According to the one-dimensional (longitudinal) conduction model of a fin, the temperature distribution along the fin, T(x), obeys the energy equation [15]

where A, h, P, and q’’’ are the fin cross-sectional area, fin-fluid heat transfer coefficient, perimeter of the fin cross section (called the wetted perimeter), and volumetric rate of heat generation. Consider the semi-infinite fin that, as shown in Fig. P1.11, is bathed by a fluid of temperature T₀ and is attached to a solid wall of temperature T₀. The heat generated by the fin is absorbed by either the fluid or the solid wall.

(a) As a system for scale analysis, select the fin section of length x, where x is measured away from the wall. Let T_∞ be the fin temperature sufficiently far from the wall. Show that if x is large enough, the longitudinal conduction term becomes negligible in the energy equation.

(b) Invoking the balance between lateral convection and internal heat generation, determine the fin temperature sufficiently far from the wall, T_∞.

(c) Determine the fin section of length δ near the wall where the heat transfer is ruled by the balance between longitudinal conduction and internal heat generation.

(d) Determine the heat transfer rate into the wall through the base of the fin.