Chapter 5, Problem 5.23P

A steel strip of thickness $\delta =12\text{mm}$ is annealed by passing it through a large furnace whose walls are maintained at a temperature $T_w$ corresponding to that of combustion gases flowing through the furnace $\left(T_w=T_{\infty }\right).$ The strip, whose density, specific heat, thermal conductivity. and emissivity are $\rho =7900{\text{kg/m}}^{\text{3}},c_p=640\text{J/kg}\cdot \text{K,}k=30\text{W/m}\cdot \text{K,}$ and $\epsilon =0.7,$ respectively, is to be heated from $300°\text{C}$ to $600°\text{C}\text{.}$

1. For a uniform convection coefficient of $h=100{\text{W/m}}^{\text{2}}\cdot \text{K}$ and $T_w=T_{\infty }=700°\text{C,}$ determine the time required to heat the strip. If the strip is moving at 0.5m/s, how long must the furnace be?
2. The annealing process may be accelerated (the strip speed increased) by increasing the environmental temperatures. For the furnace length obtained in part (a). determine the strip speed for $T_w=T_{\infty }=850°\text{C}$ and $T_w=T_{\infty }=1000°\text{C}\text{.}$ For each set of environmental temperatures (700, 850, and $1000°\text{C),}$ plot the strip temperature as a function of time over the range $\text{25°C}\le T\le 600°\text{C}\text{.}$ Over this range, also plot the radiation heat transfer coefficient, $h_r,$ as a function of time.