Page 393 - Mechanical Engineers' Handbook (Volume 4)

382 Cooling Electronic Equipment
T QL
Q
L
2
T ctr o (26)
LW 8k e 8W k e
where Q is the total heat dissipation, W, L, and are the width, length, and thickness,
respectively, and k is the effective thermal conductivity of the board.
e
This relation can be used effectively in the determination of the temperatures experi-
enced by conductively cooled substrates and conventional printed circuit boards, as well as
PCBs with copper lattices on the surface, metal cores, or heat-sink plates in the center. In
each case it is necessary to evaluate or obtain the effective thermal conductivity of the
conducting layer. As an example, consider an alumina substrate 0.20 m long, 0.15 m wide
and 0.005 m thick with a thermal conductivity of 20 W/m K, whose edges are cooled to
35 C by a cold plate. Assuming that the substrate is populated by 30 components, each
dissipating 1 W, use of Eq. (26) reveals that the substrate center temperature will equal 85 C.

External Resistance
To determine the resistance to thermal transport from the surface of a component to a fluid
in motion, that is, the convective resistance as in Eq. (11), it is necessary to quantify the
heat transfer coefficient, h. In the natural convection air cooling of printed circuit board
arrays, isolated boards, and individual components, it has been found possible to use smooth-
plate correlations, such as
h C n
k
fl
L Ra (27)
and
h 576 2.073 1/2
k
fl
12
10.5
b (El ) (El ) (28)
to obtain a first estimate of the peak temperature likely to be encountered on the populated
board. Examination of such correlations suggests that an increase in the component/board
temperature and a reduction in its length will serve to modestly increase the convective heat-
transfer coefficient and thus to modestly decrease the resistance associated with natural con-
vection. To achieve a more dramatic reduction in this resistance, it is necessary to select a
high-density coolant with a large thermal expansion coefficient—typically a pressurized gas
or a liquid.
When components are cooled by forced convection, the laminar heat-transfer coefficient,
given in Eq. (54) (page 395), is found to be directly proportional, to the square root of fluid
velocity and inversely proportional to the square root of the characteristic dimension. In-
creases in the thermal conductivity of the fluid and in Pr, as are encountered in replacing
air with a liquid coolant, will also result in higher heat-transfer coefficients. In studies of
low-velocity convective air cooling of simulated integrated circuit packages, the heat-transfer
coefficient, h, has been found to depend somewhat more strongly on Re (using channel
height as the characteristic length) than suggested in Eq. (54), and to display a Reynolds
number exponent of 0.54 to 0.72. 8–10 When the fluid velocity and the Reynolds number
increase, turbulent flow results in higher heat-transfer coefficients, which, following Eq. (56)
(page 395), vary directly with the velocity to the 0.8 power and inversely with the charac-
teristic dimension to the 0.2 power. The dependence on fluid conductivity and Pr remains
unchanged.
An application of Eq. (27) or (28) to the transfer of heat from the case of a chip module
to the coolant shows that the external resistance, R 1/hS, is inversely proportional to the
ex