Thermal Design of Electronic Equipment

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Appendix

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353

Thermal D

esign of Electronic Equipment

TABLE A1Air at Sea-Level Atmospheric Pressure

Temp.

TDensity

Coef. Exp.

10

3

Specific Heat

c

p

Thermal Cond.

k

Absolute Viscosity

10

6

Kinematic Viscosity

v

10

6

Prandtl NumberPr

F C kg/m

3

1/K J/kg K W/m K N s/m

2

m

2

/s

32 0 1.293 3.664 1003.9 0.02417 17.17 13.28 0.713141 5 1.269 3.598 1004.3 0.02445 17.35 13.67 0.712750 10 1.242 3.533 1004.6 0.02480 17.58 14.16 0.712259 15 1.222 3.470 1004.9 0.02512 17.79 14.56 0.711868 20 1.202 3.412 1005.2 0.02544 18.00 14.98 0.711377 25 1.183 3.354 1005.4 0.02577 18.22 15.40 0.710886 30 1.164 3.298 1005.7 0.02614 18.46 15.86 0.710395 35 1.147 3.244 1006.0 0.02650 18.70 16.30 0.7098

104 40 1.129 3.193 1006.3 0.02684 18.92 16.76 0.7093113 45 1.111 3.142 1006.6 0.02726 19.19 17.27 0.7087122 50 1.093 3.094 1006.9 0.02761 19.42 17.77 0.7082131 55 1.079 3.048 1007.3 0.02801 19.68 18.24 0.7077140 60 1.061 3.003 1007.7 0.02837 19.91 18.77 0.7072149 65 1.047 2.957 1008.0 0.02876 20.16 19.26 0.7067158 70 1.030 2.914 1008.4 0.02912 20.39 19.80 0.7062167 75 1.013 2.875 1008.8 0.02945 20.60 20.34 0.7057176 80 1.001 2.834 1009.3 0.02979 20.82 20.80 0.7053185 85 0.986 2.795 1009.8 0.03012 21.02 21.32 0.7048194 90 0.972 2.755 1010.3 0.03045 21.23 21.84 0.7044203 95 0.959 2.718 1010.7 0.03073 21.41 22.33 0.7041212 100 0.947 2.683 1011.2 0.03101 21.58 22.79 0.7038

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TABLE A2Water at Sea-Level Atmospheric Pressure

Temp.

TDensity

Coef. Exp.

10

3

Specific Heat

c

p

Thermal Cond.

k

Absolute Viscosity

10

6

Kinematic Viscosity

v

10

6

Prandtl NumberPr

F C kg/m

3


2

m

2

/s

32 0 999.9

0.068 4217.5 0.5580 1794 1.794 13.5641 5 1000 0.018 4202.7 0.5677 1530 1.530 11.3350 10 999.7 0.095 4192.4 0.5774 1296 1.296 9.41059 15 999.1 0.16 4185.8 0.5870 1136 1.137 8.10168 20 998.2 0.22 4181.7 0.5967 993 0.995 6.95977 25 997.1 0.26 4179.5 0.6064 880.6 0.883 6.06986 30 995.7 0.31 4178.6 0.6155 792.4 0.796 5.38095 35 994.1 0.35 4178.5 0.6243 719.8 0.724 4.818

104 40 992.2 0.39 4179.0 0.6325 658.0 0.663 4.348113 45 990.2 0.42 4179.9 0.6401 605.1 0.611 3.951122 50 988.1 0.45 4181.1 0.6472 555.1 0.562 3.586131 55 985.8 0.48 4182.6 0.6536 512.6 0.520 3.280140 60 983.5 0.51 4184.5 0.6594 470.0 0.478 2.983149 65 980.8 0.54 4186.8 0.6643 436.0 0.445 2.748158 70 978 0.57 4189.5 0.6686 402.0 0.411 2.519167 75 974.9 0.60 4192.9 0.6724 376.6 0.386 2.348176 80 971.7 0.63 4196.6 0.6753 350.0 0.361 2.175185 85 968.5 0.66 4201.0 0.6778 330.5 0.341 2.048194 90 965 0.69 4205.7 0.6797 311.0 0.322 1.924203 95 961.7 0.72 4210.6 0.6811 294.3 0.306 1.819212 100 958.4 0.75 4215.5 0.6822 277.5 0.290 1.715

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TABLE A3Perflurocarbon FC-72 at Atmospheric Pressure (Boils at 56C)

Temp.

TDensity

Coef. Exp.

10

3

Specific Heat

c

p

Thermal Cond.

k

Absolute Viscosity

10

6

Kinematic Viscosity

v

10

6

Prandtl NumberPr

F C kg/m

3


2

m

2

/s

32 0 1740 1.601 1005.0 0.0600 1009.5 0.5802 16.9141 5 1727 1.611 1016.2 0.0595 932.4 0.5399 15.9350 10 1714 1.619 1025.6 0.0590 861.6 0.5027 14.9859 15 1701 1.626 1033.2 0.0585 799.5 0.4700 14.1268 20 1688 1.633 1039.8 0.0580 743.0 0.4402 13.3277 25 1675 1.640 1046.6 0.0575 693.8 0.4142 12.6386 30 1662 1.647 1053.5 0.0570 648.2 0.3900 11.9895 35 1649 1.654 1060.8 0.0565 610.1 0.3700 11.46

104 40 1636 1.662 1068.7 0.0560 574.3 0.3510 10.96113 45 1623 1.670 1077.5 0.0555 543.9 0.3351 10.56122 50 1610 1.680 1087.0 0.0550 514.8 0.3198 10.17131 55 1597 1.689 1096.5 0.0545 486.0 0.3043 9.778

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TABLE A4Perflurocarbon FC-77 at Atmospheric Pressure (Boils at 97C)

Temp.

TDensity

Coef. Exp.

10

3

Specific Heat

c

p

Thermal Cond.

k

Absolute Viscosity

10

6

Kinematic Viscosity

v

10

6

Prandtl NumberPr

F C kg/m

3


2

m

2

/s

32 0 1838 1.399 1005 0.0649 2356 1.282 36.4841 5 1826 1.407 1016 0.0646 2117 1.159 33.3050 10 1814 1.414 1025 0.0643 1905 1.052 30.3759 15 1802 1.421 1033 0.0640 1719 0.9539 27.7568 20 1789 1.429 1041 0.0637 1554 0.8686 25.4077 25 1777 1.436 1048 0.0634 1413 0.7592 23.3686 30 1765 1.443 1056 0.0631 1288 0.7298 21.5695 35 1753 1.451 1063 0.0628 1178 0.6720 19.94

104 40 1740 1.458 1071 0.0625 1083 0.6224 18.56113 45 1728 1.466 1079 0.0621 1001 0.5793 17.39122 50 1716 1.473 1087 0.0617 927.0 0.5402 16.33131 55 1704 1.481 1096 0.0613 862.4 0.5061 15.42140 60 1691 1.489 1105 0.0609 805.0 0.4761 14.61149 65 1679 1.497 1114 0.0604 753.2 0.4486 13.89158 70 1667 1.504 1123 0.0600 706.1 0.4236 13.22167 75 1655 1.512 1131 0.0595 662.3 0.4002 12.59176 80 1642 1.520 1140 0.0590 622.1 0.3789 12.02185 85 1630 1.527 1147 0.0585 584.0 0.3583 11.45194 90 1618 1.534 1154 0.0580 548.0 0.3387 10.90203 95 1605 1.541 1159 0.0575 513.2 0.3198 10.34

Source:

Data from Fluorinert

TM

Liquids Product Manual, 3M, St. Paul, MN. With permission.

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TABLE A5Thermophysical Properties of Nonferrous Metals at 20C

Density

Coef. Exp.

10

6

Specific Heat

c

p

Thermal Cond.

k

Materials kg/m

3

1/K J/kg K W/m K

Aluminum (1100) 2713 23.6 921 222Aluminum (2014) 2796 23.0 921 192Aluminum (2024) 2768 23.2 921 189Aluminum (5052) 2685 23.8 921 139Aluminum (6061) 2713 23.4 963 180Aluminum (7075) 2796 23.6 963 121Aluminum (356) 2685 21.4 935 159Beryllium 1855 11.5 1884 151Brass (C36000) 8498 20.5 380 116Bronze (C22000) 8802 18.4 377 189Copper (C11000) 8913 17.6 383 391Copper (C12200) 8941 17.6 385 339Copper (C22000) 8802 18.4 377 189Copper (alloy MF 202) 8862 17.0 382 150Glass seal (alloy Ni 50) 8332 8.46 482 10.4Gold 19,321 14.2 129 313Inconel (625) 8442 12.8 410 9.82Kovar 8343 4.30 439 16.0Lead 11,349 29.3 130 33.9



TABLE A6Thermophysical Properties of Ferrous Metals at 20C

Density

Coef. Exp.

10

6

Specific Heat

c

p

Thermal Cond.

k

Materials kg/m

3

1/K J/kg K W/m K

Carbon steel (AISI 1010) 7830 6.60 434 64.0Carbon steel (AISI 1042) 7840 6.50 460 50.0Cast iron (ASTM A-48) 7197 10.8 544 50.2Cast iron (ASTM A-220) 7363 13.5 544 51.1Cast steels (carbon and alloy) 7834 14.7 440 46.7Stainless steel (4130) 7833 13.5 456 43.3Stainless steel (17-4 PH) 7778 10.8 461 18.0Stainless steel (304) 8027 17.3 477 16.3Stainless steel (316) 2685 16.0 468 16.3Stainless steel (440) 7750 10.1 461 24.2

TABLE A7Thermophysical Properties of Plastics at 20C

Density

Coef. Exp.

10

6

Specific Heat

c

p

Thermal Cond.

k

Materials kg/m

3

1/K J/kg K W/m K

ABS (acrylonitrile butadiene styrene)

1058 72.0 1466 2.70

Acetal 1415 82.8 1465 3.01Acrylic 1178 81.0 1466 2.49Alkyd 2206 36.0 1256 9.87Cellulose acetate 1257 121 1508 3.01Epoxy (cast) 1148 59.4 1884 4.15Epoxy (IC molding) 1820 17.0 984 4.00Fluorocarbon (PTFE) 2196 90.9 1047 2.91Polyamide (nylon type 6) 1247 89.1 1675 2.08Phenolic 1387 37.4 1570 1.74Polycarbonate 1203 67.5 1256 2.39Polybutylene Terephthalate (PBT)

1307 72.0 1905 1.90

Polyester 1287 85.5 1780 2.29Polyimide 1427 47.7 1214 8.05Polyamide-imide 1397 36.0 2.94Polyetherimide 1277 54.0 1090 2.2Polyesteretherketone 1317 40.5 2.95Polyetherketone 103 Polystyrene 1039 72.9 1361 1.54Polyethylene 933 225 2261 3.95Polypropylene 903 86.4 1884 2.22Polyvinyl chloride (PVC) 1447 54.0 1050 1.77



TABLE A8Thermophysical Properties of Ceramics at 20C

Density

Coef. Exp.

10

6

Specific Heat

c

p

Thermal Cond.

k

Materials kg/m

3

1/K J/kg K W/m K

Aluminum oxide 3982 5.67 879 30.0Aluminum nitride 3200 4.40 711 200Beryllium oxide 2900 7.00 1030 300Boron nitride (cubic) 2200 3.80 709 1300Diamond (film) 3500 2.00 510 1200Fused quartz 2200 0.50 745 1.60Glass (die attach) 2900 50.0Silicon 2300 4.20 664 83.7Silicon nitride 3300 2.00 624 21.0

TABLE A9Properties of Common Gases

Gas

p

c

10

6

T

c

(K)

10

6

T

M

W

c

p

/

c

v

atm K N s/m

2

K

Acetylene C

2

H

2

26.038 1.260 6.1404 308.3 4.033 231.8Air (a) 28.966 1.400 3.6883 132.0 19.3 3.711 78.60Ammonia NH

3

17.031 1.310 11.2777 405.6 32.7 2.900 558.3Argon Ar 39.948 1.660 4.8738 150.8 26.4 3.542 93.30Butane C

4

H

10

58.124 1.090 3.7998 425.2 25.0 4.687 531.4Carbon dioxide

CO

2

44.010 1.285 7.3766 304.2 34.3 3.941 195.2

Carbon monoxide

CO 28.010 1.399 3.4958 132.9 19.0 3.690 91.70

Chlorine Cl

2

70.096 1.355 7.7009 417.0 42.0 4.217 316.0Ethane C

2

H

6

30.070 1.183 4.8840 305.4 21.7 4.443 215.7Ethylene C

2

H

4

28.054 1.208 5.0360 282.4 21.7 4.163 224.7Fluorine F

2

37.997 5.2183 144.3 27.5 3.357 112.6Freon-12 CC1

2

F

2

120.914 1.139 4.1240 385.0 Helium He 4.003 1.667 0.2270 5.190 2.54 2.551 10.22Hydrogen H

2

2.016 1.404 1.2970 33.20 3.47 2.827 59.70Methane CH

4

16.043 1.320 4.6003 190.6 15.9 3.758 148.6Methanol CH

4

O 32.042 1.203 8.0960 512.6 39.3 3.626 481.8Neon Ne 20.183 1.667 2.7561 44.40 16.3 2.820 32.80Nitrogen N

2

28.013 1.400 3.3945 126.2 18.0 3.798 71.40Nitrous oxide

N2O 44.013 1.303 7.2449 309.6 33.2 3.828 232.4

Octane C8H18 114.232 1.044 4.2845 563.4 24.1 Oxygen O2 32.00 1.395 5.0461 154.6 25.0 3.467 106.7Pentane C5H12 72.151 1.086 3.3742 469.6 25.0 5.784 34.10Propane C3H8 44.097 1.124 4.2456 369.8 23.3 5.118 237.1



TABLE A9 (continued)Properties of Common Gases

Gas pc 106 Tc (K) 106 T

MW cp/cv atm K N s/m2 K

Steam H2O 18.015 1.329 22.049 647.3 54.1 2.641 809.1Toluene C7H8 92.140 4.1139 591.7 127.0 Xenon Xe 131.30 1.660 5.8364 289.7 53.7 4.047 231.0

Note:

Pc critical pressure (Pa)

MW molecular weight

Tc critical temperature (K)

T effective temperature characteristic force potential (K)

ratio of specfic heat at constant pressure to specific heat at constant volume at 15C to 25C

molecular collision diameter ()

c viscosity at critical pressure and critical temperature (N s/m2)

a Properties of air based on:78.084% N220.946% O20.934% Ar0.033% CO21.0% other, by volume.

TABLE A10U.S. Standard Atmosphere

Altitude H (m)

TemperaureT (K)

Density (kg/m3)

Pressure P (N/m2)

Viscosity 106 (N/m2)

100 288.80 1.237 102,534 17.920 288.15 1.225 101,327 17.89

100 287.50 1.213 100,131 17.86200 286.85 1.202 98,948 17.83400 285.55 1.179 96,613 17.76600 284.25 1.156 94,324 17.70800 282.95 1.134 92,079 17.64

1000 281.65 1.112 89,877 17.571500 278.40 1.058 84,558 17.412000 275.15 1.007 79,497 17.252500 271.90 0.9567 74,684 17.093000 268.65 0.9091 70,110 16.933500 265.40 0.8632 65,766 16.77



TABLE A10 (continued)U.S. Standard Atmosphere

Altitude H (m)

TemperaureT (K)

Density (kg/m3)

Pressure P (N/m2)

Viscosity 106 (N/m2)

4000 262.15 0.8191 61,642 16.614500 258.90 0.7768 57,730 16.445000 255.65 0.7361 54,022 16.286000 249.15 0.6597 47,183 15.957000 242.65 0.5895 41,062 15.608000 236.15 0.5252 35,601 15.269000 229.65 0.4664 30,380 14.9210,000 223.15 0.4127 26,437 14.5711,000 216.65 0.3639 22,633 14.2112,000 216.65 0.3108 19,331 14.2113,000 216.65 0.2655 16,511 14.2114,000 216.65 0.2268 14,102 14.2115,000 216.65 0.1937 12,045 14.2116,000 216.65 0.1654 10,288 14.2117,000 216.65 0.1413 8,787.1 14.2118,000 216.65 0.1207 7,505.2 14.2119,000 216.65 0.1031 6,410.4 14.2120,000 216.65 0.08803 5,475.2 14.2125,000 221.65 0.03946 2,511.2 14.4830,000 226.65 0.01801 1,172.0 14.7635,000 237.05 0.00821 558.97 15.3140,000 251.05 0.00385 277.55 16.0445,000 265.05 0.00188 143.15 16.7550,000 270.65 0.00097 75.954 17.0360,000 245.45 0.00028 20.317 15.7570,000 217.45 0.00007 4.6348 14.26



Thermal Design of Electronic EquipmentContentsAPPENDIXTABLE A1TABLE A2TABLE A3TABLE A4TABLE A5TABLE A6TABLE A7TABLE A8TABLE A9TABLE A10

Thermal_Design_of_Electronic_Equipment/Thermal_Design_of_Electronic_Equipment/Ch01.pdfHibbeler R. C. Force-System Resultants and Equilibrium Thermal Design of Electronic Equipment. Ed. Ralph RemsburgBoca Raton: CRC Press LLC, 2001


1

Introduction to Thermal Design of Electronic Equipment

1.1 INTRODUCTION TO THE MODES OF HEAT TRANSFER IN ELECTRONIC EQUIPMENT

Electronic devices produce heat as a by-product of normal operation. When electricalcurrent flows through a semiconductor or a passive device, a portion of the power isdissipated as heat energy. Besides the damage that excess heat can cause, it alsoincreases the movement of free electrons in a semiconductor, which can cause anincrease in signal noise. The primary focus of this book is to examine various waysto reduce the temperature of a semiconductor, or group of semiconductors. If we donot allow the heat to dissipate, the device junction temperature will exceed themaximum safe operating temperature specified by the manufacturer. When a deviceexceeds the specified temperature, semiconductor performance, life, and reliabilityare tremendously reduced, as shown in Figure 1.1. The basic objective, then, is tohold the junction temperature below the maximum temperature specified by thesemiconductor manufacturer.

Nature transfers heat in three ways, convection, conduction, and radiation. Wewill explore these in greater detail in subsequent chapters, but a simple definitionof each is appropriate at this stage.

1.1.1 C

ONVECTION

Convection is a combination of the bulk transportation and mixing of macroscopicparts of hot and cold fluid elements, heat conduction within the coolant media, andenergy storage. Convection can be due to the expansion of the coolant media incontact with the device. This is called free convection, or natural convection. Con-vection can also be due to other forces, such as a fan or pump forcing the coolantmedia into motion. The basic relationship of convection from a hot object to a fluidcoolant presumes a linear dependence on the temperature rise along the surface ofthe solid, known as Newtonian cooling. Therefore:

qc hc As T s T m( )

0082-01 Page 1 Wednesday, August 23, 2000 9:51 AM


where:

q

c

convective heat flow rate from the surface (W)

A

s

surface area for heat transfer (m

2

)

T

s

surface temperature (C)

T

m

coolant media temperature (C)

h

c

coefficient of convective heat transfer (W/m

2

)

This equation is often rearranged to solve for

T

, by which:

1.1.2 C

ONDUCTION

Conduction is the transfer of heat from an area of high energy (temperature) to anarea of lower relative energy. Conduction occurs by the energy of motion betweenadjacent molecules and, to varying degrees, by the movement of free electrons andthe vibration of the atomic lattice structure. In the conductive mode of heat transferwe have no appreciable displacement of the molecules. In many applications, weuse conduction to draw heat away from a device so that convection can cool theconductive surface, such as in an air-cooled heat sink. For a one-dimensional system,

FIGURE 1.1

Component failure rates with temperature for Programmable Array Logic(PAL), 256K Dynamic Random Access Memory (DRAM), and Microprocessors. Data fromMIL-HDBK-217.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

20 30 40 50 60 70 80 90 100 110 120

Junction Temperature,oC

Failure

Rate

per106h

PAL

Microprocessor

DRAM

Tqc

hc As----------



the following relation governs conductive heat transfer:

where:

q

heat flow rate (W)

k

thermal conductivity of the material (W/m K)

A

c

cross-sectional area for heat transfer (m

2

)

T

temperature differential (C)

L

length of heat transfer (m)

Since heat transfer by conduction is directly proportional to a materials thermalconductivity, temperature gradient, and cross-sectional area, we can find the tem-perature rise in an application by:

1.1.3 R

ADIATION

Radiation is the only mode of heat transfer that can occur through a vacuum and isdependent on the temperature of the radiating surface. Although researchers do not yetunderstand all of the physical mechanisms of radiative heat transfer, it appears to bethe result of electromagnetic waves and photonic motion. The quantity of heat trans-ferred by radiation between two bodies having temperatures of

T

1

and

T

2

is found by

where:

q

r

amount of heat transferred by radiation (W)

emissivity of the radiating surface (highly reflective

0, highly absorptive

1.0)

Stefan-Boltzmann constant (5.67

10

8

W/m

2

K

4

)

F

1,2

shape factor between surface area of body 1 and body 2 (

1.0)

A

surface area of radiation (m

2

)

T

1

surface temperature of body 1 (K)

T

2

surface temperature of body 2 (K)

Unless the temperature of the device is extremely high, or the difference intemperatures is extreme (such as between the sun and a spacecraft), radiation isusually disregarded as a significant source of heat transfer. To decide the importanceof radiation to the overall rate of heat transfer, we can define the radiative heat

q k AcTL

--------

TqLk Ac--------

qr F1,2 A T 14 T 2

4( )



transfer as a radiative heat transfer coefficient,

h

r

:

1.1.4 P

RACTICAL

T

HERMAL

R

ESISTANCES

The semiconductor junction temperature depends on the sum of the thermal resis-tances between the device junction and the ambient environment, which is theultimate heat sink. Figure 1.2 shows a simplified view of the primary thermalresistances:

where:

tot

total thermal resistance (K/W)

jc

junction to case thermal resistance (K/W)

cs

case to heat sink thermal resistance (K/W)

sa

heat sink to ambient thermal resistance (K/W)

FIGURE 1.2

Primary thermal resistances in a chip/heat sink assembly.

jc

is resistance from thedie junction to the device case.

cs

is resistance from the device case to the heat sink.

sa

isresistance 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.)

Lead

Die bond Chip

Heat spreader

Encapsulant

Fl

sa

cs

jc

hr F1,2 T 12 T 2

2

( ) T 1 T 2( )

tot jc cs sa



Thermal resistance between the semiconductor junction and the junctionsexternal case

This resistance is designated

jc

and is usually expressed in C orK/W. This resistance is an internal function of the design and manufacturing methodsused 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 semi-conductor manufacturer decides upon this resistance by weighing such factors asthe maximum allowable junction temperature, the cost of the device, and the powerof the device. For example, a plastic semiconductor case is often used for a low-power, inexpensive device. A typical

jc

for such a device might be 50 K/W. If thedevice operates in a 35C environment and dissipates 0.5 W, then the junctiontemperature

T

j

is found by:

For a higher-powered component, the manufacturer must use a more costly approachto dissipate the power. A typical

jc

for this type of component might be 2 K/W.Specialized chip assemblies using expensive lead forms, thermally conductiveceramics, and Diamond heat spreaders can further lower this value.

Thermal resistance from the case to the heat sink interface surface

Thisresistance is designated as

cs

and is expressed in C or K/W. Case to heat sinkthermal energy is transferred primarily by conduction across the contact interface.The field of contact interface thermal resistance is complex and is not well under-stood. No models are able to predict this value in a variety of cases. Even valuesarrived at by actual testing may vary by 20%. In any case, this value can be reducedby using thermal greases, pads, and epoxies, and by increasing the pressure at thethermal interface. In some applications, manufacturers mount the semiconductorjunction to a copper slug that extends to the surface of the case. This design resultsin a very low

jc

. In addition, they design the copper slug to be soldered to a printedcircuit board, resulting in an extremely low contact resistance.

The thermal resistance from the heat sink contact interface to the ambientenvironment is designated saLike the other resistances, it is also expressed in Cor K/W. This is often the most important resistance of the three as for susceptibilityto change by the electronic packaging engineer. The smaller this value, and thereforethe resulting total resistance tot, the more power the device can handle withoutexceeding its maximum junction temperature. For the simplified model, this valuedepends on the conductive properties of the heat sink, fin efficiency, surface area,and the convective heat transfer coefficient:

The heat transfer coefficient, hc, introduced earlier, is a complex function and cannotbe easily generalized for use. However, many empirical equations result in a reasonabledegree of accuracy when generating values of hc. As this formula shows, sa is thereciprocal of the product of the heat transfer coefficient and the sink surface area.

T j T a jcq 35C 50 K/W( ) 0.5 W( ) 60C

sa 1

hc As----------



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. Whenwe mount a semiconductor on a heat sink, the relationship between junction temper-ature rise above ambient temperature and power dissipation is given by:

The focus of the remaining chapters is to explore and expand on these basicresistances 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 electricalcurrent flows through a semiconductor or a passive device, a portion of the poweris dissipated as heat energy. The quantity of power dissipated is found by:

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 isgiven in units of mean power Pdm:

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 currentt2 upper limit of conduction for current

T q jc cs sa ( )

Pd VI

Pdm1t--- V t( )I t( ) dt

t1

t2



1.2.2 HEAT GENERATION IN ACTIVE DEVICES

1.2.2.1 CMOS Devices

The power that is dissipated by bipolar components is fairly constant with respectto frequency. The power dissipation for CMOS devices is a first-order function ofthe frequency and a second-order function of the device geometry. Switching powerconstitutes about 70 to 90% of the power dissipated by a CMOS. The switchingpower of a CMOS device can be found by:

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 thesedynamic short circuits, the number of on gates must be known. This value is usuallygiven in units of W/MHz per gate. The power dissipated is found by:

where:

Ntot total number of gatesNon percentage of gates on (%)q power loss (W/Hz per gate)f switching frequency (Hz)

1.2.2.2 Junction FET

The junction FET has three states of operation: on, off, and linear transition. Whenthe junction FET is switched on, the power dissipation is given as:

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.

PdCV 2

2---------- f

Pd Ntot Non q f

PdON ID2 RDS ON( )



1.2.2.3 Power MOSFET

The power dissipated by a power MOSFET is a combination of five sources ofcurrent 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, ande. PS: switching function losses.

Conduction losses, Pc, occurring when the device is switched on, can be found by:

where:

ID drain current (A)RDS(ON) drain to source resistance ()

Conduction losses when the device is in the linear range are found by VI, as areleakage 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 calculatedas the product of the drain-to-source voltage and the drain current; therefore:

where:

fS switching frequency (Hz)VDS MOSFET drain-to-source voltage (V)ID MOSFET drain current (A)tS1 first transition time (s)tS2 second transition time (s)

The MOSFET gate losses are composed of a capacitive load with a seriesresistance. The loss within the gate is

where:

VGS gate-to-source voltage (V)QG peak charge in the gate capacitance (coulombs)RG gate resistance ()

Pc ID2 RDS ON( )

PS f S VDS t( )ID t( ) td VDS t( )ID t( ) dt0

tS2

0

tS1

PG VGS QGRG

RS RG

--------------------



The total power dissipated by the gate structure, PG(TOT), is found by:

1.2.3 HEAT GENERATED IN PASSIVE DEVICES

1.2.3.1 Interconnects

The steady-state power dissipated by a wire interconnect is given by Joules law:

where:

I steady-state current (A)R steady-state resistance ()

The resistance of an interconnect is

where:

material resistivity per unit length (/m) (see Table 1.1)L connector length (m)Ac cross-sectional area (m2)

TABLE 1.1Resistance of Interconnect Materials

Material Resistivity, , /cm

Alloy 42 66.5Alloy 52 43.0Aluminum 2.83Copper 1.72Gold 2.44Kovar 48.9Nickel 7.80Silver 1.63

Source: King, J. A., Materials Handbook for HybridMicroelectronics, Artech House, Boston, 1988, p. 353.With permission.

PG TOT( ) V GS QG fS

PD I2R

R LAc-----



Table 1.2 shows the maximum current-carrying capacity of copper and aluminumwires in amperes:5

These values can be rerated at any anticipated ambient temperature by the equation:

TABLE 1.2Maximum Current-Carrying Capacity of Copper and Aluminum Wires (in Amperes)

Copper MIL-W-5088

Aluminum MIL-W-5088

Underwriters Laboratory

Size, AWG

Single Wire

Bundled Wirea

Single Wire

Bundled Wirea

National Electrical

Code 60C 80C

American Insurance

Association500

cmilA

30 0.2 0.4 0.2028 0.4 0.6 0.3226 0.6 1.0 0.5124 1.0 1.6 0.8122 9 5 1.6 2.5 1.2820 11 7.5 2.5 4.0 3 2.0418 16 10 6 4.0 6.0 5 3.2416 22 13 10 6.0 10.0 7 5.1614 32 17 20 10.0 16.0 15 8.2212 41 23 30 16.0 26.0 20 13.0510 55 33 35 25 20.8

8 73 46 58 36 50 35 33.06 101 60 86 51 70 50 52.64 135 80 108 64 90 70 83.42 181 100 149 82 125 90 132.81 211 125 177 105 150 100 167.50 245 150 204 125 200 125 212.0

00 283 175 237 146 225 150 266.0

Rated ambient temperatures:

57.2C for 105C-rated insulated wire

92.0C for 135C-rated insulated wire

107C for 150C-rated insulated wire

157C for 200C-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.

I I rTc TTc T r-------------------



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 Joules law:

where:

I steady-state current (A)R steady-state resistance ()

The instantaneous power, PD(t), dissipated by a resistor with a time-varyingcurrent, I(t), is

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

1.2.3.3 Capacitors

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

where:

C capacitance (F)VM peak sinusoidal voltage (V) radian frequency, 2ff frequency (Hz)

PD I2R

PD t( ) I2 t( )R

PD 0.5IM2 R

PD t( ) 0.5CV M2 2tsin



The equivalent series resistance of a capacitor in an AC circuit can lead tosignificant power dissipation. The average power in such a circuit is given as:

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,

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 dissipatedby the sinusoidal resistance of an inductor is found by:

where:

L inductance (Henry)IM peak sinusoidal current (A) radian frequency (2f )

TABLE 1.3Typical Resistances of Capacitors69

Dielectric Material Capacitance (F) RES @ 1 kHz, m

BX 0.1 19.0 kX7R 0.1 16.0 kX7R 0.18 10.0 kBX 1.0 2.0 kZ5U 3.3 0.60 kTantalum 2.2 1.0 kTantalum 22 0.20 kTantalum 33 0.20 kTantalum 33 0.26 kTantalum 68 0.168 k

PD1T--- I2 t( )RES td

t1

t2

PD I2RL

PD t( ) 0.5LIM2

2tsin



When a ferromagnetic core is used, the loss consists of two sublosses: hysteresisand eddy current. The rate of combined core power dissipation can be found by:

where:

PD(CORE ) power dissipation (W/kg)n, m constants of the core materialf switching frequency (Hz)BMAX maximum flux density (Tesla)

The power dissipation is then found by:

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 resistancenetworks 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 labo-ratories; it is now also used to provide quick answers on desktop computers. Theconvective coefficient of heat transfer, the most difficult value to assign in heattransfer, is regularly being estimated within 10%, whereas 30% was formerlythe norm.

Once we construct and verify a computer model, we can evaluate hundreds ofchanges in a short time to optimize the model. In the future, as the underlying CFDcode becomes more advanced, even the tedious model verification step may beeliminated.

As with physical designs, computer models can be a combination of conduction,convection, and radiation modes of heat transfer. Convection problems have thelargest variety of permutations, and this has given the CFD engineers the mostdifficulty: laminar flow changes to turbulent flow, energy dissipation rates changewith velocity, at slow velocity natural convection may override the expected forcedconvection effects, etc. When additional factors such as multiphase flow, compress-ibility, and fine model details such as semiconductor leads are added, it is easy tosee why convective computer modeling is so complex.

PD CORE( ) 6.51 fnBMAX

m

PD PD CORE( )M



At the core of these elaborate computer codes are the basic equations of mass,momentum, and energy conservation, shown here in the Cartesian coordinate systemfor familiarization:

Conservation of mass:

Conservation of momentum in x:

Conservation of momentum in y:

Conservation of energy:

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)

heat source per unit volume (W/m3)

These equations can take many forms and change in different coordinate systemsand under different flow conditions.

We enter the geometry of a model into a computer CFD program or, morecommonly, it is imported in a standard format from a CAD (computer-aided drafting)software program. Within the CFD program the required spatial coordinates arechosen to learn the dimensionality of the model, such as , r, and z in the polarcoordinate system. By carefully evaluating the problem, a seemingly complex three-dimensional problem can sometimes be modeled in two dimensions. An example isthe axisymmetric pipe flow model. We require a two-dimensional model to calculate

t

------ x

----- u( ) y

----- v( ) 0

t

---- u( ) x

----- uu( ) y

----- vu( ) x

----- ux

------

y-----

uy

------ p

y------ Fx

t

---- v( ) x

----- uv( ) y

----- vv( ) x

----- vx

-----

y-----

vy

----- p

y------ Fy

t

---- h( ) x

----- uh( ) y

----- vh( ) x

-----kcp----- h

x------

y

-----kcp----- h

y------

qG

qG



the radial, r, and axial, z, variations, in addition to the velocities of v and w. If werequire a more realistic and detailed model, adding a circumferential velocity can allowthe flow to swirl within the pipe, u, as a function of r and z. Although three momentumequations are used for three velocity components, the flow is still two-dimensionalbecause the flowfield variables are a function of just two space coordinates.

Once the geometry, coordinate system, and material properties are modeled ina computer, the fluid region is discretized as several smaller domains. A finer ornonuniform grid is often used in areas of greater interest or areas where the flowpatterns are so complex that a coarse solution would affect the accuracy of the entiremodel. We can classify the smaller domains into three broad methods of problemsolution:

1. Finite Element Analysis,2. Finite Difference Analysis,3. Finite Volume Analysis.

The finite element method10,11 uses a weighted residual to obtain the solutionto the discrete equations. Some methods use explicit, while others use implicit,formulations with a variety of convergence schemes. As a consequence of the explicitformulation, a solution is found in a time-sequencing manner. Time steps are takento progress toward a final flowfield solution. Usually, finite element methods areeasier to use than other methods when adapting irregular-shaped elements to complexgeometries.

The finite difference method12,13 is structured around a Taylor series expansionfor each variable adjacent to a grid point. Most codes retain only the first severalterms 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 methodsof domain solution. Usually, we obtain a full solution for a single point before werealize a solution for a subsequent point. Finite difference methods have been usedfor many years and have a history of optimized solutions.

The finite volume analysis method14 is interesting because it attempts to solvethe 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 fullyimplicit equation. Solutions are found by iterative methods with a certain flexibilityfor specific variabilities. Interestingly, different variables are solved by the point topoint method while we may solve other variables in a whole-field analysis. We knowfinite volume methods to be very stable and efficient in their use of computer resources.

1.3.1 COMMERCIAL CFD CODES

Turbulence analysis methods15The typical flow problem encountered in elec-tronic cooling is turbulence. Turbulent flows can be solved by an analysis of thecharacteristics of the mean (time-averaged) flow. The most common turbulencemodels are based on the Boussinesq concept of eddy viscosity. The use of turbulentor eddy viscosity accounts for enhanced mixing (diffusion) due to turbulence. Eddyviscosity is normally magnitudes larger than the effect of molecular viscosity and



is a flow property, not a fluid material property. The most commonly used turbulentflow model is the two-equation k ~ model. This model uses two transport equa-tionsone for turbulent kinetic energy, k, and the other for the rate of eddy dissi-pation, . We apply local calculated values of k and as turbulent viscosity values.When compared with the simpler Prandtl mixing equation, the k ~ model does notrequire prescribed scales of turbulence length. Although it is a theoretically complexequation, by extensive analysis and comparison with physical models, the k ~ method has been limited to five empirical constants. The k ~ model is being refinedand expanded16 for greater applicability in a broad range of fluid problems.

Direct Numerical SimulationA class of CFD that holds great promise isDirect Numerical Simulation (DNS). The hope for DNS is based upon the ideathat turbulence, with all its complicated large- and small-scale structures, is nothingother than a viscous flow that locally obeys the Navier-Stokes equations. If a fineenough grid is used, we can calculate all the details of this turbulent flow directlyfrom the Navier-Stokes equations with no artificial modeling of the effects ofturbulence. A current limitation of this technique is the enormous amount ofcomputer time required. To use the DNS method to directly solve the Navier-Stokes 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 aCRAY Y-MP supercomputer.

1.3.2 FLOTHERM V2.2

Several general-purpose CFD codes are available on the commercial market. Thesecodes have varying degrees of friendliness toward electronic cooling problems but,in general, are very useful. A program by Flomerics claims an 80% share of theCFD market for thermal analysis of electronic packaging. FLOTHERMTM containsa full 3-D solver for Navier-Stokes equations, built-in boundary conditions forcommon objects such as fans, vents, and filters, and an effective turbulent viscositysolver that accounts for the additional friction and heat transfer due to turbulence.This package is designed specifically for electronics cooling problems. The softwareis designed to run on personal computers and UNIX platforms. FLOTHERMTM isavailable 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 Assem-blies, 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.



6. Hopkins, D. C., Designing Power Hybrid Supplies, Powertechniques Magazine, June,3134, 1989.

7. Hopkins, D. C., Jovanovic, M. M., Lee, F. C., and Stephenson, F. W., Off-line ZCS-QRC Thick-Film Hybrid Circuit, Virginia Power Electronics Center, Sixth Annu.Power Electron. Semin., 7183, September, 1988.

8 Olean Advanced Products Data Book, S-OAP10M295-N, AVX Corporation, MyrtleBeach, 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, McGraw-

Hill, 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 Encyclo-

pedia 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 Turbulencein a Spatially Evolving Boundary Layer, AIAA paper 91-1607-CP, Proc. AIAA 10thComputer Fluid Dynamics Conf., 890914, 1991.


Thermal Design of Electronic Equipment.Contents1 Introduction to Thermal Design of Electronic Equipment1.1 INTRODUCTION TO THE MODES OF HEAT TRANSFER IN ELECTRONIC EQUIPMENT1.1.1 CONVECTION1.1.2 CONDUCTION1.1.3 RADIATION1.1.4 PRACTICAL THERMAL RESISTANCES

1.2 THEORETICAL POWER DISSIPATION IN ELECTRONIC COMPONENTS1.2.1 THEORETICAL POWER DISSIPATION1.2.2 HEAT GENERATION IN ACTIVE DEVICES1.2.2.1 CMOS Devices1.2.2.2 Junction FET1.2.2.3 Power MOSFET

1.2.3 HEAT GENERATED IN PASSIVE DEVICES1.2.3.1 Interconnects1.2.3.2 Resistors1.2.3.3 Capacitors1.2.3.4 Inductors and Transformers

1.3 THERMAL ENGINEERING SOFTWARE FOR PERSONAL COMPUTERS1.3.1 COMMERCIAL CFD CODES1.3.2 FLOTHERM V2.2

REFERENCES

Thermal_Design_of_Electronic_Equipment/Thermal_Design_of_Electronic_Equipment/Ch02.pdfHibbeler R. C. Force-System Resultants and Equilibrium Thermal Design of Electronic Equipment. Ed. Ralph RemsburgBoca Raton: CRC Press LLC, 2001


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 amaterial, or from one material to another. The energy transfer is postulated to occurbecause of kinetic energy exchange by elastic and inelastic collisions of atoms, andby electron drift. Heat energy is always transferred from a region of higher energyto an area of lower energy. The energy level, or temperature, of a material is relatedto the vibration level of the molecules within the substance. If the regions are at anequal temperature, no heat transfer occurs. Fouriers law can be used to predict therate of heat transfer.

1

The law suggests that the rate of heat transfer be proportionalto the area of transfer times the temperature gradient

dT/dx

.

In Fouriers law, the relation T(

x

) is the local temperature and

x

is the distanceof heat flow. Although this is an equation of proportionality, the actual rate of heattransfer depends on the thermal conductivity,

k

, which is a physical property ofthe heat transfer media. Thermal conductivity is generally expressed in terms ofW/m K.

Heat transfer can occur by conduction through any material: solid, liquid, orgas. Conduction cannot occur through a vacuum because there is no material toconduct through. Conduction is not usually the predominant method of heat transferthrough a gas or liquid. Usually, as we apply heat to a fluid, the heated portion ofthe fluid expands and sets up density gradients. These density gradients cause motionwithin the fluid, which leads to convective heat transfer. Convective heat transfer, amacroscopic method of energy transfer, is much more effective than conductive heattransfer.

The values used for the thermal conductivities of liquids and solids are generallyobtained by experimentation. The thermal conductivity of gases at moderate tem-peratures closely follows the kinetic theory of gases, and therefore calculated valuesmay be used.

qk AdTdx-------



2.2 THERMAL CONDUCTIVITY

Fouriers 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-stateproportionality, the thermal conductivity can be found by

Thermal conductivity is a physical property that suggests how much heat willflow per unit time across a unit area when the temperature gradient is unity, expressedin W/m K. The property of thermal conductivity is important in conduction and con-vection applications. In some natural convection applications, where we have a confinedairspace, heat transfer is actually by conduction, not convection as the designer mightassume.

The conduction of heat occurs when molecular collisions move the kineticenergy of heat from one molecule to the next. Therefore, thermal conduction canoccur only when a temperature differential exists. Usually, metals are good conduc-tors 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 collidewith other electrons, or with the larger ions and nuclei within the structure. Theidentical mode of energy transfer also occurs during electrical conduction. This iswhy most materials that are good thermal conductors are also good electrical con-ductors. The primary exception to this is diamond. Diamond has a thermal conduc-tivity value approximately 5 times higher than copper, but a dielectric strength 10times higher than rubber.

2.2.1 T

HERMAL

R

ESISTANCES

Often, the thermal resistances characterize the transmission of heat in the path ofheat transfer. Examples of this include thermal pads, dielectric insulators, and adhe-sive bonding materials. Thermal resistance is most often expressed as temperaturerise in units of C/W or K/W, and is found by:

where

A

c

is the cross-sectional area available for conduction in units of m

2

.By comparing the thermal resistances, it sometimes becomes apparent which

components in the heat transfer path are contributing most to the heat rise of thepower component. Interestingly, we can describe convective heat transfer as a ther-mal resistance by

kq

AdTdx----------------

condTqx

--------L

k Ac--------

convTqx

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



where

A

s

is the surface area in contact with the cooling media. Radiation heat transfercan be described as a thermal resistance by

where

A

F

is the area of radiation based on a geometric factor of shape andemissivity.

2.2.2 C

ONDUCTIVITY

I

N

S

OLIDS

Thermal conductivity in a solid material is based upon migration of free electronsand vibrations within the atomic lattice structure. Silver, copper, and aluminum areindicative of materials in this group. These materials have high thermal and electricalconductivity. Figure 2.1 shows how the thermal conductivity of some metals changeswith temperature.

In nonmetals, the lattice structure vibrations dominate over the movement offree electrons, and thermal conductivity may not be related to electrical conductivity.In materials with highly structured crystalline lattice structures, thermal conductivitycan be quite high, while electrical conductivity is quite low. An outstanding exampleof a material in this group is diamond. Diamond has a thermal conductivity 5 timesthat of copper, and an electrical breakdown strength of more than 2000 V of directcurrent per 0.01 mm of length.

2.2.3 C

ONDUCTIVITY

I

N

F

LUIDS

Fluids, both liquids and gases, have much greater spacing between molecules thansolids and therefore much lower thermal conductivities. The thermal conductivityof a fluid varies with pressure and temperature. Within the pressure range of fluidsused in electronic cooling, thermal conductivity variances with pressure can beignored. Temperature, however, can greatly affect the thermal conductivity of liquidsor gases.

Within the range of temperatures used in electronic cooling, the thermal con-ductivity change of a gas is linear with temperature change but is different for eachgas. The thermal conductivity change with temperature in liquids is not yet wellunderstood. Figures 2.2 and 2.3 show the thermal conductivity change with temper-ature for selected gases and liquids, respectively.

2.3 CONDUCTIONSTEADY STATE

2.3.1 C

ONDUCTION

IN

S

IMPLE

G

EOMETRIES

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 dependentvariables.

radTqx

-------- 1hr AF-----------



2.3.1.1 Conduction through a Plane Wall

In the one-dimensional form,

T

depends only on

x

. If there is no internal heatgeneration (

q

i

0), and we set the plane wall shown in Figure 2.4 to an initialtemperature and distance of

T

(

x

0)

T

1

and a final temperature and distance of

FIGURE 2.1

Comparison of the variation of thermal conductivity with temperature for typ-ical solid materials used in electronic packaging.



T

(

x

L

)

T

2

, then:

FIGURE 2.2

Comparison of the variation of thermal conductivity with temperature forcommon gases used in electronic cooling applications.

T x( )T 2 T 1

L------------------- x T 1



Using Fouriers law, we can find the rate of conductive heat transfer in the one-dimensional

x

-direction

FIGURE 2.3

Comparison of the variation of thermal conductivity with temperature for commonliquids used in electronic cooling applications. PAO represents polyalphaolefin.

qx kAdTdx-------

kAL

------ T 1 T 2( )



The heat flux, energy per unit area, is given as

Rearranging the rate of heat transfer for temperature rise, we have the familiar one-dimensional form:

More complex problems of this type may encompass one-dimensional heat flowthrough any number of series and parallel combinations of thermal resistance.Although parallel heat flow is technically a two-dimensional problem, we can usuallyreduce it to a single heat flow direction (see Figure 2.5). The general equation for

FIGURE 2.4

Conduction in a plane wall when the internal heat generation is uniform. Inthis case the temperature distribution is

T

1

T

2

.

8

8

8 L

A qgen

dx

x

Tmax

T1 T1

8

= qG ( A dx )

qxqxA---- k

L--- T 1 T 2( )

TqLk Ac--------



heat transfer for these problems, called composite walls, is

Therefore, we can describe a composite wall with three materials (

A

,

B

, and

C

) inseries and convective heat transfer along the face of material

A

and

C

as

The overall heat transfer coefficient,

U

, is sometimes used, which we describe as

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.

L1

L1

k1

,h1

hAA k1A k2A k3A hDA

,hD

k2 k3

1

x1 1

2 3

L2

L2

L3

L3

Ts,D

TB

TC

Ts,DTBqx

TC

T

T

,A

,A

8

T ,A8

T ,D8

T ,A8

TCold fluid

,D8

T ,D8

Hot fluid

s

T ,As

qxT 1, T N,

t-----------------------------

qxT 1, T 4,

1hc 1, A------------

LAkA A---------

LBkB A---------

LCkC A---------

1hc 4, A------------

------------------------------------------------------------------------------------------------

Uq

AT------------



Using the overall heat transfer coefficient, the previous expression for the compositewall of Figure 2.5 becomes

2.3.1.2 Conduction through Cylinders and Spheres

In electronic cooling, the most prevalent case of radial heat transfer is the tubecontaining a flowing coolant. Here, heat flows from the outer surface of the tube tothe center of the tube (see Figure 2.6). The rate of heat transfer in the radial directionof the tube is

Note that this shows that the distribution of the heat flow is logarithmic, not linear

FIGURE 2.6 Radial heat conduction through a cylindrical shell having no internal heatgeneration.

constant

qG

= 0

T = T (r )

L

k =

qk

r0

r1

T1

T0

U1

tot A----------- 1

1hc 1,--------

LAkA------

LBkB-----

LCkC------

1hc 4,--------

--------------------------------------------------------------------------------

qk kAdTdr------- k 2rL( )

C1r

------ 2LkTo Ti

ln rOri-----

------------------



as in the plane wall. The thermal resistance can be expressed as

Similar to the method used to calculate combined conduction and convectionheat transfer in a composite plane wall, the heat transfer equation for a compositetube (see Figure 2.7) containing three materials and a flowing fluid is

Using the overall heat transfer coefficient, the previous expression for the composite

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.

+

L

T,D

,hD

T,D,h

D

T,D

T,A,h

A

TS,D

TC

TB

TS,A

rA r

BrC

T,A,hA

T,A

Ts,A

TB

TC

Ts,D

1 2 3

T,A Ts,A TB TC Ts,D T,D

1 l

A2prALh 2p k1L 2p k2L

ln(rB/ r

A)ln(r

C/ r

B)ln(r

D/ r

C)

2p k3L hD2prDL

rD

ln rO

ri-----

2Lk--------------

qT

14 --------------

T 1, T 4,

1hc12r1L-----------------------

lnr2r1----

2kAL-----------------

lnr3r2----

2kBL-----------------

lnr4r3----

2kCL----------------- 1

hc42r4L-----------------------

-----------------------------------------------------------------------------------------------------------------



wall tube becomes

We can simplify the equation for heat conduction in spherical coordinates to

If Ti is the temperature at ri and To is the temperature at ro, then the temperaturedistribution in the sphere (see Figure 2.8) is

FIGURE 2.8 Heat conduction through a hollow sphere having a uniform surface temperatureand no internal heat generation.

qk

r0

r1

T1

T0

qG= 0

T = T (r )k = constant

U1

tot A----------- 1

1hc1-----

r1kA----ln

r2r1---

r1

kB----ln

r3r2---

r1

kC----ln

r4r3---

r1

r4--- 1

hc4-----

--------------------------------------------------------------------------------------------------------------------------

1

r2---- d

dr----- r2

dTdr-------

1r---d

2 rT( )dr2

---------------- 0

T r( ) Ti To Ti( )ro

ro ri---------------- 1

rir---



The rate of heat transfer through the sphere is then

and the thermal resistance is found by

2.3.1.3 Plane Wall with Heat Generation

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

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

T(x)TO

T0

xx-L

(a) (b)

(c)

-L

q

T,A,hA

T,B,hB

Ts,B

Ts Ts

Ts

Ts,A

+L +L

Th, ,T

h

,Th

T(x)

T(x)

.

q.

q 4r2Tr-------

To Tiro ri

4kro ri-------------------

------------------

ro ri

4krori-------------------



conductivity, k, the equation becomes

We find the temperature distribution, T(x), by

If the two surface temperatures are equal, T1 T2, the temperature distributionsimplifies to a parabolic distribution about the centerline of the plane wall,described as

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

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 cylinderor sphere with internal heat generation. Such cases occur in current-carrying busbars, wires, resistors, and a flex circuit rolled into a cylindrical shell. The followingequations apply to both cylinders and spheres (see Figure 2.10). The temperaturedistribution in a cylinder is found by

The maximum temperature is at the centerline of the cylinder, r 0; therefore,

kd2T x( )

dx2----------------- qG

T x( )qG2k------x2

T2 T1L

-------------------x qGL2k

---------x T 1

T x( )qGL

2

2k-----------

xL--- x

L---

2

T1

TqGL

2

8k-----------

T r( )qGro

2

4k---------- 1

rro----

2 T s

T max T oqGro

2

rk----------



TABLE 2.1Conduction in Plates and Walls2

Description Equations

Convectively heated and cooled plate

Convectively heated and cooled plate

q

Composite plate Composite plate

q

for 1

Plate with temperature-dependent thermal conductivity

Plate with temperature-dependent thermal conductivity3

for k k1 TTT1

q

where

km

h1T T 1 T 2

Bi1 1h1h2---

-------------------------------

T T T1

T T2 T1----------------------

h1 xk

------- 1

h1 L

k------ 1

h1h2-----

------------------------------

T T0 Tn

Liki---- 1

hi----

1h0-----

i 1

n

----------------------------------

J

T T j To

T Tn To----------------------

Liki---- 1

hi----

x jk j---- 1

h0-----k

i 1

j 1

Liki---- 1

hi----

1h0-----

i 1

n

---------------------------------------------------------------------

kmT T 1 T 2

L-----------------------=

T T T 11

---

k12

2kmT T 2 T 1 x

L k1-----------------------------------------------------------

k1 k22

-----------------



TABLE 2.1 (continued)Conduction in Plates and Walls2


Thin rectangular plate on the surface of a semi-infinite solid

Thin rectangular plate on the surface of a semi-infinite solid4

q

Infinite thin plate with heated circular hole

Infinite thin plate with heated circular hole for T T3 at r r1 and r r1

Infinite thin plate with heated circular hole

Infinite thin plate with heated circular hole for q at r r1, and r r1

where:

B

T

Finite plate with centered hole Finite plate with centered hole5

q

kwT T1 T2

ln 4wb

-------

----------------------------------

T T TT T3 T------------------------

K0Br------

K0Br1

--------

-----------------

kT T T

q---------------------------

K0Br------

2

Br1

-------- K1

Br1

--------

-------------------------------------

Bi1 Bi2

T 1 HT 21 H

------------------------

HBi1Bi2-------

kT T1 T2

d2w------- ln

w

r-----

2

------------------------------------------------------------------

(Continued)





Tube centered in a finite plateTube centered in a finite plate4 for r

q

w/d c1.00 0.16581.25 0.07931.50 0.03562.00 0.00752.50 0.00163.00 0.00034.00 1.4 105

0.0Infinite plate with internal heat generation

Infinite plate with internal heat generation2

T T1, x 0T T2, x L

where

Infinite plate with convection boundaries and internal heat generation

Infinite plate with convection boundaries and internal heat generation3

where H

d10------

2kT T1 T2

ln 4d

r------( ) c

---------------------------------

T T T1

T T2 T1---------------------- X

PoX 1 X( )2

-------------------------------

XxL---

T T T2

T T1 T2----------------------

1 Po1

Bi2-------- 1

1 Bi2 H ----------------------------------------

PoBi2------- Po

2------ 1 X2( )

Bi1 1 Po

1Bi2-------- 0.5

1 X( )

1 Bi1 H -------------------------------------------------------------------------------

h2h1-----




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 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 X length ratio (x/L)

Po coefficient of thermal expansion (C1)

q rate of heat flow, W thickness

FIGURE 2.10 Heat conduction nomenclature for a long circular cylinder with internal heatgeneration in differential element dr.

qG

qGL2

k-----------

Heat generation indifferential elementis qG L2rdr

drr r

max

s

BA

L

T

T

CL



If we evaluate the temperature distribution at the centerline of the cylinder, we findthe nondimensional temperature distribution

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

The effective heat transfer coefficient for the tube is then

Tables 2.2 and 2.3 show the solutions to a variety of conductive cylinder andsphere problems.

2.3.1.5 Critical Radius of a Cylinder

In real problems involving heat dissipation of an insulated cylinder, we must usuallyaccount for the effects of convection, whether natural or forced. When the outer radiusof the insulation is small, the surface area is also small, and the effect of convectionis not too great. As the outer radius of the insulation increases, the surface area alsoincreases. At a critical radius, the effect of convective cooling will outweigh the effectof internal conduction resistance. The rate of heat transfer per unit length of a cylinder is

where:

Ti temperature of cylinder, CT temperature of ambient air, Cro outer radius of insulation, mri inner radius of insulation, mk thermal conductivity of insulation, W/m KL length of cylinder, mhc,o external convective heat transfer coefficient, W/m2 K

We can see from this equation that we achieve a maximum heat transfer rate whenthe total thermal resistance, tot, is at a minimum. If the outer radius of the insulationequals a critical value:

T r( ) T sT max T s------------------------ 1

rrB-----

2

T s T qGr2hc--------

hcqG ro

2 ri2

( )2ri T i T ( )-------------------------------

qT i T

tot-------------------

T i T cond conv----------------------------

T i T

lnrori----

2kL--------------

12kLrohc o,----------------------------

----------------------------------------------------

ro rcritk

hc o,--------



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


Infinite hollow cylinder Infinite hollow cylinder

q

where

Bio

Composite cylinder Composite cylinder

q

for j 1

where Tj is the temperature in the jth layer

Insulated tube Insulated tube

q

where

k ktube and Bio

Maximum heat loss occurs when ro

Infinite cylinder with temperature-dependent thermal conductivity

Infinite cylinder with temperature-dependent thermal conductivity3 with:

k ko k ko at rok ki at ri

q

2kT To Ti

lnrori----( ) 1Bii

------- 1Bio--------

----------------------------------------

T T To

T Ti To---------------------

lnror----( ) 1Bio

--------

lnrori----( ) 1Bii

------- 1Bio--------

----------------------------------------

ho rok

----------

i

2T Tn T1

1ki----

lnr

i 1ri

---------------

1

rihi-------- i 1

n

i 1

n 1

------------------------------------------------------------------------------

T T j T1

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

1ki---- ln

ri 1ri

--------------- 1

rihi---------

1k j---- ln

rr j----

1r jh j----------

i 1

j 1

1ki---- ln

ri 1ri

---------------

1

rihi---------

i 1

n

i 1

n 1

--------------------------------------------------------------------------------------------------------------------------

2kT Ti T f

lnrori-----

1Bio--------

--------------------------------

hrok

--------

kh---

T T To

2kmT Ti To

lnrori-----

-----------------------------------

T T To

ko------------------------ 1

2kmko

2-------------

lnror-----

lnrori-----

---------------T Ti To

0.5

1

(Continued)



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


Pipe in semi-infinite solid Pipe in semi-infinite solid 4

q

where:

Bi1 Bi2

K D

Row of rods in semi-infinite solid Row of rods in semi-infinite solid3

For one rod

q

where:

Bi1 , Bi2 , D

Row of rods in wall Row of rods in wall3

For each rod

q

where:

Bi1 , Bi2 , D

Circular disk on the surface of a semi-infinite solid

Circular disk on the surface of a semi-infinite solid5

For

where Z and R

2k1T T2 T1

1Bi1-------- ln

r2r1-----

1K----

ln 2D2KDBi2

------------

--------------------------------------------------------------

h1r1k1

----------h2dk2

--------

k2k1---- d

r2----

2kTT 2 T 1

1Bi1-------- ln d

Dr1-------------

sinh 2 D DBi2--------

-------------------------------------------------------------------------------------

h1r1k

----------h2dk

--------ds---

4kTT 2 T 1

1Bi1-------- ln

d

Dr1--------------

sinh 2 DD

Bi2--------

------------------------------------------------------------------------------------

h1r1k

----------h2dk

--------ds---

+T1

r1

h1 +T1

r1

h1 +T1

r1

h1

T2 h2

T2

k

h2

2d+ + +

T T 1 and z q 4rokT T2 T1

T T T1

T T2 T1----------------------

2

----sin 1

2

R 1( )2 Z20.5

R 1( )2 Z20.5

--------------------------------------------------------------------------------------------------

zro---- r

ro----

T2

kz

ro

+





Circular disk in an infinite solid Circular disk in an infinite solid3

Infinite hollow square rod Infinite hollow square rod3

q

Infinite hollow square pipe Infinite hollow square pipe5

q

Vertical cylinder in a semi-infinite solid Vertical cylinder in a semi-infinite solid3

q

where Bid , D

q 8rokT T2 T1

T2

k

z

ro

+

2kT T2 T1

kh1ro---------- ln

1.08w2ro

-------------- k

2h2w-------------

-----------------------------------------------------------------

ro

T1 h1

T2 h2

+

w

w

2kT T1 T2

0.785lnwd----

---------------------------------

wd---- 1.4

2kT T1 T2

0.93lnwd----

0.0502--------------------------------------------------

wd---- 1.4

T1

T2w

d

2D

ln 2D 11

Bid--------

-------------------------------------------

Bid

D------

ro kTT 1 T o

hdk

------ dro----

To

ro

T1

h

k

+

d

(Continued)





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

Two semi-infinite regions of different conductivities connected by a circular disk4

q

where Z , R

Heat flow between two rods in an insulated infinite plate

Heat flow between two rods in an insulated infinite plate5

q

T T o z

T T 1 z

qz 0 r ro z 0

T T To

T T1 To----------------------

2 k2

k1 k2( )------------------------ 1sin 2

R 1( )2 Z20.5

R 1( )2 Z20.5

----------------------------------------------------------------------------------------------------

Z 0

T T To

T T1 To----------------------

12k1

k1 k2( )---------------------------

1sin

2

R 1( )2 Z20.5

R 1( )2 Z20.5

----------------------------------------------------------------------------------------------------

Z 0

4rok1k2k1 k2------------------

T To T1

zro---- r

ro----

k1

k2

ro

z

+

2kT T1 T2

sw

------- lnw

r------

---------------------------------

w

s2r

T1 T2+ +





Infinite cylinder with convection boundary and internal heat generation

Infinite cylinder with convection boundary and internal heat generation3

where Bi , R

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

where R , Bi , Ri

Hollow infinite cylinder with convection-cooled inside surface and internal heat generation

Hollow infinite cylinder with convection-cooled inside surface and internal heat3 generation with qr 0 and r ro

where R , Bi , Ri

T T To k

qGro2

----------------------- 0.252Bi----- 1 R2

hrok

-------- rro----

k

+

ro

qG

To h

.

T T T f k

qGro2

----------------------- 0.252Bi----- 1 Ri

2( ) 1 R2 2Ri

2lnR

rro----

hrok

--------riro----k

+

ri

ro

qG

Tf h

T T T f k

qGri2

----------------------- 0.252Bi----- Ro

2 1( ) 1 R2 2Ro2lnR

rri---

hrik

-------rori----

+

ri

ro

qGTf h.

(Continued)



A more accurate equation accounts for the variable effect of ro on the heattransfer coefficient hc,o:

where:

thermal diffusivity of the convective media, k /cpn 0.5 for laminar forced convection or 0.25 for natural convectionk 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 willexamine heat transfer in multidimensional systems. Multidimensional heat transferoccurs when we transfer the heat from different locations and the temperature mayvary in more than one dimension. One example is an active component in a pottingcompound, an irregularly shaped object, or a corner where we join three chassiswalls. Figure 2.11 shows two-dimensional conduction.



Electrically heated wire with temperature- dependent thermal and electrical conductivities

Electrically heated wire with temperature-dependent thermal and electrical conductivities1,7

with T To R ro

1 t

1 e

where B , R 1

kt thermal conductivityNotes: 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 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)

kTkTo------- T T To

kekeo------ T T To

T T To

B-------------------- R 1 B

R8---

e

T------ 2 R

16--------------

keoro2E2T

kToL2

------------------------r2

ro2

----

+

ro

To

qG

ro rcrit

1 n( )k---------------------

1

n 1--------------



TABLE 2.3Conduction in Spheres2Description Equations

Spherical shell Spherical shell

q

Composite sphere Composite sphere

q

where Tj is the local temperature in the jth layer.

Sphere with temperature-dependent thermal conductivity

Sphere with temperature-dependent thermal conductivity3

with T Ti r ri T To r ro

k ko

q

where km

with k ko T Tok ki T Ti

4rokT Ti To

rori----- 1

rori-----

khiri--------- k

horo-----------

------------------------------------------------

T T To

T Ti To---------------------

ror----- 1

kroho-----------

rori----- 1

rori-----

krihi--------- k

roho-----------

------------------------------------------------

To ho

riro

Ti hi

+

4T T1 Tn

1ki----

1ri---- 1

ri 1------------

1

ri2hi

----------i 1

n

i 1

n 1

----------------------------------------------------------

T T j T1

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

1ki---- 1

ri---- 1

ri 1------------

1

ri2hi

----------1k j----

1r j---- 1

r---

1

ri2hi

--------- i 1

j 1

1ki----

1ri---- 1

ri 1------------

1

ri2hi

----------i 1

n

i 1

n 1

---------------------------------------------------------------------------------------------

r1

T1h

1

Tn

hn

r2

r3 rj

hj+1

rn

+

kn-1

T T To

4rokmT Ti To

rori----- 1

----------------------------------------

ko ki2

----------------

T T Toko

----

-------------------- 1 2 T Ti Tokmko

2------

ror----- 1

rori----- 1

------------ 1

To

riro

Ti

+

(Continued)



TABLE 2.3 (continued)Conduction in Spheres2Description Equations

Sphere in a semi-infinite solid Sphere in a semi-infinite solid4

q

where Bi

Sphere in an infinite medium Sphere in an infinite medium5

with

Two spheres separated by a large difference in an infinite medium

Two spheres separated by a large difference in an infinite medium4

q

for s 2r, error 1%

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

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

qr qi r ri

where R , Ro

Solid sphere with internal heat generation in an infinite medium

Solid sphere with internal heat generation in an infinite medium6

with

h contact coefficient at r ro

4rokT T1 To

1 0.5dro----- 1

Bi-----

--------------------------------------

hrok

--------

T1

d ro

h1

+To

T T 2 at r q 4rokT T1 T2

ro

Ti+

4rkT T1 T2

2 1rs--

------------------------------------ s 5r2r

s

T2+

T1+

T T To k

qi ri-----------------------

qGri6qi----------

2 R Ro( )RRo

------------------------- Ro2 R2

Ro RRRo

-----------------

rri---

rori----To

riro

+qG.

qi

k ko 0 r ro

k k1 r ro

qG qGo 0 r ro

qG 0 r ro

T T r

T T T ko

qGoro2

-------------------------- 16--- 1 R2

2Bi-----

2kok1

-------- 0 R 1 h

rok1

+

qG.ko



Heat conduction in multidimensional systems can be calculated by analytic,analogic, graphical, and numerical methods. Fourier, in 1822, made the first majorcontribution to the analysis of multidimensional heat transfer.8 Fouriers method ofseparating variables led to the requirement of transform methods such as the Laplacetransform, to express the arbitrary Fourier series expansion. These methods can yieldaccurate results but are quite time-consuming.

Graphical methods (see Figure 2.12) include the flux plot method and theSchmidt9 method. Both methods involve drawing skill and considerable time but

TABLE 2.3 (continued)Conduction in Spheres2Description Equations

where Bi , R

Notes:

A area, m q rate of heat flow, W

Bi Biot number, hL/k q rate of heat flux, W/m2

d diameter, m volumetric heat flux, W/m3

h heat transfer coefficient, W/m2 K r radius, m

k thermal conductivity, W/m K s spacing, m

L length, m coefficient of thermal expansion (C1)

FIGURE 2.11 Two-dimensional conduction showing the lines of heat flow. There is nointernal heat generation.

T T T k

qGoro2

------------------------ 13R------- r ro

hroko

-------- rro----

qG

Lines of constanttemperature (isotherms)

Heat flow lines

T2 T1T1 T2

q1,q2

q1,q2

q15

q15

E

BA

C D

Heat flow lines

(a)

(c)

(b)

Isotherms

T



2.3.2.1 Multidimensional Analytic Method

In a two-dimensional system without internal heat generation and with uniformthermal conductivity, the general conduction equation has been found as

The total rate of heat transfer is a vector. The vector is dependent upon the rateof heat flow in x, which is qx, and the rate of heat flow in y, which is qy. The totalrate of heat transfer is then perpendicular to an isotherm within the boundaries ofthe geometry. Therefore, if we solve for the temperature distribution, the heat flowcan be found easily. Examine a rectangular plate that is insulated at two oppositesides (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

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.

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

2T

x2---------

2T

y2--------- 0

T x y,( ) T m

yL

------- sinh

bL

------- sinh

------------------------

xL

-------sin



FIGURE 2.14 A rectangular adiabatic plate with steady-state sinusoidal temperature distributionon the upper edge.

FIGURE 2.15 A depiction of the resulting isotherms and heat flow lines for the adiabaticplate shown in Figure 2.14.



See Ozisik10 for a more detailed explanation of these conditions. Schneider11 providesa 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 flowwithin a complex two-dimensional object when the problem is isothermal and weinsulate the boundaries. In this method the designer draws a set of lines that representconstant temperature in

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