Page 296 - Modelling in Transport Phenomena A Conceptual Approach

276 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

8.2.4 Conduction in a Fin

In the previous sections we have considered oncdimensional conduction examples.
The extension of the procedure for these problems to conduction in tw+ or three
dimensional cases is straightforward. The di%culty with multi-dimensional con-
duction problems lies in the solution of the resulting partial differential equations.
An excellent book by Carslaw and Jaeger (1959) gives the solutions of conduction
problems with various boundary conditions.
In this section first the governing equation for temperature distribution will be
developed for three-dimensional conduction in a rectangular geometry. Then the
use of area averaging4 will be introduced to simplify the problem.
Fins are extensively used in heat transfer applications to enhance the heat
transfer rate by increasing heat transfer area. Let us consider a simple rectangular
fin as shown in Figure 8.23. As an engineer we are interested in the rate of heat
loss from the surfaces of the fin. This can be calculated if the temperature distri-
bution within the fin is known. The problem will be analyzed with the following
assumptions:

1. Steady state conditions prevail.
2. The thermal conductivity of the fin is constant.

3. The average heat transfer coefficient is constant.
4. There is no heat loss from the edges and the tip of the fin.

M-A.24
Figure 8.23 Conduction in a rectangular fin.

4The first systematic use of the area averaging technique in a textbook can be attributed to
Slattery (1972).