Pagina principale Thermal design of electronic equipment

Thermal design of electronic equipment

In a field where change and growth is inevitable, new electronic packaging problems continually arise. Smaller, more powerful devices are prone to overheating, causing intermittent system failures, corrupted signals, lower MTBF, and outright system failure. Since convection cooling is the heat transfer path most engineers take to deal with thermal problems, it is appropriate to gain as much understanding about the underlying mechanisms of fluid motion as possible. Thermal Design of Electronic Equipment is the only book that specifically targets the formulas used by electronic packaging and thermal engineers. It presents heat transfer equations dealing with polyalphaolephin (PAO), silicone oils, perfluorocarbons, and silicate ester-based liquids. Instead of relying on theoretical expressions and text explanations, the author presents empirical formulas and practical techniques that allow you to quickly solve nearly any thermal engineering problem in electronic packaging.
Anno:
2001
Edizione:
1
Editore:
CRC Press
Lingua:
english
Pagine:
319
ISBN 10:
0849300827
ISBN 13:
9780849300820
Series:
Electronics handbook series
File:
PDF, 20.30 MB
Download (pdf, 20.30 MB)

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“Frontmatter”
Thermal Design of Electronic Equipment.
Ed. Ralph Remsburg
Boca Raton: CRC Press LLC, 2001

© 2001 by CRC PRESS LLC

THERMAL
DESIGN
of
ELECTRONIC
EQUIPMENT

© 2001 by CRC PRESS LLC

ELECTRONICS HANDBOOK SERIES
Series Editor:

Jerry C. Whitaker
Technical Press
Morgan Hill, California

PUBLISHED TITLES
AC POWER SYSTEMS HANDBOOK, SECOND EDITION
Jerry C. Whitaker

THE COMMUNICATIONS FACILITY DESIGN HANDBOOK
Jerry C. Whitaker

THE ELECTRONIC PACKAGING HANDBOOK
Glenn R. Blackwell

POWER VACUUM TUBES HANDBOOK, SECOND EDITION
Jerry C. Whitaker

THERMAL DESIGN OF ELECTRONIC EQUIPMENT
Ralph Remsburg

THE RESOURCE HANDBOOK OF ELECTRONICS
Jerry C. Whitaker

MICROELECTRONICS
Jerry C. Whitaker

SEMICONDUCTOR DEVICES AND CIRCUITS
Jerry C. Whitaker

SIGNAL MEASUREMENT, ANALYSIS, AND TESTING
Jerry C. Whitaker

FORTHCOMING TITLES
ELECTRONIC SYSTEMS MAINTENANCE HANDBOOK
Jerry C. Whitaker

© 2001 by CRC PRESS LLC

THERMAL
DESIGN
of
ELECTRONIC
EQUIPMENT
Ralph Remsburg
Nortel Networks
Boca Raton, Florida

CRC Press
Boca Raton London New York Washington, D.C.

© 2001 by CRC PRESS LLC

0082-FM.fm Page 4 Wednesday, August 23, 2000 9:50 AM

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 us; e.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, microfilming, and recording, or by any information storage or
retrieval system, without prior permission in writing from the publisher.
The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for
creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC
for such copying.
Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation, without intent to infringe.

© 2001 by CRC Press LLC
No claim to original U.S. Government works
International Standard Book Number 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

0082-FM.fm Page 5 Wednesday, August 23, 2000 9:50 AM

Preface
The field of electronic packaging continues to grow at an amazing rate. The electronic
packaging engineer requires analytical skills, a foundation in mechanical engineering, and access to the latest developments in the electronics field. The emphasis may
change from project to project, and from company to company, yet some constants
should continue into the foreseeable future. One of these is the emphasis on thermal
design.
Thermal analysis of electronic equipment is becoming one of the primary aspects
of many packaging jobs. An 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

0082-FM.fm Page 7 Wednesday, August 23, 2000 9:50 AM

Author
Ralph Remsburg is currently Senior Thermal Analyst at Nortel Networks, Boca
Raton, FL. Previously, he held engineering positions up to the director level. Ford
Aerospace, Chrysler Corporation, Delco, Hughes Network Systems, Loral Data
Systems, Moog Space Products, Alcon Surgical Labs, and Dell Computer have all
benefited from his expertise as a consultant.
Remsburg’s name is on over 30 patents, 5 published papers, and a previous book.
He attended New York University, received a master’s degree from Columbia University, and is completing a dissertation for a doctorate.

© 2001 by CRC PRESS LLC

0082-FM.fm Page 9 Wednesday, August 23, 2000 9:50 AM

Nomenclature and Symbology
SYMBOL DEFINITION
Symbol
a
a
A

b
c

CA
C
C
Ċ

CD
Cf

D

DAB
e
e
E
E

f
f'

Description
Velocity of sound; acceleration
Thermal diffusivity (a  k/c)
Area;
Ac, 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
Ċ c , rate of hourly heat capacity of a colder fluid
in a heat exchanger
Ċ h rate of hourly heat capacity of a warmer fluid
in a heat exchanger
Total drag coefficient
Skin friction coefficient
Cfx, local value of Cf at distance x from leading edge;
Cf , average value of Cf
Diameter
DH, hydraulic diameter
Do, outside diameter
Di, inside diameter
Mass diffusion coefficient
Base of natural or Napierian logarithm (2.71828)
Internal energy per unit mass
Internal energy
Emissive power of a radiating body
Eb, emissive power of a blackbody
E, monochromatic emissive power per micron
at wavelength 
Fanning friction coefficient for flow through a conduit
Friction coefficient for flow through 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

0082-FM.fm Page 10 Wednesday, August 23, 2000 9:50 AM

Symbol
F
F1 2

 1 2
g
gc
G
G
h
h

hl
hfg
hm
H
i
I
I
J
k

K

log
ln
l
L
Lf
ṁ
M
N
p

P
P

Description
Force
Geometric shape factor for radiation from one blackbody
to another
Geometric shape and emissivity factor for radiation from
one graybody to another
Acceleration due to gravity (9.807 m/s2)
Dimensional conversion factor
Mass velocity or flow rate per unit area (G  V )
Irradiation incident on unit surface in unit time
Enthalpy per unit mass
Combined 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

0082-FM.fm Page 11 Wednesday, August 23, 2000 9:50 AM

Symbol
q

q̇ G
q
Q
Q̇
r

R


S
SL
ST
t
T

u
u

U
U
v

Description
Rate of heat flow
qk, rate of heat flow by conduction
qr, rate of heat flow by radiation
qc, rate of heat flow by convection
qb, rate of heat flow by nucleate boiling
Rate of heat generation per unit volume
Rate of heat generation per unit area
Quantity of heat
Volumetric rate of fluid flow
Radius
rH, hydraulic radius
ri, inner radius
ro, outer radius
Electrical resistance
Perfect gas constant

Shape factor for conduction heat flow
Distance between the centerlines of 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

0082-FM.fm Page 12 Wednesday, August 23, 2000 9:50 AM

Symbol
v

Description
Time average velocity in the y direction
, instantaneous fluctuating y component of
velocity
Volume
Rate of work output
Distance from leading edge
xc, critical distance from the leading edge
(beginning of turbulent flow)
Coordinate
Coordinate
Distance from a solid boundary measured in a direction
normal to the surface
Vertical fin spacing
Coordinate
Ratio of heat exchanger hourly capacity rate

V
W˙
x

x
y
y
z
z
Z

International
Units

English
Units

m/s

ft/s

m3
W
m

ft3
Btu
ft

Dimensionless
Dimensionless
m
ft
m
ft
Dimensionless
Dimensionless

GREEK LETTERS
Symbol
(alpha)

Description

International
Units

English
Units

 (Delta)

Absorptance for radiation
, monochromatic absorptance
at wavelength 
Temperature coefficient of volume expansion
Temperature coefficient of thermal conductivity
Specific heat ratio, cp/cv
Body force per unit mass
Mass flow rate of condensate per unit breadth 
m/ D for a vertical tube
Boundary layer thickness
h, hydrodynamic boundary layer thickness
th, thermal boundary layer thickness
Difference between values

 (epsilon)

Heat exchanger effectiveness

Dimensionless

 (epsilon)

Dimensionless

H (epsilon)

Emittance for radiation
, monochromatic emittance at wavelength 
, emittance in direction of 
Thermal eddy diffusivity

m2/s

ft2/s

M (epsilon)

Momentum eddy diffusivity

m2/s

ft2/s

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

 (delta)

© 2001 by CRC PRESS LLC

Dimensionless

1/K
1/R
1/K
1/R
Dimensionless
N/kg
lbf/lbm
kg/s m
lbm/h ft
m

ft

Dimensionless

0082-FM.fm Page 13 Wednesday, August 23, 2000 9:50 AM

International
Units

Symbol

Description

(zeta)

Ratio of thermal to hydrodynamic boundary layer
thickness, th/h
Fin efficiency
Thermal resistance
c, thermal resistance to convective
heat transfer
k, thermal resistance to conductive heat
transfer
r, thermal resistance to radiative heat transfer
jc, thermal resistance from semiconductor
junction to semiconductor case
ca, thermal resistance from semiconductor
case to ambient
ja, thermal resistance from semiconductor
junction to ambient
Angle
Wavelength
max, wavelength at which monochromatic
emissive power Eb is at maximum
Latent heat of vaporization
Absolute viscosity
Kinematic viscosity, 
Frequency of radiation
Mass density; l/v
l, density of a liquid
, density of a vapor
Reflection for radiation
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
lbf/ft2
N/m2

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

DIMENSIONLESS NUMBERS
Symbol

Description

Ref.
hr
--------o
ks

Bi

hL
Biot Number  -----ks

Bo

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

or

1

2

© 2001 by CRC PRESS LLC

)
(Continued)

0082-FM.fm Page 14 Wednesday, August 23, 2000 9:50 AM

Symbol
Ec
El
Fo
Gz

Description
U
Eckert Number  ------- ( T s  T  )
cp
2
4
 g cpz T
Elenbass Number  ------------------------------ kL
t
Fourier Modulus  -----2 or
L
ṁc
Graetz Number  ---------p
kfL

Ref.
2
3

t
-----2
ro

4
5

Gr

L  gT
Grashof Number  -------------------------2


6

j

Nu
23
Colburn Factor   ------------ Pr
 RePr

7

Ja

c p ( T w  T sat )
Jakob Number  -------------------------------h fg

Le

Lewis Number  --------D AB

M

U
Mach Number  ------a

3 2

Mo
Nu

8
9

 k c
Mouromtseft Number  -----------------------
hc x
Nusselt Number  -------- at point x
kf
0.8 0.6 0.4
p
0.4

10
11

Nu

hc L
- average over surface
Nusselt Number  ------kf

11

Nu D

hc D
- average of diameter
Nusselt Number  -------kf

11

Pe

Peclet Number  Re Pr

12

Pr

cp
Prandtl Number  -------k

13

Ra

Rayleigh Number  Gr Pr

14

Re

UL
Reynolds Number  ------------

15
2


Sh

h a
Boundary Fourier Modulus  ----------2
ks
hm L
Sherwood Number  --------D AB

Sc


Schmidt Number  ------------ D AB

St

hc
Stanton Number  --------------Ucp

We

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

© 2001 by CRC PRESS LLC

2


3
16
17

or

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

18

0082-FM.fm Page 15 Wednesday, August 23, 2000 9:50 AM

UNIT CONVERSION FACTORS
SI → English

Quantity
Area (A)
Density ()
Energy (E)

Energy per
unit mass (e)
Force (F)
Heat flux
generation per
unit area (q)
Heat generation
per unit volume
( q̇ G )
Heat transfer
coefficient (hc)
Heat transfer
rate (q)
Length (L)
Mass (M)
Mass flow
rate ( ṁ )
Rate of heat (q)

Pressure and stress
(p)

Specific heat (cp)
Surface tension
( )

1
1
1
1
1
1
1
1
1

m2  10.764 ft2
m2  1550.0 in.2
kg/m3  0.06243 lbm/ft3
kg/m3  1.94032  103 slug/ft3
J  9.4787  104 Btu
J  0.73757 lbf ft
J  0.23885 cal
J  372.44  103 hp h
J/kg  4.2995  104 Btu/lbm

English → SI
1
1
1
1
1
1
1
1
1

ft2  0.09290 m2
in.2  6.452  104 m2
lbm/ft3  16.0179 kg/m3
slug/ft3  515.38 kg/m3
Btu  1055.06 J
lbf ft  1.3558 J
cal  4.1868 J
hp h  2.685  106 J
Btu/lbm  2326.0 J/kg

1 W/m2  0.3171 Btu/(h ft2)

1 lbf  4.448 N
1 lbf  1 slug ft/s2
1 Btu/(h ft2)  3.1525 W/m2

1 W/m3  0.09665 Btu/(h ft3)

1 Btu/(h ft3)  10.343 W/m3

1 W/(m2 K)  0.1761 Btu/(h ft2 °F)

1 Btu/(h ft2 °F)  5.678 W/(m2 K)

1 W  3.41213 Btu/h
1 W  0.239 cal/s
1 m  3.2808 ft
1 m  39.37 in.
1 kg  2.2046 lbm
1 kg  68.521  103 slug
1 kg/s  7936.6 lbm/h
1 kg/s  2.2046 lbm/s
1 W  3.41213 Btu/h
1 W  94.778  106 Btu/s
1 W  0.73757 lbf ft/s
1 W  1.3410  103 hp
1 N/m2  1 Pa = 0.02089 lbf /ft2
1 N/m2  0.14504  103 lbf /in.2
1 N/m2  4.015  103 in H2O
1 N/m2  9.8688 std. atmosphere
1 N/m2  0.10  106 bar
1 J/(kg K)  2.3886  104
Btu/(lbm °F)
1 N/m  0.06852 lbf/ft
1 N/m  1  103 dyn/cm

1 Btu/h  0.2931 W

1 N  0.2248 lbf

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

ft  0.30480 m
in.  0.02540 m
lbm  0.4536 kg
slug  14.594 kg
lbm/h  126.0  106 kg/s
lbm/s  0.4536 kg/s
Btu/h  0.2931 W
Btu/s  1055.1 W
lbf ft/s  1.3558 W
hp  745.7 W
lbf/ft2  47.88 N/m2
psi  1 lbf /in.2  6894.8 N/m2
in. H2O  249.066 N/m2
std. atm  0.10133  106 N/m2
bar  0.1  106 N/m2
Btu/(lbm °F)  4187.0 J/(kg K)

1 lbf/ft  14.594 N/m
1 dyn/cm  1  103 N/m
(Continued)

© 2001 by CRC PRESS LLC

0082-FM.fm Page 16 Wednesday, August 23, 2000 9:50 AM

SI → English

English → SI

T(K)  T(°C)  273.15
T(K)  T(°R)/1.8
T(K)  [T(°F)  459.67]/1.8
T(°C)  [T(°F) 32.0]/1.8
1 K  1°C
1 K  1.8°R
1 K  1.8°F
1 W/(m K)  0.57782 Btu/(h ft °F)

T(°R)  1.8T(K)
T(°R)  T(°F)  459.67
T(°F)  1.8T(°C)  32.0
T(°F)  1.8[T(K)  273.15]  32.0
1°R  1°F
1°R  (5/9)K
1°F  (5/9)K
1 Btu/(h ft °F)  1.731 W/(m K)

1 m2/s  10.7639 ft2/s
1 m2/s  38750.0 ft2/h
1 K/W  0.52750 °F h/Btu

1 ft2/s  0.0929 m2/s
1 ft2/h  25.81  106 m2/s
1 °F h/Btu  1.896 K/W

Quantity
Temperature (T)

Temperature
difference ( T)
Thermal
conductivity (k)
Thermal
diffusivity (a)
Thermal
resistance ()
Velocity (U)

Viscosity,
absolute ()
Viscosity,
kinematic ()
Volume (V)

Volumetric flow
rate ( q̇ )

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

m/s  3.2808 ft/s
m/s  196.85 ft/min
m/s  11,811 ft/h
N s/m2  0.6720 lbm/(ft2 s)
N s/m2  2419.1 lbm/(ft2 h)
N s/m2  1  103 cP
m2/s  10.7639 ft2/s
m2/s  38750.0 ft2/h
m3  35.3134 ft3
m3  61023.4 in.3
m3  264.17 gal (U.S.)
m3/s  35.3134 ft3/s
m3/s  1.2713  105 ft3/h
m3/s  2118.80 ft3/min
m3/s  15850.0 gal (U.S)/min

References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

Biot, Jean Baptiste (1774–1862)
Eckert, E.R.G.
Elenbass, W.
Fourier, Baron Jean Baptiste Joseph (1768–1830)
Graetz, Leo P. (1856–1941)
Grashof, Franz (1826–1893)
Colburn, Allan Philip (1904–1955)
Lewis, G.W.
Mach, Ernst (1838–1916)
Mouromtseff, I.E.
Nusselt, E. Wilhelm H. (1882–1957)
Peclet, Jean Claude Eugene (1793–1857)
Prandtl, Ludwig (1875–1953)
Rayleigh, Lord (1842–1919)
Reynolds, Osborne (1842–1912)
Sherwood, Thomas Kilgore (1903–1976)
Schmidt, Ernst (1892–1975)
Stanton, Sir Thomas Edward (1865–1931)

© 2001 by CRC PRESS LLC

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

ft/s  0.3048 m/s
ft/min  5.080  103 m/s
ft/h  8.4667  105 m/s
lbm/(ft2 s)  1.488 N s/m2
lbm/(ft2 h)  4.133  104 N s/m2
cP  0.001 N s/m2
ft2/s  0.0920 m2/s
ft2/h  25.81  106 m2/s
ft3  0.02832 m3
in.3  1.6387  105 m3
gal (U.S.)  3.785  103 m3
ft3/s  2.8317  102 m3/s
ft3/h  7.8658  106 m3/s
ft3/min  0.47196  103 m3/s
gal/min  63.091  106 m3/s

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Contents
Chapter 1
Introduction to Thermal Design of Electronic Equipment
1.1 Introduction to the Modes of Heat Transfer
in Electronic Equipment
1.1.1 Convection
1.1.2 Conduction
1.1.3 Radiation
1.1.4 Practical Thermal Resistances
1.2 Theoretical Power Dissipation in Electronic Components
1.2.1 Theoretical Power Dissipation
1.2.2 Heat Generation in Active Devices
1.2.2.1 CMOS Devices
1.2.2.2 Junction FET
1.2.2.3 Power MOSFET
1.2.3 Heat Generated in Passive Devices
1.2.3.1 Interconnects
1.2.3.2 Resistors
1.2.3.3 Capacitors
1.2.3.4 Inductors and Transformers
1.3 Thermal Engineering Software for Personal Computers
1.3.1 Commercial CFD Codes
1.3.2 Flotherm v2.2
References

Chapter 2
Formulas for Conductive Heat Transfer
2.1 Conduction in Electronic Equipment: Introduction
2.2 Thermal Conductivity
2.2.1 Thermal Resistances
2.2.2 Conductivity in Solids
2.2.3 Conductivity in Fluids
2.3 Conduction—Steady State
2.3.1 Conduction in Simple Geometries
2.3.1.1 Conduction through a Plane Wall
2.3.1.2 Conduction through Cylinders and Spheres
2.3.1.3 Plane Wall with Heat Generation
2.3.1.4 Cylinders and Spheres with Heat Generation
2.3.1.5 Critical Radius of a Cylinder

© 2001 by CRC PRESS LLC

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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
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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

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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
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6.3.1
6.3.2

Bubble Phenomenon
Pool Boiling
6.3.2.1 Pool Boiling Curve
6.3.2.2 Pool Boiling Correlations
6.3.2.3 Pool Boiling Critical Heat Flux Correlations
6.3.2.4 Pool Boiling Minimum Heat Flux Correlations
6.3.2.5 Pool Boiling Vapor Film Correlations
6.3.3 Flow Boiling
6.3.3.1 External Forced Convection Boiling
6.3.3.2 Internal Forced Convection Boiling
6.4 Evaporation
6.5 Condensation
6.6 Melting and Freezing
References

Chapter 7
Combined Modes of Heat Transfer for Electronic Equipment
7.1 Introduction
7.2 Conduction in Series and in Parallel
7.3 Conduction and Convection in Series
7.4 Radiation and Convection in Parallel
7.5 Overall Heat Transfer Coefficient
Appendix

© 2001 by CRC PRESS LLC

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

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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  ---------hc As

1.1.2

CONDUCTION

Conduction is the transfer of heat from an area of high energy (temperature) to an
area of lower relative energy. Conduction occurs by the energy of motion between
adjacent molecules and, to varying degrees, by the movement of free electrons and
the vibration of the atomic lattice structure. In the conductive mode of heat transfer
we have no appreciable displacement of the molecules. In many applications, we
use conduction to draw heat away from a device so that convection can cool the
conductive surface, such as in an air-cooled heat sink. For a one-dimensional system,
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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
q r   F 1,2 A ( T 1  T 2 )
4

4

where:
qr  amount of heat transferred by radiation (W)
  emissivity of the radiating surface (highly reflective  0, highly
absorptive  1.0)
  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
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Fl

θsa

θcs
Die bond
Lead

Chip

θjc

Heat spreader

Encapsulant

FIGURE 1.2 Primary thermal resistances in a chip/heat sink assembly. jc is resistance from the
die junction to the device case. cs is resistance from the device case to the heat sink. sa is
resistance from the heat sink to the ambient air. (Adapted from Kraus, A. D. and 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:
h r   F 1,2 ( T 1
2

1.1.4

2

T 2)(T 1

T 2)

PRACTICAL THERMAL RESISTANCES

The semiconductor junction temperature depends on the sum of the thermal resistances between the device junction and the ambient environment, which is the
ultimate heat sink. Figure 1.2 shows a simplified view of the primary thermal
resistances:
tot



jc

cs

sa

where:
tot
jc
cs
sa






total thermal resistance (K/W)
junction to case thermal resistance (K/W)
case to heat sink thermal resistance (K/W)
heat sink to ambient thermal resistance (K/W)

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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  Ta

jc

q  35 C

( 50 K/W ) ( 0.5 W )  60 C

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:

sa

1
 ---------hc As

The heat transfer coefficient, hc, introduced earlier, is a complex function and cannot
be easily generalized for use. However, many empirical equations result in a reasonable
degree of accuracy when generating values of hc. As this formula shows, sa is the
reciprocal of the product of the heat transfer coefficient and the sink surface area.
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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  ---  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:
 drain current (A)
ID
RDS(ON)  drain to source resistance ( )
Conduction losses when the device is in the linear range are found by VI, as are
leakage current losses, PL, and reverse current losses, Prd. Switching transition losses,
PS, occur during the transition from the on to off states. These losses can be calculated
as the product of the drain-to-source voltage and the drain current; therefore:
t

 S1
P S  f S  0 VDS ( t )I D ( t ) dt


t S2

0


VDS ( t )ID ( t ) dt


where:
fS
VDS
ID
tS1
tS2







switching frequency (Hz)
MOSFET 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 ( )
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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
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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:
PD ( t )  0.5  CV M sin 2  t
2

where:
C
VM

f






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

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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  ---  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:
PD ( t )  0.5LI M  sin 2  t
2

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

Ṗ D ( CORE )  6.51 f B MAX
where:
PD(CORE )  power dissipation (W/kg)
n, m  constants of the core material
f
 switching frequency (Hz)
BMAX  maximum flux density (Tesla)
The power dissipation is then found by:
P D  Ṗ D ( CORE ) M
where M  mass of the ferromagnetic core (kg).

1.3 THERMAL ENGINEERING SOFTWARE
FOR PERSONAL COMPUTERS
The past 10 years have seen a major change in the way we evaluate heat transfer.
Whereas mainframe computers were once used to calculate large thermal resistance
networks for conduction problems, we now perform FEA (finite element analysis)
on desktop personal computers. Ten years ago CFD (computational fluid dynamics)
was largely experimental and was almost exclusively used only in research laboratories; it is now also used to provide quick answers on desktop computers. The
convective coefficient of heat transfer, the most difficult value to assign in heat
transfer, is regularly being estimated within 10%, whereas 30% was formerly
the norm.
Once we construct and verify a computer model, we can evaluate hundreds of
changes in a short time to optimize the model. In the future, as the underlying CFD
code becomes more advanced, even the tedious model verification step may be
eliminated.
As with physical designs, computer models can be a combination of conduction,
convection, and radiation modes of heat transfer. Convection problems have the
largest variety of permutations, and this has given the CFD engineers the most
difficulty: laminar flow changes to turbulent flow, energy dissipation rates change
with velocity, at slow velocity natural convection may override the expected forced
convection effects, etc. When additional factors such as multiphase flow, compressibility, and fine model details such as semiconductor leads are added, it is easy to
see why convective computer modeling is so complex.

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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:


-----t


----- (  u )
x


----- (  v )  0
y

Conservation of momentum in x:


---- (  u )
t


----- (  uu )
x


 u
----- (  vu )  -----  ------
y
 x   x

 u  p
-----  ------  ----- y   y  y

Fx

 v  p
-----  -----  -----y  y y

Fy

Conservation of momentum in y:


---- (  v )
t


----- (  uv )
x


 v
----- (  vv )  -----  -----
y
x  x

Conservation of energy:


---- (  h )
t


----- (  uh )
x


 k h
----- (  vh )  -----  ----- ------
y
x  cp x

 k h
-----  ----- ------
y  cp y

q̇ G

where:

  fluid density (kg/m3)
t  time (s)
u, v  velocity components in x and y coordinates (m/s)
 molecular viscosity (N s/m2)
p  pressure (N/m2)
F  force per unit volume (Pa/m3)
h  specific enthalpy (J/kg)
k  thermal conductivity (W/m K)
cp  specific heat (J/kg K)
q̇ G  heat source per unit volume (W/m3)
These equations can take many forms and change in different coordinate systems
and under different flow conditions.
We enter the geometry of a model into a computer CFD program or, more
commonly, it is imported in a standard format from a CAD (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
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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
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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.

© 2001 by CRC PRESS LLC

0082-01 Page 17 Wednesday, August 23, 2000 9:51 AM

6. Hopkins, D. C., Designing Power Hybrid Supplies, Powertechniques Magazine, June,
31–34, 1989.
7. Hopkins, D. C., Jovanovic, M. M., Lee, F. C., and Stephenson, F. W., 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.

© 2001 by CRC PRESS LLC

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

T
1
 rad  --------  ----------qx
hr AF
where AF is the area of radiation based on a geometric factor of shape and
emissivity.

2.2.2

CONDUCTIVITY IN SOLIDS

Thermal conductivity in a solid material is based upon migration of free electrons
and vibrations within the atomic lattice structure. Silver, copper, and aluminum are
indicative of materials in this group. These materials have high thermal and electrical
conductivity. Figure 2.1 shows how the thermal conductivity of some metals changes
with temperature.
In nonmetals, the lattice structure vibrations dominate over the movement of
free electrons, and thermal conductivity may not be related to electrical conductivity.
In materials with highly structured crystalline lattice structures, thermal conductivity
can be quite high, while electrical conductivity is quite low. An outstanding example
of a material in this group is diamond. Diamond has a thermal conductivity 5 times
that of copper, and an electrical breakdown strength of more than 2000 V of direct
current per 0.01 mm of length.

2.2.3

CONDUCTIVITY IN FLUIDS

Fluids, both liquids and gases, have much greater spacing between molecules than
solids and therefore much lower thermal conductivities. The thermal conductivity
of a fluid varies with pressure and temperature. Within the pressure range of fluids
used in electronic cooling, thermal conductivity variances with pressure can be
ignored. Temperature, however, can greatly affect the thermal conductivity of liquids
or gases.
Within the range of temperatures used in electronic cooling, the thermal conductivity change of a gas is linear with temperature change but is different for each
gas. The thermal conductivity change with temperature in liquids is not yet well
understood. Figures 2.2 and 2.3 show the thermal conductivity change with temperature for selected gases and liquids, respectively.

2.3 CONDUCTION—STEADY STATE
2.3.1

CONDUCTION

IN

SIMPLE GEOMETRIES

In simple shapes such as a wall or cylinder, the heat flow is 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
dT
kA
q x   kA -------  ------ ( T 1  T 2 )
dx
L

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8

8

0082-02 Page 25 Wednesday, August 23, 2000 9:53 AM

L
qgen = qG ( A dx )

A
dx
x

Tmax

T1

8

T1

FIGURE 2.4 Conduction in a plane wall when the internal heat generation is uniform. In
this case the temperature distribution is T1  T2.

The heat flux, energy per unit area, is given as
q
k
q x  ----x  --- ( T 1  T 2 )
A
L
Rearranging the rate of heat transfer for temperature rise, we have the familiar onedimensional form:
qL
 T  -------k Ac
More complex problems of this type may encompass 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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,A

8

T

Ts ,A

TB
TC
Ts,D
L3

k1

k2

k3

1

2

3

T

Cold fluid
T ,D ,hD

x

T

,A

8

qx

L1
k1A
Ts ,A

L2
k 2A
TB

L3
k 3A
TC

1
h DA
Ts,D

8

1
h AA

,D

8

L2

T

,D

8

8

Hot fluid
T ,A ,h1

L1

FIGURE 2.5 Equivalent thermal circuit for heat conduction through a series composite wall.
The wall is composed of three sections, with section 2 having the lowest thermal conductivity.

heat transfer for these problems, called composite walls, is
T ,1  T ,N
q x  --------------------------- t
Therefore, we can describe a composite wall with three materials (A, B, and C) in
series and convective heat transfer along the face of material A and C as
T ,1  T ,4
q x  ----------------------------------------------------------------------------------------------1   L A   LB   LC   1 
 ----------- ---------  ---------  ---------  ----------- h c, 1 A  k A A  k B A  k C A  h c, 4 A

The overall heat transfer coefficient, U, is sometimes used, which we describe as
q
U  -----------AT
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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
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 2 dT
1 d ( rT )
----2 -----  r -------  --- --------------- 0
r dr 2
r dr  dr 
If Ti is the temperature at ri and To is the temperature at ro, then the temperature
distribution in the sphere (see Figure 2.8) is
ri
ro 
- 1  ---
T ( r )  Ti  ( To  Ti ) --------------
r
ro  ri

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-L
T s,A

x
+L

T

T (x)

¥

T ,B,hB

q
.

O

T (x) T s

Ts

T s,B
T ,A,hA

x
+L

-L

T ,h

T ,h

¥

¥

¥

(a)

(b)

T

q
.

0

T (x) T s

T ,h
¥

(c)

FIGURE 2.9 Heat conduction through a plane wall with uniform internal heat generation.
(a) Asymmetrical boundary conditions. Surface 2 has better cooling. (b) Symmetrical
boundary conditions. (c) Adiabatic surface at midplane. Only surface 2 benefits from
convection cooling.

The rate of heat transfer through the sphere is then
To  Ti
2 T
q   4 r -------  -----------------ro  ri
r
------------------4 kro ri

and the thermal resistance is found by
ro  ri
  -----------------4 kr o r i
2.3.1.3

Plane Wall with Heat Generation

In the plane wall studied previously we neglected heat generation, qG, within the
wall. If we now calculate for heat generation (see Figure 2.9) and constant thermal

© 2001 by CRC PRESS LLC

0082-02 Page 31 Wednesday, August 23, 2000 9:53 AM

conductivity, k, the equation becomes
2

d T ( x)
-   q̇ G
k ---------------2
dx
We find the temperature distribution, T(x), by
q̇ 2 T 2  T1
q̇ G L
-x  T1
x  -------T ( x )   -----G- x  ------------------2k
L
2k
If the two surface temperatures are equal, T1  T2, the temperature distribution
simplifies to a parabolic distribution about the centerline of the plane wall,
described as
2

2
q̇ G L x
- ---  x   T1
T ( x )  ---------2k L   --L-

Since the centerline, which is x  L/2, has the maximum temperature, we can find
the temperature rise by calculating
2

q̇ G L
 T  ---------8k
Table 2.1 shows the solutions to a variety of conductive plate and wall problems.
2.3.1.4

Cylinders and Spheres with Heat Generation

In this section we will examine heat transfer in a radial system such as a cylinder
or sphere with internal heat generation. Such cases occur in 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

q̇ G r o
r 2
T ( r )  ---------1   ----  T s
 r o
4k
The maximum temperature is at the centerline of the cylinder, r  0; therefore,
2

T max

© 2001 by CRC PRESS LLC

q̇ G r o
 T o  ---------rk

0082-02 Page 32 Wednesday, August 23, 2000 9:53 AM

TABLE 2.1
Conduction in Plates and Walls2
Description

Equations

Convectively heated
and cooled plate

Convectively heated and cooled plate
h1  T T 1  T 2
q  -----------------------------h1
Bi 1  1  --h
2

h1 x
T T  T
------1
k
---------------------1-  ----------------------------T T2  T1
h1 L
h1
---------

1
h
k
2

Composite plate

Composite plate
 T T0  T n
q  ---------------------------------n



for J˙

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

1

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

Plate with temperature-dependent
thermal conductivity

i 1

j 1

Li 1  x j
 ---  ----  ----  1- k
 k i h i k j ---h0
--------------------------------------------------------------------n
L
1
1   ---- ----i  --
h i h 0
i  1 ki



i1



Plate with 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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0082-02 Page 33 Wednesday, August 23, 2000 9:53 AM

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

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

Infinite thin plate with heated
circular hole

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


Infinite thin plate with heated
circular hole

Infinite thin plate with heated circular hole for q at
r  r1, and r r1
------
K 0  Br
k  T T  T

---------------------------  ------------------------------------Br
Br
q
2  -------1- K 1  -------1-





where:
B 

Bi 1  Bi 2

T1  HT2
T  -----------------------1H
Bi
H  -------1
Bi 2
Finite plate with centered hole

Finite plate with centered hole5
w
 ln  -----r
 
q  -----------------------------------------------------------------2
kT T1  T2

d
------2w

(Continued)

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0082-02 Page 34 Wednesday, August 23, 2000 9:53 AM

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

Equations
d
Tube centered in a finite plate4 for r  ----10
2 kT T1  T2
q  -------------------------------4d
ln ( ------r )  c
w/d
1.00
1.25
1.50
2.00
2.50
3.00
4.00

Infinite plate with internal
heat generation

c
0.1658
0.0793
0.0356
0.0075
0.0016
0.0003
1.4  105
0.0
Infinite plate with internal heat generation2
T  T1,
x0
T  T2,
xL

T T  T
Po X ( 1  X )
---------------------1-  X  ------------------------------2
T T  T
2
1
x
where X  --L

Infinite plate with convection
boundaries and internal heat
generation

Infinite plate with convection boundaries and internal
heat generation3
1
-  1
1  Po  ------T T  T2
Po Po
 Bi 2

----------------------  ----------------------------------------  -------  ------ ( 1  X 2 )
Bi 2
2
T T1  T2
1  Bi 2  H
1 ˙
Bi 1 1  Po  ------ 0.5 ( 1  X )
 Bi2

 ------------------------------------------------------------------------------1  Bi 1  H

h2
where H  ----h1

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0082-02 Page 35 Wednesday, August 23, 2000 9:53 AM

TABLE 2.1 (continued)
Conduction in Plates and Walls2
Notes:
Bi  Biot Number, hL/k

q

 rate of heat flux, W/m2

c  value for w/d

q

 linear heat flux, W/m

d  diameter, m

q̇ G

 volumetric heat flux, W/m3

h  heat transfer coefficient, W/m2 K

w

 width

k  thermal conductivity, W/m K

x, y, z  Cartesian coordinates

L  length, m
˙ 2
q̇ G L
Po  ----------k
q  rate of heat flow, W

X

 length ratio (x/L)
 coefficient of thermal expansion (°C1)



 thickness

CL
Ts

Tmax

rB

rA

dr

L

Heat generation in
differential element
is qG L2πrdr
FIGURE 2.10 Heat conduction nomenclature for a long circular cylinder with internal heat
generation in differential element dr.

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0082-02 Page 36 Wednesday, August 23, 2000 9:53 AM

If we evaluate the temperature distribution at the centerline of the cylinder, we find
the nondimensional temperature distribution
T (r )  T
r 2
------------------------s  1   -----
 r B
T max  T s
To find the surface temperature of a tube, Ts, having a flowing cold fluid at T ,
we evaluate with a simplified energy balance equation which yields
q̇ G r
T s  T  -------2h c
The effective heat transfer coefficient for the tube is then
2

2

q̇ G ( r o  r i )
h c  ------------------------------2r i ( T i  T )
Tables 2.2 and 2.3 show the solutions to a variety of conductive cylinder and
sphere problems.
2.3.1.5

Critical Radius of a Cylinder

In real problems involving heat dissipation of an insulated cylinder, we must usually
account for the effects of convection, whether natural or forced. When the outer radius
of the insulation is small, the surface area is also small, and the effect of convection
is not too great. As the outer radius of the insulation increases, the surface area also
increases. At a critical radius, the effect of convective cooling will outweigh the effect
of internal conduction resistance. The rate of heat transfer per unit length of a cylinder is
Ti  T
Ti  T
Ti  T
q̇  -------------------  ----------------------------  --------------------------------------------------r
 tot
 cond   conv
 ln ---r- 
1


 --------------   ----------------------------
o
i

 2 kL

2 kLr o h c, o

where:
Ti
T∞
ro
ri
k
L
hc,o









temperature of cylinder, °C
temperature of ambient air, °C
outer radius of insulation, m
inner radius of insulation, m
thermal conductivity of insulation, W/m K
length of cylinder, m
external convective heat transfer coefficient, W/m2 K

We can see from this equation that we achieve a maximum heat transfer rate when
the total thermal resistance, tot, is at a minimum. If the outer radius of the insulation
equals a critical value:
k
r o  r crit  -------h c, o
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0082-02 Page 37 Wednesday, August 23, 2000 9:53 AM

TABLE 2.2
Rods, Tubes, Cylinders, Disks, Pipes, and Wires2
Description

Equations

Infinite hollow cylinder

Infinite hollow cylinder
2 kT To  Ti
q  ---------------------------------------ro
1
1
-------ln ( ---- )  -----Bi  Bi
r
i

i

o

ro
ln ( ---- )
r

1
-------Bi o

T T  T

--------------------o-  ---------------------------------------T Ti  To
ro
1
1
-------ln ( ---- )  -----Bi  Bi
r

i

i

i

o

where
h o ro
Bio  --------k
Composite cylinder

Composite cylinder

2 T Tn  T1
q  ----------------------------------------------------------------------------n1



i1

for j

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

n



i1

1
-------r i hi

1
j1

T T j  T1
--------------------- 
T Tn  T1



1
1
r
1  r i  1
1
---- ln  ---------------  ---------  ---- ln  ----  --------k j  r j r hk i  r i  r i hi
j j
-------------------------------------------------------------------------------------------------------------------------i1

n1



i1

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

n



1
--------i hi

i  1r

where Tj is the temperature in the jth layer
Insulated tube

Insulated tube
2 kT Ti  T f
q  ------------------------------r
1
o
ln  -----  ------- r i  Bi o

where
k  ktube and

hr o
Bio  ------k

k
Maximum heat loss occurs when ro  --hInfinite cylinder with temperaturedependent thermal conductivity

Infinite cylinder with 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)

© 2001 by CRC PRESS LLC

0082-02 Page 38 Wednesday, August 23, 2000 9:53 AM

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
sinh
--------  ln   ------------
Bi 1
 Dr 1


D
2  D  -------- 

Bi 2 

where:
h2 d
h1 r 1
d
- , D  --- , Bi2  ------Bi1  --------s
k
k

Row of rods in wall3
For each rod

Row of rods in wall

T2

4 kT T  T
2
1
q  -----------------------------------------------------------------------------------

h2

 d 
1
- sinh
--------  ln   ------------
Bi 1
 Dr 1

r1

r1
T1 +h1

where:

r1
T1+h1

T1+h1
+

+

2d

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

+

k
T2

h2

Circular disk on the surface of a
semi-infinite solid

ro
+

T2

z

© 2001 by CRC PRESS LLC

D 
- 
2  D  ------Bi 2 


k

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

d
D  --s

Circular disk on the surface of a
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 1 
2
---- sin  ------------------------------------------------------------------------------------------------0.5
0.5 
 ( R  1 )2  Z 2  ( R  1 )2  Z 2 


z
r
----where Z  r
and R  r
o
o

0082-02 Page 39 Wednesday, August 23, 2000 9:53 AM

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

Equations

Circular disk in an infinite solid

Circular disk in an infinite solid3
q  8r o k  T T 2  T 1

z
ro
+

T2

k

Infinite hollow square rod3

Infinite hollow square rod

2 kT T2  T1
q  ---------------------------------------------------------------k
k
  ----------------------  ln  1.08w
-------------h1 r o
 2r  2h 2 w
o

T2

ro
w

h2

+

T1 h1

w

Infinite hollow square pipe

T2
w
T1

Infinite hollow square pipe5

2 kT T1  T2
 -------------------------------- 0.785ln  w
----

 d
q  

2 kT T1  T2
 ------------------------------------------------
w
 0.93ln  ----  0.0502
d


w
----  1.4
d
w
---d

1.4

d

Vertical cylinder in a semi-infinite solid

Vertical cylinder in a semi-infinite solid3


To



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

h



Bi



Bi d



ro
+

T1

where
d

hd
Bid  ------ ,
k

d
D  ---ro

k

(Continued)
© 2001 by CRC PRESS LLC

0082-02 Page 40 Wednesday, August 23, 2000 9:53 AM

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
0.5
0.5
(k1  k2)


2
2
2
2
 (R  1)  Z
 (R  1)  Z


0

4r o k 1 k 2
- T T  T
q   ----------------o
1
 k1  k2
z
r
where Z  ---- , R  ---ro
ro
Heat flow between two rods in an insulated infinite plate5

Heat flow between two rods in an
insulated infinite plate

2 kT T1  T2
q  -------------------------------w
s
-------  ln  ------
r
w

w
+
T1

+
T2
s

2r

© 2001 by CRC PRESS LLC

0082-02 Page 41 Wednesday, August 23, 2000 9:53 AM

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

Equations

Infinite cylinder with convection boundary
and internal heat generation

To h

Infinite cylinder with convection boundary and internal
heat generation3
T T  Tok
2
----------------------  0.25  -----  1  R 2
2
 Bi

q̇ G r o
hr
where Bi  --------o ,
k

ro

r
R  ---ro

k

+

.

qG

Hollow infinite cylinder with convection
boundary on outside surface and internal
heat generation

Hollow infinite cylinder with convection boundary on
outside surface and internal heat generation3
with qr  0 and r  ri

T T  T f k
2
----------------------  0.25 ----- ( 1  R 2i )  1  R 2  2R 2i lnR
2
Bi
q̇ G r o
ri
Tf
k

h

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

+
•

qG
ro

Hollow infinite cylinder with convectioncooled inside surface and internal
heat generation

Hollow infinite cylinder with 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
q̇ G r i
ri

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

+h
. Tf
qG
ro

(Continued)

© 2001 by CRC PRESS LLC

0082-02 Page 42 Wednesday, August 23, 2000 9:53 AM

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

Equations

Electrically heated wire with temperaturedependent thermal and electrical
conductivities

To
ro

+

Electrically heated wire with 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

q̇ G  volumetric heat flux, W/m3

k  thermal conductivity, W/m K

r

 radius, m

L  length, m

s

 spacing, m

q  rate of heat flow, W

w

 width, m
 coefficient of thermal expansion (°C1)

A more accurate equation accounts for the variable effect of ro on the heat
transfer coefficient hc,o:
1
--------------

r o  r crit

n1

  ---------------------
 ( 1  n )k

where:

  thermal diffusivity of the convective media, k /cp
n  0.5 for laminar forced convection or 0.25 for natural convection
k  thermal conductivity of insulation, W/m K

2.3.2

CONDUCTION

IN

COMPLEX GEOMETRIES

In the previous section we studied 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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0082-02 Page 43 Wednesday, August 23, 2000 9:53 AM

TABLE 2.3
Conduction in Spheres2
Description

Equations

Spherical shell

Spherical shell

ro

4 rokT Ti  To
q  -----------------------------------------------

ri

ro k
ro
k
-----  1   ----- ---------  ----------r i hi r i ho r o
ri
ro

+
T i hi

k

-----  1  ----------T T  T
r o ho
r
--------------------o-  ----------------------------------------------T Ti  To
ro k
ro
k
 
-----  1   ----- ---------  ----------r i r i hi r o ho
ri

ho

To

Composite sphere

Composite sphere

r3

rn

rj

r2

4 T T1  Tn
q  --------------------------------------------------------n
n1
1 
1
 ---1-  ---1-  -----------  ---------r


hj+1

r
i  1 ki i

kn-1

r1

i1

2
i  1 r i hi

j1

+
T1 h1



Tn

hn

1 1
----  ----

1
------------

1
----------

1
1  ------

1
---

1
---------

k  r  r i  1  r 2 h  k j  r j  r   r 2 h i
T T j  T1
i i
i
i1 i i
---------------------  -------------------------------------------------------------------------------------------n
n1
T Tn  T1
1 
1
 ---1-  ---1-  r-----------  ---------

k r
i1 i i

i1

2
i  1 r i hi

where Tj is the local temperature in the jth layer.
Sphere with 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)

© 2001 by CRC PRESS LLC

0082-02 Page 44 Wednesday, August 23, 2000 9:53 AM

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

where

ro

d

Sphere in a semi-infinite solid4

+
To

Sphere in an infinite medium
ro

+

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

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

Ti

Two spheres separated by a large
difference in an infinite medium
T1

T2

+

Two spheres separated by a large difference in an infinite
medium4
4 rk  T T 1  T 2
q  -----------------------------------

2r

+

r
2  1  -s



s

for s

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

ri

+
qi

2r, error  1%

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

.

qG

r
where R  --- ,
ri

Solid sphere with internal heat generation
in an infinite medium
k1

ko

5r

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

To

ro

+

s

.

qG
h

© 2001 by CRC PRESS LLC

ro
Ro  ---ri

Solid sphere with internal heat generation in an infinite
medium6
k  ko
0  r  ro
with
k  k1
r ro
q̇ G  q̇ Go
0  r  ro
q̇ G  0
r ro
h  contact coefficient at r  ro
T  T
r→

 T T  T ko
2k o
2
2
-------------------------  1
---  1  R  -----  --------
2
k1 
Bi
6
q̇ Go r o

0R1

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TABLE 2.3 (continued)
Conduction in Spheres2
Description

Equations

T T  T k
1
-----------------------  ------2
3R
q̇ Go r o
where

hr o
Bi  -------- ,
ko

r

ro

r
R  ---ro

Notes:
A  area, m

q

Bi  Biot number, hL/k

q  rate of heat flux, W/m2

d  diameter, m

q̇ G  volumetric heat flux, W/m3

h  heat transfer coefficient, W/m2 K

r

 radius, m

k  thermal conductivity, W/m K

s

 spacing, m

L  length, m

 rate of heat flow, W

 coefficient of thermal expansion (°C1)

y
qy"

Lines of
constant T1
temperature
(isotherms)

T2

<

q" = iq"x + jqy"

T1

qx"

Heat flow
lines
x

Isotherm

FIGURE 2.11 Two-dimensional conduction showing the lines of heat flow. There is no
internal heat generation.

Heat conduction in multidimensional systems can be calculated by analytic,
analogic, graphical, and numerical methods. Fourier, in 1822, made the first major
contribution to the analysis of multidimensional heat transfer.8 Fourier’s method of
separating variables led to the requirement of transform methods such as the Laplace
transform, to express the arbitrary Fourier series expansion. These methods can yield
accurate results but are quite time-consuming.
Graphical methods (see Figure 2.12) include the flux plot method and the
Schmidt9 method. Both methods involve drawing skill and considerable time but

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0082-02 Page 46 Wednesday, August 23, 2000 9:53 AM

T2
F

∆q1, ∆q2

Heat flow lines

E

B

A
T1
T1 > T2

C

∆q1, ∆q2

D

∆q15
Isotherms

∆q15

(a)

(b)

∆ι

∆ι
q"
∆T
(c)
FIGURE 2.12 Construction of a network of curvilinear squares for an adiabatic corner section
with no internal heat generation. (a) Scale model. (b) Heact flux plot. (c) Typical curvilinear
square.

can yield accurate results. These techniques are generally used only for very simple
geometries with simple boundary conditions. Nevertheless, these solutions can be used
to provide exact and relatively quick answers to complex geometries that we can
simplify. These equations can also be used to find partial solutions to simple areas
of very complex geometries.
Analytic techniques provide a solution at every point in time and space within
the prescribed boundaries of the problem. Finite difference methods provide a
solution only at a finite number of points (see Figure 2.13) within the problem and
are an approximation of the analytic solution. Using a finite number of points
simplifies the calculation to repetitive arithmetic instead of the complex calculations
involved with the analytic solutions. Although many texts devote considerable space
to numerical and finite difference methods, computers are now used to solve problems of this complexity. For this reason, we will concentrate only on a broad view
of the analytic techniques.

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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 

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

4 r1

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

4 r1

r1

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

h  distance below surface
if h

3r1

1

2 L
------------------
ln  2h
r 
1

h
if --- → , s → 0
ri
Buried rectangular box

L
h,a,b
h
h  distance below surface 2.756L ln  1  ---

a
a  box width
b  box height

Edge of adjoining walls

L  wall thickness
W  length of attachment
W L/5

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 ------
r 

 0.59

0.54W

 h---
 b

 0.078

0082-02 Page 53 Wednesday, August 23, 2000 9:53 AM

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

Restrictions

Corner of three adjoining walls

W

Disk on semi-infinite medium

r  radius of disk

Vertical cylinder in semi-infinite medium

L

Conduction between two parallel cylinders

L/5

0.15L

2.3.2.4

4r

D

L

2 L
-------------------
ln  4L
D

D1, D2

z  axial centerline spacing
D1  diameter of cylinder 1
D2  diameter of cylinder 2
Buried thin horizontal disk

Conduction Shape Factor

D  diameter of disk
z  distance below surface

2 L
------------------------------------------------2
2
2
cosh

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

 1  4z

4.45D
-----------------D
1  ------------5.67z

Finite Difference Method

Finite difference equations are constructed of nodal networks. These networks are
composed of discrete points placed on the surface of an object or, in the case of a
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 w