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A straight fin of uniform cross section is fabricated from a material of thermal conductivity k = 5W/m ∙ K, thickness w = 20 mm, and length L = 200 mm. The fin is very long in the direction normal to the page. The base of the fin is maintained at Tb = 200Â°C and the tip condition allows for convection (case A of Table 3.4), with h = 500W/m2 ∙ K and T∞, = 25Â°C.

(a) Assuming one-dimensional heat transfer in the fin, calculate the fin heat rate, q; (W/m), and the tip temperature TL Calculate the Biot number for the fin to determine whether the one-dimensional assumption is valid.

(b) Using the finite-element method of FEHT, perform a two-dimensional analysis on the fin to determine the fin heat rate and tip temperature. Compare your results with those from the one-dimensional, analytical solution of part (a). Use the View Temperature Contours option to display isotherms, and discuss key features of the corresponding temperature field and heat flow pattern. Hint: In drawing the outline of the fin, take advantage of symmetry. Use a fine mesh near the base and a coarser mesh near the tip. Why?

(c) Validate your FEHT model by comparing predictions with the analytical solution for a fin with thermal conductivities of k = 50W/m ∙ K and 500 W/m ∙ K. Is the one-dimensional heat transfer assumption valid for these conditions?

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