ENGINEERINGA plane wall of thickness 2L=40 mm and thermal conductivity $k=5 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ experiences uniform volumetric heat generation at a rateq $\dot{q}$, while convection heat transfer occurs at both of its surfaces (x=-L, +L), each of which is exposed to a fluid of temperature $T_{\infty}=20^{\circ} \mathrm{C}$. Under steady-state conditions, the temperature distribution in the wall is of the form $T(x)=a+b x+c x^{2}$ where $a=82.0^{\circ} \mathrm{C}, b=-210^{\circ} \mathrm{C} / \mathrm{m}, c=-2 \times 10^{4} \mathrm{C} / \mathrm{m}^{2}$, and x is in meters. The origin of the x-coordinate is at the midplane 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, $q_{x}^{\prime \prime}(-L)$ and $q_{x}^{\prime \prime}(+L)$. 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 $q_{x}^{\prime \prime}(x)$. 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=0? How much energy must be removed by the fluid per unit area of the wall $\left(\mathrm{J} / \mathrm{m}^{2}\right)$ to reach this state? The density and specific heat of the wall material are $2600 \mathrm{kg} / \mathrm{m}^{3}$ and $800 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$, respectively.