Chapter 5, Problem 5.72P

A cold air chamber is proposed for quenching steel ball bearings of diameter $D=0.2\text{m}$ and initial temperature $T_i=400°\text{C}\text{.}$ Air in the chamber is maintained at $-15°\text{C}$ by a refrigeration system, and the steel balls pass through the chamber on a conveyor belt. Optimum bearing production requires that $70\%$ of the initial thermal energy content of the ball above $-15°\text{C}$ be removed. Radiation effects may be neglected, and the convection heat transfer coefficient within the chamber is $1000{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ Estimate the residence time of the balls within the chamber, and recommend a drive velocity of the conveyor. The following properties may be used for the steel: $k=50\text{W/m}\cdot \text{K,}\alpha =2\times {10}^{-5}{\text{m}}^{\text{2}}\text{/s,}$ and $c=450J/kg\cdot \text{K}\text{.}$