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26
Analytical Heat Transfer
In order to get the higher fin effectiveness and fin efficiency, we need to have
a thin fin (larger P/A c ratio) with larger thermal conductivity k (aluminum or
copper) and low working fluid heat transfer coefficient h (air cooling).
q = q fin + q non-fin
q = Nη f q max + h(T b − T ∞ )A non-fin
where N is the number of fins and
q max = h(T b − T ∞ )A s,fin
dT �
q fin = −kA c
�
dx x=0
It is important to point out that the temperature distribution through a
fin varies depending on the aforementioned fin tip boundary conditions. In
general, these temperatures are a decay curve from the fin base to the fin tip
as shown in Figure 2.8. These decay curves are the combination of sinh and
cosh functions shown above. In addition, the heat transfer rate through the
fin depends on the temperature gradient at the fin base and the fin thermal
conductivity. For example, the temperature gradient at the fin base is greater
for the steel fin than the aluminum fin. However, heat transfer rate through
the fin base is higher for an aluminum fin than for a steel fin for the same fin
geometry and working fluid conditions. This is because the aluminum fin has
a much larger thermal conductivity than the steel fin.
2.3.2 Radiation Effect
""
If we also consider radiation flux q , the energy balance equation 2.29 can be
r
rewritten as
dq x
""
q x − q x +
dx − hA s (T − T ∞ ) + A s q = 0
dx
r
""
""
where q = radiation gain from solar = constant, or q = radiation loss =
r
r
""
−εσ(T 4 − T 4 ). If we consider q = constant, the solution of above equation
sur
r
can be obtained by Equation 2.34 by setting
""
q r
θ = T − T ∞ − h
""
However, if we consider q = −εσ(T 4 − T 4 ), the above energy balance
r
sur
equation can be rewritten as
dq x
q x − q x +
dx − hA s (T − T ∞ ) − εσA s (T
4
− T
4 ) = 0
dx
sur