Consider a straight fin with the trapezoidal shape sketched in Fig. P.4.9. Assuming that the temperature in the fin is only a function of the axial coordinate x (i.e., it is uniform at each vertical location):

(a) Write the differential balances needed to describe conductive heat transfer in the fin and simultaneous convective heat transfer at the fin surfaces and the boundary conditions assuming that the fin is long and thin. Note that the cross-section A and the perimeter p of the fin vary as a function of x. (b)

Solve your equations numerically (note that this is a boundary value problem) to find the temperature profile in the fin and the fin efficiency. (c)

Plot the temperature profile in the fin and determine the fin efficiency and the heat transfer rate for a 316 stainless-steel fin with T⁰ =100°C, T^a =

20°C, L = 0.05 m, B⁰ = 0.005 m, B^L = 0.001 m, and W = 0.1 m, if the convective heat transfer coefficient is h = 50 W/m² K.

FIGURE P.4.9:

Transcribed Image Text:

2B L Surface at To Fluid at Ta X 2B