Page 414 - Mechanical Engineers' Handbook (Volume 4)

3 Thermal Control Techniques 403
q 2 rkm I (mr )K (mr ) K (mr )I (mr )
1
a
1
b
1
b
1
a
b
b
b
I (mr )K (mr ) I (mr )K (mr ) (80)
a
1
0
0
b
a
1
b
and the finefficiency is
I (mr )K (mr ) K (mr )I (mr )
2r b 1 a 1 b 1 a 1 b (81)
ƒ
m(r r ) I (mr )K (mr ) I (mr )K (mr )
2
2
1
0
b
0
a
b
b
a
a
1
Tables of the finefficiency are available, 36 and they are organized in terms of two
parameters, the radius ratio
r
b (82a)
r a
and a parameter
(r r ) 1/2
2h
a
b
kA (82b)
p
where A is the profile area of the fin:
p
A (r r ) (82c)
p a b
For air under forced convection conditions, the correlation for the heat-transfer coeffi-
37
cient developed by Briggs and Young is applicable:
0.200 0.1134
0.681
1 / 3
s
h 2 Vr b c s
p
2rk k r r b (83)
a
b
where all thermal properties are evaluated at the bulk air temperature, s is the space between
the fins, and r and r pertain to the fins.
a
b
The Cylindrical Spine
With the origin of the height coordinate x taken at the spine tip, which is presumed to be
adiabatic, the temperature excess at any point on the spine is given by Eq. (72), but for the
cylindrical spine
m 1/2
4h
kd (84)
where d is the spine diameter. The heat dissipated by the spine is given by Eq. (76), but in
this case
3 1 / 2
2
Y ( hkd ) /2 (85)
0
and the spine efficiency is given by Eq. (78).
Algorithms for Combining Single Fins into Arrays
The differential equation for temperature excess that can be developed for any fin shape can
be solved to yield a particular solution, based on prescribed initial conditions of fin base
temperature excess and fin base heat flow, that can be written in matrix form 38,39 as
[ ] 11 12

b
b
a
q a q b 21 22 q b (86)