The bottom half of an I-beam providing support for a furnace roof extends into the heating zone. The web is well insulated, while the flange surfaces experience convection with hot gases at T_∞. = 400°C and a convection coefficient of h = 150 W/m² ∙ K. Consider the symmetrical element of the flange region (inset a), assuming that the temperature distribution across the web is uniform at T_w = 100°C. The beam thermal conductivity is 10W/m ∙ K, and its dimensions are w_f = 80mm, w_w = 30 mm, and L = 30 mm.

(a) Calculate the heat transfer rate per unit length to the beam using a 5 X 4 nodal network.

(b) Is it reasonable to assume that the temperature distribution across the web-flange interface is uniform? Consider the L-shaped domain of inset (b) and use a fine grid to obtain the temperature distribution across the web-flange interface. Make the distance w_o > w_w/2.

-Oven roof -l-beam Insulation Flange Web Assume uniform Gases -Uniform? W. 2 (b) Wf 2 ,h (a)