A plane wall that is insulated on one side (x = 0) is initially at a uniform temperature T_i, when its exposed surface at x = L is suddenly raised to a temperature T_s'

(a) Verify that the following equation satisfies the heat equation and boundary conditions: where C) is a constant and a is the thermal diffusivity.

(b) Obtain expressions for the heat flux at x = 0 and x = L.

(c) Sketch the temperature distribution T(x) at t = 0, at t → ∞, and at an intermediate time. Sketch the variation with time of the heat flux at x = L, q_L (t).

(d) What effect does a have on the thermal response of the material to a change in surface temperature?

7 at Tx,) , T;- T, cos 2 L) - xp| 4 L2