2r   1 ---- 2r 

15

h   ----2r

11

h h 2 F 1, 2  2 -----  1  -----  2r 2r 

0.5

11

r 2 1 h 2 F 1, 2  ---  1   ----2   ----     r1 2 r 1 r 2 r 2 h 2 2

 1  ----2  ----   4  ----2   r 1  r 1   r 1

0.5

 

(Continued)

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TABLE 5.3 (continued) View Factors for Two- and Three-Dimensional Geometries Ref. Annular ring on cylinder base or top to inside of right circular cylinder

 r 2 h h 2 0.5 1 1 -  ----  4  ----2  ----  F 1, 2  ---  1 ---------------------2       r r r 2 r 1 1  ----2  1  1   r 1 h 2 2 r 2 r 2   1  ----2  -----1    4  ----2  r 1  R 1   r 1 

0.5

11

  

FIGURE 5.13 A depiction of Hottel and Sarofim’s16 “string rule” for shape factors of twodimensional configurations.

By solving this equation, we find the six unknown variables of Fi,j. We call a variation of diffuse view factor algebra the cross-string method, or Hottel and Sarofim’s rule.16 To use this method, we must first measure the sum of the lengths of the strings attached to the ends of lines that represent the surfaces. We then subtract the sum of the lengths of the strings that are not crossed when attached to the same ends. Next we divide this value by two. The result will be the view factor between the two surfaces. Figure 5.13 shows the lines used for Hottel and Sarofim’s rule; therefore, 1 F 1, 2  -------- [ AD BC  AC  BD ] 2L 1

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5.5 ENVIRONMENTAL EFFECTS Some electronic component chassis are exposed to the earth’s ambient environment. Designers often seal these enclosures to prevent the hazards associated with rain and airborne contaminants. Since these boxes are not inside a building, they are exposed to solar radiation and the radiation effects of the atmosphere. Another class of component is mounted on space vehicles. These electronic boxes are exposed to solar radiation flux unattenuated by our atmosphere. In earth orbit, a space vehicle can be exposed to radiation from the sun, the moon, and the earth. During interplanetary travels the component enclosure may be exposed to planetary radiation plus solar radiation. Much of this radiation is in the ultraviolet and the visible light range, but a significant amount contains wavelengths in the infrared region.

5.5.1

SOLAR RADIATION

We call the average heat flux from solar radiation the solar constant, designated G0. The solar constant is measured unattenuated by the earth’s atmosphere at a mean distance from the sun of 1.495  1011 m, a distance called one astronomical unit. Under these conditions the solar constant has a mean value of 1353 W/m2. Because the earth moves in an elliptical orbit around the sun, the solar constant varies from about 1310 W/m2 in June to 1400 W/m2 in January. When we measure the solar radiation on the earth’s surface, scattering and absorption by gas molecules and dust have reduced the solar constant from G0 to Gs (see Figure 5.14). The solar flux at the earth’s surface is then the sum of direct and

FIGURE 5.14 Schematic of clear sky absorption and scattering of solar radiation, and the components of irradiation incident on the earth’s surface.

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scattered radiation, called Gs, and has a value of about 1300 W/m2. The actual value varies by time of day, time of year, and atmospheric conditions. Large nearby surfaces such as buildings can also affect the value of Gs. When engineers require exact values, they should consult NASA publications. Table 5.1 shows values for the ratio of the absorptance of a surface to its emittance, /. At infrared wavelengths color is not important. In the visible light region color is very important. Therefore, the steady-state temperature of a surface is based on the ratio of solar absorptance, , to the infrared emittance, . A high ratio of / shows that the surface will absorb a high level of solar energy, resulting in a higher temperature in space. Therefore, we can control the temperature of a surface in space by controlling the ratio /. A surface in space absorbs solar radiation and radiation from the planets and emits radiation to the vacuum of space, which is at absolute zero, 273.15°C. If we call the internal energy developed by an electronic box qi then we can express the total of the absorbed energy and the internal energy as

sG0 As qi Therefore,

sG0 As qi   AsT 4 Rearranging the equation to find the temperature of a surface in space exposed to the sun, we have

s G 0 A s 0.25 T   ----------------  A s 

5.5.2

ATMOSPHERIC RADIATION

Dust particles and gas molecules, mostly H2O and CO2, absorb and scatter solar radiation and the radiation from the earth’s surface (see Figure 5.15). Because of the absorption of radiation at different wavelengths the radiation within the atmosphere does not follow blackbody distribution and is therefore quite complex. Nevertheless, we can estimate the emittance of the atmosphere that corresponds to the effective temperature of the air at the earth’s surface by 4

J sky   sky  T e

An approximate correlation for clear sky emittance recommended by Brunt17 accounts for the effects of H2O, CO2, and dust particles: P H2O 0.5   sky  0.55 1.8  --------- P 

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FIGURE 5.15 Solar spectra outside the earth’s atmosphere and at sea level. (Adapted from References 7 and 20.)

where: P H 2 O  partial pressure of water vapor in atmosphere P  total atmosphere pressure Berdahl and Fromberg18 correlated equations to account for a variety of clear sky emittance measurements made in the U.S. For a nighttime sky,

nite sky  0.741 0.0062ΤDP and for a daytime sky,

day sky  0.727 0.0060ΤDP where TDP  dewpoint temperature (oC).

REFERENCES 1. Planck, M., The Theory of Heat Radiation, Dover Publishers, New York, 1959. 2. Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley & Sons, New York, 1990. 3. Touloukian, Y. S. and Ho, C. Y., Eds., Thermophysical Properties of Matter, Vol. 7, Plenum Press, New York, 1972.

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4. Mallory, J. F., Thermal Insulation, Van Nostrand Reinhold, New York, 1969. 5. Gubareff, G. G., Janssen, J. E., and Torborg, R. H., Thermal Radiation Properties Survey, Minneapolis-Honeywell Regulator Co., Minneapolis, MN, 1960. 6. Kreith, F. and Kreider, J. F., Principles of Solar Energy, Hemisphere, New York, 1978. 7. Mills, A. F., Heat and Mass Transfer, Irwin, Chicago, 1995, 1148. 8. Bilitzky, A., The Effect of Geometry on Heat Transfer by Free Convection from a Fin Array, M.S. thesis, Ben-Gurion University, Israel, 1986. 9. Harper, C. Handbook of Electronic Packaging, McGraw-Hill, New York, 1995. 10. Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 2nd ed., McGrawHill, New York, 1981. 11. Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1985. 12. TRASYS User’s Manual, Version P22, Manual prepared for NASA Johnson Space Center by Lockheed Engineering Manual Services, April 1988. National Aeronautics and Space Administration, Washington, D.C., 1988. 13. Hottel, H. C., Radiant Heat Transmission Between Surfaces, Separated by NonAbsorbing Media, Trans. ASME, FSP-53-196, 265, 1931. 14. Wong, H. Y., Handbook of Essential Formulae and Data on Heat Transfer for Engineers, Longman, London, 1977. 15. Martin, J. G. and Muriel, M. J. B., Radiation Heat Transfer, in Handbook of Applied Thermal Design, Guyer, E. C., Ed., McGraw-Hill, New York, 1989. 16. Hottel, H. C. and Sarofim, A. F., Radiative Transfer, McGraw-Hill, New York, 1967. 17. Brunt, D., Radiation in the Atmosphere, Q. J. R. Meteorol. Soc., 66 (Suppl.), 34, 1940. 18. Berdahl, P. and Fromberg, R., The Thermal Radiance of Clear Skies, Sol. Energy, 29, 299, 1982. 19. Kraus, A. D. and Bar-Cohen, A., Design and Analysis of Heat Sinks, John Wiley & Sons, New York, 1995. 20. Garg, H. P., Treatise on Solar Energy, Vol. 1, John Wiley & Sons, New York, 1982.

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6

Heat Transfer with Phase Change

6.1 INTRODUCTION Heat transfer of a phase change coolant is much more complex than the previous modes of heat transfer that we have studied. By phase change we denote the following processes: a. b. c. e. f. g.

Solid changing to a liquid—fusion, or melting, Liquid changing to a vapor—evaporation, also boiling, Vapor changing to a liquid—condensation, Liquid changing to a solid—crystallization, or freezing, Solid changing to a vapor—sublimation, Vapor changing to a solid—deposition.

Because these processes involve a fluid medium, we generally classify them as convection processes. Beyond all of the parameters that we previously examined in convection, we must now add the variables of surface tension, ; ambient pressure, P; and either the enthalpy or latent heat of evaporation, hfg; the latent heat of solidification, hfs; or the latent heat of sublimation, hsg. The latent heat is the amount of heat required to convert a unit mass of a substance from one phase to another phase. There is also a difference in density between the liquid phase and the vapor phase that induces a buoyancy force,   l  v , when bubbles are present. Also, we will note that heat transfer during phase change does not always occur with a change in temperature of the media. In fact, a very large rate of heat transfer can be achieved with very little change in temperature. This is one of the attractions of phase change heat transfer. Furthermore, in contrast to natural or forced convection, increasing the T may result in a decrease in the heat transfer coefficient. Because of the number of variables, there are no accurate general equations or correlations to use. Of the usable equations, most have an empirical value that changes with the surface characteristics and must be evaluated by experimentation. The accuracy of these correlations without experimental verification may be 50%. Although heat transfer by phase change is not yet widely used in electronics cooling, as the component heat flux rises the laws of physics dictate that high-end cooling technologies will progress from air-cooled to liquid-cooled to phase change. Our studies in this section will be largely theoretical.

© 2001 by CRC PRESS LLC

6.1.1

DEFINITIONS

OF

PHASE CHANGE PARAMETERS

Vapor pressure—At any temperature above absolute zero, the molecules in a liquid are in constant motion. Some of these molecules will have a higher velocity than the average. If the energy in one of these high-speed molecules is greater than the cohesive forces, the molecule can “escape” through the surface of the liquid. We know this as evaporation. Since these higher-speed molecules contain more energy than the remaining molecules, the reduction in energy causes the bulk liquid to cool. If the liquid is in an airtight enclosure, the escaped molecules will fill the airspace. Some “escaped” liquid molecules will even reenter the liquid. Eventually, the number of molecules escaping the liquid equals the number of molecules reentering the liquid. When this occurs, we call the air space a saturated vapor. Since the molecules exert a pressure within this enclosure, we call this the saturated vapor pressure. If we increase the pressure within the airspace, or decrease the volume of the airspace, the vapor will contain more fluid particles than it can hold. We can then say that we have supersaturated the vapor. Supersaturation conditions can occur only temporarily. Phase change—In the previous discussion, the liquid molecules changed phase, from a liquid to a vapor. If the walls of the enclosure (from the previous discussion) are suddenly cooled, we would again supersaturate the vapor. Randomly moving liquid molecules that strike the cool enclosure walls would leave their vapor state and return to the liquid state, forming condensation. This is also a change in phase—from vapor phase to liquid phase. We see an identical process when water, in the solid form of ice, melts and becomes a liquid. Another process, known as sublimation, occurs when a material changes from a solid phase directly to a vapor phase. The opposite of sublimation is deposition, where the vapor phase changes directly to a solid. The triple point of a material is the combination of temperature and pressure conditions at which the material can exist in a solid, liquid, and vapor state simultaneously. Water has a triple point of 0.01°C at 4.58 torr. The sublimation point is the temperature and pressure at which the material can exist as solid and vapor. The critical point is the temperature at which no amount of pressure will cause the vapor to liquefy. This occurs when the molecules are moving so fast that the internal cohesive forces are not strong enough to form a surface. Figure 6.1 shows the phase diagram for H2O. A is the triple point. B is the critical point. Curve AB shows the points at which H2O can exist in both the liquid and vapor phase. Line AC shows where H2O can exist as a solid and a liquid, and AD shows where it can exist as both a solid and a vapor. Standard atmospheric pressure is 760 torr (760 mmHg). Moving along the 760 torr line we see that H2O has a phase change from solid to liquid at 0°C, and changes from a liquid to a gas at 100°C. Sublimation occurs along line efg, where the solid may change directly into a vapor without turning into a liquid first. At point B, temperatures of 374°C and above, H2O cannot liquefy at any pressure. This is called the critical temperature and occurs when the gas molecules are moving so fast that cohesive forces are unable to form a surface. Technically, a vapor is called a gas when it exceeds the critical temperature. As previously noted, a liquid will evaporate if the vapor pressure is lower than saturation pressure. Saturation pressure increases with temperature. If the saturation

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FIGURE 6.1 The phase diagram for H2O.

pressure is equal to the surrounding atmospheric pressure, the liquid will form vapor bubbles throughout and boil. Latent heat—As an object absorbs energy, the object will display a temperature increase, defined by the specific heat of the object. We call this temperature increase sensible heat. Another form of heat is latent heat, by which an object will absorb energy but will not increase in temperature. The latent heat of vaporization is higher than the latent heat of fusion because the molecules are spaced farther apart in a gas than in a liquid.

6.2 DIMENSIONLESS PARAMETERS IN BOILING AND CONDENSATION For boiling, the Jakob (Ja) number is the ratio of maximum sensible heat absorbed by the liquid to the latent heat absorbed by the vapor, or the reverse for condensation. The Jakob number can also be described in terms of temperature difference as c p ( T s  T sat ) Ja  ------------------------------h fg This dimensionless group characterizes the heat transfer during a phase change. The Jakob number is usually quite small. For example, we can let Ts equal the temperature of the liquid surface of an ice block Ti equal the internal temperature of the ice block. If we say the temperature difference between an ice block and a

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liquid surface is 10°C, then Ja  0.058. The Bond number is the ratio of the gravitational buoyancy force to the surface tension.

6.3 MODES OF BOILING LIQUIDS Boiling occurs when the temperature of a surface wall, Tw, exceeds that of the liquid saturation temperature, Tsat. The heat transferred from the heated wall to the liquid can be found by qs  h ( T w  T sat )  h  T e where qs is the rate of heat generation per unit of surface (W/m2) and Te is the excess temperature. As we know from experience, bubbles within the liquid characterize the boiling process. The dynamics of the bubbles reaching the surface affect the fluid motion and, therefore, the convection. The boiling heat transfer process consists of two basic types: pool boiling, which occurs in an initially stagnant liquid, and flow boiling, which occurs in the presence of liquid velocity. Boiling may occur in both process when no bubbles are visible. We call this subcooled boiling. In this process, the temperature of the liquid is below the saturation temperature. In subcooled boiling, bubbles form in the superheated liquid at the wall but are condensed when they grow large enough to extend into the subcooled bulk liquid. When the vapor extends into the subcooled liquid, it loses its heat and collapses. Figure 6.2 shows the phases in subcooled boiling. Saturated boiling

FIGURE 6.2 Fluid flow pattern induced by a bubble in a subcooled liquid in different stages of development collapse.

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is the type of boiling with which we are most familiar: the temperature of the liquid exceeds the saturation temperature, bubbles form, and they rise to the liquid surface.

6.3.1

BUBBLE PHENOMENON

Bubbles are an important characteristic of saturated boiling. Bubbles form individually in microscopic surface imperfections on the heated body. The bubbles grow until they separate from the heated surface and reach the surface of the liquid. At higher heat fluxes bubbles form, separate, and reform so quickly that continuous streams or vapor columns are seen. At still higher heat flux levels, the process occurs so quickly that the liquid cannot reach the surface of the heated body. In this case the heated body is continually blanketed by a vapor film. A vapor bubble forms and grows in a liquid as long as the bubble vapor pressure is higher than the ambient liquid pressure, that is, p pl. Stability is reached when 2 p  p l  ------Rb where Rb is the radius of the vapor bubble. If we assume that the bubble and the liquid are at identical temperatures, the  p between the vapor and the liquid can be translated into a temperature by the Clausius-Clapeyron relation. Therefore, from Hetsroni1 we see that 2  fg  ----------------T  T l  T s 1  ------- -----R b h fg  l   While this equation is for an isothermal condition, Bergles and Rohsenow2 found the theoretical temperature difference between the wall surface and the liquid saturation temperature that will form the first bubble, or the incipience of boiling

 T b  T w  T sat 

8qT  ---------------- v h fg k f

This equation assumes that temperature decreases linearly with increasing distance from the heated wall. Since this is not strictly true in actual occurrences of wall superheat, the equation underpredicts the wall superheat, Tb. The familiar boiling process is actually called nucleate boiling. When a bubble becomes large enough to detach from a heated surface, the bubble is said to have nucleated. In nucleate boiling the characteristic length is the size of the bubble when it separates from the surface, Lb. To find this dimension, we balance the bubble surface tension to the bubble buoyancy force, as shown in Figure 6.3; therefore, Surface Tension  Buoyancy Force 2 3 2 R b   --- R b (  l   )g 3 where Rb is the bubble radius. © 2001 by CRC PRESS LLC

FIGURE 6.3 Force balance on vapor bubbles in a fluid: (a) unattached forces, (b) attached vapor bubble forces.

The bubble radius is therefore found by 3  gc R b  -----------------------(  l   )g

0.5

and the characteristic length is then related to the radius by

 gc L b  -----------------------(  l   )g

0.5

Cole and Rohsenow3 correlated vapor bubble departure diameters in a saturated water pool by a dimensionless Eustis number 4

E 0  [ 1.5 10 ( Jaⴱ )

1.25 2

]

and in saturated pools of other liquids by 1.25 4 E 0  [ 4.65 10 ( Jaⴱ ) ]

where: 2

E0  Eustis number,

g (  l   )D b -------------------------------gc 

 cT s Ja*  modified Jakob number -----------vh f s

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2

Now that we have correlated the bubble departure diameter, we can find the frequency of departure. Although current experimental data are irregular, Hetsroni1 found that bubbles that grow very slowly depart at a frequency of about 2

0.5625  Ja f  ------------------------------2 Db Cole4 roughly correlated the frequency of departure for rapidly growing bubbles as g ( l   ) f  --------------------------C D l Db 4 --3

0.5

where CD is the dimensionless drag coefficient. Theoretically, now that we can calculate the diameter of the bubbles, Db, and the frequency of bubble departure, f, we can determine the bubble, or vapor flux, Qv, if we can find the number and size of nucleation sites per unit area. Since this value is usually unknown, another way to find the vapor flux is by the equation q Q   -----------h fg  where  is the dimensionless fraction of the heat flux that will result in a net generation of bubbles. Unfortunately,  is not a constant. Graham and Hendricks5 found that  varies in a complex manner from about 0.01 to 0.02 at low heat fluxes, to 0.5 at about 20% of the critical heat flux, and approaches 1.0 at the critical heat flux. If the vapor flux can be determined, then we can find the amount of agitation caused by nucleate boiling, which is described as the bubble Reynolds number Db Q Re b  ------------l A good estimate of the maximum bubble velocity can be found by balancing the vapor kinetic energy against the buoyancy force of the bubble in terms of the characteristic length. Therefore, Vapor Kinetic Energy  Buoyancy 1 2 ---  V max  g (  l   )L 2 Since we know the characteristic length is

 gc L b  -----------------------(  l   )g

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0.5

the maximum bubble velocity can be described as

 (  l   )gg c V max ~  --------------------------------2   

0.25



The maximum bubble velocity can also be described using what is known as the Helmholtz* theory of instability. That is, the bubble vapor column is disrupted by a wavelength disturbance of Lb, the characteristic length, and becomes unstable at VH, or 2  g 0.5 V H ~  ----------------c   Lb  If the vapor bubble diameter is limited to an experimental range found by Bromley et al.,6 that is gc  gc  3.14 ------------------------  D b  5.45 -----------------------g ( l   ) g ( l   ) then we may use a quick estimate of the velocity of an undisturbed bubble given by Zuber et al.7 1 V b  --- 4gr 3 Similar to bubble growth in nucleate boiling, vapor bubbles collapse when subcooled. Florschuetz and Chao8 used a temperature integral developed by Plesset and Zwick9 to determine the rate of collapse, H, of a bubble:

H

  ˙ 2  Rb  2 1  ----------4 2   -t  ---   R˙   --------  3  ---- Ja ------2   b R b, i 3   -------- R b, i     R b, i 

where Rb is the bubble radius and Rb,i is the initial bubble radius. If we define the bubble collapse period as the time when the bubble volume is 1% of the departure bubble volume, then the ratio Rb/Rb,i  0.2. We now define the collapse period, c, as   2  Rb  2 1  ----------  c  ---   R˙    --------  3 R b, i 3  --------b      R b, i 

Rb ----------  0.2 R b, i

⬇ 2.32

* Hermann von Helmholtz (1821–1894) described the theory of instability and also provided a physical proof of Fourier’s Theorem by producing complex musical tones using individual tuning forks.

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and substituting and rearranging, we see that 2

2

R b, i 2.32 R b, i -  0.580 ----------- c  ---------- -----------2 2 4 Ja  Ja  We can now determine the relationship among the rise velocity, Vb, the rate of collapse, c, and the departure diameter, Db , to the collapse length, Lc, as c

Lc 

冮0 V b dt

Substituting the equations for the collapse rate, H, and the undisturbed bubble velocity, Vb, and letting the ratio of the bubble radius, Rb /Rb,i, equal the ratio of the bubble diameter, Db /Db,i, we have an equation for the collapse length, Lc, as 2.5

2.5

D b 2g D b 2g - ------  0.0292 ----------------------L c  1.4 ------------------2 2 48 Ja  Ja 

6.3.2

POOL BOILING

There are two different types of boiling: pool, and flow. The type of boiling known as saturated pool boiling is depicted in Figure 6.4. Note that the temperature of the boiling liquid is nearly constant except at the surface of the heated wall. At the heated surface the liquid temperature increases sharply.

FIGURE 6.4 Temperature distribution of the heated solid surface, Ts, and the boiling liquid, Tsat , during saturated pool boiling.

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FIGURE 6.5 The boiling curve generated by Nukiyama10 for saturated water. Adapted from Incropera, F. P., and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley & Sons, New York, 1995, 722, using the data of Nukiyama, S., “The Maximum and Minimum Values of Heat Transmitted from Metal to Boiling Water Under Atmospheric Pressure,” J. Japan Soc. Mech. Eng., 37, 367, 1934. (Translation: Int. J. Heat Mass Transfer, 1419, 1966).

6.3.2.1

Pool Boiling Curve

In 1934, Nukiyama10 published experiments defining the different regimes of pool boiling. Nukiyama used a nichrome wire immersed in a bath of water. As he increased the power to the wire, he noticed bubbles forming at an excess temperature,  Te , of about 5°C. As he increased power further, he noticed that the power supplied to the wire could be increased greatly without a large increase in temperature. As he increased the power further still, the temperature suddenly increased dramatically, and the wire reached its melting point. After experimenting with a platinum wire with a higher melting point, he noted that the heat flux and temperature were related, as shown in Figure 6.5. Nukiyama believed that a method of temperature control of Te, instead of a power-controlled method, would yield a better curve. In 1937, Drew and Mueller11 performed the experiment using a steam pipe and obtained a curve similar to that shown in Figure 6.6. This figure shows the relationship of heat flux and temperature for water at sea-level atmosphere. The relationship is similar for most other liquids. We call the range when  Te   Te, A, free convection boiling. Bubbles do not form in this region because there is not enough vapor in contact with the liquid. Fluid motion in this range is caused by natural convection. Nucleate boiling occurs when individual bubbles form on the heated surface and rise to the liquid surface. This point on the curve is designated A. The portion of the curve designated AB is characterized by isolated bubbles. As the temperature of the heated surface is increased further, the bubbles are generated faster and sometimes merge to form vapor columns. This region is shown by the portion of the curve designated BC. Nucleate boiling occurs in the © 2001 by CRC PRESS LLC

{

{

Film

Jets and columns

Isolated bubbles

107

106

Nucleate

{

{

Free convection

Boiling regimes transition

qmax 

qmax  max C Critical heat flux, q"

q"s (W / m2 )

P B

105

qmin 

D

A

10 4

Leindenfrost point, qq"min  min

ONB

∆Te,A ∆Te,B

∆Te,D

∆Te,C

10 3

1

5

10

120

30

1000

∆Te = Ts Tsat( C) _

o

FIGURE 6.6 Characteristics of the boiling curve for a heated horizontal surface in water.

temperature range of  Te,A   Te   Te,C where  Te,C is about 30°C. This is the range of operation for most heat transfer work. High levels of power can be dissipated without a large increase in temperature. Point P indicates the point of the maximum heat transfer coefficient. Ideally, equipment should operate at this point. In water, the convection coefficient in this region can exceed 104 W/m2 K. The region when  Te,C   Te   Te,D, where  Te,D is about 120°C, is called the transition boiling, partial film boiling, or unstable film boiling range. In the previous range, as each bubble left the surface, liquid covered the surface until a new bubble formed. In the transition boiling range, new bubbles are formed before the liquid can reach the surface. A continuous vapor film forms on the surface. The entire heated surface oscillates between a liquid and vapor blanket. As the temperature differential  Te increases, the entire surface is more often covered by a vapor layer than the liquid layer. Also, as  Te increases, hc and therefore qs decrease, because the thermal conductivity of the vapor layer is much lower than when the adjacent layer was a liquid. The region when  Te  Te,D is called film boiling. Point D of the boiling curve is called the Leidenfrost point. In 1756, Leidenfrost noticed that when water droplets are placed on a hot surface, the droplets dart about the surface, supported by a vapor layer. We know that during the transition phase, as  Te increases, a higher percentage of the surface is covered by vapor at any point in time. The Leidenfrost point occurs when the entire surface is covered by a vapor layer, and the heat flux reaches a minimum, qs,D  qmin. In the film boiling range, heat transfer can only occur by conduction through the vapor layer. After the Leidenfrost point, radiation heat transfer through the vapor layer becomes more important, and the heat flux increases with increasing  Te. © 2001 by CRC PRESS LLC

Researchers have experimented with the region after point C, but in actual engineering applications this region is difficult to control. Any increase in the heat flux after point C creates a marked increase in temperature. The size of this increase may cause destruction of the heat flux surface. For this reason, point C is often called the burnout point or, more commonly, the Critical Heat Flux (CHF). 6.3.2.2

Pool Boiling Correlations

Researchers have calculated a number of correlations to explain the actions of pool boiling, based on the pool boiling curve. For Nusselt numbers up to the Te, A point, standard free convection correlations can be used. In the region of nucleate boiling, where  Te,A   Te   Te,C and where  Te,C is about 30°C, the Nusselt number is highly dependent upon the rate of bubble formation. Although no exact models are available to describe this phenomenon, Yamagata et al.12 correlated the influence of nucleation sites on the heat flux by a b

qs  C  T e n where:

qs  surface heat flux (W/m2) C  constant dependent upon liquid/surface combination Te  excess temperature (Ts  Tsat) (°C) a ⬇ 1.2 6 n  bubble site density (N/m2) (   T e ) b ⬇ 1/3 Currently, the Rohsenow13 correlation is the most useful: g ( l   ) qs   l h fg ----------------------- gc

0.5

c p, l  T e  3  --------------------- C h Pr n sf

fg

l

where:

  absolute viscosity (N s/m2) hfg  latent heat of vaporization (J/kg) g  gravitational acceleration (9.806 m/s2)   mass density (kg/m3)   surface tension (N/m) gc  gravitational constant (1.0 kg m/N s2)

cp  specific heat (J/kg K) Cs f  coefficient of liquid/surface combination Pr  Prandtl number, cp/k n  exponent for liquid l  liquid phase  vapor phase

Figure 6.7 shows pool boiling data points for water that were correlated by the Rohsenow method. Collier14 recommends the following correlation as being simpler

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FIGURE 6.7 Pool boiling data for water correlated by the Rohsenow method. (Adapted from References 13 and 36.)

to use than the Rohsenow correlation: 3.33

qs  0.000481  T

3.33 e

p

2.3 cr

p 10 p 1.2 p 0.17 1.8  ------  4  ------  10  ------  p cr  p cr  p cr

where:

Te  excess temperature, °C pcr  critical pressure, atmosphere (101,325 N/m2) p  operating pressure, atmosphere (101,325 N/m2) The Rohsenow correlation13 can be manipulated to find the nucleate boiling Nusselt number in terms of the Jakob number c p, l ( T w  T sat ) -----------------------------------h fg

2

2

Ja -  --------------Nu  ----------------------------------3 m 3 m C C sf Pr l sf Pr l where the exponent m is 2.0 for water and 4.1 for other liquids. Danielson et al.15 report a number of values for Csf . The value of Csf can be assumed to equal ~0.013 when the experimental value is unknown. Hetsroni1 reports that this value correlates a wide spectrum of experimental data to within 20%. The most important variables that affect Csf are the surface roughness of the heated © 2001 by CRC PRESS LLC

TABLE 6.1 Values of the Surface/Liquid Coefficients Liquid/Surface Combination Benzene–Chromium Carbon Tetrachloride–Copper Carbon Tetrachloride–Polished Copper Ethyl Alcohol–Chromium Isopropyl Alcohol–Copper n-Pentane–Chromium n-Pentane–Polished Copper n-Pentane–Lapped Copper Water–Brass Water–Copper Water–Scored Copper Water–Polished Copper Water–Nickel Water–Chemically Etched Stainless Steel Water–Mechanically Polished Stainless Steel Water–Ground and Polished Stainless Steel

Csf 0.010 0.013 0.007 0.0027 0.0023 0.015 0.0154 0.0049 0.006 0.013 0.0068 0.013 0.006 0.0133 0.0132 0.008

surface and the angle of contact between the vapor bubble and the heated surface. The surface roughness affects the number of nucleation sites, and the angle of contact is a measure of the wettability of the surface. Smaller contact angles represent greater wettability. A totally wetted surface has the least amount of vapor and represents the greatest heat transfer coefficient. Values of the coefficients of liquid/surface combinations are shown in Table 6.1. Table 6.2 presents important boiling point thermophysical data for coolants commonly used in electronic cooling. 6.3.2.3

Pool Boiling Critical Heat Flux Correlations

The critical heat flux, point C in Figure 6.6, is determined by the maximum rate that vapor bubbles can leave the heated surface. This can also be described as the maximum speed that the liquid can re-wet the heated surface after a bubble leaves the wall. We can now relate the bubble velocity to the critical heat flux by qmax ⬃  V max h fg or qmax ----------------------  C max  V max h fg

© 2001 by CRC PRESS LLC

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TABLE 6.2 Fluid Properties at Respective Boiling Points (1.0 atm) Coolant Property Boiling Point, °C Density, Liquid, l, kg/m3 Density, Vapor,  , kg/m3 Absolute Viscosity, , N s/m2 Specific Heat, cp, J/kg K Heat Vaporization, hfg, J/kg Thermal Conductivity, k, W/m K Surface Tension, , N/m Coefficient of Expansion, , 1/K

FC-72

FC-77

FC-84

FC-87

L-1402

R-12

R-113

Water

52.0 1592 12.68 0.00045 1088 87927 0.0545 0.0085 0.0016

100.0 1590 14.31 0.00045 1172 83740 0.0570 0.0080 0.0014

83.0 1575 13.28 0.00042 1130 79553 0.0535 0.0077 0.0015

30.0 1633 11.58 0.00042 1088 87927 0.0551 0.0089 0.0016

51.0 1635 11.25 0.00052 1059 104675 0.0596 0.0109 0.0016

30.0 1487 6.34 0.00036 – 165065 0.0900 0.0118 –

48.0 1511 7.40 0.00050 979 146824 0.0702 0.0147 0.0017

100.0 958.0 0.59 0.00027 4184 2257044 0.683 0.0589 0.0002

where: qmax  critical heat flux (W/m2 ) v  vapor density (kg/m3) Vmax  maximum bubble velocity (m/s) hfg  latent heat of vaporization (J/kg) Cmax  coefficient of critical heat flux geometry Kutateladze17 defined the equation for the critical heat flux in 1948 by dimensional analysis. Later, in 1959, Zuber18 found the equation for critical heat flux by a hydrodynamic stability analysis. The critical heat flux can be found by 2

qmax  C max h fg [  v (  l   v )gg c ]

0.25

where Cmax is ⬃ /24, also called the Zuber constant. Table 6.3 presents the experimental data of Lienhard and Dhir19 for the value of Cmax using a dimensionless parameter called L*. This parameter is the ratio of the characteristic length of the heated surface, Ls, to the characteristic bubble dimension, Lb. Pressure affects the maximum heat flux because of the influence on vapor density and the boiling point of the liquid. As the boiling point changes, so does the heat of vaporization and the surface tension. Therefore, each fluid has a specific pressure that will yield a maximum heat flux. Cichelli and Bonilla20 have experimentally demonstrated that the critical heat flux increases with pressure up to 1/3 of the critical pressure. After this peak, the critical heat flux falls to zero at the critical pressure, as shown in Figure 6.8.

TABLE 6.3 Values of Cmax (Zuber Constant) for Critical Heat Flux Surface Geometry Infinite flat heater Small flat heater Large horizontal cylinder Small horizontal cylinder Large sphere Small sphere Large arbitrary body

Cmax 0.15 2 12 L 0.15 ----------------b As 0.12

Range of Applicability

Width or Diameter Width or Diameter

L* 27 9  L*  20

Radius

L 1.2

0.12L* 0.25

Radius

0.15  L*  1.2

0.11 0.227L* 0.5 ⬃0.12

Radius Radius —

4.26  L* 0.15  L*  4.26 —

Ls Ls -  ------------------------------------Note: L*  ---0.5  gc Lb -------------------------(  l   )g

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Characteristic Length

FIGURE 6.8 Critical heat flux in nucleate boiling as a function of pressure. (From Cichelli, M. T. and Bonilla, C. F., Trans. AIChE, 41, 755, 1945. With permission.)

When the liquid is subcooled, Zuber et al.7 have found that the critical heat flux is correlated to a reasonable estimation by 2  2k l ( T sat  T l ) 24  - ---------------- ---------------------------------qmax  qmax sat  1  -------------------------------- h fg  v g  g c (  l   )  l  

  

0.25

where: qmax sat kl Tl l

 critical heat flux in saturated pool boiling (W/m2)  thermal conductivity of the liquid (W/m K)  temperature of the bulk liquid (C)  thermal diffusivity of the liquid, k/cp

 gc   ---- 2 -----------------------g ( l   ) 3

0.5

2

 -------------------------------- gg c (  l   v )

0.25

The effect of subcooling on the critical heat flux was studied by Ellion.21 All of the previous correlations are for clean horizontal surfaces. Bernath22 has shown that a vertical surface may have a critical heat flux as much as 25% less than a horizontal surface. © 2001 by CRC PRESS LLC

6.3.2.4

Pool Boiling Minimum Heat Flux Correlations

There are no adequate theories that describe the transition regime between critical heat flux and vapor film boiling. This region corresponds to qs , D of Figure 6.6 and is called the minimum heat flux, or the Leidenfrost point. Fortunately, this area of the boiling curve has little practical value in electronic cooling. At lower temperatures within this region, fluid motion is characterized by periodic, unstable liquid and heater contact. At the higher-temperature region, a stable vapor film forms and the heat flux reaches a minimum. If the heat flux drops below the minimum required to initiate a stable vapor film, the liquid will again contact the heater surface and nucleate boiling will be reestablished. Zuber23 used stability theory to derive an equation for minimum heat flux, qmin gc  g ( l   ) qmin  C  h fg ---------------------------------2 ( l  v )

0.25

Kovalev24 notes that this correlation is accurate to only about 50% at moderate pressures and has even less accuracy at high pressures. Nevertheless, Berenson,25 for a large flat horizontal heated surface, and Lienhard and Wong,26 for a horizontal heated cylinder, found that C  ~0.09. 6.3.2.5

Pool Boiling Vapor Film Correlations

The region of vapor film boiling, like minimum heat flux boiling, is not of great concern in applications involving electronic cooling. Above the Leidenfrost point the heat flux increases, but at a dramatic increase in surface temperature. In this range the heated body is enveloped in a continuous blanket of vapor; therefore, heat transfer is by conduction through the vapor. At temperature ranges much higher than those seen in electronics, radiation heat transfer across the vapor film becomes important. Because the vapor film is thin, the Rayleigh number is low, which indicates heat transfer by conduction only. Because of this, analysis of the heat flux for vapor film boiling is relatively simple. The minimum wall superheat that sustains the vapor film was found by Berenson25 to be

 , f h fg g (  l   ) - ------------------------T min  0.127 --------------k , f l  

23

gc  ----------------g ( l  )

0.5

 , f -----------------------------gc  ( l   )

13

where the subscript f is that property at the film temperature. For film boiling, the following correlation for the average heat transfer coefficient is commonly used: 3

g (  l   )  k ( h fg  0.35c p,  T e ) h c  C fb ------------------------------------------------------------------------------------D   T e

0.25

where the constant Cfb is 0.62 for a horizontal cylinder (Bromley27), 0.67 for a sphere (Frederking and Daniels28), and replacing D with L is 0.71 for a plane vertical surface (Mills29). When the vapor flow rate is high, that is, when 3

L (  l   )g ( h fg  0.50c p, v  T e ) -------------------------------------------------------------------------------- 5 10 7 k   T e © 2001 by CRC PRESS LLC

Frederking and Clark30 recommend the following equation for the average film boiling heat transfer coefficient 13 2

 (  l   )g ( h fg  0.50c p,  T e )k v h  0.15 -------------------------------------------------------------------------------------  T e

6.3.3

FLOW BOILING

Similar to normal convection, boiling has two modes: pool boiling and forced convection boiling. Pool boiling is dominated by fluid motion caused by buoyancy forces, both liquid and vapor, within the fluid. In forced convection boiling, fluid motion is characterized by an external force such as a pump. Also, similar to normal convection, flow can be divided into two major categories: external flow and internal flow. In electronics cooling, external flow usually occurs in large enclosures directly on the surfaces of hot components. Internal flow occurs through tubes and ducts and requires a complex analysis because the bulk flow is part liquid and part vapor. In instances of pool boiling, thermal overshoot is a problem. Thermal overshoot is the initial resistance of a liquid to boil. The heated surface may reach 30°C above the normal boiling temperature before actual nucleate boiling occurs. This problem does not normally occur in forced convection boiling. In two-phase flow, a variable known as the vapor mass quality, , is often used. The vapor mass quality describes the ratio of the vapor mass flow to the total mass flow and is found by m˙   ------------------m˙  m˙ l where: m˙  mass vapor flow rate (kg/s) m˙ l  mass liquid flow rate (kg/s) Another term often used in two-phase internal flow is the mass velocity, G, which is the mass flow per cross-sectional area. The mass velocity is further divided into vapor mass flow, Gv , and liquid mass flow, Gl. The quantities can be shown as G  G  Gl where: m˙ G v  ----Ac m˙ l G l  ----Ac

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6.3.3.1

External Forced Convection Boiling

In electronic cooling external flow is usually a relative statement. The flow may be external over hot components, but it is still within a conduit. The pressure drop in this type of flow is not at all well understood. Some correlations are available, but the range of applicability is so narrow as to be nearly useless for engineering design. In cases of conduit flow which passes over exposed components, the engineer should always build a scale model to determine pressure drop. At temperatures before boiling incipience, standard correlations for forced convection may be used, with attention to temperature-induced property variations within the single phase. The maximum heat flux can be increased substantially by using forced convection in the boiling regime. Researchers have recorded heat flux values of 35 MW/m2. Lienhard and Eichhorn31 developed a correlation for the maximum heat flux using the Weber number, which is similar to the Reynolds number. Their correlation is usually accurate to about 20%. The Weber number, We, is the ratio of inertia to surface tension and can be described as 2

 V D We  -------------- For a heated cylinder in a low-velocity liquid cross flow, the critical heat flux is estimated by 1 4 13 qmax   h fg V ---- 1   ---------  We D where low velocity is described as: qmax  0.275  0.5  --------------- -------------  -----l   1   h fg V     For a heated cylinder in a high-velocity liquid cross flow, the critical heat flux is estimated by l   ---  

0.75

l   ---  

0.5

q m ax   h fg V -------------- -----------------------------169  19.2 We 1D 3 where high velocity is described as: qmax  0.275  0.5  --------------- -------------  -----l   1   h fg V    

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FIGURE 6.9 The fluid flow regime progression from liquid to vapor for two-phase vertical internal flow.

6.3.3.2

Internal Forced Convection Boiling

Forced convection boiling in a tube is quite complex. Researchers have identified six regions of heat transfer in a vertical tube. These regions may blend into each other and are not well defined. Approximate views of the modes of vertical internal two-phase flow are shown in Figure 6.9. When a liquid flow enters a vertical tube having a surface temperature above Tsat, the liquid initially is heated to Tsat and initiates nucleate boiling. When nucleate boiling occurs, the single-phase liquid (mode 1) flow becomes a two-phase bubbly flow (mode 2), composed of vapor bubbles and liquid. As the fluid temperature increases, the vapor bubbles merge and form large bubbles called slug flow (mode 3). Churn flow (mode 4) is highly irregular and unsteady, and is composed of large bubbles that continually break apart and then merge together. Higher in the tube, annular flow (mode 5) occurs. Annular flow occurs when liquid is only in contact with the tube walls, and a core of vapor and a liquid mist fills the center of the tube. Still higher in the tube, flow is composed of pure vapor and the liquid mist called droplet flow (mode 6), until finally the mist evaporates and only the single-phase vapor (mode 7) is present. Horizontal tubes show the same modes of heat transfer but have more of a sloshing effect which tends to blend the modes even more, as shown in Figure 6.10. The calculation of a two-phase pressure drop within a tube is extremely complex. Recently, this has been an area of extensive research, with only some success. The most generally applicable model is based on the assumption of homogeneous flow. That is, this model ignores the mode of flow and treats the flow as an average of vapor and liquid. For a simplistic case of flow through a round tube, the total pressure gradient is the sum of the pressure gradients related to wall friction, gravitational resistance, and momentum changes:

    ( 1   )  l © 2001 by CRC PRESS LLC

FIGURE 6.10 The fluid flow regime progression from liquid to vapor for two-phase horizontal internal flow.

Researchers have proposed many different and quite complex heat transfer correlations to describe the regimes of internal forced boiling flow. Klimenko32 proposed a methodology which has sufficient accuracy, is not overly complex, and applies to heat transfer stages before mist flow. To use the Klimenko method, we must first determine if the heat transfer is dominated by film evaporation or by nucleate boiling, by evaluating the dimensionless parameter Φ.

l Gh   1  3 -  1  ---  ----------fg- 1    ---    l  q If Φ 1.6 104, film evaporation is the dominant mode of heat transfer; if Φ  1.6 104, nucleate boiling dominates. The next step in the Klimenko method is to determine the Nusselt number resulting from the two-phase heat transfer coefficient, hTP , in either film evaporation or by nucleate boiling. The characteristic length for the Nusselt number is based on bubble size, which is found by

 gc L b  -----------------------g ( l   )

0.5

If the heat transfer mode is nucleate boiling, Φ  1.6 104, the Nusselt number is found by 3

0.6

0.5

Nu  7.4 10 q* P* Pr l

1  3

where: qL b q*  ---------------h fg   l pL p ---------b p*  -------------------------------------0.5   [  g ( l   ) ] kw  wall material thermal conductivity (W/m K) © 2001 by CRC PRESS LLC

 k----w-  kl 

0.15

If the heat transfer mode is film evaporation , Φ 1.6 104, the Nusselt number is found by 0.6

1  6

Nu  0.087Re Pr l

  0.2  k w 0.09  ------- l   k l 

where:

 V Lb Re  -----------l l G V  ---- 1    -----  1    pl Finally, knowing the Nusselt number for the heat transfer mode, we can calculate a single-phase heat transfer coefficient, which is based on the Reynolds number of the liquid portion of the flow. Usually the single-phase heat transfer coefficient, hFC, is so much smaller than the two-phase coefficient, hTP , that it can be ignored. However, if it is significant, the overall heat transfer coefficient can be found by 3

3

h c  ( h TP  h FC )

13

Klimenko’s correlations,32 shown in Figure 6.11 and 6.12, match experimental data with a mean absolute deviation of about 13% for most common coolants used in electronic cooling. The limitations are 6.1 103 N/m2  p  3.04 106 N/m2 50 kg/m2 s  G  2690 kg/m2 s 0.017    1.00 0.00163 m  D  0.0413 m

6.4 EVAPORATION Liquid may exist on a surface if the surface is below the saturation temperature, Tsat. Evaporation, like boiling, is considered a convection heat transfer process. Evaporation occurs when the molecules in a liquid gain enough energy to escape the liquid binding energy and enter the vapor state. Since the molecules leaving the liquid contain excess energy, the net effect is a loss of energy of the liquid. Since temperature is a measure of energy, the temperature of the liquid decreases when evaporation occurs. The convection of vapor from a surface is related to the mass transfer coefficient, hm. The magnitude of evaporative cooling can be expressed as the equation h T   T s  h fg  -----m [  A, sat ( T s )   A,  ]  hc  © 2001 by CRC PRESS LLC

© 2001 by CRC PRESS LLC

FIGURE 6.11 Klimenko data for the effect of wall conductivity on the Nusselt number for internal forced convection boiling flow. (From Klimenko, V. V., Int. J. Heat Mass Transfer, 31, 541, 1988. With permission.)

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FIGURE 6.12 Klimenko data showing the transition from nucleate boiling to film evaporation for internal forced convection flow. (From Klimenko, V. V., Int. J. Heat Mass Transfer, 31, 541, 1988. With permission.)

where: hm  mass transfer convection coefficient (m/s) hc  convection heat transfer coefficient (W/m2 K) A,sat(Ts)  saturated vapor density (kg/m 3)  infinite vapor density (kg/m3) A,∞ The ratio of the heat transfer coefficient to the mass transfer coefficient, hc/hm, can also be expressed in terms of a dimensionless Lewis number, Le, h k -----c  ------------------n   c Le l  n p hm D AB Le where: DAB  binary mass diffusion coefficient (m2/s) Le  Lewis number, /DAB n ⯝ 13 Using this relationship we see that the cooling effect can also be described as p A, sat ( T s ) p A,  ᏹ A h fg -  ---------- ---------------------T   T s  -------------------------23 T Ts ᑬ c p Le where:

ᏹ  molecular weight of species (kg/kmol) ᑬ  universal gas constant (8.315 kJ/kmol) pA,sat(Ts)  saturated vapor pressure (N/m2) pA,  infinite vapor pressure (N/m2) Gilliland and Sherwood33 correlated the mass transfer coefficient from liquids to air in a wetted-wall column as D AB 0.83 0.44 - Re Sc h m  0.023  ------- D L where: D  tube diameter (m) Sc  Schmidt number, /DAB  2000  Re  35,000 0.6  Sc  2.5

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Rohsenow and Choi34 indicate that by using the Chilton-Colburn factor, j, the mass transfer coefficient of evaporation from a flat, wetted surface in laminar flow can be found by V h m  0.664 ------------------------0.67 Re L Sc and for turbulent flow by V h m  0.037 ----------------------0.2 0.67 Re L Sc where V is the velocity in m/s, and the Reynolds number is based on the length of the surface. A Reynolds number based on the thickness of a falling evaporative film has also been proposed. In this case, the falling film Reynolds number, Re, is defined in terms of the hydraulic diameter, DH, and the mean velocity, um, of the film as  - 4  l  ----- l  l um DH 4 -------------------------------------  ------ Re   l l l

where: 4 Ac 4 -----D H  -------P  P  u m  -----l  

 mass flow per unit film width (kg/m s)







冮0 l u dy

3

g l  ( l  v )  ---------------------------------3 l

In evaporation from falling films of water, Chun and Seban35 determined the following correlations for the Nusselt number 1  3 3 Nu   --- Re  4 

0  Re   30

 0.22

Nu  0.822Re 

3

30  Re   Re tr 0.4 

Nu  3.8 10 Re Pr

0.65 l

Re tr  Re 

where Retr is the turbulent Reynolds number, which is equal to 5800Prl1.06.

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6.5 CONDENSATION Condensation will form on a surface if the vapor contacting the surface is saturated, and the surface temperature is lower than the temperature of the saturated vapor. When this occurs, the latent heat within the vapor is transferred to the cooler surface. Normally, condensation occurs as drops on a surface. If the conditions that caused the condensation are steady state, and the surface is clean, the droplets will coalesce and form a condensate film on the surface. In a gravity field, the laminar fluid film will flow. If the surface is long enough, the laminar flow of condensate may become wavy and then turbulent. Contrary to other forms of heat transfer, as T increases the heat transfer coefficient decreases. This is because as the temperature difference increases, more vapor becomes condensate, which causes a thicker layer of film. Because condensation heat transfer is a conduction process, the thicker film impedes the transfer of heat. Because of the heat transfer impedance caused by the thick fluid film, applications using condensation often have short flow paths or use small horizontal cylinders. The short flow paths do not allow the film to become very thick. Because of the action of film condensation, droplet condensation provides a higher heat transfer coefficient. Heat transfer rates of droplet condensation can be more than a magnitude higher than film condensation. Therefore, most applications use some type of coating on the condensation surfaces which promotes droplet formation. The condensation film begins at point x  0 and continues down a surface. The thickness of the film increases as x increases, as shown in Figure 6.13. Because the

FIGURE 6.13 The geometry and nomenclature used to describe film condensation on a vertical surface.

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flow moves at a constant velocity, and because the film is thicker at larger values of x, the mass flow rate also increases with x. The following correlations for condensation heat transfer are most accurate when evaluated at the film temperature recommended by Addoms36: T eval  T w  0.33 ( T sv  T w ) Nusselt37 obtained the first relationships for the condensation variables in 1916. Nusselt used simplifying assumptions to arrive at the basic relationship of the thickness of the film, , 4  l k l x ( T sat  T w )   ------------------------------------------g  l (  l   )hf g

0.25

the local heat transfer coefficient, hx, 0.25 3

 l (  l   )hf g k l h x  g-------------------------------------------4  l x ( T sat  T w ) and the local Nusselt number, Nu, h x x g  (    )h x 3 l l fg Nu  ------k l -------------------------------------------4  l k l ( T sat  T w )

0.25

For a vertical surface of unit width and a height of L, the average heat transfer coefficient can be written as 3

g  l (  l   )hf g k l sin  h c  0.943 ------------------------------------------------------- l L ( T sat  T w )

0.25

In the previous equations by Nusselt, the modified latent heat of vaporization is hf g  hfg0.375cp,l(Tsat  Tw). Rohsenow38 modified the relationship to match experimental data by equating the hf g  hfg0.68cp,l(Tsat  Tw), for the range when Pr

0.5 and when cp,l(Tsat  Tw) hf g  1.0. Sadasivan and Lienhard39 calculated the modified latent heat without Nusselt's simplifying assumptions as hf g  hfg(0.6830.228/Prl)cp,l(Tsat  Tw). Chen40 further modified Nusselt's equation to account for interfacial shear and momentum forces. Chen corrected the heat transfer coefficient as c p, l ( T sat  T w ) k l ( T sat  T w ) p, l ( T sat  T w )  1  0.68 c----------------------------  0.02 ---------------------------- -------------------------  l h fg  h fg h fg - hc  hc  ----------------------------------------------------------------------------------------------------------------------------k l ( T sat  T w ) c p, l ( T sat  T w ) k l ( T sat  T w )  1  0.85 --------------------------  0.15 ----------------------------- --------------------------   l h fg  l h fg  h fg 

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0.25

where: c p, l ( T sat  T w ) -----------------------------------  2.0 h fg k l ( T sat  T w ) -------------------------------  20  l h fg 1  Pr l  10.05 Chun and Seban35 found that the following correlations for the local and the average Nusselt number are applicable for water condensation flowing down a flat surface. When the Reynolds number, based on the film thickness, Reδ , represents laminar film condensation, and 0  Reδ  30, 1  3 3 Nu   --- Re  4 

When the Reynolds number indicates wavy laminar film condensation, and 30  Reδ  Retr,  0.22

Nu  0.822Re 

When the Reynolds number indicates turbulent film condensation, and Retr  Reδ, 3

0.4

0.65

Nu  3.8 10 Re  Pr l

 1.06

where Retr is the turbulent Reynolds number, which is equal to 5800 Pr l . To find the average Nusselt number, we use an equation based on the distance the condensate has traveled down the surface. Using the Reynolds number based on that length, we have the equation for laminar film condensation, when ReL  30, written as 0.25

2-l  1  3 4 Pr  -----Nu  --g l 3 --------- --------------4Ja l L When the Reynolds number indicates wavy laminar condensation flow, that is, when 30  ReL  Retr 0.18

2l- 1  3  ------

Pr l  g  Nu  -------- --------------4Ja l L

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FIGURE 6.14 Local Nusselt number as a function of the Reynolds number for laminar and turbulent film condensation of water. Adapted from References 29 and 35.

When the Reynolds number indicates turbulent film condensation, that is, when ReL Retr, then the average Nusselt number is described as 10  6

2 13

-l   ----- g

3

9.12 10 Ja l ( L  x tr ) Pr ----------------------------------------------------------  Re 0.6 13 Nu  --------l- --------------tr  ˙2l  0.35 4Ja l L  ------- Pr l  g

Figure 6.14 shows the local Nusselt number as a function of the Reynolds number for laminar and turbulent film condensation data gathered by Chun and Seban.35 The average heat transfer coefficient of a pure saturated vapor condensing on a horizontal tube can be found by 0.25 3

g  l (  l   )hf g k l h c  0.725 -------------------------------------------D  l ( T sat  T w ) Some applications have vertical rows of horizontal tubes; that is, the tubes are arranged so that the film is falling from an upper horizontal tube to a lower horizontal tube. If the condensation flow is continuous, the tube diameter, D, in the previous equation can be replaced by DN, where N is the number of tubes. This method gives conservative results because the condensation flow is rarely continuous. Chen40 suggested that since the condensation flow is subcooled, additional condensation occurs when the condensate is between the horizontal tubes. Assuming that all of the subcooling is used for additional condensation, Chen obtained the

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average heat transfer coefficient as 3

g  l (  l   )hf g k l c p ( T sv  T w ) ( N  1 ) --------------------------------------------h c  0.728 1  0.2 ------------------------------DN  l ( T sat  T w ) h fg

0.25

This correlation is very accurate when the following condition is met: ( N  1 )c p ( T sv  T w )  2.0 ---------------------------------------------------h fg

6.6 MELTING AND FREEZING Phase change materials are used in many specialized electronic cooling applications. Most often, transient power applications such as those used in missiles realize these benefits. As we have seen, during a phase change a material may absorb great amounts of power with only a small increase in temperature. For an application such as a missile, an onboard package of phase change material may absorb the heat given off by an electronic package, without the need for a dedicated cooling system. The phase change material absorbs the heat and melts. Depending on the quantity of heat and the mass of phase change material, the cooling effect will last until all of the material has melted. The opposite effect, freezing, occurs when the material gives up its latent heat to the surroundings. Applications for the freezing effect might be seen in missiles that travel outside the atmosphere for a short time, where the electronics package may need to be protected from the extreme cold of space until re-entry. The quantity of heat that a phase change material can absorb can be found by the equation q  m a m  h m  c p, s ( T m  T i )  c p, l ( T 2  T m ) where: q  quantity of heat stored (W) m  mass of phase change material (kg/m3) am  fraction melted (%) hm  latent heat of fusion (J/kg) T  absolute temperature (K) Ti  initial absolute temperature (K) Tm  melting point absolute temperature (K) T2  final absolute temperature (K) cp  specific heat (J/kgK) cp,s  average specific heat between Ti and Tm (J/kgK) cp,l  average specific heat between Tm and T2 (J/kgK)

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FIGURE 6.15 Temperature distribution for ice forming on water, with ambient air as a heat sink.

When a phase change material freezes, the problem can be considered to be one of boundary condition values, as shown in Figure 6.15. We can see that the solution to this problem is quite complex. Researchers have obtained only solutions for simple cases. We can instead use an approximate solution that was obtained by assuming that the heat capacity of the subcooled solid phase is negligible relative to the latent heat of solidification. Another simplifying assumption is that the physical properties are uniform and that the conductance and the heat sink temperature are constant throughout the process. To find the time to form a solid phase of a specific thickness we have:  L 2- ---L-  ----2k  U t  --------------------s  T o  T------------------  h fs 

REFERENCES 1. Hetsroni, G., Ed., Handbook of Multiphase Systems, Hemisphere, Washington, D.C., 1982. 2. Bergles, A. E. and Rohsenow, W. M., The Determination of Forced Convection Surface Boiling Heat Transfer, J. Heat Transfer, 86, 365, 1964. 3. Cole, R. and Rohsenow, W. M., Correlation of Bubble Diameters of Saturated Liquids, Chem. Eng. Prog. Symp. Ser., 65(92), 211, 1969. 4. Cole, R., Photographic Study of Boiling in Region of Critical Heat Flux, AIChE J., 6, 533, 1960.

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Thermal Design of Electronic Equipment

5. Graham, R. W. and Hendricks, R. C., Assessment of Convection, Conduction and Evaporation in Nucleate Boiling, NASA TND-3943, National Technical Information Service, Springfield, VA, 5/1967. 6. Bromley, L. A., Leroy, N. R., and Robbers, A., Heat Transfer in Forced Convection Film Boiling, Ind. Eng. Chem., 45, 2639, 1953. 7. Zuber, N., Tribus, M., and Westwater, J. W., The Hydrodynamic Crisis in Pool Boiling of Saturated and Subcooled Liquids, Paper 27, Int. Dev. in Heat Transfer, ASME, New York, 1961. 8. Florschuetz, L. W. and Chao, B. T., On the Mechanics of Vapor Bubble Collapse, J. Heat Transfer, 87, 209, 1965. 9. Plesset, M. S. and Zwick, S. A., A Non-Steady Heat Diffusion Problem with Spherical Symmetry, J. Appl. Phys., 23, 95, 1952. 10. Nukiyama, S., The Maximum and Minimum Values of Heat Transmitted from Metal to Boiling Water Under Atmospheric Pressure, J. Jpn. Soc. Mech. Eng., 37, 367, 1934. (Translation: Int. J. Heat Mass Transfer, 9, 1419, 1966.) 11. Drew, T. B. and Mueller, C., Boiling, Trans. AIChE, 33, 449, 1937. 12. Yamagata, K., Kirano, F., Nishiwaka, K., and Matsuoka, H., Nucleate Boiling of Water on the Horizontal Heating Surface, Mem. Fac. Eng. Kyushu Imp. Univ., 15, 98, 1955. 13. Rohsenow, W. M., A Method of Correlating Heat Transfer Data for Surface Boiling of Liquids, Trans. ASME, 74, 969, 1952. 14. Collier, J. G., Convective Boiling and Condensation, 2nd ed., McGraw-Hill, New York, 1981. 15. Danielson, R. D., Tousignant, L., and Bar-Cohen, A., Saturated Pool Boiling of Commercially Available Perfluorinated Inert Liquids, Proc. ASME-JJME Therm. Eng. Joint Conf., 1987, 419. 16. Hetsroni, G., Ed., Handbook of Multiphase Systems, Hemisphere, Washington, D.C., 1982. 17. Kutateladze, S. S., On the Transition to Film Boiling Under Natural Convection, Kotloturbostroenie, 3, 10, 1948. 18. Zuber, N., Hydrodynamic Aspects of Nucleate Boiling, Ph.D. dissertation, University of California, Los Angeles, 1959. 19. Lienhard, J. H. and Dhir, V. K., Hydrodynamic Prediction of Peak Pool Boiling Heat Fluxes from Finite Bodies, J. Heat Transfer, 95, 152, 1973. 20. Cichelli, M. T. and Bonilla, C. F., Heat Transfer to Liquids Boiling Under Pressure, Trans. AIChE, 41, 755, 1945. 21. Ellion, M. E., A Study of the Mechanism of Boiling Heat Transfer, Mem. 20–88, Jet Propulsion Laboratory, Pasadena, CA, 3/1954. 22. Bernath, L., A Theory of Local-Boiling Burnout and Its Application to Existing Data, Chem. Eng. Prog. Symp. Ser., 56(30), 95, 1960. 23. Zuber, N., On the Stability of Boiling Heat Transfer, Trans. ASME, 80, 711, 1958. 24. Kovalev, S. A., An Investigation of Minimum Heat Fluxes in Pool Boiling of Water, Int. J. Heat Mass Transfer, 9, 1219, 1966. 25. Berenson, P. J., Film Boiling Heat Transfer for a Horizontal Surface, J. Heat Transfer, 83, 351, 1961. 26. Lienhard, J. H. and Wong, P. T. Y., The Dominant Unstable Wavelength and Minimum Heat Flux During Film Boiling on a Horizontal Cylinder, J. Heat Transfer, 86, 220, 1964. 27. Bromley, L. A., Heat Transfer in Stable Film Boiling, Chem. Eng. Prog., 46, 221, 1950.

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Heat Transfer with Phase Change

341

28. Frederking, T. H. K. and Daniels, D. J., The Relation Between Bubble Diameter and Frequency of Removal From a Sphere During Film Boiling, J. Heat Transfer, 88, 87, 1966. 29. Mills, A. F., Heat and Mass Transfer, Irwin, Chicago, 1995, 640. 30. Frederking, T. H. K. and Clark, J. A., Natural Convection Film Boiling on a Sphere, Adv. Cryog. Eng., 8, 501, 1962. 31. Lienhard, J. H. and Eichhorn, R., Peak Boiling Heat Flux on Cylinders in a Cross Flow, Int. J. Heat Mass Transfer, 19, 1135, 1976. 32. Klimenko, V. V., A Generalized Correlation For Two-Phase Forced Flow Heat Transfer, Int. J. Heat Mass Transfer, 31, 541, 1988. 33. Gilliland, E. R. and Sherwood, T. K., Diffusion of Vapors into Air Streams, Ind. Eng. Chem., 26, 516, 1934. 34. Rohsenow, W. M. and Choi, H., Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, NJ, 1961. 35. Chun, K. R. and Seban, R. A., Heat Transfer to Evaporating Liquid Films, J. Heat Transfer, 93, 391, 1971. 36. Addoms, J. N., Heat Transfer at High Rates to Water Boiling Outside Cylinders, D. Sc. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1948. 37. Nusselt, W., Die Oberflachenkondensation des Wasserdampfes, Z. Vergle. D-Ing., 60, 541, 1916. 38. Rohsenow, W. M., Heat Transfer and Temperature Distribution in Laminar Film Condensation, Trans. ASME, 78, 1645, 1956. 39. Sadasivan, P. and Lienhard, J. H., Sensible Heat Correction in Laminar Film Boiling and Condensation, J. Heat Transfer, 109, 545, 1987. 40. Chen, M. M., An Analytical Study of Laminar Film Condensation. I. Flat Plates. II. Single and Multiple Horizontal Tubes, Trans. ASME, Ser. C., 83, 48, 1961.

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7

Combined Modes of Heat Transfer for Electronic Equipment

7.1 INTRODUCTION Although we have examined the three modes of heat transfer in great detail, in engineering practice we usually see cases where modes are combined simultaneously. Radiation is not usually a significant factor, but to achieve the best result we should calculate the importance of each mode. In a computer chip, for example, heat is conducted in parallel paths from the junction to the case and leads. Heat is then conducted from the leads to the circuit board, and from the case to a heat sink. Simultaneously, the heat in the leads and in the heat sink is convected to the air and radiated to the ambient environment. Table 7.1 is a reminder of the equations used for heat transfer and thermal resistance in the three modes. The simplest way to solve most problems of combined modes is to set up a resistance network. In this manner we can graphically examine the paths of each mode for simultaneous, parallel, and series heat transfer.

7.2 CONDUCTION IN SERIES AND IN PARALLEL Because combined modes of heat transfer often occur in parallel, reviewing the theory of series and parallel combinations may be helpful as applied to conduction. When heat is conducted through a single wall of a single material, the rate of heat conduction and the thermal gradient are constant. However, when heat is conducted in a series path of different materials, the temperature gradient is different for each material. Examining the composite wall of three materials in series, as shown in Figure 7.1, the rate of heat conduction through the entire block, qk, can be found by T1  T4 q k  ----------------------------------------------------L L L ------   ------   ------       kA

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A

kA

B

kA

C

TABLE 7.1 Thermal Resistances and Heat Transfer Rates Heat Transfer Mode

Rate of Heat Transfer, q

Thermal Resistance, ␪

Conduction

k q k  ( T 1  T 2 ) --L

L  k  ----kA

qc  hc A ( T s  T  )

1  c  --------hc A

Convection

Radiation

4

4

q r  A 1 ᑠ 1, 2  ( T 1  T 2 )

T1  T2  r  --------------------------------------------4 4 A 1 ᑠ 1, 2  ( T 1  T 2 )

If we have an indeterminate number, N, of individual materials, n, in series, the rate of heat conduction is T1  TN  1 T -  -----------------------------q k  ----------n N  L  L ----- kA n

冱n  1

----- kA n

Since  T is the overall temperature difference across the entire composite, we often call it the potential temperature. If we describe the thermal resistance to heat

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FIGURE 7.1 Thermal conduction through a three-layer system in series.

conduction as  k  L/ Ak , then we can rewrite the equation as T1  TN  1 T -  -------------------------q k  -------------------------n N n N  冱 n  1 k,n 冱 n  1 k,n From this equation we see that the flow of heat is proportional to the temperature potential. From this basic understanding of series heat flow and resistance, we can determine the characteristics of parallel heat flow through different materials. We know that the total rate of heat flow, qk, through two different materials, A and B, in a parallel path is the sum of the heat flows, qk  qA  qB. More comprehensively, to account for each area of heat flow we have T1  T2 T1  T2 T1  T2 -------------------  -------------------q k  ------------------ L L 1 2   ------  ------  ----------------- kA B  kA A   1   2 For the more common problem of combinations of series and parallel heat flows, such as shown in Figure 7.2, we see that the thermal resistance for the material in parallel, Section 2, is

B C  2  -----------------B  C

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FIGURE 7.2 Thermal conduction through a wall consisting of series and parallel paths of heat flow.

and the rate of heat flow is found by

T q k  ---------------------n3 冱 n  1 n For the composite shown in Figure 7.2, N  3; for the number of layers in series, n is the thermal resistance of the nth layer; and T is the overall temperature difference across the exterior walls.

7.3 CONDUCTION AND CONVECTION IN SERIES In electronics cooling problems involving conduction and convection modes of heat transfer in series, such as shown in Figure 7.3, we must usually determine the temperature increase of a device when we know the heat rate. We can easily add convection to a series conduction model by using the convection resistance term 1  c  --------hc A Figure 7.4 depicts an example of heat being transferred from a hot fluid to a cold fluid, through a wall. Both sides of the wall receive a different rate of convective heat transfer. We can describe the rate of heat transfer as Th  Tc T -  ------------------------------q  --------------------n3   2  3 1  冱n  1 i © 2001 by CRC PRESS LLC

FIGURE 7.3 Conduction and convection in an electronic module. The silicon die is encapsulated in an epoxy foam insulator case. The majority of the heat transfer is through the die surface.

FIGURE 7.4 Heat transfer from a hot gas to a cold gas through a plane wall.

where: 1  1  ------------------( h c A ) hot L  2  ----kA 1  3  -------------------( h c A ) cold © 2001 by CRC PRESS LLC

qk L

Wall

FIIGURE 7.5 Heat transfer from a hot gas to a cold gas through a plane wall, showing a parallel path for radiation.

7.4 RADIATION AND CONVECTION IN PARALLEL Many problems in electronic cooling involve the combined effect of simultaneous radiation and convection heat transfer. The total rate of heat transfer for the system shown in Figure 7.5 is the sum of the radiation and convection effects, q  qr  qc, which we can also write as q  hr A ( T 1  T 2 )  hc A ( T 1  T 2 )  ( hr  hc ) A ( T 1  T 2 ) The radiation heat transfer coefficient is the equation 4 4 qr (T 1  T 2) --------------------------------------------------hr   ᑠ 1, 2 A1 T 1  T 2 T1  T2

In most systems, determining the radiant heat transfer coefficient directly is very difficult. Since the temperature factor ᑠT contains the temperatures of the emitter and the receiver, we can evaluate it only when we know both. In electronics cooling, the temperature of the emitter usually varies with power; therefore, we must estimate a value for the emitter and then reiterate until the solution converges in steady state.

7.5 OVERALL HEAT TRANSFER COEFFICIENT The overall heat transfer coefficient, U, is used to describe the result of multiple convection coefficients as a single value. A common form of heat transfer used in heat exchangers is to transfer heat from a higher-temperature fluid to a lower-temperature © 2001 by CRC PRESS LLC

fluid when the fluids are separated by a wall. If we know that the wall is plane and there is only convection on both sides, we can find the rate of heat transfer using the following equation: Th  Tc T -  ------------------------------q  ----------------------------------------------------1  L 1 1  2  3  --------   ------   ---------       hc A

h

kA

hc A

c

This equation describes the flow of heat only in terms of a temperature potential difference and the thermal transfer characteristics of each section in the heat flow path. In some instances it is more helpful to describe the heat flow as a single value in terms of the resistances (or conductances). Rewriting the equation in terms of an overall value, we obtain q  UA  T where:

T  total temperature difference (˚C) 11 -  ---------UA  ------------------------------1  2  3

 TOT

The area that UA is based on should be stated in the problem to avoid uncertainty. If the chosen area is small, it may not reflect the overall value of the system but, instead, of only a local area. We can describe the overall heat transfer coefficient on the outside area, Ao, of a tube by the equation: 1 U o  ----------------------------------------------------------------r A o ln  ----o- ri Ao 1 ------------  -------------------  -----Ai hi ho 2 kL Conversely, we can determine the heat transfer coefficient on the inside area, Ai, of a tube with the equation: 1 U i  ----------------------------------------------------------------r A i ln  ----o- ri Ai 1 --------------  -------------------  ----2 kL Ao ho hi The overall heat transfer coefficient can also be used to describe radiation heat transfer. In this case, the fluid would be a gas. If we examine a plane wall that separates a hot gas from a cold gas, as shown in Figure 7.4, we see that heat is transferred into the wall by both convection and radiation. We can describe this © 2001 by CRC PRESS LLC

section of the heat transfer model as q  q c  q r  h c A ( T g  T sg )  h r A ( T g  T sg ) T g  T sg  ( h c1  h r 1 ) A ( T g  T sg )  -------------------1 where: Tg  T1  temperature of hot gas (˚C) Tsg  T2  temperature of hot wall surface (˚C) 4 4  A ( T g  T sg )

 radiation heat transfer coefficient (W/m2 K), where 1.0 hr1  ------------------------------------T g  T sg hc1  convective coefficient from the gas to the wall surface ( W/m 2 K ) 1 -  combined thermal resistance of Section 1, Figure 7.2 (K/W)  1  ---------------------------(h  h ) A r

cl

After the heat is transferred into the wall, it is transferred through the wall by conduction. The heat transfer through this second section can be written as T sg  T sc kA q  q k  ------ ( T sg  T sc )  ---------------------2 L where: Tsc  temperature of the cool wall surface (˚C) 1  thermal resistance of Section 2 (K/W) After the heat is transferred through the wall, it is transferred into the cooler gas by convection. The convection coefficient through this third section can be written as T sc  T c q  q c  h c3 A ( T sc  T c )  -------------------3 where: Tc  temperature of the cooler gas (˚C) 3  thermal resistance of Section 3 (K/W)

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Appendix

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353

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TABLE A1 Air at Sea-Level Atmospheric Pressure Temp. T

Density °C

kg/m3

32 41 50 59 68 77 86 95 104 113 122 131 140 149 158 167 176 185 194 203 212

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

1.293 1.269 1.242 1.222 1.202 1.183 1.164 1.147 1.129 1.111 1.093 1.079 1.061 1.047 1.030 1.013 1.001 0.986 0.972 0.959 0.947

3.664 3.598 3.533 3.470 3.412 3.354 3.298 3.244 3.193 3.142 3.094 3.048 3.003 2.957 2.914 2.875 2.834 2.795 2.755 2.718 2.683

Specific Heat cp J/kg K

Thermal Cond. k W/m K

Absolute Viscosity  106 N s/m2

Kinematic Viscosity v 106 m2/s

Prandtl Number Pr –

1003.9 1004.3 1004.6 1004.9 1005.2 1005.4 1005.7 1006.0 1006.3 1006.6 1006.9 1007.3 1007.7 1008.0 1008.4 1008.8 1009.3 1009.8 1010.3 1010.7 1011.2

0.02417 0.02445 0.02480 0.02512 0.02544 0.02577 0.02614 0.02650 0.02684 0.02726 0.02761 0.02801 0.02837 0.02876 0.02912 0.02945 0.02979 0.03012 0.03045 0.03073 0.03101

17.17 17.35 17.58 17.79 18.00 18.22 18.46 18.70 18.92 19.19 19.42 19.68 19.91 20.16 20.39 20.60 20.82 21.02 21.23 21.41 21.58

13.28 13.67 14.16 14.56 14.98 15.40 15.86 16.30 16.76 17.27 17.77 18.24 18.77 19.26 19.80 20.34 20.80 21.32 21.84 22.33 22.79

0.7131 0.7127 0.7122 0.7118 0.7113 0.7108 0.7103 0.7098 0.7093 0.7087 0.7082 0.7077 0.7072 0.7067 0.7062 0.7057 0.7053 0.7048 0.7044 0.7041 0.7038

Thermal Design of Electronic Equipment

°F

Coef. Exp.  103 1/K



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TABLE A2 Water at Sea-Level Atmospheric Pressure Temp. T °F

°C

32 41 50 59 68 77 86 95 104 113 122 131 140 149 158 167 176 185 194 203 212

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Density  kg/m3 999.9 1000 999.7 999.1 998.2 997.1 995.7 994.1 992.2 990.2 988.1 985.8 983.5 980.8 978 974.9 971.7 968.5 965 961.7 958.4

Coef. Exp.   103 1/K 0.068 0.018 0.095 0.16 0.22 0.26 0.31 0.35 0.39 0.42 0.45 0.48 0.51 0.54 0.57 0.60 0.63 0.66 0.69 0.72 0.75

Specific Heat cp J/kg K

Thermal Cond. k W/m K

Absolute Viscosity  106 N s/m2

Kinematic Viscosity v 106 m2/s

4217.5 4202.7 4192.4 4185.8 4181.7 4179.5 4178.6 4178.5 4179.0 4179.9 4181.1 4182.6 4184.5 4186.8 4189.5 4192.9 4196.6 4201.0 4205.7 4210.6 4215.5

0.5580 0.5677 0.5774 0.5870 0.5967 0.6064 0.6155 0.6243 0.6325 0.6401 0.6472 0.6536 0.6594 0.6643 0.6686 0.6724 0.6753 0.6778 0.6797 0.6811 0.6822

1794 1530 1296 1136 993 880.6 792.4 719.8 658.0 605.1 555.1 512.6 470.0 436.0 402.0 376.6 350.0 330.5 311.0 294.3 277.5

1.794 1.530 1.296 1.137 0.995 0.883 0.796 0.724 0.663 0.611 0.562 0.520 0.478 0.445 0.411 0.386 0.361 0.341 0.322 0.306 0.290

Prandtl Number Pr – 13.56 11.33 9.410 8.101 6.959 6.069 5.380 4.818 4.348 3.951 3.586 3.280 2.983 2.748 2.519 2.348 2.175 2.048 1.924 1.819 1.715

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TABLE A3 Perflurocarbon FC-72 at Atmospheric Pressure (Boils at 56°C) Temp. T °F 32 41 50 59 68 77 86 95 104 113 122 131

°C 0 5 10 15 20 25 30 35 40 45 50 55

Density  kg/m3

Coef. Exp.   103 1/K

1740 1727 1714 1701 1688 1675 1662 1649 1636 1623 1610 1597

1.601 1.611 1.619 1.626 1.633 1.640 1.647 1.654 1.662 1.670 1.680 1.689

Specific Heat cp J/kg K 1005.0 1016.2 1025.6 1033.2 1039.8 1046.6 1053.5 1060.8 1068.7 1077.5 1087.0 1096.5

Thermal Cond. k W/m K 0.0600 0.0595 0.0590 0.0585 0.0580 0.0575 0.0570 0.0565 0.0560 0.0555 0.0550 0.0545

Absolute Viscosity  106 N s/m2 1009.5 932.4 861.6 799.5 743.0 693.8 648.2 610.1 574.3 543.9 514.8 486.0

Kinematic Viscosity v  106 m2/s 0.5802 0.5399 0.5027 0.4700 0.4402 0.4142 0.3900 0.3700 0.3510 0.3351 0.3198 0.3043

Prandtl Number Pr – 16.91 15.93 14.98 14.12 13.32 12.63 11.98 11.46 10.96 10.56 10.17 9.778

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TABLE A4 Perflurocarbon FC-77 at Atmospheric Pressure (Boils at 97°C) Temp. T

Density

°F

°C

kg/m3

Coef. Exp.  103 1/K

32 41 50 59 68 77 86 95 104 113 122 131 140 149 158 167 176 185 194 203

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

1838 1826 1814 1802 1789 1777 1765 1753 1740 1728 1716 1704 1691 1679 1667 1655 1642 1630 1618 1605

1.399 1.407 1.414 1.421 1.429 1.436 1.443 1.451 1.458 1.466 1.473 1.481 1.489 1.497 1.504 1.512 1.520 1.527 1.534 1.541



Specific Heat cp J/kg K

Thermal Cond. k W/m K

Absolute Viscosity  106 N s/m2

Kinematic Viscosity v 106 m2/s

Prandtl Number Pr –

1005 1016 1025 1033 1041 1048 1056 1063 1071 1079 1087 1096 1105 1114 1123 1131 1140 1147 1154 1159

0.0649 0.0646 0.0643 0.0640 0.0637 0.0634 0.0631 0.0628 0.0625 0.0621 0.0617 0.0613 0.0609 0.0604 0.0600 0.0595 0.0590 0.0585 0.0580 0.0575

2356 2117 1905 1719 1554 1413 1288 1178 1083 1001 927.0 862.4 805.0 753.2 706.1 662.3 622.1 584.0 548.0 513.2

1.282 1.159 1.052 0.9539 0.8686 0.7592 0.7298 0.6720 0.6224 0.5793 0.5402 0.5061 0.4761 0.4486 0.4236 0.4002 0.3789 0.3583 0.3387 0.3198

36.48 33.30 30.37 27.75 25.40 23.36 21.56 19.94 18.56 17.39 16.33 15.42 14.61 13.89 13.22 12.59 12.02 11.45 10.90 10.34

Source: Data from FluorinertTM Liquids Product Manual, 3M, St. Paul, MN. With permission.

TABLE A5 Thermophysical Properties of Nonferrous Metals at 20°C

Materials

Density  kg/m3

Aluminum (1100) Aluminum (2014) Aluminum (2024) Aluminum (5052) Aluminum (6061) Aluminum (7075) Aluminum (356) Beryllium Brass (C36000) Bronze (C22000) Copper (C11000) Copper (C12200) Copper (C22000) Copper (alloy MF 202) Glass seal (alloy Ni 50) Gold Inconel (625) Kovar Lead

2713 2796 2768 2685 2713 2796 2685 1855 8498 8802 8913 8941 8802 8862 8332 19,321 8442 8343 11,349

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Coef. Exp.   106 1/K 23.6 23.0 23.2 23.8 23.4 23.6 21.4 11.5 20.5 18.4 17.6 17.6 18.4 17.0 8.46 14.2 12.8 4.30 29.3

Specific Heat cp J/kg K 921 921 921 921 963 963 935 1884 380 377 383 385 377 382 482 129 410 439 130

Thermal Cond. k W/m K 222 192 189 139 180 121 159 151 116 189 391 339 189 150 10.4 313 9.82 16.0 33.9

TABLE A6 Thermophysical Properties of Ferrous Metals at 20°C

Materials Carbon steel (AISI 1010) Carbon steel (AISI 1042) Cast iron (ASTM A-48) Cast iron (ASTM A-220) Cast steels (carbon and alloy) Stainless steel (4130) Stainless steel (17-4 PH) Stainless steel (304) Stainless steel (316) Stainless steel (440)

Density  kg/m3 7830 7840 7197 7363 7834 7833 7778 8027 2685 7750

Coef. Exp.   106 1/K 6.60 6.50 10.8 13.5 14.7 13.5 10.8 17.3 16.0 10.1

Specific Heat cp J/kg K 434 460 544 544 440 456 461 477 468 461

Thermal Cond. k W/m K 64.0 50.0 50.2 51.1 46.7 43.3 18.0 16.3 16.3 24.2

TABLE A7 Thermophysical Properties of Plastics at 20°C

Materials ABS (acrylonitrile butadiene styrene) Acetal Acrylic Alkyd Cellulose acetate Epoxy (cast) Epoxy (IC molding) Fluorocarbon (PTFE) Polyamide (nylon type 6) Phenolic Polycarbonate Polybutylene Terephthalate (PBT) Polyester Polyimide Polyamide-imide Polyetherimide Polyesteretherketone Polyetherketone Polystyrene Polyethylene Polypropylene Polyvinyl chloride (PVC)

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Density  kg/m3

Coef. Exp.   106 1/K

Specific Heat cp J/kg K

Thermal Cond. k W/m K

1058

72.0

1466

2.70

1415 1178 2206 1257 1148 1820 2196 1247 1387 1203 1307

82.8 81.0 36.0 121 59.4 17.0 90.9 89.1 37.4 67.5 72.0

1465 1466 1256 1508 1884 984 1047 1675 1570 1256 1905

3.01 2.49 9.87 3.01 4.15 4.00 2.91 2.08 1.74 2.39 1.90

1287 1427 1397 1277 1317 – 1039 933 903 1447

85.5 47.7 36.0 54.0 40.5 103 72.9 225 86.4 54.0

1780 1214 – 1090 – – 1361 2261 1884 1050

2.29 8.05 2.94 2.2 2.95 – 1.54 3.95 2.22 1.77

TABLE A8 Thermophysical Properties of Ceramics at 20°C

Materials

Density  kg/m3

Coef. Exp.   106 1/K

Specific Heat cp J/kg K

3982 3200 2900 2200 3500 2200 2900 2300 3300

5.67 4.40 7.00 3.80 2.00 0.50 50.0 4.20 2.00

879 711 1030 709 510 745

30.0 200 300 1300 1200 1.60

664 624

83.7 21.0

Aluminum oxide Aluminum nitride Beryllium oxide Boron nitride (cubic) Diamond (film) Fused quartz Glass (die attach) Silicon Silicon nitride

Thermal Cond. k W/m K

TABLE A9 Properties of Common Gases

Acetylene Air Ammonia Argon Butane Carbon dioxide Carbon monoxide Chlorine Ethane Ethylene Fluorine Freon-12 Helium Hydrogen Methane Methanol Neon Nitrogen Nitrous oxide Octane Oxygen Pentane Propane

  106 N s/m2

Å

T

K

308.3 132.0 405.6 150.8 425.2 304.2

– 19.3 32.7 26.4 25.0 34.3

4.033 3.711 2.900 3.542 4.687 3.941

231.8 78.60 558.3 93.30 531.4 195.2

3.4958

132.9

19.0

3.690

91.70

1.355 1.183 1.208 – 1.139 1.667 1.404 1.320 1.203 1.667 1.400 1.303

7.7009 4.8840 5.0360 5.2183 4.1240 0.2270 1.2970 4.6003 8.0960 2.7561 3.3945 7.2449

417.0 305.4 282.4 144.3 385.0 5.190 33.20 190.6 512.6 44.40 126.2 309.6

42.0 21.7 21.7 27.5 – 2.54 3.47 15.9 39.3 16.3 18.0 33.2

4.217 4.443 4.163 3.357 – 2.551 2.827 3.758 3.626 2.820 3.798 3.828

316.0 215.7 224.7 112.6 – 10.22 59.70 148.6 481.8 32.80 71.40 232.4

1.044 1.395 1.086 1.124

4.2845 5.0461 3.3742 4.2456

563.4 154.6 469.6 369.8

24.1 25.0 25.0 23.3

– 3.467 5.784 5.118

– 106.7 34.10 237.1

MW

 cp/cv

pc  106 atm

C2H2 (a) NH3 Ar C4H10 CO2

26.038 28.966 17.031 39.948 58.124 44.010

1.260 1.400 1.310 1.660 1.090 1.285

6.1404 3.6883 11.2777 4.8738 3.7998 7.3766

CO

28.010

1.399

Cl2 C2H6 C2H4 F2 CC12F2 He H2 CH4 CH4O Ne N2 N2O

70.096 30.070 28.054 37.997 120.914 4.003 2.016 16.043 32.042 20.183 28.013 44.013

C8H18 O2 C5H12 C3H8

114.232 32.00 72.151 44.097

Gas

© 2001 by CRC PRESS LLC

Tc (K) K

TABLE A9 (continued) Properties of Common Gases MW

 cp/cv

pc  106 atm

Tc (K) K

  106 N s/m2

Å

T

K

18.015 92.140 131.30

1.329 – 1.660

22.049 4.1139 5.8364

647.3 591.7 289.7

54.1 127.0 53.7

2.641 – 4.047

809.1 – 231.0

Gas Steam Toluene Xenon

H2O C7H8 Xe

Note: Pc  critical pressure (Pa) MW  molecular weight Tc  critical temperature (K) T  effective temperature characteristic force potential (K)



 ratio of specfic heat at constant pressure to specific heat at constant volume at 15°C to 25°C



 molecular collision diameter (Å)

c  viscosity at critical pressure and critical temperature (N s/m2) a

Properties 78.084% 20.946% 0.934% 0.033% 1.0%

of air based on: N2 O2 Ar CO2 other, by volume.

TABLE A10 U.S. Standard Atmosphere Altitude H (m) 100 0 100 200 400 600 800 1000 1500 2000 2500 3000 3500

© 2001 by CRC PRESS LLC

Temperaure T (K)

Density  (kg/m3)

Pressure P (N/m2)

Viscosity   106 (N/m2)

288.80 288.15 287.50 286.85 285.55 284.25 282.95 281.65 278.40 275.15 271.90 268.65 265.40

1.237 1.225 1.213 1.202 1.179 1.156 1.134 1.112 1.058 1.007 0.9567 0.9091 0.8632

102,534 101,327 100,131 98,948 96,613 94,324 92,079 89,877 84,558 79,497 74,684 70,110 65,766

17.92 17.89 17.86 17.83 17.76 17.70 17.64 17.57 17.41 17.25 17.09 16.93 16.77

TABLE A10 (continued) U.S. Standard Atmosphere Altitude H (m) 4000 4500 5000 6000 7000 8000 9000 10,000 11,000 12,000 13,000 14,000 15,000 16,000 17,000 18,000 19,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 60,000 70,000

© 2001 by CRC PRESS LLC

Temperaure T (K) 262.15 258.90 255.65 249.15 242.65 236.15 229.65 223.15 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 221.65 226.65 237.05 251.05 265.05 270.65 245.45 217.45

Density  (kg/m3)

Pressure P (N/m2)

Viscosity   106 (N/m2)

0.8191 0.7768 0.7361 0.6597 0.5895 0.5252 0.4664 0.4127 0.3639 0.3108 0.2655 0.2268 0.1937 0.1654 0.1413 0.1207 0.1031 0.08803 0.03946 0.01801 0.00821 0.00385 0.00188 0.00097 0.00028 0.00007

61,642 57,730 54,022 47,183 41,062 35,601 30,380 26,437 22,633 19,331 16,511 14,102 12,045 10,288 8,787.1 7,505.2 6,410.4 5,475.2 2,511.2 1,172.0 558.97 277.55 143.15 75.954 20.317 4.6348

16.61 16.44 16.28 15.95 15.60 15.26 14.92 14.57 14.21 14.21 14.21 14.21 14.21 14.21 14.21 14.21 14.21 14.21 14.48 14.76 15.31 16.04 16.75 17.03 15.75 14.26