The plane wall with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature T_i Suddenly the surface at x = L is heated by a fluid at T_∞ having a convection heat transfer coefficient h. The boundary at x = 0 is perfectly insulated.

(a) Write the differential equation and identify the boundary and initial conditions that could be used to determine the temperature as a function of position and time in the wall.

(b) On T - x coordinates, sketch the temperature distributions for the following conditions: initial condition (t ≤ 0), steady-state condition (t → ∞), and two intermediate times.

(c) On q_x – t coordinates, sketch the heat flux at the locations x = 0 and x = L. That is, show qualitatively how q_x (0, t) and qx (L, t) vary with time.

(d) Write an expression for the total energy transferred to the wall per unit volume of the wall (J/m³).

T, h Insulation L.