Functionally graded materials are intentionally fabricated to establish a spatial distribution of properties in the final product. Consider an L × L two-dimensional object with L = 20 mm. The thermal conductivity distribution of the functionally graded material is k(x) = 25 W/m·K +(9010 W/m5/2·K) x3/2. Two sets of boundary conditions, denoted as cases 1 and 2, are applied. (a) Determine the spatially averaged value of the thermal conductivity k¯ . Use this value to estimate the heat rate per unit length for cases 1 and 2.
(b) Using a grid spacing of 2 mm, determine the heat rate per unit depth for case 1.
(c) Using a grid spacing of 2 mm, determine the heat rate per unit depth for case 2.