Chapter 2, Problem 2.65P

A plane wall of thickness $L=0.1\text{m}$ experiences uniform volumetric heating at a rate $\stackrel{.}{q}.$ One surface of the wall $\left(x=0\right)$ is insulated, and the other surface is exposed to a fluid at $T_{\infty }=20°\text{C,}$ with convection heat transfer characterized by $h=1000{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ Initially, the temperature distribution in the wall is $T\left(x,0\right)=a+b{x}^2,$ where $a=300°\text{C,}b=-1.0\times {10}^{40}{\text{C/m}}^2,$ and x is in meters. Suddenly, the volumetric heat generation is deactivated $\left(\stackrel{.}{q}=0\text{for}t\ge 0\right),$ while convection heat transfer continues to occur at $x=L.$ The properties of the wall are $\rho =7000{\text{kg/m}}^{\text{3}},c_p=450\text{J/kg}\cdot \text{K,}$ and $k=90\text{W/m}\cdot \text{K}\text{.}$

1. Determine the magnitude of the volumetric energy generation rate $\stackrel{.}{q}$ associated with the initial condition $\left(t<0\right).$
2. On $T-x$ coordinates, sketch the temperature distribution for the following conditions: initial condition $\left(t<0\right),$ steady-state condition $\left(t\to \infty \right),$ and two intermediate conditions.
3. On $q_x^n-t$ coordinates, sketch the variation with time of the heat flux at the boundary exposed to the convection process, $q_x^n\left(L,t\right).$ Calculate the corresponding value of the heat flux at $t=0,q_x^n\left(L,0\right).$
4. Calculate the amount of energy removed from the wall per unit area $\left({\text{J/m}}^{\text{2}}\right)$ by the fluid stream as the wall cools from its initial to steady-state condition.