Chapter 2, Problem 2.54P

The one-dimensional system of mass M with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature $T_i.$ The electrical heater is suddenly energized, providing a uniform heat flux $q_o^{"}$ at the surface $x=0.$ The boundaries at $x=L$ and elsewhere are 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 system.
2. On $T-x$ coordinates, sketch the temperature distributions for the initial condition $\left(t\le 0\right)$ and for several times after the heater is energized. Will a steady-state temperature distribution ever be reached?
3. On $q_x^{"}-t$ coordinates, sketch the heat flux $q_x^{"}\left(x,t\right)$ at the planes $x=0,x=L/2,$ and $x=L$ as a function of time.
4. After a period of time $t_e$ has elapsed, the heater power is switched off. Assuming that the insulation is perfect, the system will eventually reach a final uniform temperature $T_f.$ Derive an expression that can be used to determine $T_f$ as a function of the parameters $q_o^{"},t_e,T_i,$ and the system characteristics $M,c_p,$ and $A_s$ (the heater surface area).