Chapter 5, Problem 5.22P

A plane wall of a furnace is fabricated from plain carbon steel $\left(k=60\text{W/m}\cdot \text{K},\rho =7850{\text{kg/m}}^{\text{3}},c=430\text{J/kg}\cdot \text{K}\right)$ and is of thickness $L=10\text{mm}\text{.}$ To protect it from the corrosive effects of the furnace combustion gases, one surface of the wall is coated with a thin ceramic film that, for a unit surface area. has a thermal resistance of $R_{t,f}^{"}=0.01{\text{m}}^{\text{2}}\cdot \text{K/W}\text{.}$ The opposite surface is well insulated from the surroundings.

At furnace start-up the wall is at an initial temperature of $T_i=300\text{K,}$ and combustion gases at $T_{\infty }=1300\text{K}$ enter the furnace, providing a convection coefficient of $h=25{\text{W/m}}^{\text{2}}\cdot \text{K}$ at the ceramic film. Assuming the film to have negligible thermal capacitance. how long will it take for the inner surface of the steel to achieve a temperature of $T_{s,i}=1200\text{K?}$ What is the temperature  $T_{s,o}$ of the exposed surface of the ceramic film at this time?