Page 388 - Mechanical Engineers' Handbook (Volume 4)

1 Thermal Modeling 377

Forced Convection
For forced flow in long, or very narrow, parallel-plate channels, the heat-transfer coefficient
attains an asymptotic value (a fully developed limit), which for symmetrically heated channel
surfaces is equal approximately to
4k
2
h fl (W/m K) (14)
d
e
where d is the hydraulic diameter defined in terms of the flow area, A, and the wetted
e
perimeter of the channel, P
w
4A
d
e
P w
Several correlations for the coefficient of heat transfer in forced convection for various
configurations are provided in Section 2.2.
Phase Change Heat Transfer
Boiling heat transfer displays a complex dependence on the temperature difference between
the heated surface and the saturation temperature (boiling point) of the liquid. In nucleate
boiling, the primary region of interest, the ebullient heat-transfer rate can be approximated
by a relation of the form
q CA(T T ) 3 (W) (15)

sat
s
sf
where C is a function of the surface/fluid combination and various fluid properties. For
sf
comparison purposes, it is possible to define a boiling heat-transfer coefficient, h ,

2
h C (T T ) 2 [W/m K]

s
sf
sat
which, however, will vary strongly with surface temperature.
Finned Surfaces
A simplified discussion of finned surfaces is germane here and what now follows is not
inconsistent with the subject matter contained Section 3.1. In the thermal design of electronic
equipment, frequent use is made of finned or ‘‘extended’’ surfaces in the form of heat sinks
or coolers. While such finning can substantially increase the surface area in contact with the
coolant, resistance to heat flow in the fin reduces the average temperature of the exposed
surface relative to the fin base. In the analysis of such finned surfaces, it is common to define
a fin efficiency, , equal to the ratio of the actual heat dissipated by the fin to the heat that
would be dissipated if the fin possessed an infinite thermal conductivity. Using this approach,
heat transferred from a fin or a fin structure can be expressed in the form
q hS (T T ) (W) (16)
ƒ
b
s
ƒ
where T is the temperature at the base of the fin and where T is the surrounding temperature
b
s
and q is the heat entering the base of the fin, which, in the steady state, is equal to the heat
ƒ
dissipated by the fin.
The thermal resistance of a finned surface is given by
1
R (17)
ƒ
hS
ƒ
where , the fin efficiency, is 0.627 for a thermally optimum rectangular cross-section fin, 11