Page 413 - Mechanical Engineers' Handbook (Volume 4)

402 Cooling Electronic Equipment

6. There is no bond resistance to the flow of heat at the base of the fin.
7. The temperature at the base of the fin is uniform and constant.
8. There are no heat sources within the fin itself.
9. There is a negligible flow of heat from the tip and sides of the fin.
10. The heat flow from the fin is proportioned to the temperature difference or temper-
ature excess, (x) T(x) T , at any point on the face of the fin.
s
The Fin Efficiency
Because a temperature gradient always exists along the height of a fin when heat is being
transferred to the surrounding environment by the fin, there is a question regarding the
temperature to be used in Eq. (72). If the base temperature T (and the base temperature
b
excess, T T ) is to be used, then the surface area of the fin must be modified by
s
b
b
the computational artifice known as the finefficiency, defined as the ratio of the heat actually
transferred by the fin to the ideal heat transferred if the fin were operating over its entirety
at the base temperature excess. In this case, the surface area A in Eq. (54) becomes
A A A (73)
b ƒ ƒ
The Longitudinal Fin of Rectangular Profile
With the origin of the height coordinate x taken at the fin tip, which is presumed to be
adiabatic, the temperature excess at any point on the finis

cosh mx
(x) b (74)
cosh mb
where
m 1/2
2h
k (75)
The heat dissipated by the finis
q Y tanh mb (76)
b
0
b
where Y is called the characteristic admittance
0
Y (2hk ) 1/2 L (77)
0
and the finefficiency is
tanh mb
(78)
f
mb
The heat-transfer coefficient in natural convection may be determined from the sym-
metric isothermal case pertaining to vertical plates in Section 2.1. For forced convection, the
London correlation described in Section 2.2 applies.

The Radial Fin of Rectangular Profile
With the origin of the radial height coordinate taken at the center of curvature and with the
fin tip at r r presumed to be adiabatic, the temperature excess at any point on the finis
a
(r) K (mr )I (mr) I (mr )K (mr)
1
a
0
a
1
0
b
I (mr )K (mr ) I (mr )K (mr ) (79)
1
a
b
0
0
a
b
1
where m is given by Eq. (75). The heat dissipated by the finis