In the conventional fin efficiency analysis, the ambient temperature T8 is idealized as constant…

In the conventional fin efficiency analysis, the ambient
temperature T_ is idealized as constant along the fin length ‘. However,
for the case of laminar flow, the transverse mixing along the fin length ‘ may
be negligible after a short distance along the heat exchanger flow length Lf .
In such a case, the difference between the fin temperature and the ambient
temperature (T- T_) at any x will be constant, independent of x. Consider
the straight, thin fin of uniform thickness shown in Fig. P4.10.

(a) Starting with Eq. (4.61), show that the temperature
distribution within this fin is given
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In the conventional fin efficiency analysis, the ambient
temperature T_ is idealized as constant along the fin length ‘. However,
for the case of laminar flow, the transverse mixing along the fin length ‘ may
be negligible after a short distance along the heat exchanger flow length Lf .
In such a case, the difference between the fin temperature and the ambient
temperature (T- T_) at any x will be constant, independent of x. Consider
the straight, thin fin of uniform thickness shown in Fig. P4.10.

(a) Starting with Eq. (4.61), show that the temperature
distribution within this fin is given by

(b) Derive an expression for the actual heat transfer q0
through the base.

(c) For the fin efficiency, consider the integrated average
temperature T_  to obtain qmax, where

(d) Define the fin efficiency for this problem and obtain

(e) If we would have considered T_ as a constant, the
fin efficiency would have been

as summarized in Eq. (4.134). For the fin of Example 4.4, =
0:896. How much error is introduced in this f , if we would have considered (T-
T_) as constant instead of T_ as constant?

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