A plane wall of thickness L = 0.1 m experiences uniform volumetric heating at a rate q. One surface of the wall (x = 0) is insulated, while the other surface is exposed to a fluid at Too = 20°C, with convection heat transfer characterized by h = 1000 W/m² ∙ K. Initially, the temperature distribution in the wall is T(x, 0) = a + bx², where a = 300°C, b = -1.0 X 10⁴ °C/m² and x is in meters. Suddenly, the volumetric heat generation is deactivated (q = 0 for t > 0), while convection heat transfer continues to occur at x = L. The properties of the wall are p = 7000 kg/m³, c p = 450 J/kg. K, and k = 90 W/m ∙ K.

(a) Determine the magnitude of the volumetric energy generation rate q associated with the initial condition (t < 0).

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

(c) On q_x – t coordinates, sketch the variation with time of the heat flux at the boundary exposed to the convection process, qx (L, t). Calculate the corresponding value of the heat flux at t = 0, q_x (L, 0).

(d) Calculate the amount of energy removed from the wall per unit area (J/m²) by the fluid stream as the wall cools from its initial to steady-state condition.

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