Page 383 - Mechanical Engineers' Handbook (Volume 4)

372 Cooling Electronic Equipment

dT
q kA (W) (1)
dx
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 dT/dx 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. (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
L
(T T ) q (K) (2)
1 2 cd
kA
The form of Eq. (2) suggests that, by analogy to Ohm’s Law governing electrical current
as
flow through a resistance, it is possible to define a thermal resistance for conduction, R cd,
T T 2 L
1
R cd (3)
q kA
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 var-
iation in a slim, internally heated conductor whose edges (ends) are held at a temperature
T is found to equal
o
T T q
2
2
L
x
x
o
g
2k L L
3
,inW/m is uniform throughout, the peak
When the volumetic heat generation rate, q g
temperature is developed at the center of the solid and is given by
L 2
T max T q g (K) (4)
o
8k
is the volumetric heat generation, q q/LW , the center–
Alternatively, because q g g
edge temperature difference can be expressed as
L 2 L
T max T q q (5)
o
8kLW 8kA
where the cross-sectional area, A, is the product of the width, W, and the thickness, .An
examination of Eq. (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 as-
sociated 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 en-
gineering approximation for the spreading resistance for a small heat source on a thick