Chapter 2, Problem 2.55P

Consider a one-dimensional plane wall of thickness 2L.The surface at $x=-L$ is subjected to convectiveconditions characterized by $T_{\infty ,1},h_1,$ while the surfaceat $x=+L$ is subjected to conditions $T_{\infty ,2},h_2.$ The initialtemperature of the wall is $T_o=\left(T_{\infty ,1}+T_{\infty ,2}\right)/2$ where $T_{\infty ,1}>T_{\infty }{}_{,2}.$

1. Write the differential equation, and identify theboundary and initial conditions that could be usedto determine thetemperature distribution $T\left(x,t\right)$ asa function of position and time.
2. On $T-x$ coordinates, sketch the temperature distributionsfor the initial condition, the steady-statecondition, and for twointermediate times for thecase $h_1=h_2.$ On $q_x^{"}-t$ coordinates, sketch the heat flux $q_x^{"}\left(x,t\right)$ at the planes $x=0,-L,$ and $+L.$ The value of $h_1$ is now doubled with all other conditionsbeing identical as in parts (a) through (c).On $T-x$ coordinates drawnto the same scale asused in part (b), sketch the temperature distributionsfor the initial condition, the steady-state condition,and for two intermediate times. Compare thesketch to that of part (b).
3. Using the doubled value of $h_1,$ sketch the heat flux $q_x^{"}\left(x,t\right)$ at the planes $x=-0,-L,$ and $+L$ on the sameplot you prepared for part (c). Compare the tworesponses.