Page 416 - Mechanical Engineers' Handbook (Volume 4)

3 Thermal Control Techniques 405

Y 21.e (92)
in
11.e
If the tip of the most remote fin is not adiabatic, the heat flow to temperature excess
ratio at the tip, which is designated as ,

q a (93)
a
will be known. For example, for a fin dissipating to the environment through its tip desig-
nated by the subscript a:
hA (94)
a
In this case, Y may be obtained through successive use of what is termed the reflection
in
relationship (actually a bilinear transformation):
(q / )
Y 21,k 1 22,k 1 a a (95)
in,k 1
11,k 1 12,k 1 (q / )
a
a
The Cluster Algorithm. For n fins in cluster, as shown in Fig. 15b, the equivalent thermal
transmission ratio will be the sum of the individual fin input admittances:
Y (96)
n
n
q
b
e
k 1 in,k k 1 b k
Here, Y in,k can be determined for each individual fin via Eq. (93) if the fin has an
adiabatic tip or via Eq. (95) if the tip is not adiabatic. It is obvious that this holds if subarrays
containing more than one fin are in cluster.
The Parallel Algorithm. For n fins in parallel, as shown in Fig. 15c, an equivalent thermal
admittance matrix [Y] can be obtained from the sum of the individual thermal admittance
e
matrices:
[Y] [Y] (97)
n
e
k 1 k
where the individual thermal admittance matrices can be obtained from
[Y] y 11 y 12
11 1 22
22 1 21

12
12
12
12
y 21 y 22 1 1 (98)
12 12 12 22
If necessary, [ ] may be obtained from [Y] using
[ ] 11 12
y 22 y 1 11
y
21
21
21 22
y (99)
y 21 y 21
where
y y y y
12 21
11 22
Y