Chapter 2, Problem 2.55P

Consider a one-dimensional plane wall of thickness 2L. The surface at $x=-L$ is subjected to convective conditions characterized by $T_{\infty ,1},h_1,$ while the surface at $x=+L$ is subjected to conditions $T_{\infty ,2},h_2.$ The initial temperature 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 the boundary and initial conditions that could be used to determine the temperature distribution $T\left(x,t\right)$ a function of position and time.
2. On $T-x$ coordinates, sketch the temperature distributions for the initial condition. the steady-state condition, and for two intermediate times for the case $h_1=h_2.$
3. On $q_x^n-t$ coordinates, sketch the heat flux $q_x^n\left(x,t\right)$ at the planes $x=0,-L$ and $+L.$
4. The value of $h_1$ is now doubled with all other conditions being identical as in parts (a) through (c). On $T-x$ coordinates drawn to the same scale as used in part (b), sketch the temperature distributions for the initial condition, the steady-state condition, and for two intermediate times. Compare the sketch to that of part (b).
5. Using the doubled value of $h_1,$ sketch the heat flux $q_x^n\left(x,t\right)$ at the planes $x=0,-L,$ and $+L$ on the same plot you prepared for part (c). Compare the two responses.