Chapter 2, Problem 2.59P

A plane wall has constant properties, no internal heat generation, and is initially at a uniform temperature $T_i.$ Suddenly, the surface at $x=L$ is heated by a fluid at $T_{\infty }$ having a convection coefficient h. At the same instant, the electrical heater is energized, providing a constant heat flux $q_o^{"}$ at $x=0.$

1. On $T-x$ coordinates, sketch the temperature distributions for the following conditions: initial condition $\left(t\le 0\right),$ steady-state condition $\left(t\to \infty \right),$ and for two intermediate times.
2. On $q_x^{"}-t$ coordinates, sketch the heat flux corresponding to the four temperature distributions of part (a).
3. On $q_x^{"}-t$ coordinates, sketch the heat flux at the locations $x=0$ and $x=L.$ That is, show qualitatively how $q_x^{"}\left(0,t\right)$ and $q_x^{"}\left(L,t\right)$ vary with time.
4. Derive an expression for the steady-state temperature at the heater surface, $T\left(0,\infty \right),$ in terms of $q_o^{"},T_{\infty },$ k, h, and L.