Find the temperature distribution in the tapered fin shown in Fig. P4.14. Assume that the temperature at the root of the fin is 250◦F, the conductivity k = 120 Btu/(h ft ◦F), and the film coefficient β = 15 Btu/(h ft2◦F), and use three linear elements. The ambient temperature at the top and bottom of the fin is T∞ = 75◦F.
The governing differential equation for this problem can be derived by making the following assumptions: The fin thickness at the base x = L = 3 is small enough so that the temperature is uniform in the transverse direction (i.e., y-direction) to the fin; the heat transfer from the fin edges (see the cross hatched part) may be neglected in comparison to that from the top surface of the fin (into the plane of the paper); and there is no temperature variation along the z-direction (into the plane of the paper). Then the equation governing the one—dimensional heat transfer in the fin is given by
The element coefficient matrix needed here is already evaluated and recorded in Eq.(3.2.35), page 122 (set a = 1 and c = N2). The assembled coefficient matrix for three-element mesh is
The assembled source vector is given by
The assembed equations of the three linear element mesh (he = 1/12 ft., T∞ = 75)