In the analytical treatment of the fin with uniform cross-sectional area, it was assumed that the convection heat transfer coefficient is constant along the length of the fin. Consider an AISI 316 steel fin of 6-mm diameter and 50-mm length (with insulated tip) operating under conditions for which Tb = 125°C, T∞ = 27°C, Tsur = 27°C, and ε = 0.6.

(a) Estimate average values of the fin heat transfer coefficients for free convection (hc and radiation exchange (hr). Use these values to predict the tip temperature and fin effectiveness.

(b) Use a numerical method of solution to estimate the foregoing parameters when the convection and radiation coefficients are based on local, rather than average, values for the fin.