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Analytical Heat Transfer
3.15. Given a very long and wide fin with a height of 2H. The base of
the fin is maintained at a uniform temperature of T b . The top and
bottom surfaces of the fin are exposed to a fluid whose temperature is T ∞ (T ∞ < T b ). The convective heat transfer coefficient
between the fin surfaces and the fluid is h.
a. Derive an expression for the steady 2-D local temperature in
the fin, in terms of the thermal conductivity of the fin, k, the
convective heat transfer coefficient, h, the half-height of the fin,
H, and the base and fluid temperature, T b and T ∞ .
b. Sketch the steady 2-D temperature and heat flux distribution
in the fin.

Note that

x
� �
2
�
1
cos (ax) dx = a [2ax
4
+ sin(2ax)] and
0
x
�
[cos(ax) · cos(bx)] dx = 0 when a = b.
0
References
1. V. Arpaci, Conduction Heat Transfer, Addison-Wesley Publishing Company, Reading, MA, 1966.
2. A. Mills, Heat Transfer, Richard D. Irwin, Inc., Boston, MA, 1992.
3. F. Incropera and D. Dewitt, Fundamentals of Heat and Mass Transfer, Fifth Edition,
John Wiley & Sons, New York, NY, 2002.
4. W. Rohsenow and H. Choi, Heat, Mass, and Momentum Transfer, Prentice-Hall, Inc.,
Englewood Cliffs, NJ, 1961.