Page 387 - Mechanical Engineers' Handbook (Volume 4)

376 Cooling Electronic Equipment

and the Reynolds number, Re, which relates the momentum in the flow to the viscous dis-
sipation,
VL
Re

Natural Convection
In natural convection, fluid motion is induced by density differences resulting from temper-
ature 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,
2
gc p
3
Ra L
T
k
where the fluid properties, , , c , , and k, are evaluated at the fluid bulk temperature and
p

T is the temperature difference between the surface and the fluid.
Empirical correlations for the natural convection heat-transfer coefficient generally take
the form
h C n 2
k
fl
L (Ra) (W/m K) (12)
3
9
where n is found to be approximately 0.25 for 10 Ra 10 , representing laminar flow,
9
12
0.33 for 10 Ra 10 , the region associated with the transition to turbulent flow, and
0.4 for Ra 10 , when strong turbulent flow prevails. The precise value of the correlating
12
coefficient, C, depends on the 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. 42
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 appear to have little influence upon one another and the Nusselt number
in this case achieves its isolated plate limit. On the other hand, for closely spaced plates or
for relatively long channels, the fluid velocity attains its fully developed value and the Nusselt
number reaches its fully developed limit. Intermediate values of the Nusselt number can be
obtained from a form of a correlating expression for smoothly varying processes and have
been verified by detailed experimental and numerical studies. 19,20
Thus, the correlation for the average value of h along isothermal vertical placed sepa-
rated by a spacing, z
h 576 2.873 1/2
k
fl
z (El) 2 (El) 1 / 2 (13)
where El is the Elenbaas number
2
4
gc z
T
p
El
kL
fl
and
T T T .
s
fl
Several correlations for the coefficient of heat transfer in natural convection for various
configurations are provided in Section 2.1.