Heat Conduction in a Cooling Fin
Section 9.7; Bird, Stewart and Lightfoot

---

The calculation of the efficiency of a cooling fin is another practical application of heat conduction. The area available for heat transfer between metal walls and poorly conducting fluids, like gases, is increased by cooling fins.



Figure 1

The model in Figure 1 makes the following assumptions:

a) Temperature is a function of z only.
b) Neither the end nor the edges lose heat.
c) q=h(T-Ta) is the heat flux at the surface, where h is the heat transfer coefficient and T=T(z).

A thermal energy balance on a segment dz of the bar proceeded by the insertion of Fourier's law yields the following differential equation and its solution.


Solving this equation directly requires the following boundary conditions:
B.C. 1: at z=0, T=Tw
B.C. 2: at z=L, dT/dz=0
However, the answer is very ugly, and so its evaluation is left to the discretion of the reader. The use of the following dimensionless quatities cleans up the answer significantly.

---

SAMPLE PROBLEM