Chapter 5, Problem 5.58P

The strength and stability of tires may be enhanced by heating both sides of the rubber $\left(k=0.14\text{W/m}\cdot \text{K,}\alpha =6.35\times {10}^{-8}{\text{m}}^{\text{2}}\text{/s}\right)$ in a steam chamber for which $T_{\infty }=200°\text{C}\text{.}$ In the heating process, a 20-mm-thick rubber wall (assumed to be untreaded) is taken from an initial temperature of $25°\text{C}$ to a midplane temperature of $150°\text{C}\text{.}$

1. If steam flow over the tire surfaces maintains a convection coefficient of $h=200{\text{W/m}}^{\text{2}}\cdot \text{K,}$ how long will it take to achieve the desired midplane temperature?
2. To accelerate the heating process. it is recommended that the steam flow be made sufficiently vigorous to maintain the tire surfaces at $200°\text{C}$ throughout the process. Compute and plot the mid-plane and surface temperatures for this case, as well as for the conditions of part (a).