This problem concerns the solution of heat transfer from a radial fin. Although this problem has a solution in terms of Bessel functions, we are going to see how to compute it numerically directly from the boundary value problem.

(a) Consider a radial fin of thickness t and length L. The fin has a thermal conductivity k and a heat transfer coefficient (from the sides) of h to a far-field temperature T_∞. The shell balance for the fin is

where A_c = 2πrt is the cross-sectional area of the fin and A_s = 4πrΔr is the surface area on the top and the bottom of the fin in the shell volume. Complete the derivation of the differential equation and find an appropriate change of variables to cast the problem as the modified Bessel equation

(b) Write a MATLAB program that uses centered finite differences to compute the fin temperature profile for a temperature T = 100°C at the hot end of the fin and a far-field temperature of T_∞ = 25°C. You can treat the cold end of the fin as insulating. For the physical parameters, you can use k = 210 W/mK, h = 20W/m²K, t = 5 cm, and L = 25 cm. Your program should automatically produce a plot of the temperature (in celsius) versus the distance along the fin (in centimeters).

(c) Write a MATLAB program that finds the fin size (to a tolerance of 1 mm) that produces a temperature within 1 degree of T_∞ using the physical parameters from part (b). Your program should provide the size of this fin to the screen. (It would be helpful to display more information to the screen so that we can understand your program and give you full credit!) Does this seem like a practical design?

Transcribed Image Text:

-kAc dT dr = -kAc dT dr Ir+dr + hAs(T - T∞)