A plane wall of thickness 2L = 40 mm and thermal conductivity k = 5 W/m ∙ K experiences uniform volumetric heat generation at a rate q, while convection heat transfer occurs at both of its surfaces (x = - L, + L), each of which is exposed to a fluid of temperature T_∞ = 20°C. Under steady-state conditions, the temperature distribution in the wall is of the form T(x) = a + bx + cx², where a = 82.0°C, b = -210°C/m, c = -2 X 10⁴ °C/m², and x is in meters. The origin of the x-coordinate is at the mid plane of the wall.

(a) Sketch the temperature distribution and identify significant physical features.

(b) What is the volumetric rate of heat generation q in the wall?

(c) Determine the surface heat fluxes, q_x (- L) and q_x (+ L). How are these fluxes related to the heat generation rate?

(d) What are the convection coefficients for the surfaces at x = - L and x = + L?

(e) Obtain an expression for the heat flux distribution, q_x (x). Is the heat flux zero at any location? Explain any significant features of the distribution.

(f) If the source of the heat generation is suddenly deactivated (q = 0), what is the rate of change of energy stored in the wall at this instant?

(g) What temperature will the wall eventually reach with q = O? How much energy must be removed by the fluid per unit area of the wall (J/m²) to reach this state? The density and specific heat of the wall material are 2600 kg/m³ and 800 J/kg ∙ K, respectively.