Chapter 2, Problem 2.65P

A plane wall of thickness $L=0.1\text{m}$ experiences uniformvolumetric heating at a rate $\stackrel{.}{q}.$ One surface of the wall $\left(x=0\right)$ is insulated, and the other surface is exposed to afluid at $T_{\infty }=20°\text{C,}$ with convection heat transfer characterizedby $h=1000{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ Initially, the temperaturedistribution in the wall is $T\left(x,0\right)=ab{x}^2,$ where $a=300°\text{C,}b=-1.0\times {10}^4°{\text{C/m}}^{\text{2}},$ 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 continuesto occur at $x=L.$ The properties of the wall are $\rho =7000{\text{kg/m}}^3,c_p=450\text{J/kg}\cdot \text{K,}$ and $k=90\text{W/m}\cdot \text{K}\text{.}$

1. Determine the magnitude of the volumetric energygeneration rate $\stackrel{.}{q}$ associated with the initial condition $\left(t<0\right).$ On $T-x$ coordinates, sketch the temperature distributionfor the following conditions: initial condition $\left(t<0\right),$ steady-state condition $\left(t\to \infty \right),$ andtwo intermediate conditions.
2. On $q_x^{"}-t$ coordinates, sketch the variation withtime of the heat flux at the boundary exposed to theconvection process, $q_x^{"}\left(L,t\right).$ Calculate the correspondingvalue of the heat flux at $t=0,q_x^{"}\left(L,0\right).$ Calculate the amount of energy removed from thewall per unit area $\left({\text{J/m}}^{\text{2}}\right)$ by the fluid streamas the wall cools from its initial to steady-statecondition.