30079_54.pdf

CHAPTER 54

COOLING ELECTRONIC EQUIPMENT

Allan Kraus

Allan D. Kraus Associates

Aurora, Ohio

54.1 THERMAL MODELING

1649

54.2.2 Forced Convection

1662

54. 1 . 1 Introduction

1 649

54.3 THERMAL CONTROL

TECHNIQUES

54. 1 .2 Conduction Heat

Transfer

1667

1649

54.3.1 Extended Surface and

Heat Sinks

54.1.3 Convective Heat

Transfer

1672

1652

54.3.2 The Cold Plate

1672

54.1.4 Radiative Heat Transfer

1655

54.3.3 Thermoelectric Coolers

1674

54.1.5 Chip Module Thermal

Resistances

1656

54.2 HEAT-TRANSFER

CORRELATIONS

FOR ELECTRONIC

EQUIPMENT COOLING

1661

54.2.1 Natural Convection in

Confined Spaces

1661

54.1 THERMAL MODELING

54.1.1 Introduction

To determine the temperature differences encountered in the flow of heat within electronic systems,

it is necessary to recognize the relevant heat transfer mechanisms and their governing relations. In a

typical system, heat removal from the active regions of the microcircuit(s) or chip(s) may require the

use of several mechanisms, some operating in series and others in parallel, to transport the generated

heat to the coolant or ultimate heat sink. Practitioners of the thermal arts and sciences generally deal

with four basic thermal transport modes: conduction, convection, phase change, and radiation.

54.1.2 Conduction Heat Transfer

One-Dimensional Conduction

Steady thermal transport through solids is governed by the Fourier equation, which, in one-

dimensional form, is expressible as

q=-kAj^

(W)

(54.1)

where q is the heat flow, k is the thermal conductivity of the medium, A is the cross-sectional area

for the heat flow, and dTldx is the temperature gradient. Here, heat flow produced by a negative

temperature gradient is considered positive. This convention requires the insertion of the minus sign

in Eq. (54.1) to assure a positive heat flow, q. The temperature difference resulting from the steady

state diffusion of heat is thus related to the thermal conductivity of the material, the cross-sectional

area and the path length, L, according to

(T 1 ~ T 2 ) c d = qj^

(K)

(54.2)

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

The form of Eq. (54.2) suggests that, by analogy to Ohm's Law governing electrical current flow

through a resistance, it is possible to define a thermal resistance for conduction, R cd as

*•- 21 T^-C

One-Dimensional Conduction with Internal Heat Generation

Situations in which a solid experiences internal heat generation, such as that produced by the flow

of an electric current, give rise to more complex governing equations and require greater care in

obtaining the appropriate temperature differences. The axial temperature variation in a slim, internally

heated conductor whose edges (ends) are held at a temperature T 0 is found to equal

r=r - + *.2*lAzHzJJ

M 2 I

^ \( X \

T +

When the volumetic heat generation rate, q g , in W/m 3 is uniform throughout, the peak temperature

is developed at the center of the solid and is given by

r ma x = T 0 + q g ^

(K)

(54.4)

Alternatively, because q g is the volumetric heat generation, q g = q/LWd, the center-edge tem-

perature difference can be expressed as

7I -- r - = «8^ra"«5S

(54 ' 5 )

where the cross-sectional area, A, is the product of the width, W, and the thickness, 8. An examination

of Eq. (54.5) reveals that the thermal resistance of a conductor with a distributed heat input is only

one quarter that of a structure in which all of the heat is generated at the center.

Spreading Resistance

In chip packages that provide for lateral spreading of the heat generated in the chip, the increasing

cross-sectional area for heat flow at successive"layers" below the chip reduces the internal thermal

resistance. Unfortunately, however, there is an additional resistance associated with this lateral flow

of heat. This, of course, must be taken into account in the determination of the overall chip package

temperature difference.

For the circular and square geometries common in microelectronic applications, an engineering

approximation for the spreading resistance for a small heat source on a thick substrate or heat spreader

(required to be 3 to 5 times thicker than the square root of the heat source area) can be expressed

as 1

f ,

0.475 - 0.62e + 0.13e 2

R sp =

—

(K/W)

(54.6)

kvA c

where e is the ratio of the heat source area to the substrate area, k is the thermal conductivity of the

substrate, and A c is the area of the heat source.

For relatively thin layers on thicker substrates, such as encountered in the use of thin lead-frames,

or heat spreaders interposed between the chip and substrate, Eq. (54.6) cannot provide an acceptable

prediction of R sp . Instead, use can be made of the numerical results plotted in Fig 54.1 to obtain the

requisite value of the spreading resistance.

Interface/Contact Resistance

Heat transfer across the interface between two solids is generally accompanied by a measurable

temperature difference, which can be ascribed to a contact or interface thermal resistance. For per-

fectly adhering solids, geometrical differences in the crystal structure (lattice mismatch) can impede

the flow of phonons and electrons across the interface, but this resistance is generally negligible in

engineering design. However, when dealing with real interfaces, the asperities present on each of the

surfaces, as shown in an artist's conception in Fig 54.2, limit actual contact between the two solids

to a very small fraction of the apparent interface area. The flow of heat across the gap between two

solids in nominal contact is thus seen to involve solid conduction in the areas of actual contact and

fluid conduction across the "open" spaces. Radiation across the gap can be important in a vacuum

environment or when the surface temperatures are high.

Fig. 54.1 The thermal resistance for a circular heat source on a

two layer substrate (from Ref. 2).

The heat transferred across an interface can be found by adding the effects of the solid-to-solid

conduction and the conduction through the fluid and recognizing that the solid-to-solid conduction,

in the contact zones, involves heat flowing sequentially through the two solids. With the total contact

conductance, h co , taken as the sum of the solid-to-solid conductance, h c , and the gap conductance,

h co = h c + h g

(W/m 2 • K)

(54.7a)

the contact resistance based on the apparent contact area, A a , may be defined as

- Intimate contact

Gap filled with fluid with

thermal conductivity Ay

Fig. 54.2 Physical contact between two nonideal surfaces.

(54 - 8fl )

R co - -^-

(K/W)

(54.7/7)

n co A a

*• = 54 - 25 *< (?) (S) 09 5

In Eq. (54.7«), /z c is given by

where k s is the harmonic mean thermal conductivity for the two solids with thermal conductivities,

k l and & 2 ,

1 Jk k

k * = T^TT

(W/m-K)

/T 1 + £ 2

(j is the effective rms surface roughness developed from the surface roughnesses of the two materials,

(T 1 and O 2 ,

cr = VcrfT~af

(/^ • m)

and m is the effective absolute surface slope composed of the individual slopes of the two materials,

M 1 and m 2 ,

m = Vm 2 + ra 2

where P is the contact pressure and H is the microhardness of the softer material, both in NVm 2 . In

the absence of detailed information, the aim ratio can be taken equal to 5-9 microns for relatively

smooth surfaces. 1 ' 2

In Eq. (54.70), h g is given by

( 54 - 8& )

*' = FT^

where k g is the thermal conductivity of the gap fluid, Y is the distance between the mean planes (Fig.

54.2) given by

Y f / PM 0 - 54 7

- = 54.185 [-in (3.132 -JJ

and M is a gas parameter used to account for rarefied gas effects

M = a0A

where a is an accommodation parameter (approximately equal to 2.4 for air and clean metals), A is

the mean free path of the molecules (equal to approximately 0.06 fjum for air at atmospheric pressure

and 15 0 C), and ft is a fluid property parameter (equal to approximately 54.7 for air and other diatomic

gases).

Equations (54.80) and (54.Sb) can be added and, in accordance with Eq. (54.Ib), the contact

resistance becomes

54.1.3 Convective Heat Transfer

The Heat Transfer Coefficient

Convective thermal transport from a surface to a fluid in motion can be related to the heat transfer

coefficient, h, the surface-to-fluid temperature difference, and the "wetted" surface area, S, in the

form

q = hS(T s - T fl )

(W)

(54.10)

The differences between convection to a rapidly moving fluid, a slowly flowing or stagnant fluid,

(54 - n )

as well as variations in the convective heat transfer rate among various fluids, are reflected in the

values of h. For a particular geometry and flow regime, h may be found from available empirical

correlations and/or theoretical relations. Use of Eq. (54.10) makes it possible to define the convective

thermal resistance as

Rc ^~ns (K/w )

Dimensionless Parameters

Common dimensionless quantities that are used in the correlation of heat transfer data are the Nusselt

number, Nu, which relates the convective heat transfer coefficient to the conduction in the fluid where

the subscript, f/, pertains to a fluid property,

Nu = — = —

Kfi/L

k fl

the Prandtl number, Pr, which is a fluid property parameter relating the diffusion of momentum to

the conduction of heat,

ft -^ K a

the Grashof number, Gr, which accounts for the bouyancy effect produced by the volumetric expan-

sion of the fluid,

Grs £^AT

M 2

and the Reynolds number, Re, which relates the momentum in the flow to the viscous dissipation,

R.-fi M

Natural Convection

In natural convection, fluid motion is induced by density differences resulting from temperature

gradients in the fluid. The heat transfer coefficient for this regime can be related to the buoyancy

and the thermal properties of the fluid through the Rayleigh number, which is the product of the

Grashof and Prandtl numbers,

R a = £^ L 3 A r

Mf/

where the fluid properties, p, /3, c p , /i, and k, are evaluated at the fluid bulk temperature and Ar is

the temperature difference between the surface and the fluid.

Empirical correlations for the natural convection heat transfer coefficient generally take the form

Ik \

/z = C — (Ra)"

(W/m 2 • K)

(54.12)

\L/

where n is found to be approximately 0.25 for 10 3 < Ra < 10 9 , representing laminar flow, 0.33 for

10 9 < Ra < 10 12 , the region associated with the transition to turbulent flow, and 0.4 for Ra > 10 12 ,

when strong turbulent flow prevails. The precise value of the correlating coefficient, C, depends on

fluid, the geometry of the surface, and the Rayleigh number range. Nevertheless, for common plate,

cylinder, and sphere configurations, it has been found to vary in the relatively narrow range of

0.45-0.65 for laminar flow and 0.11-0.15 for turbulent flow past the heated surface. 4 2

Natural convection in vertical channels such as those formed by arrays of longitudinal fins is of

major significance in the analysis and design of heat sinks and experiments for this configuration

have been conducted and confirmed. 4 ' 5

These studies have revealed that the value of the Nusselt number lies between two extremes

associated with the separation between the plates or the channel width. For wide spacing, the plates

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