Chapter 2, Problem 2.57P

The plane wall with constant properties and no internal heat generation shown in the figure 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 heat transfer coefficient h. The boundary at $x=0$ is perfectly insulated.

1. Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature as a function of position and time in the wall.
2. 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 two intermediate times.
3. On $q_x^n-t$ coordinates, sketch the heat flux at the locations $x=0,x=L.$ That is, show qualitatively how $q_x^n\left(0,t\right)$ and $q_x^n\left(L,t\right)$ vary with time.
4. Write an expression for the total energy transferred to the wall per unit volume of the wall $\left({\text{J/m}}^{\text{3}}\right).$