A very thick slab with thermal diffusivity 5.6 x 10^-6 m²/s and thermal conductivity 20 W/m ∙ K is initially at a uniform temperature of 325°C. Suddenly, the surface is exposed to a coolant at 15°C for which the convection heat transfer coefficient is 100 W/m² ∙ K.

(a) Determine temperatures at the surface and at a depth of 45 mm after 3 min have elapsed.

(b) Compute and plot temperature histories (0 < t < 300 s) at x = 0 and x = 45 mm for the following parametric variations: (i) a = 5.6 X 10^-7, 5.6 x 10^-6, and 5.6 X 10^-5 m²/s; and (ii) k = 2, 20, and 200 W/m ∙ K.