IEEE TRANSACTIONS ON PARTS, HYBRIDS, AND PACKAGING, VOL. PHP-10, NO. 4, DECEMBER 1974

Free Convection Heat Transfer from Vertical Fin-Arrays

DAVID W. VAN DE POLAND JAMES K. TIERNEY

267

Abstract-A simple but relatively accurate, approximate technique is presented for analyzing the heat transfer due to

natural convection and radiation from parallel-fin heat sinks. This technique accounts for a nonuniform base plate temper-

ature. The finned surfaces and the base plate have been con- sidered to be vertical U-shaped channels and a relationship for

the NusseJt number has been used which has a suitable form for both very long and very short fins. The accuracy of the

technique has been demonstrated by a testing program on a

representative sample of heat sinks.

NOMENCLATURE

a = channel aspect ratio = S/L; b = base thickness; C = center space;

CP = specific heat at constant pressure; D = source diameter; 9 = acceleration due to gravity;

Gr, = Grashof number = g/3(T,+,-T,)r3/v2;

G W = Grashof number = g/?(T,-T,)H3/v2; Ii = average heat transfer coefficient; H = height of heat sink; k = thermal conductivity; L = fin length;

Nu, = Nusselt number = E/k; NUH = Nusselt number = EH/k; Pr = Prandtl number = pCp/k; 0 = rate of heat transfer; r = characteristic length = 2LS/(2L+S); Ra+ = modified Rayleigh number = (r/H)GrFr; S = space between fins; SF = source factor; T = absolute temperature;

V = constant in (2) = -11.8 (l/in.).

Greek

P = volumetric coefficient of expansion;

P = dynamic viscosity;

V = kinematic viscosity;

ti = channel configuration factor;

@J = angle in Fig. 3.

Subscripts

f = heat sink at base of first fin;

D = heat source;

LS = line source;

Manuscript received October 11, 1973; revised May 6, 1974. The authors are with Bell Laboratories, Whippany, N. J. 07981.

HS = heat sink; 00 = ambient condition; S = sector;

9 = quadrant; w = wall condition.

Other Notation

P = convective losses for parallel plates; lJ = convective losses for U-shaped channels.

INTRODUCTION

Finned surfaces are frequently used as an efficient method

of rejecting waste heat from electronic equipment. These

finned surfaces, commonly known as heat sinks, are econom- ical and highly reliable when cooling is by natural convection

and radiation. Several authors have developed thermal relation-

ships for closed channels and parallel plates, but there appears

to be only one general analytical model for fin-arrays, that described by Fritsch [ 11. He neglects temperature variation in the base plate and uses parallel flat plate relations for the convective film coefficient. However, many practical heat sink

designs may consist of a series of relatively short fins attached to a heated base plate and cannot be accurately approximated

by parallel flat plates. The base plate creates additional surface area and a corner geometry having a detrimental effect on heat transfer rates. Fin-arrays of this type, which can best be described as a series of U-shaped channels (Fig. I), have

experimentally been investigated by Starner and McManus 121 and Welling and Wooldridge [3]. Additional experimental

investigations have been conducted by lzume and Nakamura

[4] who have also developed a mathematical relationship describing heat transfer from fin-arrays. Their relationship,

however, does not hold in the limiting cases of very large or

very small fin length to fin spacing ratios (L/S). Donovan and Rohrer [5] have theoretically investigated the radiative and

Fig. 1. U-shaped channel and heat sink configuration.

268 IEEE TRANSACTIONS ON PARTS, HYBRIDS, ANDPACKAGING, DECEMBER 1974

convective heat transfer characteristics of heat-conducting fins

on a plane wall, but they were mainly concerned with the effectiveness of the extended surfaces and the individual con-

tributions of radiation and convection for a film coefficient

which was not a function of the fin geometry. The authors have developed a simple but relatively ac-

curate, approximate technique for analyzing the heat transfer

from heat sinks due to natural convection and radiation with a

nonuniform base plate temperature. This method utilizes a

model of finite sized elements solved iteratively, where the

element size is of the order of the fin spacing. The model is

applicable to heat sinks having fins which are vertical,

rectangular, and mounted normal to a vertical, rectangular, constant thickness base plate as shotin in Fig. 1 and having a

single source of heat at the center of the base plate. This heat sink configuration is common in electronic equipment, and

thus the authors have limited the scope of this paper to that

family of heat sinks. Techniques for extending the method to analyze heat sinks having multiple heat sources are currently being developed.

The authors have considered the finned surfaces and the

base plate to be vertical U-shaped channels and have used a relationship for the Nusselt number for that geometry which

has the proper behavior for both very long and very short fins.

Therefore, fin length to fin spacing ratios (L/S) of from zero to infinity may be considered.

Experimental results for various sizes of heat sinks are

presented which show good correlation with the mathematical

model. The effect of considering the fins to be parallel flat plates instead of the walls of three-sided open ducts is also

illustrated.

MATHEMATICAL MODEL

The basic model views a heat sink as a series of adjacent, U-shaped channels formed by the heat sink fins. The individual fin temperatures are used to determine the thermal dissipa- tions from each of the channels, which in turn are used in an energy -balance of the entire heat sink to obtain an iterative

solution. The model assumes the following. 1. A constant temperature, vertical line source which is

made approx/mately equivalent to the real source and which is

positioned through the heat sink base centerline.

2. One dimensional heat conduction through the base plate

(i.e., a nonconstant base plate temperature in the horizontal

direction). 3. A series of constant thermal gradients between adjacent

fins (the heat lost from the base plate being combined with

that from the fins).

4. Gray, diffuse surfaces. 5. A constant ambient fluid temperature which is equal to

the radiative environment temperature. 6. Heat losses neglected from the heat sink ends. The model is based on an iterative scheme using a finite

sized element which is suitable for programming on a digital computer. Due to the bilateral symmetry assumed by the model, heat transfer need be computed for only one half of the heat sink.

The iterative process can be used if any one of the follow-

ing parameters is unknown and the rest are known: source

dissipation, device temperature, and the principle heat sink

dimensions (height, depth or width). In all cases, the ambient

temperature, material conductivity, and surface emissivity are

known. The procedure is as follows when the device temper- ature and dimensions are known and the source dissipation is

unknown. 1. A value is assumed for the source dissipation.

2. Using the source factor which relates the equivalent line

source temperature to that of the real source (see section on

source factor), the temperature of the fin closest to the source

is computed assuming one-dimensional conduction in the base

and no intermediate heat loss from the base area; therefore a

constant temperature gradient. 3. The convective film coefficients for the U-shaped chan-

nels on either side of the fin are computed using the relation-

ship for the Nusselt number developed by the authors [6]:

(see nomenclature for definition of terms).

Nu, = +{I - exp [-$(s)34]}

where

dJ= 24(1-0.483e-0.7a)

(2) {[I +3 [I + (l-e~0~83a)(9.14a1~2eVs-Cl.6~)] 1 3.

Through the use of Newtons law of cooling, the convective thermal dissipation is calculated for the fin area plus one-half

of the base area on either side of the fin (local front area, see Fig. 2). The fin efficiency given by Gardner [7] is used to compensate for thermal gradients in lpng fins.

FIN 2 X/l SVRFACES -FRONT LOCAL AREA // SURFACES-REAR LOCAL AREA

FIN *3 \\ 1 SURFACES FRONT LOCAL AREA

Xl\\ SURFACES REAR LOCAL AREA

ENTIRE HEIGHT OF SINK (HI USED IN LOCAL AREAS ERMOCOUPLE

L-FIN END

DEVICE, CHEAT SINK END

L-HEAT SINK END

Fig. 2. Surface area notation and thermocouple location.

4. The radiative thermal dissipation is calculated for the

U-shaped channels using relationships developed for finned surfaces by Fritsch [l] and the appropriate geometric view factors for L shaped surfaces given by Kreith [8]. The same

VAN DE POL AND TIERNEY: FREE CONVECTION HEAT TRANSFER 269

local front area and fin efficiency as in the previous step are

used. 5. Using the free convection Nusselt number for a constant

temperature vertical flat plate given for laminar flow by

McAdams 191

NUH = 0.59 [CrH Prl I4 (3)

and Newtons law of cooling, the convective thermal dissipa- tion from the rear of the heat sink is computed for one-half of

the base area on either side of the fin plus the fin base area (local rear area, see Fig. 2). The radiative heat loss from the

local rear area is evaluated using the Stefan-Boltzmann law. 6. The thermal dissipation from the local front and rear

areas is totaled, subtracted from the dissipation conducted to that fin, and the temperature of the next fin then computed

by again assuming a constant temperature gradient between the two fins.

7. Steps 3, 4, 5, and 6 are repeated until the end fin is reached. At that point, the dissipation from the outside of the end fin is considered as that from a flat plate.

8. After the thermal dissipation of the end fin has been computed, the total computed dissipation is compared to the

assumed value. If they are not within a specified percentage,

the assumed dissipation is modified and steps 2 through 8 are repeated until the assumed and calculated dissipations differ

by less than that prescribed percentage. At this point the

solution is reached.

A similar series of steps is used if a parameter other than the source dissipation is unknown.

DERIVATION OF SOURCE FACTOR

The source factor is used to approximately account for the

differences between the basically radial heat flux from an actual electronic device located at the center of the heat sink

base and the one dimensional heat flux from a constant temperature, vertical line source through the center of the

base. It partially accounts for temperature variations in the vertical direction. The method used to determine the source

factor is to calculate, with the real source, the average temper- ature of the fin which is closest to the device assuming no

intermediate heat loss from the base area and uniform radial

heat conduction in the heat sink base constrained within the sector -02 < @ < $2 (see Fig. 3 showing half of the heat sink). Symmetry is assumed about the vertical centerline. Then the theoretical line source temperature required to obtain a temperature equal to the above average for the same thermal

0, = (TD-Tf)kHsb $6 -

In( -1 d$ D cos $J

. It is apparent that this equation also holds for

C> D by defining $1 = 0.

1 (7) dissipation is computed. The source factor is the multiplica- For the case of a vertical constant temperature line source, tion factor which converts the actual device temperature above

first fin temperature to the line source temperature above first Q; = (TLS-Tf)Kjjsb; . (8) fin temperature. This is defined by

Fig. 3. Geometry for determination of wurce factor.

temperature at the base of the first fin at any 4 in the sector &+$I) can be found to be

(5)

where

Q,=Qs-. #2-h

The temperature (rf) averaged over (J at the base of the first

fin in sector $2 is then given by

0, In(C) Dcos@ @hid 1 d@+$lTD

(6) ($2

Compared to averaging over fin length, averaging over (I has been found experimentally by the authors to give good results

for heatsinks, particularly in the case where H >> C because it reduces the effects of distant parts of the fin. Solving for 0,.

TLs - T,= SF(TD-Tf) . (4) Since 0;. must equal 0, and using the definition of the source factor (4),

In particular, the source factor is determined in the follow- ing way. Consider the fin closest to the device on the half heat

sink shown in Fig. 3, where C < D (device overlaps first fin). In the sector $1, the temperature at the base of the first fin is equal to the source temperature, assuming a uniform source

SF= & $2

H c

In(m) d@ cos @

temperature. Using the radial heat flux assumption, the

279 IEEE TRANSACTIONS ON PARTS, HYBRIDS, AND PACKAGING, DECEMBER 1974

I 0 C>D @1=, cos-($I C-CD. (9) EXPERIMENTAL RESULTS,

Various configurations of commercially available aluminum heat sink extrusions were experimentally tested to verify the mathematical model. These heat sinks were suspended in free air using ceramic standoffs to minimize the effects of the

supporting structure. The tests were performed in a shroud

which protected the test specimen from stray room air cur- rents. The heat source, an actual semiconductor device ir

either a TO-3 or TO-66 case and operated as a variable resistor, was located at the center of the heat sink base. All

surfaces were painted with 3-M Corporations Nextelblack

velvet paint to assure an emissivity near unity.

Temperatures were measured using a thermocouple made in the heat sink by drilling a small diameter hole in the base plate opposite the semiconductor device, inserting 30 gauge copper and constantan wires separately into the hole, and making the

thermocouple junction by forcing a taper pin into the hole between the wires (see Fig. 21. Temperatures were recorded on a Digitec@ 590TC Thermocouple Thermometer calibrated with a Leeds and Northrup 8686 Millivolt Potentiometer.

Temperature error due to thermocouple wire and measuring

instrument were calculated to be less than 1.3C. Error in

power input measurement was estimated to be between 0 and

+5%. Out of the large variety of sizes of heat sinks available, the

authors have chosen what they feel to be a representative

sample for experimental testing. Figs. 4 through 8 present

observations for nine of the.heat sinks tested, giving temper- ature rise above ambient as a function of source dissipation for the heat sink configurations whose dimensions (in inches) are

shown. The solid lines labeled U represent the results from a

computer program which employs the mathematical model

presented here. The solid lines labeled P were determined in the same way except the convective losses were calculated

assuming the fins to be parallel flat plates. These computed

results show good correlation with the experimental data. In every case examined, including the ones not presented here, considering the heat sinks as three-sided open channels gave

results closer to the experimental data than the parallel flat

Fig. 4.

0 5 10 15 20 25 30

SOURCE DISSIPATION -WATTS

Correlation of theoretical and experimental heat sink data.

t JEDEC (Joint Electron Device Engineering Council) standard case size [lo].

Fig. 5.

Fig. 6.

Fig. 7.

1 0

U&P

0 H=Z.O 0 H = 5.99

-0 5 10 15 20 25

SOURCE DISSIPATION -WATTS

Correlation of theoretical and experimenta; heat sink data.

. c! 6Or .W ,

,I P. u a lo- I Y H : 4.90

5 zo- Gi

0 H= 4.96

I 0 I I I I

0 5 10 15 20

SOURCE DISSIPATION-WATTS

Correlation of theoretical and experimental heat sink data.

- 2.58 -

0 5 10 15 20

SOURCE DISSIPATION - WATTS

Correlation of theoretical and experimental heat sink data.

0 5 10 15 20 SOURCE DISSIPATION-WATTS

Fig. 8, Correlation of theoretical and experimental heat sink data.

VAN DE POLAND TIERNEY: FREE CONVECTION HEAT TRANSFER 271

plate analogy where they differed. The error was reduced in one case from 6C to 2C as shown for the 3-in high heat sink of Fig. 7 which exhibits relatively short fins and a relatively

small value of the characteristic length r. Also, as illustrated in Fig. 8, a similar but unexpected improvement was found for

heat sinks with relatively long fins and a relatively small value

of the characteristic length r (narrow fin spacing). For very small r, however, the P and U solutions must coincide.

CONCLUSION

A practical and relatively accurate technique has been pre- sented for analyzing heat transfer from heat sinks which have vertical, rectangular fins mounted normal to a vertical, rec- tangular, constant thickness base plate. This method is appli-

cable to heat sinks having fin length to fin spacing ratios ranging from zero to infinity, and allows for piecewise-varying

base plate temperatures.

The parametric extremes have not been investigated com- pletely to define the limits of applicability of the method. However, some general limitations can be stated. The Nusselt

I number was developed for laminar flow .in a constant temper-

ature channel, so the heat sink must be short enough to insure

that only laminar flow is encountered. A partial correction for

temperature variations in the vertical direction is provided by the source factor, at least through the maximum height heat

sink tested (IO inches). The consistent accuracy observed with

increasing height (see Fig. 5) implies that this correction

should apply to significantly higher heat sinks, except when

the derivation of the source factor itself does not hold (when D>>C).

The mathematical model has been. written into a users

computer program by the authors which includes the use of

optimization routines for most dimensions. Thjs program has

been an invaluable tool used in the evaluation of over 50 heat sinks, most of which were examined for actual design applica-

tions.

ACKNQWLEDGMENT

The authors wish to express their thanks to Mr. D. R. Asher for conducting the experimental program and to Miss A. L.

Alexander for her assistance in that program.

REFERENCES

111

[21

[31

[41

[51

[61

[71

Bl

I91 [lOI

Fritsch, C. A., Radiative heat transfer, Physical Design of Electronic Systems, Vol. I, Prentice Hall, 1970, pp. 248-254 and pp. 282-285. Starner, K. E., and McManus, H. N., An experimental investiga- tion of free convection heat transfer from rectangular fin arrays, Journal of Heat Transfer, Trans. A.S.M.E., Vol. 85, pp. 273-8, 1963. Welling, J. R., and Wooldridge, C. B., Free convection heat transfer coefficients from rectangular vertical fins, Journal of Heat Transfer, Trans. A.S.M.E., Vol. 87, pp. 439-44. 1965. Izume, K., and Nakamura, H., Heat transfer by convection on the Heated Surface with Parallel Fins, Jap. Sot. Me& Eng., 34 (261). pp. 909-14, 1968. Donovan, R. C., and Rohrer, W. M., Radiative and convective conducting fins on a plane wall, including mutual irradiation, Journal of Heat Transfer, Trans. A.S.M.E., Vol. 93, pp 41-46, 1971. Van de Pol, D. W., and Tierney, J. K,, Free convection Nusselt number for vertical U-shaped channels, Journal of Heat Trans- fer, Trans. A.S.M.E., Vol. 95, pp. 542-43, 1973. Gardner, K. A., Efficiency of extended surfaces, Journal of Heat Transfer, Trans. A.S.M.E., Vol. 67, pp. 621-31, 1945. Kreith, F., Radiation Heat Transfer. fbr Spacecraft and Solar Power P/ant Design, Internationa! Textbook Co., p. 211, Con- figuration 19, 1962. McAdams, W. H., Heat Transmission, McGraw-Hill, 1954, p. 172. Electronic Industries Association, 2001 Eve St. N. W., Washing- ton, Cl. C. 20006.