Chapter 5, Problem 5.14P

A copper sheet of thickness $2L=2\text{mm}$ has an initial temperature of $T_i=118°\text{C}\text{.}$ It is suddenly quenched in liquid water, resulting in boiling at its two surfaces. For boiling, Newton's law of cooling is expressed as $q^n=h\left(T_s-T_{\text{sat}}\right),$ where $T_s$ is the solid surface temperature and $T_{\text{sat}}$ is the saturation temperature of the fluid (in this case $T_{\text{sat}}=100°\text{C)}\text{.}$ The convection heat transfer coefficient may be expressed as $h=1010{\text{W/m}}^{\text{2}}\cdot {\text{K}}^{\text{3}}{\left(T-T_{\text{sat}}\right)}^2.$ Determine the time needed for the sheet to reach a temperature of $T=102°\text{C}\text{.}$ Plot the copper temperature versus time for $0\le t\le 0.5\text{s}\text{.}$ On the same graph, plot the copper temperature history assuming the heat transfer coefficient is constant. evaluated at the average copper temperature $\bar{T}=100°\text{C}\text{.}$ Assume lumped capacitance behavior.